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\newcommand{\statusAuthorsNote}{
\ifupperformal
The formal proofs of the five results
\cref{thm:barrier-730,b:cor:subcritical,thm:upper,f:main,cor:mean-bias}
have completed the independent checks described in \cref{sec:formal}.
That section gives their precise statements and distinguishes them from
the stronger conclusions proved only in the written argument.
\else
The formal proofs of the four results
\cref{thm:barrier-730,b:cor:subcritical,f:main,cor:mean-bias}
have completed the independent checks described in \cref{sec:formal}.
The upper-bound statement and its verification record are described in
\cref{sec:formal-upper}.
\fi}

\newcommand{\statusAbstract}{
\ifupperformal
We have also formalized the upper bound in continuous time.
\else
The formal treatment of the upper bound is described in
\cref{sec:formal-upper}.
\fi}

\newcommand{\statusIntroShort}{
\ifupperformal
We have formalized \cref{thm:upper}; see \cref{sec:formal-upper}.
\else
For the formal statement corresponding to \cref{thm:upper}, see
\cref{sec:formal-upper}.
\fi}

\newcommand{\statusIntroFormal}{
\ifupperformal
\Cref{thm:upper} is also formalized, as the statement described in
\cref{sec:formal-upper}, and is included in the combined five-claim
verification described in \cref{sec:formal}.
\Cref{cor:upper-discrete,cor:two-sided} are not separately formalized.
\else
\Cref{sec:formal-upper} gives the formal statement corresponding to
\cref{thm:upper}, the proof declaration, and the verification record.
\fi}

\newcommand{\statusFormalUpper}{
The proposition
\path{SKConstants.OptimizedSKTypicalStartStatement}, defined in
\path{SKConstants/SKModel.lean}, states that, for every $\beta>1$ and
every $\omega\in(0,1/4)$,
\[
 \PP_{J,X_0\sim\pi^J}
 \bigl\{d^J\bigl(X_0,\exp(n^{1-\omega})\bigr)>1/4\bigr\}
 \longrightarrow0.
\]
The declaration uses $c$ for the parameter denoted $\omega$ here, with
both $\beta$ and $c$ fixed before $n\to\infty$.

In the formal model, the variables $g_{ij}$ are independent standard
Gaussians, and the completed matrix and Hamiltonian are
\[
 W_{ij}=\frac{g_{ij}+g_{ji}}{\sqrt{2n}},\qquad
 H_g^c(x)=\frac12x^{\mathsf T}Wx,
\]
including its diagonal, with $\beta$ outside the Hamiltonian.
For $i<j$, we define $J_{ij}=\sqrt n\,W_{ij}$. These couplings are
independent standard Gaussians, and
\[
 H_g^c(x)=H_n^J(x)+\frac12\tr W,\qquad
 Z_n^c=e^{\beta\tr W/2}Z_n.
\]
The trace term is independent of $x$ and cancels in the Gibbs
normalization, as in \cref{lem:model-completion}(b), and in every energy
difference. The Gibbs probabilities and flip rates coincide with those
of \cref{sec:model}. The same holds for the semigroup and the event
defined by the total variation distance from a fixed initial spin.
The formal generator is $n(K^J-I)$, so each site has rate one, as in
\eqref{eq:common-semigroup}. Its joint law samples the completed matrix
and then a Gibbs spin. By integrating out the independent diagonal, we
obtain the joint law of \cref{def:model-joint}. The primitive representation
in \cref{u:sec:primitive} uses the same completed matrix law. The initial
spin is fixed before its transition law and total variation distance
are formed. By the complementary form of \cref{lem:model-joint}, the
displayed limit is \cref{thm:upper}. The formal sequence is also defined
at sizes zero and one, which does not affect the limit.

\ifupperformal
The theorem
\path{SKConstants.Completion.Main.optimizedSKTypicalStartStatement_holds}
proves this proposition without additional hypotheses. The original
upper-bound package at commit \path{352426b9007b6bbb31a3c8cfe6887caa0badf3fd}
was checked on 19 October 2026, using
Lean version \texttt{v4.35.0-rc2} and Mathlib at commit
\path{065356127b1dc0016f66b7283ce0ce2c4055aa55}.
A fresh build succeeded at this revision. The package's \path{Challenge.lean}
contains 47 independently stated claims and imports only Mathlib. Its
definitions explicitly include the completed disorder law, Gibbs kernel,
semigroup, total variation distance, and joint probability. The official
Comparator confirmed equality of the proved statements and these
independent statements, including their definitions. The axiom audit of
all 20,281 exported development declarations in 1,605 modules found only
\path{propext}, \path{Classical.choice}, and \path{Quot.sound}; the main
theorem has exactly these three axiom dependencies. The same Comparator
run checked all 47 claims and their proof dependencies with Lean's kernel,
NanoDa, and con-ron. All three accepted them. The upper-bound theorem
is also one of the five final statements checked in the combined package
described at the start of \cref{sec:formal}.

The formal proof follows the route of \cref{part:upper}. It constructs
Brownian motion, proves the scalar and Gaussian-query laws, establishes
the fine entropy and sharp coarse estimates, and combines coverage and
the connecting sequence with the finite-chain mixing reduction. The
steps are organized as separately proved statements and assembly
theorems, whose hypotheses are supplied by the final proof. The
parameter choice used by that proof is described in
\cref{sec:formal-scope}. The development includes Ezzeri Esa's
StochasticCalculus library and adapted OpenAI Lean modules, both under
the Apache-2.0 license. Intermediate estimates of the written proof are
checked only insofar as they enter the formal proof or are separately
stated formal results.
\else
The proof declaration and verification records are available with the
sources described in \cref{sec:formal-scope}.
\fi}

\newcommand{\statusFormalUpperScope}{
\ifupperformal
\Cref{cor:upper-discrete,cor:two-sided} are not separately formalized.
The development proves admissibility of the affine family in
\cref{u:prop:range} throughout
$0<\varepsilon_\omega<1/80$, including the choice
$\varepsilon_\omega=(1/4-\omega)/20$. The final proof of
\cref{thm:upper} instead uses the same affine family at
$\min\{1/2000,(1/4-\omega)/20\}$ and proves the target for every
$\omega\in(0,1/4)$. A separate formal result characterizes the open
range of the development's strict estimate system, and another checks
an explicit rational instance. The ancillary calculation described
after \cref{u:prop:range}, including every identity for
\cref{u:tab:budget}, is not a separately certified formal statement.
\else
The scope of the upper-bound verification is recorded with the sources
described below.
\fi}
\allowdisplaybreaks[2]

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\begin{document}
\frontmatter
\npptitlepage{Lower and Upper Bounds for Glauber\\ Dynamics from Typical States\\ of the Low-Temperature\\ Sherrington--Kirkpatrick Model}{P.\ M.\ Aronow\\ Patrick Lopatto}
\begin{authorsnote}
With the exception of this note, the entire
document is machine-written. We have carefully supervised its preparation
to ensure that the proofs are readable and that
previous literature is appropriately cited. We are
releasing this manuscript on Hexagon to disseminate the result as quickly as
possible because it may be of interest to the mathematical
community. 
We welcome suggestions concerning mathematical corrections, omitted citations, and attribution of ideas.

P.\ L.\ was partially supported by NSF grant DMS-2450004.
\end{authorsnote}
\begin{abstractpage}
We study the heat-bath (Glauber) dynamics of the Sherrington--Kirkpatrick
model at zero external field and fixed inverse temperature $\beta>1$. We
sample the disorder $J$ and then a configuration $X_0$ from the Gibbs
measure, and run the continuous-time dynamics with rate one per site from
$X_0$. Let $T^J(X_0)$ be the first time at which the law of the configuration,
conditional on $J$ and the realized $X_0$, is within total-variation distance
$1/4$ of the Gibbs measure. We prove lower
and upper bounds: for every fixed $\omega\in(0,1/4)$, with probability tending
to one,
\[
 A_\beta n^{7/30}<\log T^J(X_0)\le n^{1-\omega},
\]
where $A_\beta>0$ depends only on $\beta$. Hence, if
$\log T^J(X_0)=n^{\zeta+o(1)}$, then $7/30\le\zeta\le3/4$; the physics
literature suggests $\zeta=1/3$. Both bounds also hold for the discrete-time
chain.

These results improve on OpenAI's work in the same setting in two ways.
First, OpenAI proved a lower bound with exponent $1/10000$~\cite[Theorem~1.1]{oai-barriers} and an upper bound
with exponent $1-1/40000000$~\cite[Theorem~1]{oai-typical-upper}. We follow
the architecture of both proofs. The lower bound adds a three-replica
comparison whose pressure deficit is cubic in the width of the overlap window,
together with the $n^{-7/10}$ upper bound of Aronow and
Lopatto~\cite[Theorem~1.1]{hexagon-bias} on the mean bias. The upper bound adds a pathwise cancellation between information and
energy, concentration estimates on a growing mesh, and a choice of exponents
that reaches every saving below $1/4$. Second, OpenAI formalized its lower
bound in the Lean~4 proof assistant but not its upper bound; we have
formalized both of our bounds in Lean.

We also formalize two known results that we present. The first is OpenAI's
theorem that the variance of the logarithm of the partition function is
$n^{1/3+o(1)}$. OpenAI's own formalization of this theorem assumes a bound on
positive logarithmic moments that is proved in their paper but not
formalized; our formal proof establishes this bound. The second is the
consequence, recorded by Aronow and Lopatto~\cite[Section~1.6]{hexagon-bias},
that the mean bias of the pressure is at least of order $n^{-5/6-\varepsilon}$ for every fixed
$\varepsilon>0$.
\end{abstractpage}
\tableofcontents
\mainmatter

\chapter{Introduction}\label{sec:intro}

The Sherrington--Kirkpatrick (SK) model is the mean-field Ising spin glass,
in which every pair of spins interacts through an independent Gaussian
coupling~\cite{sk-original}. Its equilibrium theory is well developed.
Parisi's hierarchical ansatz~\cite{parisi-1979} predicted the limiting free
energy, and this prediction was proved through Guerra's interpolation
bound~\cite{guerra-2003} and Talagrand's matching lower
bound~\cite{parisi-formula}. At every fixed inverse temperature $\beta>1$
and zero external field, the minimizing Parisi measure is supported on an
interval $[0,q_\beta]$; this was proved by Zhou for $\beta$ close to
one~\cite[Theorem~1]{zhou-2026} and in~\cite[Theorem~1.1]{full-rsb} for every
$\beta>1$. Much less is known about the
dynamics of the model at low temperature, and about the finite-size
corrections to its equilibrium quantities. This paper addresses both.

Our main results concern the heat-bath dynamics, also called Glauber
dynamics~\cite{glauber-1963}, in which one spin at a time is resampled from
its conditional law. For fixed $\beta>1$, we sample the couplings and then one
configuration from the Gibbs measure, and run the dynamics from this
configuration. We prove that, with probability tending to one, the law of
the configuration at time $\exp(A_\beta n^{7/30})$ is still at total-variation
distance greater than $\tau$ from the Gibbs measure, for every fixed $\tau<1/2$
and a coefficient $A_\beta>0$ that does not depend on $\tau$
(\cref{thm:barrier-730}). In the other direction, for every fixed
$\omega\in(0,1/4)$, the distance is at most $1/4$ at time $\exp(n^{1-\omega})$,
again with probability tending to one (\cref{thm:upper}). OpenAI proved bounds
of both kinds in the same setting, with the same joint law of the couplings
and the initial configuration, the same clocks of rate one per site, and the
threshold $1/4$: a lower bound at time
$\exp(n^{1/10000})$~\cite[Theorem~1.1]{oai-barriers} and an upper bound at
time $\exp(n^{1-1/40000000})$~\cite[Theorem~1]{oai-typical-upper}. We follow
the overall strategy of both proofs. Let $T^J(X_0)$ be the first time at which
the distance from the Gibbs measure is at most $1/4$. Combining our two
bounds, we find that if $\log T^J(X_0)=n^{\zeta+o(1)}$, then
$7/30\le\zeta\le3/4$ (\cref{cor:two-sided}). The proof of the lower bound
uses the upper bound $n^{-7/10}$ of Aronow and Lopatto on the mean bias of
the pressure~\cite[Theorem~1.1]{hexagon-bias}.

Our second contribution is a complete formal proof of
\cref{thm:barrier-730} in the Lean~4 proof assistant~\cite{lean4,mathlib},
together with formal proofs of two results that are not new. The first is the
fluctuation theorem $\Var(F_n)=n^{1/3+o(1)}$ for the logarithm $F_n$ of the
partition function (\cref{f:main}), which was stated and proved by OpenAI
in~\cite[Theorem~1.1]{oai-fluctuations}. OpenAI's public formalization of
this theorem assumes a two-sided bound on positive logarithmic moments of
the centered completed log partition function, which is proved on paper
in~\cite[Proposition~7.5]{oai-fluctuations}; our formal proof establishes
this bound. We restate the theorem with attribution,
describe the structure of its proof, and reproduce the short final step of
that proof; we do not claim a new mathematical proof of \cref{f:main}.
\statusIntroShort

The second known result is a lower bound with exponent $5/6$ on the mean bias
$P_\beta-n^{-1}\E F_n$, where $P_\beta=\lim_{n\to\infty}n^{-1}\E F_n$ is the
limiting pressure. It follows from \cref{f:main} and a variance--bias
inequality of Aronow and Lopatto
\cite[Lemma~7.4 and Remark~7.14]{quantitative-parisi} that the mean bias is
at least of order $n^{-5/6-\varepsilon}$ for every fixed $\varepsilon>0$
(\cref{cor:mean-bias}). This deduction was already recorded
in~\cite[Section~1.6]{hexagon-bias}; what is new here is its formal proof.
Together with the upper bound of~\cite{hexagon-bias}, it shows that if the
mean bias is $n^{-a+o(1)}$, then $7/10\le a\le5/6$.

\section{Main results}\label{sec:intro-results}

For an integer $n\ge2$, let $\Sigma_n=\{-1,1\}^n$, and let
$J=(J_{ij})_{1\le i<j\le n}$ be independent standard Gaussian variables. We
set $J_{ji}=J_{ij}$ and $J_{ii}=0$. The Hamiltonian, the partition function,
and the Gibbs measure at inverse temperature $\beta$ are
\begin{equation}\label{eq:intro-model}
 H_n(x)=\frac1{\sqrt n}\sum_{1\le i<j\le n}J_{ij}x_ix_j,\qquad
 Z_n=\sum_{x\in\Sigma_n}e^{\beta H_n(x)},\qquad
 \pi^J(x)=\frac{e^{\beta H_n(x)}}{Z_n}.
\end{equation}
We write $R(x,y)=n^{-1}\sum_{i=1}^n x_iy_i$ for the overlap of
$x,y\in\Sigma_n$, and $F_n=\log Z_n$.

In the heat-bath dynamics, each site carries an independent exponential
clock of rate one. When the clock of site $i$ rings at a configuration $x$,
the spin $x_i$ is replaced by $\sigma\in\{-1,1\}$ with probability
\[
 p_i^J(\sigma\mid x)=\frac{e^{\beta\sigma h_i(x)}}{2\cosh(\beta h_i(x))},\qquad
 h_i(x)=\frac1{\sqrt n}\sum_{j\ne i}J_{ij}x_j,
\]
which is the conditional law of the $i$th spin under $\pi^J$ given the
other spins. An update may leave the configuration unchanged. We write
$P_t^J(x,\cdot)$ for the law at time $t$ of this process started at $x$, and
$K^J$ for the transition matrix of the discrete chain that chooses a site
uniformly at random and performs one such update. One unit of continuous
time contains $n$ update attempts on average. Both chains are reversible with
respect to $\pi^J$. For $x\in\Sigma_n$, $t\ge0$, and integers $k\ge0$, we
set
\[
 d^J(x,t)=\norm{P_t^J(x,\cdot)-\pi^J}_{\TV},\qquad
 d^J_{\mathrm{disc}}(x,k)=\norm{(K^J)^k(x,\cdot)-\pi^J}_{\TV},
\]
where $\norm{\nu-\zeta}_{\TV}=\frac12\sum_x\abs{\nu(x)-\zeta(x)}$. We also
write
\[
 T^J(x)=\inf\{t\ge0:d^J(x,t)\le1/4\},\qquad
 T^J_{\mathrm{disc}}(x)=\min\{k\ge0:d^J_{\mathrm{disc}}(x,k)\le1/4\}
\]
for the mixing times of the two chains started at $x$. Since
$t\mapsto d^J(x,t)$ is continuous and nonincreasing
(\cref{lem:model-dynamics}), we have $T^J(x)\le t$ if and only if
$d^J(x,t)\le1/4$, and similarly in discrete time. Finally,
$\PP_{J,\,X_0\sim\pi^J}$ denotes the joint law of the couplings $J$ and of a
configuration $X_0$ whose conditional law given $J$ is $\pi^J$; see
\eqref{eq:common-joint}. These definitions are given in detail in
\cref{sec:model}.

\begin{theorem}[Barrier from typical equilibrium states]\label{thm:barrier-730}
For every fixed $\beta>1$ there exists $A_\beta>0$ such that, for every
fixed $\tau\in(0,1/2)$,
\begin{align}
 \PP_{J,\,X_0\sim\pi^J}
   \bigl\{d^J\bigl(X_0,e^{A_\beta n^{7/30}}\bigr)>\tau\bigr\}
   &\longrightarrow1,\label{eq:barrier-730-continuous}\\
 \PP_{J,\,X_0\sim\pi^J}
   \bigl\{d^J_{\mathrm{disc}}\bigl(X_0,\lfloor e^{A_\beta n^{7/30}}\rfloor\bigr)>\tau\bigr\}
   &\longrightarrow1.\label{eq:barrier-730-discrete}
\end{align}
\end{theorem}

The configuration $X_0$ is sampled once and then held fixed, so that
$d^J(X_0,t)$ measures the distance from equilibrium of the chain started at
this particular configuration. If the initial configuration were instead
averaged over $\pi^J$ before the distance is taken, the law at time $t$
would be $\pi^J$ by stationarity, and the distance would vanish. Since the
Gibbs mass of the set of initial configurations in the event of
\eqref{eq:barrier-730-continuous} takes values in $[0,1]$ and has expectation
equal to the displayed probability, \eqref{eq:barrier-730-continuous} is
equivalent to the convergence of this Gibbs mass to one in probability over
the disorder; this observation is made after
\cite[Theorem~1.1]{oai-barriers} and is recalled in \cref{sec:model}. The
coefficient $A_\beta$ depends only on $\beta$: the same coefficient serves
for every fixed threshold $\tau<1/2$ and for both clocks, while the rate of
convergence in \eqref{eq:barrier-730-continuous} and
\eqref{eq:barrier-730-discrete} may depend on $\tau$. The restriction
$\tau<1/2$ comes from spin-flip symmetry: for every configuration $v$, the
half-space $\{y:R(v,y)\le0\}$ has Gibbs mass at least $1/2$, and the proof
shows that, for a suitable configuration $v$ with $R(v,X_0)$ bounded away
from zero, the chain started at $X_0$ is unlikely to enter this half-space
before the stated time.

The exponent $7/30$ is one third of the exponent $7/10$ in the bound
$0\le P_\beta-n^{-1}\E F_n\le C_b n^{-7/10}$ on the mean bias
proved in~\cite[Theorem~1.1]{hexagon-bias}, where
we recall that $P_\beta$ is the limiting pressure. The proof needs overlap constraints on a scale
$\rho$ whose pressure deficit, of order $\rho^3$ per spin, dominates the
mean bias, and our counting argument allows a time $t$ with $\log t$ of
order $1/\rho$. With the bound $n^{-7/10}$, we can take
$\rho\asymp n^{-7/30}$. \Cref{sec:intro-methods} explains this balance in
more detail. A union bound over polynomially many constraints in the
argument requires $n\rho^4$ to be large compared with $\log n$, so the
present argument gives no exponent of $1/4$ or more, whatever the upper
bound on the mean bias. \Cref{thm:barrier-730} is proved in
\cref{sec:barrier-proof}.

Since total variation contracts under a Markov kernel that preserves
$\pi^J$, the distances $d^J(x,t)$ and $d^J_{\mathrm{disc}}(x,k)$ are
nonincreasing in $t$ and $k$. For every fixed $A>0$ and $\kappa<7/30$, we
have $n^\kappa<An^{7/30}$ for all sufficiently large $n$.
\Cref{thm:barrier-730} implies the following statement with
coefficient one, which is also proved in \cref{sec:barrier-proof}.

\begin{corollary}[Unit coefficient below the exponent $7/30$]\label{b:cor:subcritical}
For every fixed $\beta>1$, every fixed $\kappa\in(0,7/30)$, and every
fixed $\tau\in(0,1/2)$,
\[
 \PP_{J,\,X_0\sim\pi^J}\bigl\{d^J\bigl(X_0,e^{n^\kappa}\bigr)>\tau\bigr\}
   \longrightarrow1,\qquad
 \PP_{J,\,X_0\sim\pi^J}
   \bigl\{d^J_{\mathrm{disc}}\bigl(X_0,\lfloor e^{n^\kappa}\rfloor\bigr)>\tau\bigr\}
   \longrightarrow1.
\]
\end{corollary}

The corollary does not assert coefficient one at the exponent $7/30$
itself.

The proof gives more than the convergence of the probabilities in
\cref{thm:barrier-730}; we record the stronger forms in
\cref{thm:barrier-strong}. There exist $A>0$ and, for each $n$, a disorder
event of probability at least $1-Ce^{-cn^{1/15}}$ on which the following
holds for all sufficiently large $n$. The set of initial configurations $x$
with $d^J(x,t)\ge1/2-Ce^{-cn^{7/30}}$ for every $t\le e^{An^{7/30}}$, and
with the analogous bound for the discrete chain, has Gibbs mass at least
$1-Ce^{-cn^{7/30}}$. The same holds for the Metropolis dynamics and, more
generally, for every single-site dynamics with uniformly bounded rates that
leaves $\pi^J$ stationary. For every reversible such dynamics, the
relaxation time is at least $e^{An^{7/30}}/(Cn^{7/30})$. Finally, the Gibbs
mass of the slow initial configurations tends to one almost surely along the
full sequence of system sizes, however the disorders of different sizes are
coupled. The main change in the proof is a lower bound on $F_n$ from the
small-deviation inequality of Paouris and Valettas for convex functions of
Gaussian vectors~\cite[Theorem~2.1 and
Remark~2.4(3)]{paouris-valettas-2018}, in place of Chebyshev's inequality.
We have formalized \cref{thm:barrier-730} and \cref{b:cor:subcritical} in
Lean, but not \cref{thm:barrier-strong}; see \cref{sec:intro-formal}.

\begin{theorem}[Mixing from typical equilibrium states]\label{thm:upper}
For every fixed $\beta>1$ and every fixed $\omega\in(0,1/4)$,
\[
 \PP_{J,\,X_0\sim\pi^J}\bigl\{d^J\bigl(X_0,e^{n^{1-\omega}}\bigr)\le 1/4\bigr\}\longrightarrow1;
 \quad\text{equivalently}\quad
 \PP_{J,\,X_0\sim\pi^J}\bigl\{T^J(X_0)\le e^{n^{1-\omega}}\bigr\}\longrightarrow1 .
\]
\end{theorem}

As in \cref{thm:barrier-730}, the configuration $X_0$ is sampled once and
then held fixed when the distance $d^J(X_0,t)$ is formed, and the clocks
have rate one per site, as in \cref{sec:model-dynamics}. By the argument of
\cref{lem:model-joint}, applied to the complementary events
$\{d^J(X_0,t)>1/4\}$, \cref{thm:upper} is equivalent to the convergence to
zero, in probability over the disorder, of the Gibbs mass of the initial
configurations $x$ with $d^J(x,e^{n^{1-\omega}})>1/4$. The theorem asserts
nothing at $\omega=1/4$. It improves the exponent $1-1/40000000$
of~\cite[Theorem~1]{oai-typical-upper}, and it is proved in
\cref{part:upper}. \statusIntroShort

For the discrete chain, we prove the same bound after
$\lfloor e^{n^{1-\omega}}\rfloor$ update attempts. Since one unit of
continuous time contains $n$ update attempts on average, this number of
attempts corresponds to the time $e^{n^{1-\omega}}/n$. Together with
\cref{thm:barrier-730}, we obtain two-sided bounds in both time
conventions.

\begin{corollary}[Discrete time]\label{cor:upper-discrete}
For every fixed $\beta>1$ and $\omega\in(0,1/4)$,
\[
\PP_{J,X_0\sim\pi^J}\{d^J_{\mathrm{disc}}(X_0,\lfloor e^{n^{1-\omega}}\rfloor)\le1/4\}\to1.
\]
\end{corollary}

\begin{corollary}[Two-sided typical-start bounds]\label{cor:two-sided}
For every fixed $\beta>1$ and $\omega\in(0,1/4)$, with $A_\beta$ from \cref{thm:barrier-730},
\begin{gather*}
\PP_{J,X_0\sim\pi^J}\bigl\{e^{A_\beta n^{7/30}}<T^J(X_0)\le e^{n^{1-\omega}}\bigr\}\to1,\\
\PP_{J,X_0\sim\pi^J}\bigl\{\lfloor e^{A_\beta n^{7/30}}\rfloor<T^J_{\mathrm{disc}}(X_0)\le \lfloor e^{n^{1-\omega}}\rfloor\bigr\}\to1 .
\end{gather*}
\end{corollary}

In the notation of \cref{sec:model-asymptotics}, \cref{cor:two-sided}
implies that if $\log T^J(X_0)=n^{\zeta+o_\PP(1)}$ under
$\PP_{J,\,X_0\sim\pi^J}$ for some $\zeta$, then $7/30\le\zeta\le3/4$, and the
same holds for $T^J_{\mathrm{disc}}(X_0)$. The value $\zeta=1/3$ suggested by
the physics literature (\cref{sec:intro-dynamics}) lies in this interval.
\Cref{cor:upper-discrete,cor:two-sided} are proved in \cref{u:sec:proof}.

Both arguments stop at the exponent $1/4$. The proof of
\cref{thm:barrier-730} needs $n\rho^4/\log n\to\infty$ for the overlap scale
$\rho$, and the proof of \cref{thm:upper} needs the chain of inequalities
$\omega<\mathsf q<\mathsf s_{\rm b}<\mathsf s_{\rm seq}<\mathsf e<\mathsf u_{\rm E}<\mathsf k_{\rm f}/2\le1/4$
of \cref{u:sec:budget}. We do not know whether this coincidence has a common
explanation.

\section{Two known results: fluctuations and the mean bias}\label{sec:intro-known}

The other two results that we present concern the fluctuations of $F_n$ and
the mean bias. The first was proved by OpenAI. The
paper~\cite{oai-fluctuations} uses the uniform probability measure on
$\Sigma_n$ instead of the counting measure in \eqref{eq:intro-model}. This
changes $F_n$ by the constant $n\log2$ and leaves the following statement
unchanged (see \cref{sec:model-dictionary}).

\begin{theorem}[{\cite[Theorem~1.1]{oai-fluctuations}}]\label{f:main}
For every fixed $\beta>1$ and every $\eta>0$, there exists $n_0(\beta,\eta)$
such that, for all $n\ge n_0$,
\begin{align}
 n^{1/6-\eta}&\le\sqrt{\Var(F_n)}\le n^{1/6+\eta},\label{f:sd}\\
 n^{1/3-\eta}&\le\Var(F_n)\le n^{1/3+\eta}.\label{f:variance}
\end{align}
Moreover, for every $\eta>0$,
\begin{equation}\label{f:typical}
 \PP\bigl\{n^{1/6-\eta}\le\abs{F_n-\E F_n}\le n^{1/6+\eta}\bigr\}
 \longrightarrow1.
\end{equation}
\end{theorem}

The proof of \cref{f:main} in~\cite{oai-fluctuations} reduces it to a
two-sided bound on positive logarithmic moments of the centered completed
log partition function, which we state as \cref{f:window} and discuss in
\cref{sec:intro-fluct}. Our formal proof of
\cref{f:main} includes a formal proof of this bound
(\cref{sec:intro-formal}).

Let $p_n(\beta)=n^{-1}\E F_n$, so that the mean bias is
$\Delta_{n,\beta}=P_\beta-p_n(\beta)$. By Guerra's
bound~\cite[Theorem~3]{guerra-2003} and the Parisi
formula~\cite[Theorem~1.1]{parisi-formula}, we have $\Delta_{n,\beta}\ge0$
for every $n$. A lower bound on the variance implies a lower bound on the
mean bias. Aronow and Lopatto proved that for every fixed $\beta>1$ there
exists $c_\beta>0$ such that
\begin{equation}\label{eq:intro-variance-deficit}
 \Delta_{n,\beta}\ge c_\beta\left(\frac{\Var(F_n)+\beta^2/2}{n}\right)^{5/4}
\end{equation}
for every $n\ge2$~\cite[Lemma~7.4 and Remark~7.14, equation~(7.68)]{quantitative-parisi};
we state it as \cref{prop:bias-variance-deficit}. The term $\beta^2/2$ is
the variance of the Gaussian variable that completes the covariance of
$\beta H_n$ (\cref{sec:model-completion}). Inserting \eqref{f:variance}
into \eqref{eq:intro-variance-deficit}, we obtain the following corollary,
which is already recorded in~\cite[Section~1.6, p.~11]{hexagon-bias}.

\begin{corollary}[Mean bias]\label{cor:mean-bias}
For every fixed $\beta>1$ and every $\varepsilon>0$, there exist
$c_{\beta,\varepsilon}>0$ and $n_0(\beta,\varepsilon)$ such that
\begin{equation}\label{eq:mean-bias-lower}
 \Delta_{n,\beta}\ge c_{\beta,\varepsilon}n^{-5/6-\varepsilon}
 \qquad(n\ge n_0(\beta,\varepsilon)).
\end{equation}
Equivalently, $nP_\beta-\E F_n\ge c_{\beta,\varepsilon}n^{1/6-\varepsilon}$ for
$n\ge n_0(\beta,\varepsilon)$.
\end{corollary}

Together with the upper bound of~\cite{hexagon-bias}, \cref{cor:mean-bias}
implies that if $\Delta_{n,\beta}=n^{-a+o(1)}$ for some $a$, then
$7/10\le a\le5/6$. \Cref{sec:bias} proves \cref{cor:mean-bias}, which does
not use \cref{thm:barrier-730}, and discusses what it leaves open. If
$\liminf_{n\to\infty}n^{-1/3}\Var(F_n)>0$, the same inequality implies that
$\Delta_{n,\beta}\ge c_\beta n^{-5/6}$ for all sufficiently large $n$, as
observed in~\cite[Section~1.6, equation~(1.4)]{hexagon-bias}; this
hypothesis follows from the convergence of $n^{-1/3}\Var(F_n)$ to a positive
constant, which is stated in~\cite[Theorem~1.2]{oai-limiting-law}. Our formal
proof of \cref{cor:mean-bias} combines the formal proof of \cref{f:main}
with a formal proof of \eqref{eq:intro-variance-deficit}.

\section{Related results on the dynamics}\label{sec:intro-dynamics}

\paragraph{Physics.}
The slow relaxation of finite SK systems has been studied in physics since
the early 1980s. Sompolinsky and Zippelius developed the dynamical mean-field
theory of the spin-glass
phase~\cite{sompolinsky-zippelius-1981,sompolinsky-zippelius-1982}, in which
the relaxation times of a finite system diverge with its size $n$. In
simulations of the Ising SK model, Mackenzie and Young reported two families of
time scales $\tau_n$. For observables invariant under global spin reversal,
the behavior of $\log\tau_n$ was consistent with $n^{1/4}$; for the reversal
itself, the disorder average of $\log\tau_n$ was consistent with
$n^{1/2}$~\cite{mackenzie-young-1982,mackenzie-young-1983}. Later work
revised this picture. Vertechi and Virasoro computed energy barriers between
metastable states numerically and found growth like $n^\alpha$ with
$\alpha=0.34\pm0.08$~\cite{vertechi-virasoro-1989}. Colborne's Monte Carlo
study gave the exponent $0.25\pm0.05$ for the barrier that governs the
ergodic time~\cite{colborne-1990}. Kinzelbach and Horner predicted
free-energy barriers of order $n^{1/3}$ from a resummed perturbative
analysis of the dynamics of finite systems~\cite{kinzelbach-horner-1991}.
The simulations of Billoire and Marinari~\cite{billoire-marinari-2001}, of
Bittner and Janke~\cite{bittner-janke-2006}, and of
Billoire~\cite{billoire-2010} are consistent with $\log\tau_n\propto n^{1/3}$,
and so is the computation of spectral gaps of small systems by Monthus and
Garel~\cite{monthus-garel-2009}. Aspelmeier and Moore studied barriers
between solutions of the Thouless--Anderson--Palmer
equations~\cite{thouless-anderson-palmer-1977,bray-moore-1981}. They found
that barriers of order $n^{1/3}$ are one of several possibilities below a
threshold for the free-energy density~\cite{aspelmeier-moore-2022}.

These works concern different quantities: energy barriers, free-energy
barriers along an overlap coordinate, autocorrelation times, and spectral
gaps. They use binary as well as Gaussian couplings, Metropolis as well as
heat-bath updates, and moderate system sizes. They suggest that the logarithm
of the relaxation time from a typical configuration grows like $n^{1/3}$,
but they do not determine a mixing exponent. A different question concerns
times that stay bounded as $n$ grows. When $n\to\infty$ is taken first, the
dynamics from a random configuration ages and does not reach
equilibrium~\cite{cugliandolo-kurchan-1994}; we refer
to~\cite[Section~3.7]{bouchaud-cugliandolo-kurchan-mezard-1997} for a
discussion of the two orders of limits.

\paragraph{High and critical temperature.}
Rigorous results on the dynamics depend strongly on the temperature. At high
temperature, functional inequalities imply fast mixing from every initial
configuration. Bauerschmidt and Bodineau proved a logarithmic Sobolev
inequality, uniform in $n$, for a Dirichlet form of gradient type under a
condition on the spectral width of the interaction matrix, which applies to
SK when $\beta<1/4$~\cite[Theorem~1 and Corollary~2]{bauerschmidt-bodineau-2019}.
Eldan, Koehler, and Zeitouni proved a Poincar\'e inequality for the
heat-bath Dirichlet form itself under a spectral condition of the same kind,
and deduced a polynomial bound on the mixing time from every initial
configuration~\cite[Theorems~1 and~11]{eldan-koehler-zeitouni-2022}. Anari,
Jain, Koehler, Pham, and Vuong obtained mixing in $O(n\log n)$ updates in the
same range through entropic
independence~\cite[Theorem~12]{anari-jain-koehler-pham-vuong-2022}.
Anari, Koehler, and Vuong extended this range to a threshold of approximately
$0.295$ using localization schemes~\cite[Theorem~1 and Section~1.3]{anari-koehler-vuong-2024}. Wang
extended mixing in $O(n\log n)$ updates to every $\beta<1/2$
\cite[Theorem~1.1]{wang-2026}, and Boban, Li, and Oveis Gharan proved
polynomial mixing for $\beta<1/2+\varepsilon_0$, for an absolute constant
$\varepsilon_0>0$~\cite[Theorem~1.5 and Corollary~1.6]{boban-li-oveis-gharan-2026}.
With external fields sufficiently large at every site and their average
bounded by a fixed constant $H$, Paliy and Sohn proved a modified
logarithmic Sobolev inequality and mixing in $O_{\beta,H}(n\log n)$ update
attempts at every fixed $\beta$, with probability tending to
one~\cite[Corollary~1.5]{paliy-sohn-2026}.
For every fixed $\beta<1$, OpenAI proved that the spectral gap of the
continuous-time chain is bounded below by a positive constant with
probability tending to one~\cite[Theorem~1.1]{oai-spectral-gap}, and that
the chain started from the worst initial configuration exhibits cutoff at
time $(\log n)/(2\lambda(\beta))$, for a deterministic
$\lambda(\beta)>0$~\cite[Theorem~1.1]{oai-cutoff}. At the critical
temperature $\beta=1$, OpenAI proved that, for every deterministic $t_n$
with $t_n=o(n^{2/3})$, the Gibbs mass of initial configurations $x$ with
$d^J(x,t_n)>1/4$ tends to one in probability~\cite[Theorem~1.1]{oai-critical-slowing},
and that the mixing time is $n^{2/3+o(1)}$~\cite[Theorem~1.1]{oai-critical-mixing}.

\paragraph{Low temperature.}
At low temperature, Ben Arous and Jagannath showed that a free-energy barrier
in the overlap of two replicas forces an exponentially small spectral gap
in mean-field spin glasses~\cite[Theorem~1.3]{ben-arous-jagannath-2018}.
Their sufficient conditions for such a barrier include the vanishing of the
second derivative at zero of the covariance
function~\cite[Theorem~1.9]{ben-arous-jagannath-2018}, so they do not apply
to the quadratic covariance of SK. For the pure spherical $p$-spin models
with $p\ge3$, Gheissari and Jagannath proved that the spectral gap of the
Langevin dynamics is exponentially small at sufficiently low
temperature~\cite[Theorem~1]{gheissari-jagannath-2019}. For SK, Sellke proved
that there exist absolute constants $\beta_*,c>0$ such that, for every
$\beta>\beta_*$, the mixing time from the worst initial configuration is at
least $e^{c\beta n}$ with probability tending to
one~\cite[Theorem~1.1]{sellke-2025}. The slow initial configurations in his
proof are configurations at which almost every spin is aligned with its
local field by a uniform margin. He notes after his theorem that the chain may mix much
faster from a random initial configuration. El Alaoui, Montanari, and Sellke
proved that for $\beta>1$ no family of sampling algorithms that is stable
under small perturbations of the disorder approximates the Gibbs measure in
normalized Wasserstein distance~\cite[Theorem~2.6]{el-alaoui-montanari-sellke-sampling}.
Neither result determines the Gibbs mass of the initial configurations from
which the heat-bath chain mixes slowly.

OpenAI proved a lower bound and an upper bound for initial configurations
sampled from the Gibbs measure that hold throughout $\beta>1$.
By~\cite[Theorem~1.1]{oai-barriers}, the conclusions of \cref{thm:barrier-730}
hold with the time $\exp(n^{1/10000})$ and the threshold $\tau=1/4$. In a
companion preprint, OpenAI proved that
$\PP_{J,\,X_0\sim\pi^J}\{T^J(X_0)\le\exp(n^{1-c_\beta})\}\to1$ for some
$c_\beta\in(0,1)$, and the proof permits
$c_\beta=1/40000000$~\cite[Theorem~1]{oai-typical-upper}. Combining the two
bounds, and using that $t\mapsto d^J(x,t)$ is continuous and nonincreasing,
they obtained a two-sided bound on $T^J(X_0)$ with these
exponents~\cite[Corollary~2]{oai-typical-upper}. The upper bound
of~\cite{oai-typical-upper} is stated only in continuous time.

\Cref{thm:barrier-730} improves the exponent $1/10000$ of~\cite{oai-barriers}
to $7/30$, and it holds for every fixed threshold below $1/2$ with a
coefficient that does not depend on the threshold. \Cref{thm:upper} improves
the exponent $1-1/40000000$ of~\cite{oai-typical-upper} to $1-\omega$ for
every fixed $\omega<1/4$, and \cref{cor:upper-discrete} gives the same bound
for the discrete chain. For $\beta>\beta_*$, we find from \cref{thm:upper}
that the mixing time from a typical initial configuration is much smaller
than the lower bound
$e^{c\beta n}$ of~\cite{sellke-2025} on the mixing time from the worst initial
configuration. Our results do not determine the exponent of the typical
mixing time, which by \cref{cor:two-sided} lies in $[7/30,3/4]$ if it exists.
They are compatible with the value $1/3$ suggested by the physics literature,
but they do not decide whether this value is correct. The value $1/3$ also
appears as the exponent of $\Var(F_n)$ in \cref{f:main}; we know of no
argument relating the two exponents. The coefficient $A_\beta$ depends on
$\beta$, and our proof does not control it as $\beta\downarrow1$ or
$\beta\to\infty$. Similarly, the rate of convergence in \cref{thm:upper}
depends on $\beta$ and $\omega$, and we assert no uniformity in either
parameter.

\section{The fluctuation scale and the mean bias}\label{sec:intro-fluct}

For a survey of free-energy fluctuations in the SK and spherical SK models,
including their multi-species variants and the effects of an external field,
see Collins-Woodfin and Le~\cite{collins-woodfin-le-survey-2025}.

\paragraph{Physics.}
The prediction of the fluctuation scale of $F_n$ has a separate history in
physics. In a truncated replica theory near the critical temperature, Kondor
found that the replica-normalized free energy depends on a small positive
replica number $s$ first at order $s^5$~\cite{kondor-1983}. Crisanti,
Paladin, Sommers, and Vulpiani interpreted the replica number as a tilt of
the disorder distribution, and predicted that the free energy per spin
fluctuates on the scale $n^{-5/6}$ in the low-temperature phase at zero
field~\cite{crisanti-paladin-sommers-vulpiani-1992}. At fixed $\beta$, this
is the scale $n^{1/6}$ for $F_n$. Parisi and Rizzo extended the fifth-order
calculation throughout the low-temperature phase and computed the resulting
large deviations of the free energy~\cite{parisi-rizzo-2008,parisi-rizzo-2009}.
Passing from these large deviations to the width of typical fluctuations
requires matching the tail of the distribution to its center. This is an
additional assumption, which they discuss
in~\cite[Section~VI]{parisi-rizzo-2009}; the limits $n\to\infty$ and
$s\downarrow0$ cannot simply be interchanged.
Aspelmeier, Billoire, Marinari, and Moore argued that a system of $n$ spins
resolves about $n^{1/6}$ levels of replica symmetry breaking. From this they
derived both a variance of order $n^{1/3}$ for the total free energy and a
correction of order $n^{-2/3}$ to its mean per
spin~\cite[Section~2]{aspelmeier-billoire-marinari-moore-2008}. Aspelmeier
derived exact identities relating the variance of the free energy to
overlaps between systems with correlated disorder~\cite{aspelmeier-2008}. At
zero temperature, numerical studies of the width of the distribution of the
ground-state energy have produced exponents between $3/4$ and $5/6$ for the
standard deviation per spin, and the fits are sensitive to corrections to
scaling~\cite{bouchaud-krzakala-martin-2003,boettcher-2005,boettcher-2010,palassini-2008}.

\paragraph{Rigorous results.}
Rigorous results on the fluctuations of $F_n$ depend on the temperature and
the field. For $0<\beta<1$ at zero field, $F_n$ has Gaussian fluctuations of
order one, as shown by Aizenman, Lebowitz, and Ruelle~\cite{alr-1987}. Comets
and Neveu studied these fluctuations by stochastic
calculus~\cite{comets-neveu-1995}. At $\beta=1$ the variance grows
logarithmically. Chen and Lam proved an upper bound of order
$(\log n)^2$~\cite[Theorem~1.1]{chen-lam-2019}, Schertzer proved the upper
bound $(\log n)/4+C$ and a lower bound of order
$\log\log\log n$~\cite[Theorems~1 and~2]{schertzer-2026}, Tierz bounded the
variance for every $n$ in terms of the spin-glass
susceptibility~\cite[Theorem~1.2]{tierz-critical-2026}, and Du and Huang
proved that $\Var(F_n)=(\log n)/6+O(1)$ and that the fluctuations have a
Gaussian limit~\cite[Theorem~1.3]{du-huang-2026}. Dey and Kang obtained the
variance asymptotics and a Gaussian limit along sequences $\beta_n\uparrow1$
for which $n^{1/3}(1-\beta_n^2)$ converges to a limit in
$(0,\infty]$~\cite[Theorem~1.1]{dey-kang-2026}. With a fixed nonzero external field,
Chen, Dey, and Panchenko proved that the variance is of order $n$ and that a
central limit theorem
holds~\cite[Theorems~1 and~2]{chen-dey-panchenko-2017}. Dey and Wu identify
a crossover at field strength $n^{-1/4}$ in the high-temperature
regime~\cite[Theorem~1.3]{dey-wu-2023}. The spin space also
matters: for the spherical SK model at low temperature, Baik and Lee proved
that the free energy per spin has Tracy--Widom fluctuations of order
$n^{-2/3}$~\cite[Theorems~2.10 and~2.11]{baik-lee-2016}, so that the total
free energy fluctuates on the scale $n^{1/3}$ rather than $n^{1/6}$.

For fixed $\beta>1$ at zero field, the Gaussian Poincar\'e inequality implies
that $\Var(F_n)\le C_\beta n$. Chatterjee improved this to $O_\beta(n/\log n)$ and
related this superconcentration to disorder
chaos~\cite[Theorem~1.5]{chatterjee-2009}. He also proved that the fluctuations
of $F_n$ are at least of constant order at every fixed $\beta>0$ and zero
field, in the sense of nonconcentration on sufficiently short
intervals~\cite[Theorem~2.4]{chatterjee-lower-2019}. Aronow and Lopatto proved
$c_\beta n^{4/15}\le\Var(F_n)\le C_v n^{7/15}$ for all sufficiently large
$n$~\cite[Theorem~1.4]{quantitative-parisi}, using quantitative comparisons
between positive fractional moments of $Z_n$ and constrained Parisi
functionals~\cite[Theorem~1.1]{quantitative-parisi}. The sharp exponent was
then obtained by OpenAI~\cite[Theorem~1.1]{oai-fluctuations}, as stated in
\cref{f:main}.

\Cref{f:main} is a statement about exponents. It identifies neither a
limiting law nor a positive limit of $n^{-1/3}\Var(F_n)$. Both are stated in
the companion preprint~\cite[Theorems~1.1 and~1.2]{oai-limiting-law}: there,
$n^{-1/3}\Var(F_n)$ converges to a constant $a_2(\beta)\in(0,\infty)$, and
$(F_n-\E F_n)/\sqrt{\Var(F_n)}$ converges in law to a nondegenerate
probability measure with mean zero and variance one. The proofs
in~\cite{oai-limiting-law} use the exponent of \cref{f:main} and the
comparison estimates of~\cite{oai-fluctuations};
see~\cite[Appendix~A]{oai-limiting-law}. We do not use~\cite{oai-limiting-law} in
this paper. On the side of large deviations, Pei identified the coefficient
of the fifth-order term in the constrained Parisi functional and the
resulting rate for upward deviations of $F_n/n$, with $n\to\infty$ taken
before the deviation tends to zero~\cite[Theorem~1]{pei-fifth-order-2026}.

The proof of \cref{f:main} in~\cite{oai-fluctuations} reduces it to a
two-sided bound on a positive logarithmic moment. Let $F_n^c=F_n+\beta G$,
where $G$ is a centered Gaussian variable of variance $1/2$, independent of
the disorder. The variable $\beta G$ completes the covariance of $\beta H_n$ to
$n\beta^2R(x,y)^2/2$, and we call $F_n^c$ the completed log partition
function (\cref{sec:model-completion}). For every fixed $m\in(0,1/300)$ and
$s_n=n^{-1/6+5m}$, it is shown in~\cite[Proposition~7.5]{oai-fluctuations}
that $\log\E\exp\{s_n(F_n^c-\E F_n^c)\}$ lies between constant multiples of
$n^{20m}$ and $n^{100m}$ for all sufficiently large $n$. We state this bound
precisely as \cref{f:window}. The deduction of \cref{f:main} from this bound
in~\cite[Lemmas~7.6 and~7.7]{oai-fluctuations} uses an argument of Aronow
and Lopatto~\cite[Proposition~5.1]{quantitative-parisi}, based on the
log-concavity of an exponential augmentation of $F_n^c$. We reproduce it in
\cref{sec:fluct}.

\paragraph{The mean bias.}
For fixed $\beta>1$, Aronow and Lopatto proved that
$c_\beta n^{-11/12}\le\Delta_{n,\beta}\le C_\beta n^{-2/3}$ for all
sufficiently large $n$~\cite[Theorem~1.4]{quantitative-parisi}, and later
that $\Delta_{n,\beta}\le C_b n^{-7/10}$ for every
$n\ge2$~\cite[Theorem~1.1]{hexagon-bias}. Their lower bound $n^{-11/12}$
follows from \eqref{eq:intro-variance-deficit} and the lower bound
$c_\beta n^{4/15}$ on the variance; with \eqref{f:variance} in place of this
bound, the exponent becomes $5/6$, as in \cref{cor:mean-bias}. The proof of
\eqref{eq:intro-variance-deficit} balances a gain that is linear in a small
positive tilt and proportional to the variance against the fifth-order cost
of constraining the Parisi measure to have mass at least $s$ at
zero~\cite[Lemma~7.2]{quantitative-parisi}. In physics, the correction is
expected to be of order $n^{-a}$ with $a<1$ at low temperature, and
two values have been proposed. The value $a=2/3$ was supported by the
zero-temperature numerics of Bouchaud, Krzakala, and
Martin~\cite{bouchaud-krzakala-martin-2003} and of
Boettcher~\cite{boettcher-2005}, by Billoire's numerical integration along
the interpolation of Guerra and Toninelli~\cite{guerra-toninelli-2002}
in~\cite{billoire-2006}, and by the
argument of Aspelmeier, Billoire, Marinari, and Moore mentioned above.
Franz, Parisi, and Ricci-Tersenghi argued that the correct value is
$a=3/4$, possibly up to logarithmic
factors~\cite[equations~(13)--(17)]{franz-parisi-ricci-2026}. The bound
of~\cite{hexagon-bias} excludes a nonzero correction of order $n^{-2/3}$ at
every fixed $\beta>1$ and is consistent with the value $3/4$. A fuller account
of these predictions is given in~\cite[Section~1.4]{hexagon-bias}.

\paragraph{Normalizations.}
We have converted the results quoted in this section and the previous one to
the normalization \eqref{eq:intro-model}. Some of the cited works sum over
ordered pairs of sites, which changes the inverse temperature by a constant
factor; for instance, the critical inverse temperature
of~\cite{chen-lam-2019} is $1/\sqrt2$ in its own normalization.
\Cref{sec:model-dictionary} compares our normalization with those of the
works on which our proofs rely.

\section{Formal verification}\label{sec:intro-formal}

OpenAI has published formalizations of~\cite{oai-barriers}
and~\cite{oai-fluctuations} in the Lean~4 proof
assistant~\cite{lean4,mathlib,oai-math-lean}. The main formal theorem on the
fluctuation exponents takes the two-sided bound of \cref{f:window} as an
explicit hypothesis. In~\cite{oai-fluctuations}, that bound is the
conclusion of Proposition~7.5, which is proved there on paper. We have
completed this formal proof: our Lean development proves the three
conclusions of \cref{f:main} with no hypothesis beyond the definition of the
model and $\beta>1$. It follows the mathematical route
of~\cite{oai-fluctuations}; the work consisted in carrying out that route in
full detail for the comparison problems of~\cite[Section~3]{oai-fluctuations}
whose minimizers have a positive lower radius, and for the successive
comparisons built from them.

We have also formalized \cref{thm:barrier-730}, \cref{b:cor:subcritical},
and \cref{cor:mean-bias}. In these formal proofs, the upper bound on the
mean bias of~\cite[Theorem~1.1]{hexagon-bias} and the variance--bias
inequality \eqref{eq:intro-variance-deficit} are proved, not assumed;
their formal proofs come from our Lean development~\cite{hexagon-bias-lean}
for~\cite{hexagon-bias}. The formal proof of \cref{cor:mean-bias} also
proves the identities that relate the normalizations of \cref{f:main} and
\eqref{eq:intro-variance-deficit}. The formal proof of
\cref{thm:barrier-730} has the structure of the written proof, but it uses
a weaker form of the local expansion of \cref{b:prop:os-expansion}, which
suffices for the same exponent. \Cref{thm:barrier-strong} is not
formalized.

In their source packages, each of these four formal theorems was compared with a separately written
statement of the same result that depends only on Mathlib, and its proof was
checked to use only the standard axioms \texttt{propext},
\texttt{Classical.choice}, and \texttt{Quot.sound}. The declarations on
which its statement and proof depend were then checked again by three
kernels: the kernel of Lean and the independent kernels nanoda and con-ron.
\Cref{sec:formal} states the formal theorems, explains how they correspond
to the statements of this paper, describes what the completion of the proof
of \cref{f:main} had to establish, and describes these checks.

\statusIntroFormal{} The proofs in this paper are complete without reference
to the formalization.

\section{Methods and their sources}\label{sec:intro-methods}

We describe the proofs of \cref{thm:barrier-730,thm:upper} and the sources
of their ideas. A more detailed overview is given in \cref{sec:overview}.

\subsection{The lower bound}

\paragraph{From overlap constraints to slow mixing.}
We follow the strategy of~\cite{oai-barriers}. Its first step reduces the
problem to a statement about the stationary
chain~\cite[Section~8.1]{oai-barriers}. Let $v$ be a reference
configuration with $R(v,X_0)$ bounded away from zero. By spin-flip symmetry,
the half-space $\{y:R(v,y)\le0\}$ has Gibbs mass at least $1/2$. If
$d^J(X_0,t)\le\tau<1/2$, the chain started at $X_0$ is in this
half-space at time $t$ with probability at least $1/2-\tau$. A single update
changes $R(v,\cdot)$ by at most $2/n$, so the chain must cross every
intermediate overlap level. It suffices to show that the chain started from
$X_0\sim\pi^J$ crosses these levels before time $t$ only with small
probability. At each deterministic time the stationary chain has law
$\pi^J$, and a union bound over times controls the discrete chain; an
independent Poisson clock of rate $n$ then gives the continuous-time chain.
This argument is related to bottleneck estimates for Markov chains
(see~\cite[Theorem~7.4]{levin-peres-2017}). Here, however, the bottleneck
depends on the initial configuration, and it must separate almost every
initial configuration in Gibbs measure from equilibrium.

To show that crossings are rare, we build, as in~\cite{oai-barriers}, a
growing family of reference configurations along the path. The references
are chosen from independent Gibbs samples, and their overlaps with the
current configuration are constrained simultaneously. At each crossing
level, we find a new reference with large overlap with the current configuration
using a bound on the Gibbs mass of overlap
neighborhoods~\cite[Proposition~6.1]{oai-barriers}. This bound is derived
from an upper tail bound for the energy and an estimate for small Gaussian
rotations of the disorder~\cite[Lemmas~5.1 and~5.2]{oai-barriers}. We
then search backward for the crossing of the overlap level of the new
reference, while overlap constraints keep the overlaps with the earlier
references almost unchanged. Iterating this step creates many simultaneous
constraints at a single time. We arrange the references in blocks of fixed
length, so that a conflict deletes only a short final segment of a block, and
we draw the overlap thresholds at random, one level at a time; these features
allow us to obtain the final
probability estimate~\cite[Sections~8.3--8.5]{oai-barriers}.

The overlap constraints have the following form. Let $\varepsilon$ be a
tolerance. If the mean $r$ of the overlaps $R(v,x)$ and $R(v,w)$ lies in a
fixed compact subinterval of $(0,q_\beta)$ and $R(w,x)\ge r+\rho$, then
$\abs{R(v,x)-R(v,w)}$ does not lie in $[2\varepsilon,4\varepsilon]$. In words, a large overlap between $w$ and $x$
forces their overlaps with $v$ to agree. Since overlaps change by at most
$2/n$ in one update, excluding this band keeps the two overlaps close along
a path. To show that such a constraint fails with small Gibbs probability, we show
that the partition function restricted to the triples $(v,x,w)$ with
violating overlaps is small compared with $Z_n^3$. In physics, free energies constrained by the overlap
with an equilibrium reference configuration go back to Franz and
Parisi~\cite{franz-parisi-1995}. The work of Ben Arous and
Jagannath~\cite{ben-arous-jagannath-2018} is an earlier rigorous argument
that passes from such constrained free energies to the dynamics, there for
the spectral gap.

\paragraph{Constrained comparisons and the scalar problem.}
We bound constrained partition functions by Gaussian interpolation.
Guerra's hierarchical interpolation~\cite{guerra-2003} gives the upper bound
in the Parisi formula. Talagrand extended it to systems of replicas with
prescribed overlaps, in the proof of the matching lower
bound~\cite[Section~3]{parisi-formula} and in the two- and three-replica
bounds of~\cite[Propositions~3.4 and~4.4]{talagrand-2007}. These bounds
already retain a cost when the prescribed overlaps differ from the endpoint
of the interpolation. Bovier and Klimovsky developed the interpolation for
multidimensional spins~\cite{bovier-klimovsky-2009}. Chen studied the
two-dimensional Guerra--Talagrand bound and its variational
representation~\cite{chen-2017}, and Panchenko proved the vector-spin
variational formula and the asymptotic sharpness of the constrained
bounds~\cite{panchenko-2018}. We use the matrix comparison for finite hierarchies
of~\cite[Proposition~2.3]{oai-barriers} for the quadratic covariance of SK,
which allows any finite number of replicas and endpoint matrices whose
diagonal entries need not equal one. The hierarchies themselves descend from Parisi's
recursion~\cite{parisi-1979} and from Ruelle's probability
cascades~\cite{ruelle-1987}; see~\cite{panchenko-2013}. For the mixed even
$p$-spin models with exponentially summable coefficients, including SK,
Aronow and Lopatto identified the limiting overlap array averaged over the
disorder~\cite[Theorems~1.1 and~1.2]{al-overlap-structure}. At zero field, the
absolute overlaps have the law of the overlap array $(U_{ab})_{a<b}$ of the
corresponding Ruelle probability cascade, and the signed array has the law of
$(\epsilon_a\epsilon_bU_{ab})_{a<b}$, where the replica signs $\epsilon_a$
are independent symmetric signs, independent of the cascade. These are
limiting statements, while the proof below needs estimates at finite $n$ on
shrinking scales.

We obtain the pressure deficits from the scalar Parisi problem. Auffinger and
Chen proved that the Parisi functional has a unique minimizer, and they
expressed it through a stochastic control
problem~\cite[Theorems~2 and~3 and Corollary~1]{auffinger-chen-uniqueness-2015}.
They also proved regularity properties of the Parisi equation and of its
minimizers~\cite[Propositions~1 and~2 and Theorems~1 and~2]{auffinger-chen-2015}.
Jagannath and Tobasco gave a dynamic-programming treatment of the
functional~\cite[Lemma~18]{jagannath-tobasco-2016} and characterized its
minimizers through their first
variation~\cite[Proposition~1.1 and Corollary~3.10]{jagannath-tobasco-2017};
Chen obtained the same characterization~\cite[Theorem~2 and
Proposition~1]{chen-2017}. Toninelli proved that replica symmetry is broken
at zero field for $\beta>1$~\cite{toninelli-2002}. Let $u$ be the spatial
derivative of the solution of the Parisi equation at the minimizer, and let
$Y$ be the associated diffusion. The identity $\E u(t,Y_t)^2=t$ at every
point $t$ of the support is due to
Talagrand~\cite{talagrand-2006-parisi-measures}; see
also~\cite[Theorem~5]{auffinger-chen-2015}, \cite[Proposition~1]{chen-2017},
and~\cite[Lemma~2.2]{full-rsb}. OpenAI proved it in the normalization used
here, together with the regularity of the scalar evolution that it
requires~\cite[Lemma~3.1 and Proposition~3.2]{oai-barriers}; the argument
of~\cite{oai-barriers} does not assume a description of the support, and it
treats separately the case of a forbidden overlap
interval~\cite[Section~8.2]{oai-barriers}. By~\cite[Theorem~1.1]{full-rsb}, the identity holds on the whole
interval $(0,q_\beta)$, and this case does not arise. Differentiating the
identity once and twice along this interval, we obtain two stationarity
identities,
\[
 \beta^2\E\chi(t,Y_t)^2=1,\qquad
 \E w(t,Y_t)^2=2\alpha(t)\E\chi(t,Y_t)^3\qquad(0<t<q_\beta),
\]
where $\chi$ and $w$ are the second and third spatial derivatives and
$\alpha$ is the distribution function of the Parisi measure
(\cref{sec:scalar}). Both identities appear, in the normalization of energy
units, in the proof of~\cite[Lemma~7.2]{quantitative-parisi}
(see~\cite[equation~(7.12)]{quantitative-parisi} and the identity preceding
it), and the differentiated moment identity is also
in~\cite[Lemma~2.5]{full-rsb}. The first identity expresses, in scalar form, the
marginal stability of the Parisi solution, which De Dominicis and Kondor
found in a truncated replica model~\cite{de-dominicis-kondor-1983}; Crisanti
and Rizzo wrote it in physical
variables~\cite[equations~(22) and~(23)]{crisanti-rizzo-2002}.

Positive fractional moments $\E Z_n^s$ have their own genealogy. Talagrand
computed the limits of their normalized logarithms~\cite{talagrand-2007-large-deviations}.
For SK at zero field and fixed $s\in(0,1)$, the limit of
$(ns)^{-1}\log\E Z_n^s$ is the minimum of the Parisi functional over measures
with mass at least $s$ at zero. This formula is contained in the multiscale
formula of Contucci and Mingione~\cite[Theorem~2.1]{contucci-mingione-2019},
and it was stated in this form by Chen, Guionnet, Ko, Lacroix-A-Chez-Toine,
and Mourrat~\cite[Theorem~2.1]{chen-guionnet-ko-lacroix-mourrat-2026}. Aronow
and Lopatto made the comparison quantitative when $s$ depends on
$n$~\cite[Theorem~1.1]{quantitative-parisi}, and they proved that the
constraint raises the Parisi value by an amount of order
$s^5$~\cite[Lemma~7.2 and Proposition~7.3]{quantitative-parisi}. This is a
rigorous counterpart of the replica calculations of Kondor and of Parisi and
Rizzo, and Pei identified its leading
coefficient~\cite[Theorem~1(a)]{pei-fifth-order-2026}.

\paragraph{What is new.}
Four parts of the proof are new. Together they allow overlap constraints at
the scale $\rho\asymp n^{-7/30}$, with tolerance comparable to $\rho$.
\begin{enumerate}
\item \emph{A three-replica covariance perturbation with a cubic deficit}
(\cref{sec:cubic}). In~\cite[Section~7]{oai-barriers}, the overlap
constraints near a level $r$ come from a local perturbation of the covariance
path of width $\rho^8$ with tolerance $\rho^{20}$, and the resulting pressure
deficit is of order $\rho^{43}$. We keep the block hierarchy and the
antisymmetric Gaussian contrast of~\cite[Section~7, Steps~1 and~2]{oai-barriers},
but we perturb the covariance path differently. It is positive
semidefinite at every stage, ends exactly at the prescribed overlap matrix
after a terminal correction, and its expansion uses both stationarity
identities to cancel the lower-order terms. The resulting
deficit is of order $\rho^3$ per spin, with tolerance comparable to $\rho$,
uniformly for $r$ in a compact subinterval of $(0,q_\beta)$. Its coefficient
involves $\E\chi(r,Y_r)^3$, which is positive (\cref{sec:scalar}).
\item \emph{A positive fractional-moment comparison} (\cref{sec:fractional}).
For constraints near the level $r$, the deficit $n\rho^3$ in the logarithm
of a constrained partition function is smaller than the scale $O_\beta(\sqrt n)$
supplied by the standard Gaussian concentration bound for Lipschitz
functions, so that bound is too coarse here. Instead, we bound the moment $\E Z^s$ of a constrained
partition function $Z$, with $s$ comparable to $\rho$, by a fractional-moment
version of the matrix comparison of~\cite[Proposition~2.3]{oai-barriers},
in which the outer level of the hierarchy has mass $s$, as in the
positive-moment interpolation of~\cite[Section~2]{quantitative-parisi}. The
comparison uses the profile $\max\{\alpha,3s\}$, whose cost is of order
$s^5$ by the trial bound of~\cite[Lemma~7.2]{quantitative-parisi}, together
with a bound on the response of the local perturbation to this change of
profile. Markov's inequality then bounds the disorder probability of
failure of each of these bounds on constrained partition functions by
$e^{-cn\rho^4}$.
\item \emph{One disorder event for all constraints} (\cref{sec:static}). The
denominator $Z_n^3$ is bounded below on the event
$F_n/n\ge P_\beta-c_0\rho^3$, for a fixed $c_0>0$. Let $\delta\in(0,c_0)$ be
fixed such that $\Delta_{n,\beta}\le(c_0-\delta)\rho^3$ for all
sufficiently large $n$; this margin is obtained by choosing the constant $R$
in $\rho=Rn^{-7/30}$ large. Chebyshev's inequality and the variance bound
of~\cite[Theorem~1.4]{quantitative-parisi} give
\[
 \PP\bigl\{F_n/n<P_\beta-c_0\rho^3\bigr\}
 \le\frac{\Var(F_n)}{\delta^2n^2\rho^6}
 \le C_{\beta,\delta}\,n^{-23/15}\rho^{-6},
\]
which is $O(n^{-2/15})$ for $\rho=Rn^{-7/30}$. We intersect this event with
the bounds on polynomially many constrained partition functions, and the
complement of the resulting event, whose probability tends to zero, is
discarded once and for all. On this event, the number
of reference configurations and path steps, which is exponential in
$1/\rho$, multiplies only the conditional Gibbs probability of a violation,
which is at most $e^{-cn\rho^3}$. This replaces the probability estimate
of~\cite[Lemma~4.2]{oai-barriers}.
\item \emph{Control of crossing errors through the next retained reference}
(\cref{sec:induction,sec:barrier-proof}). In~\cite[Section~8.4]{oai-barriers},
the overlap of an earlier reference with the current configuration may drift
by up to $6j\varepsilon$ after $j$ levels, and the final count requires
$p\varepsilon/\rho$ to be small for $p$ levels. Since $p$ is of order
$1/\rho$, this forces $\varepsilon$ to be much smaller than $\rho^2$.
Instead, we control the overlap of each retained reference with the next
retained reference. This overlap changes only when a deletion occurs, and the
error stays of order $\varepsilon$ at every level, rather than accumulating.
The first reference of each block keeps its next retained
reference, and at one common time the overlap of the path with the first
reference of each of the $M$ blocks lies within a fixed multiple of
$\varepsilon$ of its threshold. A count over the $m+1$ path steps of these
$M$ windows of predictable center gives the bound $(m+1)\gamma_n^M$, where
$\gamma_n$ is the ratio of the window length to the length of a threshold
interval, and $\gamma_n$ stays below one when $\varepsilon$ is a small fixed
multiple of $\rho$. This allows $\varepsilon$ comparable to $\rho$.
\end{enumerate}
The other overlap constraints use the inherited pressure bounds:
the pair bound of~\cite[Lemma~4.3]{oai-barriers}, the three-replica bound at
fixed scales of~\cite[equation~(4.20)]{oai-barriers}, and, for large
overlaps $R(w,x)$, the comparison of~\cite[Section~7, Steps~5--8]{oai-barriers}.
We also extend the bound on the Gibbs mass of overlap neighborhoods
of~\cite[Proposition~6.1]{oai-barriers} to every scale $L\to\infty$ with
$L=o(n)$; its proof is that of~\cite{oai-barriers}
(\cref{sec:references}).

\paragraph{The exponent.}
We determine the exponent by balancing these estimates. The lower
bound on $Z_n^3$ in item~3 requires the mean bias $\Delta_{n,\beta}$ to be
smaller than a small multiple of the deficit $\rho^3$. The comparison proved
in~\cite[Theorem~2.4]{oai-barriers} has error $O(n^{-1/24})$. This is much
smaller than the deficit $\rho^{43}$ at the scale $\rho=n^{-1/10000}$ used
there, but not than $\rho^3$ at the scale used here. For this reason we use
the upper bound $\Delta_{n,\beta}\le C_b n^{-7/10}$ of Aronow and
Lopatto~\cite{hexagon-bias}, and we take $\rho=Rn^{-7/30}$ for a large
constant $R$. A fixed overlap interval inside $(0,q_\beta)$ then contains
of order $1/\rho\asymp n^{7/30}$ crossing levels. By counting, we find
that the probability of crossing all of them before time $t$ is small
when $\log t$ is a small multiple of the number of levels. This leads to the
time $\exp(A_\beta n^{7/30})$. Two further conditions concern the two kinds
of union bound in the proof. For the union over reference configurations and
path steps, the conditional Gibbs error is at most $\exp(C/\rho-cn\rho^3)$,
which tends to zero if $cn\rho^3-C/\rho\to\infty$. For the disorder event,
a union over at most $Cn^d$ constrained partition functions has failure
probability at most $Cn^de^{-c'n\rho^4}$, which tends to zero if
$c'n\rho^4-d\log n\to\infty$. The
condition $n\rho^4/\log n\to\infty$ implies both. For $\rho=Rn^{-\lambda}$
with fixed $R$, it holds when $\lambda<1/4$, in particular at
$\lambda=7/30$; at $\lambda=1/4$, the second union bound fails. This is the
restriction to exponents below $1/4$ mentioned in \cref{sec:intro-results}.

\paragraph{Inputs for the lower bound.}
Besides standard tools, the proof of \cref{thm:barrier-730} uses the following
public results, which are stated in \cref{sec:inputs} with their hypotheses:
the Parisi formula and Guerra's bound~\cite{guerra-2003,parisi-formula}, the
uniqueness of the Parisi minimizer~\cite{auffinger-chen-uniqueness-2015}, the
interval support of the Parisi measure~\cite{full-rsb}, the upper bound on
the mean bias of~\cite[Theorem~1.1]{hexagon-bias}, the upper bound on the
variance and the trial bound of~\cite{quantitative-parisi}, and results
of~\cite[Sections~2--8]{oai-barriers}. The proof of
\cref{thm:barrier-strong} uses in addition the small-deviation inequality
of~\cite{paouris-valettas-2018}. All other steps are proved in this paper.

\subsection{The upper bound}

\paragraph{From flows to mixing.}
We follow the strategy of~\cite{oai-typical-upper}. Its last step reduces the
problem to a lower bound on stationary flows across
cuts~\cite[Section~7, Lemma~8]{oai-typical-upper}. For $S\subseteq\Sigma_n$,
the flow $\sum_{x\in S,\,y\notin S}\pi^J(x)\mathcal L^J(x,y)$ is the stationary
rate at which the continuous heat-bath process jumps from $S$ to its
complement. Let $0<\epsilon<1/3$ and $\phi>0$, and suppose that the flow
across $S$ is greater than $\phi$ for every $S$ with
$\epsilon\le\pi^J(S)\le1-\epsilon$. Then for every $t>0$,
\[
 \sum_{x\in\Sigma_n}\pi^J(x)\,d^J(x,t)\le\frac{n\sqrt{2\log2}}{\phi\sqrt t}+2\epsilon.
\]
The proof removes a set of small Gibbs mass, proves a Poincar\'e inequality
on the remainder by the coarea argument for the reversible Cheeger
bound~\cite[Theorem~3.5]{lawler-sokal-1988}, and then bounds the Gibbs
average of the distances through the entropy dissipation of the whole
chain~\cite[Lemmas~2.5 and~2.7]{diaconis-saloff-coste-1996}. It uses only
cuts that separate appreciable Gibbs masses, and it does not require the
trajectory to stay in the retained set. We reproduce it in
\cref{u:lem:pruning}. With $\epsilon=n^{-\mathsf d}$,
$\phi=e^{-n^{1-\mathsf q}}$, and $t=e^{n^{1-\omega}}$, where $\mathsf d$ and
$\mathsf q$ are positive exponents of the proof (\cref{u:sec:parameters}),
the right side tends to zero when $\omega<\mathsf q$, and using Markov's
inequality for the Gibbs average, we obtain \cref{thm:upper}. It therefore suffices to show
that, with probability tending to one over the disorder, the flow across
every $S$ with $\min\{\pi^J(S),\pi^J(S^{\rm c})\}\ge n^{-\mathsf d}$ is greater
than $e^{-n^{1-\mathsf q}}$.

\paragraph{The proposal.}
To bound these flows, OpenAI constructs a random configuration correlated
with the disorder, called a proposal~\cite[Section~4]{oai-typical-upper}.
Each coordinate of the proposal follows the magnetization martingale of the
scalar Parisi diffusion. The martingale is driven by successive queries of
the interaction matrix: over each cell of a time mesh, the increment of the
magnetizations, orthogonalized against the earlier query directions, is
multiplied by the matrix. By orthogonal invariance, the part of the matrix
that has not been revealed is a compressed block of the Gaussian orthogonal
ensemble (GOE) at each step. Beyond the top $q_\beta$ of the support of the
Parisi measure, the martingale is continued until the average uncertainty of
the coordinates is small, and the query noise is mixed with an independent
Brownian noise. At the final time, the spins are drawn independently with
the magnetizations as means. Successive Gaussian conditioning along
orthogonalized directions is used in Bolthausen's iterative construction of
solutions of the TAP equations~\cite[Sections~2--3]{bolthausen-2014}, and
Montanari's incremental approximate message passing algorithm uses a Parisi
diffusion and normalized increments of the
magnetizations~\cite[Sections~2--3]{montanari-2021}.

The equilibrium input is the stationarity of the Parisi measure along the
whole interval $[0,q_\beta]$. The stationarity conditions on the support were
obtained by Chen~\cite[Theorem~2 and Proposition~1]{chen-2017} and by
Jagannath and Tobasco~\cite[Proposition~1.1 and
Corollary~3.10]{jagannath-tobasco-2017}, and the support is the whole interval
by~\cite[Theorem~1]{zhou-2026} for $\beta$ close to one and
by~\cite[Theorem~1.1]{full-rsb} for every $\beta>1$. Along the scalar
diffusion, an exact identity balances the information about the matrix that
the queries reveal against the energy gained by the
proposal~\cite[Section~3, Lemma~4]{oai-typical-upper}. It yields the two
estimates of~\cite[Proposition~5]{oai-typical-upper}. The relative entropy of
the joint law of the matrix and the proposal with respect to the joint law of
the disorder and a Gibbs sample is sublinear in $n$, and outside an event of
small probability, the law of the pair is dominated by $e^{o(n)}$ times this
joint law.

\paragraph{Coverage and the connecting sequence.}
The information inequalities used in these comparisons---the chain rule,
data processing, and the variational formula for relative entropy---are
recalled in \cite[Chapter~4]{boucheron-lugosi-massart-2013}.
Forward relative entropy alone does not show that the proposal reaches a
prescribed set of positive Gibbs mass. Under the joint law of the disorder
and a Gibbs sample, the conditional law of the disorder given the spin is
strongly log-concave. A transport bound for such
laws~\cite[Corollary~3.10]{bogachev-kolesnikov-medvedev-2005} and Gaussian
smoothing of the disorder~\cite[Lemma~11]{lugosi-neu-2022} reverse the entropy
comparison. They give a lower bound on the probability that the proposal lands
in any set of appreciable Gibbs mass~\cite[Section~5,
Proposition~6]{oai-typical-upper}. Two proposals generated from a common
smoothed disorder then land on the two sides of a balanced cut with a
probability that is not too small. Following~\cite{oai-typical-upper}, these
two proposals are run to a longer, fine horizon, and they are joined by a
sequence of proposals run to a shorter, coarse horizon, obtained by rotating
the Gaussian inputs, with small consecutive Hamming distances. Near-pinning of
the magnetizations joins the two horizons. If the cut had small flow, its
enlarged vertex boundary would have tiny Gibbs mass, and the likelihood bound
keeps the coarse proposals away from this boundary. But a sequence with short
steps that crosses the cut must come close to its boundary. This
contradiction bounds the flows from below~\cite[Section~6,
Proposition~7]{oai-typical-upper}.

\paragraph{What is new.}
In~\cite{oai-typical-upper}, the parameters of the proof are fixed numbers,
and the saving $1/40000000$ is not optimized~\cite[Section~1, after
Theorem~1]{oai-typical-upper}. To reach every saving below $1/4$, the
estimates for the coarse proposal must hold outside events of probability
close to $e^{-n^{3/4}}$, with a likelihood bound close to $e^{n^{3/4}}$ times
the joint law. The following refinements yield every fixed saving below $1/4$.
\begin{enumerate}
\item \emph{Sharp estimates for the coarse proposal}
(\cref{u:sec:concentration,u:sec:coarse}). We strengthen the scalar
entropy--energy identity of~\cite[Section~3, Lemma~4]{oai-typical-upper}
to a pathwise terminal identity. Along the scalar diffusion beyond the
support, the information revealed by the queries and $\beta$ times the
linearized energy of the proposal cancel up to a terminal term. The centered fluctuation
of this term has variance at most $\beta^4p(V)^3$, where $p(V)$ is the mean
conditional variance of a spin given its magnetization at the final time $V$
(\cref{u:lem:joint}). The coarse likelihood estimate absorbs the
deterministic amplitude of this term. The centered difference between this cancellation
and its version on the mesh has a moment generating function bounded by
$e^{C_\beta\theta^2h}$ for $\abs\theta\le c_\beta/h$, where $h$ is the mesh
parameter (\cref{u:lem:lag}). We combine these bounds with concentration of
weighted dyadic sums over cells of unequal length, normalized by the
durations of the cells (\cref{u:lem:weighted}), with a bound on the deviation
of the Gram matrix of the queries from the identity that uses the variance of
the squared projections and not only their tails (\cref{u:lem:gram}), and
with sharper bounds on the variances of the Gaussian terms that the proposal
does not use (\cref{u:lem:weighted-energy}). The dyadic triangular and
Cholesky arguments build on~\cite[Section~4, proof of
Proposition~5]{oai-typical-upper}.
\item \emph{The growing mesh} (\cref{u:sec:proposal}). The cells of the mesh
in~\cite{oai-typical-upper} have a common length. Ours have length $he^{t/4}$
at time $t$, with $h=n^{-\mathsf a}$, so that the horizon $L\log n$ is
reached after at most $8n^{\mathsf a}$ cells when $\mathsf a>L/4$
(\cref{u:lem:quad}). The quadrature error is $O_\beta(h)$, uniformly in the
horizon. On this mesh, adjacent cells have different lengths, and
the estimates distinguish the duration of the previous cell, which governs
the conditional law of the spins, from that of the next cell, which
normalizes the query. For instance, the bound on the conditional
correlations of the query entries with the spins in \cref{u:sec:fine-energy}
loses a factor two compared with equal cells.
\item \emph{Direct expected energy at the fine horizon}
(\cref{u:sec:fine-energy}). We compute the expected queried energy through
the exact one-query conditional identity of \cref{u:lem:onequery}, including
fallback outcomes. This avoids the Gram-event restriction in the energy
calculation of~\cite[Section~4, Proposition~5]{oai-typical-upper} and gives
the fine relative-entropy bound of \cref{u:prop:fine-kl}, with saving
$\mathsf k_{\rm f}<\min(\mathsf a,\mathsf f,1-\mathsf a)$.
\item \emph{The parameter system} (\cref{u:sec:budget}). The exponents of the
proof are affine functions of $\varepsilon_\omega=(1/4-\omega)/20$, and this
single family satisfies every condition of the proof for every
$\omega\in(0,1/4)$ (\cref{u:prop:range}).
\end{enumerate}

The connecting sequence, including fine proposals only at the two
endpoints, stability from coarse Gram events, and pinning without a fine
Gram exception, follows~\cite[Section~6, proof of
Proposition~7]{oai-typical-upper}.

\paragraph{The exponent.}
The proof of \cref{u:prop:range} shows that, for every choice of the
exponents, the estimates of \cref{part:upper} require
\[
 \omega<\mathsf q<\mathsf s_{\rm b}<\mathsf s_{\rm seq}<\mathsf e<\mathsf u_{\rm E}
 <\mathsf k_{\rm f}/2\le1/4 .
\]
The mixing step needs $\omega<\mathsf q$, since the flow threshold is
$e^{-n^{1-\mathsf q}}$. In the connecting sequence, $e^{-n^{1-\mathsf s_{\rm b}}}$
bounds the failure probability of one coarse proposal,
$e^{-n^{1-\mathsf s_{\rm seq}}}$ that of the whole sequence, and
$e^{-2n^{1-\mathsf e}}$ bounds from below the probability that the two fine
endpoints lie on opposite sides of the cut, and the proof needs
$\mathsf q<\mathsf s_{\rm b}<\mathsf s_{\rm seq}<\mathsf e$. Smoothing and
coverage need $\mathsf e+\mathsf d<\mathsf u_{\rm E}
<\min\{\mathsf D,\mathsf k_{\rm f}-\mathsf D\}$, where $n^{-\mathsf D}$ is the
smoothing parameter and $n^{1-\mathsf k_{\rm f}}$ bounds the forward relative
entropy at the fine horizon. Finally, this relative entropy contains terms of
order $nh=n^{1-\mathsf a}$ and $n^{\mathsf a}$, the second from the number of
queries, so that $\mathsf k_{\rm f}<\min\{\mathsf a,1-\mathsf a\}\le1/2$.

\paragraph{Inputs for the upper bound.}
Besides standard tools of Gaussian analysis and stochastic
calculus~\cite{tsirelson-ibragimov-sudakov-1976,karatzas-shreve-1991,boucheron-lugosi-massart-2013},
the proof of \cref{thm:upper} uses the following results, which are stated in
\cref{sec:inputs} and \cref{u:sec:scalar} with their hypotheses: the Parisi
formula and Guerra's bound (\cref{inp:parisi}), the uniqueness of the Parisi
minimizer (\cref{inp:unique}), the interval support of the Parisi measure
(\cref{inp:full-rsb}) and the identity $\Gamma(t)=t$ on $(0,q_\beta)$
(\cref{cor:inp-identity}), the regularity and scalar evolution of
\cref{inp:ob-scalar}, the scalar estimates of~\cite[Section~3, Lemmas~3
and~4]{oai-typical-upper}, and the transport, smoothing, and finite-chain
results cited above. All other steps are proved in this paper.

\subsection{The fluctuation theorem}

The proof of \cref{f:window} in~\cite{oai-fluctuations} combines finite
replica trees and covariance differentiation for Gaussian
hierarchies~\cite[Section~2]{oai-fluctuations}, optimized short comparisons
over overlap quantiles~\cite[Section~3]{oai-fluctuations}, a one-site cavity
estimate, which compares an overlap moment of a comparison with the
corresponding moment of a scalar recursion~\cite[Proposition~4.1]{oai-fluctuations},
a rigidity theorem, by which rescaled overlap quantiles of minimizing
comparisons have linear limits at small masses~\cite[Theorem~5.1]{oai-fluctuations},
finite propagation of regularity through successive
comparisons~\cite[Proposition~6.6]{oai-fluctuations}, and a low-field
perturbation that yields the upper bound~\cite[Section~7]{oai-fluctuations}.
As explained in~\cite[Section~1]{oai-fluctuations}, its closest analytic
predecessor is the combination in~\cite{quantitative-parisi} of a pinned
Brascamp--Lieb estimate for the variance~\cite{brascamp-lieb-1976}, optimized
covariance interpolation, and positive
moments~\cite[Propositions~2.2 and~2.3 and Section~3]{quantitative-parisi}.
The single-site analysis of Thouless, Anderson, and
Palmer~\cite{thouless-anderson-palmer-1977} is an early antecedent of the cavity
method in spin glasses, and M\'ezard, Parisi, and Virasoro developed
the cavity construction of the solution with broken replica
symmetry~\cite{mezard-parisi-virasoro-1986}. The cavity variational principle
of Aizenman, Sims, and Starr~\cite{aizenman-sims-starr-2003} is a rigorous
antecedent of comparisons that add or remove sites. \Cref{sec:fluct} summarizes this route, with
references to the corresponding statements.

\section{Organization of the paper}\label{sec:intro-organization}

\Cref{sec:overview} gives an overview of the proofs of both bounds.
\Cref{part:prelim} contains preliminaries: \cref{sec:model} defines the model,
the dynamics, and the joint law of the disorder and the initial
configuration, and \cref{sec:inputs} defines Gaussian hierarchies and the
Parisi functional and states the imported results. \Cref{part:static}
contains the static estimates for the proof of \cref{thm:barrier-730}.
\Cref{sec:scalar} treats the scalar
problem near a point of the support of the Parisi measure,
\cref{sec:flow} proves short-time estimates for Gaussian hierarchies,
\cref{sec:cubic} constructs the three-replica covariance perturbation and
proves the cubic deficit, \cref{sec:fractional} proves the fractional-moment
comparison, and \cref{sec:static} combines these into a single estimate for
all overlap constraints. \Cref{part:dynamics} contains the dynamical part of
the proof of the lower bound. \Cref{sec:references} treats the Gibbs mass of
overlap neighborhoods and the families of reference configurations,
\cref{sec:induction} proves that the overlap constraints persist along a
trajectory, and \cref{sec:barrier-proof} completes the proof of
\cref{thm:barrier-730} and \cref{b:cor:subcritical} and proves
\cref{thm:barrier-strong}.

\Cref{part:upper} proves the upper bound. \Cref{u:sec:setup} introduces the
Gaussian matrix from which the disorder is built, the scalar Parisi process
and its continuation beyond the support, and the exponents of the proof.
\Cref{u:sec:proposal} constructs the Gaussian-query proposal on a growing
mesh, and \cref{u:sec:fine} bounds its forward relative entropy at the fine
horizon. \Cref{u:sec:concentration} proves the concentration estimates on the
growing mesh, and \cref{u:sec:coarse} proves the joint cancellation and the
likelihood bound for the coarse proposal. \Cref{u:sec:connections} proves the
coverage estimate and constructs the connecting sequence, and
\cref{u:sec:proof} proves \cref{thm:upper,cor:upper-discrete,cor:two-sided}
and verifies the conditions on the exponents.

\Cref{part:fluct} concerns the fluctuations and the mean bias:
\cref{sec:fluct} states the bound of \cref{f:window} and deduces
\cref{f:main} from it, and \cref{sec:bias} proves \cref{cor:mean-bias}.
\Cref{part:formal} contains \cref{sec:formal}, which describes the formal
proofs, and \cref{sec:open}, which lists open problems.
\chapter{Overview of the proofs}\label{sec:overview}

This section gives an overview of the proofs of \cref{thm:barrier-730} and
\cref{thm:upper}, and a shorter account of the parts of the paper that concern
\cref{f:main} and \cref{cor:mean-bias}. Every claim made here is proved later in
the paper, or quoted there with its source.
\Cref{sec:ov-sign,sec:ov-architecture,sec:ov-scales} explain the reduction to
paths, the architecture of the proof in \cite{oai-barriers}, which we follow,
and the balance of scales that gives the exponent $7/30$.
\Cref{sec:ov-cubic,sec:ov-static,sec:ov-dynamics} describe the cubic pressure
deficit (\cref{sec:scalar,sec:flow,sec:cubic}), the passage to Gibbs
probabilities (\cref{sec:fractional,sec:static}), and the dynamical argument
(\cref{sec:references,sec:induction,sec:barrier-proof}). \Cref{sec:ov-upper}
concerns \cref{part:upper}, and \cref{sec:ov-fluct} concerns
\cref{part:fluct}. \Cref{tab:ov-dependencies} shows how the sections depend on
each other.

Throughout this section, $\beta>1$ is fixed. In
\cref{sec:ov-sign,sec:ov-architecture,sec:ov-scales,sec:ov-cubic,sec:ov-static,sec:ov-dynamics},
$\pi=\pi^J$ is the Gibbs measure at a fixed disorder. We write $\lambda$ for
the exponent of the barrier, and we use an upper bound
$\Delta_{n,\beta}\le C_bn^{-3\lambda}$ on the mean bias
$\Delta_{n,\beta}=P_\beta-n^{-1}\E F_n$. For \cref{thm:barrier-730}, we take
$\lambda=7/30$ and the bound $\Delta_{n,\beta}\le C_bn^{-7/10}$ of
\cref{inp:hb-rate}. The exponent of the upper bound is $1-\omega$, for a fixed
$\omega\in(0,1/4)$, and \cref{sec:ov-upper} uses the notation of
\cref{part:upper}.

\begin{table}[!t]
\centering
\scriptsize
\caption{The main results of
\cref{sec:model,sec:inputs,sec:scalar,sec:flow,sec:cubic,sec:fractional,sec:static,sec:references,sec:induction,sec:barrier-proof,u:sec:setup,u:sec:proposal,u:sec:fine,u:sec:concentration,u:sec:coarse,u:sec:connections,u:sec:proof,sec:fluct,sec:bias},
the sections whose results they use, and the sections in which they are used.
}\label{tab:ov-dependencies}
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.17\textwidth}>{\raggedright\arraybackslash}p{0.5\textwidth}>{\raggedright\arraybackslash}p{0.1\textwidth}>{\raggedright\arraybackslash}p{0.13\textwidth}@{}}
\toprule
Section & Main results & Uses & Used in\\
\midrule
\ref{sec:model} Model & realized-start probability, \cref{lem:model-joint};
 heat-bath dynamics, \cref{lem:model-dynamics}; completed Hamiltonian,
 \cref{lem:model-completion} & --- & \ref{sec:inputs}, \ref{sec:fractional},
 \ref{sec:references}, \ref{sec:barrier-proof}, \ref{u:sec:setup},
 \ref{u:sec:connections}, \ref{u:sec:proof}, \ref{sec:fluct}, \ref{sec:bias}\\
\ref{sec:inputs} Inputs & matrix comparison, \cref{inp:ob-matrix}; upper
 bound on the mean bias, \cref{inp:hb-rate}; variance and profile bounds,
 \cref{inp:qp-rate,inp:qp-floor}; pair, fixed-scale, and separated-overlap bounds,
 \cref{inp:ob-pair-gap,inp:ob-fixed-gap,inp:ob-remote}; overlap-neighborhood
 masses, \cref{inp:ob-coverage} & \ref{sec:model}, literature &
 \ref{sec:scalar}--\ref{sec:references}, \ref{sec:barrier-proof},
 \ref{u:sec:setup}, \ref{u:sec:coarse}, \ref{u:sec:connections}\\
\ref{sec:scalar} Scalar problem & stationarity identities, \cref{b:lem:os-scalar-jets};
 the constant $m_\beta$, \cref{cor:sc-mbeta} & \ref{sec:inputs} & \ref{sec:cubic},
 \ref{sec:fractional}, \ref{sec:references}, \ref{u:sec:setup},
 \ref{u:sec:coarse}\\
\ref{sec:flow} Flows & uniform short-time bounds, \cref{prop:flow-uniform};
 response to the terminal function, \cref{b:lem:os-prefix,cor:flow-prefix} &
 \ref{sec:inputs} & \ref{sec:cubic}\\
\ref{sec:cubic} Cubic deficit & local expansion, \cref{b:prop:os-expansion};
 deficit on a band, \cref{prop:cub-band} & \ref{sec:inputs}--\ref{sec:flow} & \ref{sec:fractional}, \ref{sec:references}\\
\ref{sec:fractional} Fractional moments & fractional comparison,
 \cref{b:prop:fractional-matrix}; local fractional bound, \cref{lem:fr-local-gap}
 & \ref{sec:model}--\ref{sec:scalar}, \ref{sec:cubic} & \ref{sec:static}\\
\ref{sec:static} Static event & simultaneous disorder event, \cref{b:lem:static-event}
 & \ref{sec:inputs}, \ref{sec:fractional} & \ref{sec:references}, \ref{sec:barrier-proof}\\
\ref{sec:references} References & overlap-neighborhood masses,
 \cref{b:lem:coverage-extension}; fixed families of references,
 \cref{lem:ref-families} & \ref{sec:model}--\ref{sec:scalar}, \ref{sec:cubic}--\ref{sec:static} & \ref{sec:induction}, \ref{sec:barrier-proof}\\
\ref{sec:induction} Induction & constraints along a path, \cref{prop:ind-induction};
 a common time, \cref{cor:ind-witness} & \ref{sec:references} & \ref{sec:barrier-proof}\\
\ref{sec:barrier-proof} Barrier proof & sign obstruction, \cref{prop:bp-obstruction};
 \cref{thm:barrier-730,b:cor:subcritical,thm:barrier-strong} &
 \ref{sec:model}, \ref{sec:inputs}, \ref{sec:static}--\ref{sec:induction} &
 \ref{u:sec:proof}\\
\ref{u:sec:setup} Upper-bound setup & primitive disorder and Gaussian bounds,
 \cref{u:lem:goe-norm,u:lem:pressure-tail}; the scalar process beyond the
 support, \cref{u:prop:scalar,u:lem:cross,u:lem:tails} & \ref{sec:model},
 \ref{sec:inputs}, \ref{sec:scalar} & \ref{u:sec:proposal}--\ref{u:sec:proof}\\
\ref{u:sec:proposal} Proposal & Gaussian assembly, \cref{u:lem:assembly};
 spin laws, \cref{u:def:laws}; growing mesh, \cref{u:lem:quad} &
 \ref{u:sec:setup} & \ref{u:sec:fine}--\ref{u:sec:connections}\\
\ref{u:sec:fine} Fine entropy & reference law, \cref{u:lem:reference};
 forward entropy, \cref{u:lem:forward-kl,u:prop:fine-kl} &
 \ref{u:sec:setup}, \ref{u:sec:proposal} & \ref{u:sec:coarse},
 \ref{u:sec:connections}\\
\ref{u:sec:concentration} Concentration & fixed Gram stability,
 \cref{u:lem:gram}; weighted concentration, \cref{u:lem:weighted} &
 \ref{u:sec:setup}, \ref{u:sec:proposal} & \ref{u:sec:coarse}\\
\ref{u:sec:coarse} Coarse domination & weighted energy,
 \cref{u:lem:weighted-energy}; joint cancellation,
 \cref{u:lem:joint,u:lem:lag}; likelihood bound, \cref{u:prop:coarse} &
 \ref{sec:inputs}, \ref{sec:scalar}, \ref{u:sec:setup}--\ref{u:sec:concentration} &
 \ref{u:sec:connections}\\
\ref{u:sec:connections} Connections & smoothing and coverage,
 \cref{u:lem:smoothing,u:lem:coverage}; boundary mass, \cref{u:lem:boundary};
 connecting sequence, \cref{u:prop:sequence}; witness decay,
 \cref{u:prop:witness} & \ref{sec:model}, \ref{sec:inputs},
 \ref{u:sec:setup}--\ref{u:sec:fine}, \ref{u:sec:coarse}
 & \ref{u:sec:proof}\\
\ref{u:sec:proof} Upper-bound proof & pruning, \cref{u:lem:pruning};
 \cref{thm:upper,cor:upper-discrete,cor:two-sided}; parameters,
 \cref{u:prop:range} & \ref{sec:model}, \ref{sec:barrier-proof},
 \ref{u:sec:setup}, \ref{u:sec:connections} & ---\\
\ref{sec:fluct} Fluctuations & the bound of \cite{oai-fluctuations},
 \cref{f:window}; log-concave augmentation, \cref{f:logconcave}; \cref{f:main}
 & \ref{sec:model} & \ref{sec:bias}\\
\ref{sec:bias} Mean bias & variance--bias inequality,
 \cref{prop:bias-variance-deficit}; \cref{cor:mean-bias} & \ref{sec:model}, \ref{sec:fluct} & ---\\
\bottomrule
\end{tabular}
\end{table}

\section{The sign test}\label{sec:ov-sign}

Recall from \cref{lem:model-joint} that \cref{thm:barrier-730} states that
the Gibbs mass of the initial configurations from which the chain is at total
variation distance greater than $\tau$ from $\pi$ at time
$e^{A_\beta n^{7/30}}$ tends to one in probability over the disorder. Since $\pi$ is stationary, the chain started from a Gibbs sample has
law $\pi$ at every deterministic time (\cref{lem:model-dynamics}), and the law
averaged over the initial configuration carries no information. A lower bound
on the distance must come from a property of almost every individual initial
configuration that the dynamics forgets slowly.

The property we use is the sign of an overlap. As in
\cite[Section~8]{oai-barriers}, we choose for each $x\in\Sigma_n$ a reference
configuration $v(x)$, with $R(v(x),x)\ge q_*$ for all $x$ outside a set of small
Gibbs mass, where $q_*\in(0,1)$ is a fixed overlap (\cref{def:ind-algorithm}). Since $H_n(-y)=H_n(y)$, the half-space
$\{y:R(v(x),y)\le0\}$ has Gibbs mass at least $1/2$. If the law $\nu_x$ of the
chain started at $x$ is within $\tau$ of $\pi$ in total variation, then the
half-space has mass at least $1/2-\tau$ under $\nu_x$. Multiplying by
$\pi(x)$ and summing over $x$, we find that
\begin{equation}\label{eq:ov-half-space}
 \Bigl(\frac12-\tau\Bigr)\,\pi\bigl\{x:d^J_{\rm disc}(x,k)\le\tau\bigr\}
 \le\PP\bigl\{R(v(X_0),X_k)\le0\bigr\},
\end{equation}
where $(X_k)_{k\ge0}$ is the discrete chain with $X_0\sim\pi$
(see \eqref{eq:bp-half-space}). This is the reduction of
\cite[(8.1)--(8.3)]{oai-barriers}. The half-space depends on the
initial configuration, so that the distance on the left side is evaluated
before the average over $X_0$, while the right side is a probability for the
stationary chain. Since one update changes every overlap by at most $2/n$, a
chain that enters the half-space must cross every overlap level between $q_*$
and $0$.

We show that these crossings cannot occur along paths that avoid a fixed set
of small Gibbs mass. We call a sequence $x_0,\dots,x_m$ a path if all overlaps
change by at most $2/n$ from one element to the next \eqref{eq:ind-jump}, so
that every trajectory of the discrete chain is a path. Let $L=An^\lambda$ be
the logarithmic time scale, for a coefficient $A>0$ fixed below, so that $e^L$
is the time horizon, and let $T_n=\lceil n^2e^L\rceil$ be the budget of update
attempts. For every
disorder in an event $\mathcal G_n$ with $\PP(\mathcal G_n)\to1$, we construct
the map $x\mapsto v(x)$ and a set $V\subseteq\Sigma_n$ with
$\pi(V)\le3e^{-4L}$ such that
\begin{equation}\label{eq:ov-obstruction}
 R\bigl(v(x_0),x_m\bigr)>0\qquad\text{for every path }x_0,\dots,x_m
 \text{ with }m\le T_n\text{ that avoids }V
\end{equation}
(\cref{lem:ref-families,prop:bp-obstruction,lem:bp-coefficient}). Given
\eqref{eq:ov-obstruction}, a chain with $R(v(X_0),X_k)\le0$ for some
$k\le T_n$ visits $V$ at one of the times $0,\dots,k$. Since $X_a$ has law
$\pi$ for every deterministic $a$, a union bound implies that
\[
 \PP\bigl\{R(v(X_0),X_k)\le0\bigr\}\le(k+1)\pi(V)\le3(k+1)e^{-4L}
\]
(see \eqref{b:dyn:stationary-sign}). This is the only place where we use
stationarity, and we use it only at deterministic times. For this reason the
map $v$ and the set $V$ must be fixed before the chain is run, and they may not
depend on its trajectory. In continuous time, the number of updates up to time
$e^L$ is a Poisson variable with mean $ne^L$, and by Markov's inequality it
exceeds $T_n$ with probability at most $1/n$. We conclude that
\[
 \pi\bigl\{x:d^J(x,e^L)\le\tau\bigr\}\le\frac{3(T_n+1)e^{-4L}+1/n}{1/2-\tau}
\]
for every disorder in $\mathcal G_n$ (\cref{prop:bp-quenched}), and the right
side tends to zero. When we integrate over the disorder, $\PP(\mathcal G_n^c)$
is added once. The threshold $\tau$ enters only through the factor
$1/(1/2-\tau)$, which is why one coefficient serves every fixed $\tau<1/2$ and
both clocks.

\section{Overlap constraints and reference configurations}\label{sec:ov-architecture}

We follow the architecture of the proof of \cite[Theorem~1.1]{oai-barriers},
which proves a barrier at the time $\exp(n^{1/10000})$ and the threshold
$\tau=1/4$.

\paragraph{Overlap constraints.} For $(v,x,{\widetilde v})\in\Sigma_n^3$, we write
\[
 r=\frac{R(v,x)+R(v,{\widetilde v})}2,\qquad\delta=\frac{R(v,x)-R(v,{\widetilde v})}2,\qquad q=R(x,{\widetilde v}),
\]
so that the overlap matrix of $(v,x,{\widetilde v})$ is the matrix $D(r,q,\delta)$ of
\eqref{eq:inp-triple}. We use two kinds of constraints. The constraints of the
first kind act at fixed scales. For fixed cutoffs $0<b_2<b_1<1$
(\cref{tab:ref-parameters}), they state that if $R(x,{\widetilde v})\ge q_*$ and
$\abs{R(v,x)},\abs{R(v,{\widetilde v})}\le b_1$, then $\abs{R(v,x)-R(v,{\widetilde v})}\le2\eta_n$, and
similarly with the cutoffs $(b_1,b_2)$ in place of $(q_*,b_1)$. The triples
that violate them form the sets $\mathcal B_n(q_*,b_1)$ and
$\mathcal B_n(b_1,b_2)$ of \eqref{eq:ref-fixed-violation}. In words, a large
overlap between $x$ and ${\widetilde v}$ forces their small overlaps with
$v$ to agree. The
constraints of the second kind act at the scale $\rho$, near a level $r$ in a
compact interval $J_\beta\subset(0,q_\beta)$. With $\varepsilon=E\rho$ and
$E=1/100$, they state that no triple satisfies
\[
 r\in J_\beta,\qquad\varepsilon\le\abs\delta\le2\varepsilon,\qquad q\ge r+\rho.
\]
Such triples are called narrow, and they form the set $\mathcal N_n$
(\cref{sec:ref-parameters}). Equivalently, if $R(x,{\widetilde v})\ge r+\rho$, then
$\abs{R(v,x)-R(v,{\widetilde v})}$ does not lie in $[2\varepsilon,4\varepsilon]$. We
exclude a band, rather than a half-line, for the following reason. Along a
path, every overlap changes by at most $2/n<2\varepsilon$ in one step, and the
difference $\abs{R(v,x_k)-R(v,{\widetilde v})}$ cannot pass from below $2\varepsilon$ to
above $4\varepsilon$ without lying in the band at some time. As long as the path
keeps a large overlap with ${\widetilde v}$, a narrow constraint keeps its overlap with $v$
within $2\varepsilon$ of $R(v,{\widetilde v})$ (\cref{lem:ind-search}).

\paragraph{Constrained pressures.} In
\cref{sec:inputs,sec:scalar,sec:flow,sec:cubic,sec:fractional,sec:static}, we
write the three configurations as $(v,x,y)$. For a feasible overlap matrix
$D$, the probability that three independent samples from $\pi$ have overlap
matrix $D$ is $Z_{3,n}(D)/Z_n^3$, where $Z_{3,n}(D)$ is the partition function
restricted to the triples with overlap matrix $D$ (\cref{def:inp-constrained}).
There are at most $(n+1)^3$ feasible matrices, and it suffices to bound each
numerator from above and the denominator from below. Upper bounds for the
numerators come from Guerra's hierarchical interpolation~\cite{guerra-2003}
and its extension by Talagrand to replicas with prescribed overlaps
\cite[Section~3]{parisi-formula}, \cite[Propositions~3.4 and~4.4]{talagrand-2007}.
We use it in the form of \cite[Proposition~2.3]{oai-barriers}, stated as
\cref{inp:ob-matrix}. For every admissible path $(Q,m)$ of $3\times3$
covariance matrices, with positive semidefinite increments and nondecreasing
masses, the comparison with zero terminal multiplier bounds the constrained
pressure $p_{3,n}(D)=n^{-1}\E\log Z_{3,n}(D)$ by the trial value
\[
 p_{3,n}(D)\le\mathcal T_D(Q,m,0)
 =\Psi(Q,m,0)-\frac{\beta^2}{4}\int m\dd\norm Q_F^2
   +\frac{\beta^2}{4}\norm{D-Q_N}_F^2
\]
(\cref{def:inp-trial,inp:ob-matrix}). The last term is the cost of an endpoint
$Q_N$ that differs from $D$.

The baseline is the block hierarchy of
\cite[Section~4.2 and Section~7, Step~1]{oai-barriers}, described in
\cref{lem:inp-blocks} and built from the Parisi
distribution function $\alpha=\alpha_\beta$. Below the level $r$, the three
fields share their increments, at mass $\alpha/3$. Between $r$ and $q$, the
fields of $x$ and $y$ share their increments, at mass $\alpha/2$, while the
field of $v$ evolves alone, and above $q$ the three fields evolve
independently. The endpoint of this path is $D(r,q,0)$, and its trial value is
exactly $3P_\beta$. A perturbation of the baseline with endpoint
$D(r,q,\delta)$ and a smaller trial value proves a pressure deficit. Whether
such a perturbation exists is decided by the scalar Parisi problem. Let $u$ be
the spatial derivative of the solution of the Parisi equation, let $Y$ be the
associated diffusion, and let $\Gamma(t)=\E u(t,Y_t)^2$. It was proved in
\cite[Proposition~3.2]{oai-barriers} that $\Gamma(q)=q$ at every point $q$ of
the support of the Parisi measure, and by
\cite[Theorem~1.1]{full-rsb}, the support is $[0,q_\beta]$. We conclude that
$\Gamma(t)=t$ on $[0,q_\beta]$ (\cref{cor:inp-identity}). For two
configurations, we have the pair bound
$p_{2,n}(D_2(q))\le2P_\beta-(\Gamma(q)-q)^2/2$ of
\cite[Lemma~4.3]{oai-barriers} (\cref{inp:ob-pair-gap}). This deficit
vanishes on the support, and
constraints at overlaps inside $(0,q_\beta)$ require the three-replica
perturbations of \cite[Sections~4 and~7]{oai-barriers} and of
\cref{sec:cubic}.

\paragraph{Reference configurations and crossings.} In the dynamical argument,
$v$ and ${\widetilde v}$ are reference configurations and $x$ is a configuration of the
path. We take the references from lists of independent Gibbs samples, which
are fixed before any path is considered (\cref{sec:ref-families}). Suppose that
a path $x_0,\dots,x_m$ that avoids $V$ satisfies $R(v(x_0),x_m)\le0$. We
proceed through levels $i=0,1,\dots,N_{\rm lev}-1$, with thresholds
$r_0<r_1<\cdots$ in an interval $[t_0,2t_0]\subset J_\beta$
(\cref{def:ind-algorithm}). At level $0$, the reference is $v_0=v(x_0)$, and
$k_0$ is the first time with $R(v_0,x_k)\le r_0$. At level $j\ge1$, a bound
on the Gibbs mass of overlap neighborhoods supplies a new reference $v_j$ with
$R(v_j,x_{k_{j-1}})\ge q_*$ (\cref{b:lem:coverage-extension}, which extends
\cite[Proposition~6.1]{oai-barriers}). We then search backward in time from
$k_{j-1}$ for the last time $k_j$ with $R(v_j,x_k)\le r_j$. During the
search, the constraints keep the overlaps of the path with the earlier
references close to their values at $k_{j-1}$. The overlap
with $v_0$ stays close to $r_0<q_*$, while $R(v_0,x_0)\ge q_*$, so that the
search finds the crossing before it reaches time $0$ (\cref{lem:ind-crossing}).
After the last level, at the single time $k_{N_{\rm lev}-1}$, the overlaps
of the path with many references lie close to their thresholds. The
thresholds are drawn at random, and a count over the time index shows that this
is impossible when $m\le T_n$.

As in \cite[Sections~8.3 and~8.4]{oai-barriers}, the references are arranged
in blocks of $B$ levels. Within a block, a new reference has small overlaps with
the earlier references of the block, and it deletes every retained reference
with which its overlap exceeds $b_1$ in absolute value. Such conflicts occur
only at the first level of a block, and they never reach the first $B-H\ge2$
levels of the preceding block (\cref{lem:ind-deletion}).

\section{The exponent}\label{sec:ov-scales}

Three requirements fix the scales, which are listed in \cref{tab:ov-scales}.
The order in which the parameters are chosen is recorded in
\cref{tab:ref-parameters,tab:bp-order}.

\begin{table}[!t]
\centering
\small
\caption{The scales of the proof of \cref{thm:barrier-730} and their roles.
Here $R$, $E=1/100$, $S$, and $A_\beta$ are constants fixed before $n\to\infty$. The
last five rows are the rates of decay of the error terms of
\cref{b:lem:static-event}, without constants and polynomial factors; see
\cref{tab:st-errors,tab:ref-exponents}.}\label{tab:ov-scales}
\begin{tabular}{@{}ll>{\raggedright\arraybackslash}p{0.5\textwidth}@{}}
\toprule
Quantity & $\lambda=7/30$ & Role\\
\midrule
$\rho_n$ & $Rn^{-7/30}$ & gap $q-r$ of the narrow constraints;
 length of a threshold interval\\
$\varepsilon_n=E\rho_n$ & $ERn^{-7/30}$ & overlap imbalance
 $\varepsilon_n\le\abs\delta\le2\varepsilon_n$; radius of the windows\\
$\eta_n$ & $n^{-5/21}$ & tolerance of the fixed-scale constraints\\
$s_n=S\rho_n$ & $SRn^{-7/30}$ & order of the fractional moment\\
$L_n$ & $A_\beta n^{7/30}$ & logarithmic time scale; logarithm of
 the list sizes\\
\midrule
$n\rho_n^3$ & $R^3n^{3/10}$ & Gibbs probability of narrow triples\\
$n\eta_n^2$ & $n^{11/21}$ & Gibbs probability of fixed-scale triples\\
$n\rho_n^4$ & $R^4n^{1/15}$ & disorder failure of a local or
 separated-overlap bound\\
$n\eta_n^4$ & $n^{1/21}$ & disorder failure of a fixed-scale bound\\
$n^2\rho_n^6/n^{7/15}$ & $R^6n^{2/15}$ & disorder failure of the
 denominator (polynomial)\\
\bottomrule
\end{tabular}
\end{table}

\paragraph{The denominator.} On the disorder event of \cref{sec:static}, we
have $F_n\ge n(P_\beta-c_0\rho^3)$, so that
$Z_n^3\ge e^{3n(P_\beta-c_0\rho^3)}$. To obtain this bound from
Chebyshev's inequality with a fixed margin, we choose constants
$a_0<c_0$ such that $\Delta_{n,\beta}\le a_0\rho^3$
for all large $n$. Since $\E F_n=n(P_\beta-\Delta_{n,\beta})$, the distance from
the mean of $F_n$ to the threshold is then at least
$n(c_0-a_0)\rho^3$. The bounds on the numerators keep a deficit
$\varkappa g\rho^3$, where $g\rho^3$ is a lower bound, uniform over the band, for the
deficit per spin of the three-replica pressure near a level $r$
(\cref{sec:ov-cubic,sec:ov-static}) and $\varkappa\in(0,1)$ is fixed. The ratio
$Z_{3,n}(D)/Z_n^3$ is then at most $e^{-n(\varkappa g-3c_0)\rho^3}$, which is
small when $3c_0<\varkappa g$. Constants $a_0<c_0<\varkappa g/3$
with these properties exist if and only if
\[
 \limsup_{n\to\infty}\frac{\Delta_{n,\beta}}{\rho^3}<\frac{\varkappa g}3
\]
(see \eqref{eq:st-S1}). With $\Delta_{n,\beta}\le C_bn^{-7/10}$, this holds
for $\rho=Rn^{-7/30}$ if $R^3>3C_b/(\varkappa g)$ (see
\eqref{b:dyn:R-condition}). By Chebyshev's inequality and the bound
$\Var F_n\le C_vn^{7/15}$ of \cref{inp:qp-rate}, the lower bound on $F_n$ fails
with probability at most a constant times $n^{7/15}/(n^2\rho^6)$, which is
$O(n^{-2/15})$. We note that \cref{f:main} is not used in the proofs of
\cref{thm:barrier-730,thm:barrier-strong}.

\paragraph{The count.} The thresholds lie in the intervals
$I_i=[t_0+3i\rho,t_0+(3i+1)\rho]$, of length $\rho$ and spacing $3\rho$. There
are $N_{\rm lev}=MB$ levels, arranged in $M=\lfloor t_0/(3B\rho)\rfloor$
blocks (see \eqref{b:dyn:threshold-intervals}). In \cref{sec:ov-dynamics}, we
explain why a path of length $m$ that avoids $V$ and violates
\eqref{eq:ov-obstruction} produces, for every array of thresholds, a time
$k\le m$ at which $M$ overlaps lie in windows of radius
$\Psi_0=2\varepsilon+2\eta_n+2/n$ around their thresholds
(\cref{cor:ind-witness}). For independent thresholds, uniform on their
intervals, this event has probability at most
\begin{equation}\label{eq:ov-count}
 (m+1)\gamma_n^M,\qquad
 \gamma_n=\frac{2\Psi_0}\rho\longrightarrow4E=\frac1{25}
\end{equation}
(see \eqref{b:dyn:window-weights} and \eqref{b:dyn:prefix-count}). If this quantity
is less than one, such a path cannot exist. Note that $M$ is of order
$1/\rho$, so that we may take $\log m$ of order $n^\lambda$. This leads to the
time horizon $e^L$ with the logarithmic time scale $L=An^\lambda$, for every
$A<t_0\log25/(3BR)$ (see \eqref{b:dyn:coefficient}). The count requires the
ratio $\gamma_n$ of the length of a window to the length $\rho$ of a threshold
interval to stay below one. With $\varepsilon=E\rho$, the overlap
imbalance is of the largest permitted order, and this ratio stays below a
fixed bound. The gap satisfies $q-r\ge\rho$, and the weakest local deficit
occurs at the corner $q-r=\rho$, $\abs\delta=E\rho$ of the band of
\cref{sec:ov-cubic}.

In summary, the upper bound $n^{-3\lambda}$ on the mean bias allows
$\rho\asymp n^{-\lambda}$, there are then of order $n^\lambda$ levels, and the
count allows $\log t$ of order $n^\lambda$: the exponent of the barrier is one
third of the exponent of the upper bound on the mean bias. For the upper bound
$n^{-7/10}$ of \cref{inp:hb-rate}, this gives the exponent $7/30$ of
\cref{thm:barrier-730}.

\paragraph{The union bounds and the exponent $1/4$.} The proof uses two kinds
of union bound. In the disorder event, each of at most $(n+1)^3$ bounds on
constrained partition functions near a level $r$ fails with probability at
most $e^{-cn\rho^4}$ (\cref{sec:ov-static}). The sufficient condition
$n\rho^4/\log n\to\infty$ makes this polynomial union bound vanish (see
\eqref{eq:st-S3}). At a fixed disorder, we multiply a count $e^{CL}$ of reference
configurations and path positions by Gibbs probabilities bounded by
$C'(n+1)^3e^{-cn\rho^3}$, and it suffices that $cn\rho^3-CL-3\log(n+1)\to\infty$.
Under the stronger condition $n\rho^3/L\to\infty$, we have this for every fixed $C$.
For $\rho=Rn^{-\lambda}$ and $L=An^\lambda$, both sufficient conditions hold
when $\lambda<1/4$. At $\lambda=1/4$, the quantity $n\rho^4=R^4$ is bounded, and
the polynomial union bound in the disorder event is ineffective, whether or
not the conditional union bound can be absorbed by a choice of constants. The
argument gives no exponent of $1/4$ or more, whatever the upper bound on
the mean bias (\cref{tab:bp-exponents}). We note that a
mean bias of order $n^{-3/4}$, as predicted by Franz, Parisi, and
Ricci-Tersenghi~\cite{franz-parisi-ricci-2026}, would correspond exactly to this
threshold (\cref{sec:open}).

\section{The cubic deficit}\label{sec:ov-cubic}

We now describe the pressure deficit for the narrow constraints. We fix the
compact intervals $J_\beta\subset\operatorname{int}I_\beta$ with
$I_\beta\subset(0,q_\beta)$ of \cref{sec:ref-parameters}; the general
statements of \cref{sec:fractional,sec:static} write them as
$\mathcal J\subset\operatorname{int}I$. For a center $r\in J_\beta$, the gap
$d=q-r$, and the overlap imbalance $\delta$, we need a trial value for
$D(r,q,\delta)$ below $3P_\beta$ by a fixed multiple of $\rho^3$, uniformly
over $r\in J_\beta$ and over the band
$\{\rho\le d\le d_0,\ E\rho\le\abs\delta\le2E\rho\}$. The local cutoff $d_0$ is
chosen so small that every local window lies in $I_\beta$.

\paragraph{The obstacle.} In \cite[Section~7]{oai-barriers}, the field of $x$
receives $\delta$ times a Gaussian combination of the field increments in a
window $[r-h,r+h]$, and the field of $y$ receives $-\delta$ times the same
combination. The resulting excess covariance in the antisymmetric direction
$\mathbf e_x-\mathbf e_y$ contributes to the endpoint cost $(\beta^2/4)\norm{D-Q_N}_F^2$, and the expansion of
the trial value has the remainder $C(\abs\delta^3h^{-3/2}+\delta^4h^{-2})$
(see \eqref{eq:inp-remote}), which is small compared with the gain only if
$\abs\delta$ is much smaller than $h^{3/2}$, while $h$ is smaller than $q-r$.
In \cite{oai-barriers}, $h=\rho^8$ and $\abs\delta$ is of order $\rho^{20}$,
and the resulting deficit is of order $\rho^{43}$ \cite[(7.32)]{oai-barriers}.
When $\abs\delta$ and $q-r$ are both of order $\rho$, however, the mismatch and
the remainder are as large as the gain.

\paragraph{The observation.} For the perturbation constructed below, at
centers in the interior of the support, the terms of order at most two in the
width parameter $h$ of the window cancel exactly. The reason is the first stationarity identity
$\beta^2\E\chi(t,Y_t)^2=1$, where $\chi$ is the second spatial derivative of
the Parisi solution. This identity is the derivative of $\Gamma(t)=t$, and it
expresses the marginal stability of the Parisi solution
\cite[(22)--(23)]{crisanti-rizzo-2002}; its stochastic form is recorded in
\cite[Lemma~2.5]{full-rsb}. The deficit is then of
third order in $h$. To find it, we need a perturbed path whose endpoint
is exactly $D(r,q,\delta)$, so that we incur no mismatch cost, and an expansion of
its trial value to third order with a remainder of fourth order.

\paragraph{The perturbed path.} We fix a ratio $\vartheta\in(0,1/2]$; the
dynamical argument uses $\vartheta=1/2$. Given $(r,d,\delta)$, let $h=d/\vartheta$
be the width parameter of the window, $q_0=r-h$ its left endpoint, and $k=\delta/h$ the
relative imbalance (see \eqref{b:eq:os-parameters}). The perturbed path has the five
stages of \cref{tab:cub-stages}. In the baseline, the first four stages are
the following: the three fields share their increments on $[q_0,r]$, the
fields of $x$ and $y$ share theirs on $[r,q]$ while $v$ evolves alone, and all
three are independent on $[q,q_0+2h]$. To perturb them, we add $\delta\,\Xi$ to the
field of $x$ and subtract it from the field of $y$, where
\[
 \Xi=\frac1h\bigl\{\omega_0(g_1-g_2-g_x-g_y)+\omega_1g_v\bigr\},\qquad
 \omega_0=\frac1{2-\vartheta},\qquad\omega_1=\frac{1-\vartheta}{2-\vartheta},
\]
and $g_1,g_2,g_v,g_x,g_y$ are the field increments of the window
(see \eqref{eq:cub-contrast}). The weights are fixed, and the contrast extends past
the time $q$ at which $x$ and $y$ separate. By \eqref{eq:cub-contrast-moments},
the perturbation changes the covariance of the window in two ways: it shifts
the overlaps of $v$ with $x$ and with $y$ by $+\delta$ and $-\delta$, and it
adds an excess covariance of order $\delta^2/h$ in the antisymmetric
direction $\mathbf e_x-\mathbf e_y$. A fifth
stage, the terminal correction, removes this excess exactly. It lasts a time
proportional to $hk^2$, and it uses the positive semidefinite rate
$\mathrm{Id}-(\mathbf e_x-\mathbf e_y)(\mathbf e_x-\mathbf e_y)^{\mathsf T}/4$ in place of
$\mathrm{Id}$. Every increment of the path is positive semidefinite, the masses are
nondecreasing, and the endpoint is exactly $D(r,q,\delta)$
(\cref{lem:cub-identities,lem:cub-path}). From
\cite[Section~7]{oai-barriers} we keep the hierarchy in which the masses are
divided by the sizes of the common blocks, the antisymmetric Gaussian
contrast, and the source of the deficit in the identity
$\beta^2\E\chi(r,Y_r)^2=1$.

\paragraph{The reduction.} We split the trial value $\mathcal T(q_0,h,k)$
into the initial segment $[0,q_0]$, on which the three fields share every
increment, and the local window. Let $V(q_0,h,k,z)$ be the difference between
the value of the local window on the diagonal $\{z_v=z_x=z_y=z\}$ and its
unperturbed value $3\Phi(q_0,z)$ (see \eqref{eq:cub-V}), and let $\Delta\Pi$ be
the change of the penalty $\int m\dd\norm{Q}_F^2$. The initial segment responds
to the change $V$ of its terminal function linearly, up to an error
$\norm{V}_\infty^2/2$ that does not depend on the number of levels
(\cref{b:lem:os-prefix,cor:flow-prefix}). Since $\norm{V}_\infty\le Ch^2k^2$,
we have
\[
 \mathcal T(q_0,h,k)-3P_\beta=\E V(q_0,h,k,Y_{q_0})-\frac{\beta^2}4\Delta\Pi
 +O(h^4k^4)
\]
(\cref{cor:cub-prefix}). The expectation is taken under the law of the scalar
diffusion at the fixed time $q_0$, which does not depend on $(h,k)$; this is
why we expand at a fixed left endpoint. The local window is a composition of
five short semilinear parabolic flows on $\R^3$, some with singular diffusion
matrices. In \cref{sec:flow} we bound their mixed derivatives in $(h,k)$,
uniformly in the field and including $h=0$ (\cref{prop:flow-uniform}). Using
these bounds, we find that
\begin{equation}\label{eq:ov-pointwise}
 V(q_0,h,k,z)=\frac{\beta^4\chi_0^2}4\Delta\Pi
 +\beta^6h^3\bigl\{\alpha(q_0)^2\chi_0^3\,C_\chi(k)
 +\alpha(q_0)\,w_0^2\,C_w(k)\bigr\}+O(h^4k^2)
\end{equation}
(\cref{b:lem:os-flow,prop:cub-pointwise}). Here $\chi_0=\chi(q_0,z)$ and
$w_0=w(q_0,z)$ are the second and third spatial derivatives of the Parisi
solution, and $C_\chi$ and $C_w$ are the trace polynomial and the Wick
polynomial of \eqref{b:eq:os-R} and \eqref{b:eq:os-W}. When the mass is frozen
at $\alpha(q_0)$ on the window, the stages are Gaussian integration steps, and
three formal steps with explicit coefficient matrices give the coefficients of
order two and three (\cref{lem:cub-three-steps,lem:cub-centered,prop:cub-frozen}).
The variation of $\alpha$ on the window contributes, at order three, exactly
the corresponding part of $\beta^4\chi_0^2\Delta\Pi/4$ (\cref{lem:cub-mass}).

\paragraph{The cancellations.} The penalty enters the trial value twice:
directly, with the factor $-\beta^2/4$, and through $V$, with the factor
$\beta^4\chi(q_0,Y_{q_0})^2/4$. After the expectation, the sum is
$(\beta^2/4)(\beta^2\E\chi(q_0,Y_{q_0})^2-1)\Delta\Pi$, which vanishes by the
first stationarity identity. Since $\Delta\Pi$ is of order $h^2$,
this removes every term of order $h^2$, as well as the terms of order $h^3$ that involve
$\alpha'(q_0)$. The second stationarity identity
$\E w(t,Y_t)^2=2\alpha(t)\E\chi(t,Y_t)^3$ implies that the expectation of the
braces in \eqref{eq:ov-pointwise} equals
$\alpha(q_0)^2\E\chi(q_0,Y_{q_0})^3\{C_\chi(k)+2C_w(k)\}$. We conclude that
\begin{equation}\label{eq:ov-expansion}
 \Bigl|\mathcal T(q_0,h,k)-3P_\beta
 -\alpha(r)^2\beta^6\E\chi(r,Y_r)^3\,h^3P_\vartheta(k)\Bigr|\le Ch^4k^2,
 \qquad P_\vartheta=C_\chi+2C_w
\end{equation}
(\cref{b:prop:os-expansion}, with $q_0+h=r$). Both identities hold on
$(0,q_\beta)$ (\cref{b:lem:os-scalar-jets}), and both were known; see
\cite[proof of Lemma~7.2]{quantitative-parisi} and \cite[Lemma~2.5]{full-rsb}.
The use of the second identity in a covariance expansion is new. The
polynomial $P_\vartheta$ is even, with coefficients
$c_2(\vartheta)=2\vartheta(\vartheta-1)/(\vartheta-2)^2<0$,
$c_4(\vartheta)<0$, and $c_6(\vartheta)>0$ (see \eqref{eq:cub-c}). For
$\vartheta=1/2$,
\[
 P_{1/2}(k)=-\frac29k^2-\frac{62}{243}k^4+\frac{6952}{19683}k^6
\]
(see \eqref{eq:cub-P-half}), and \cref{sec:cub-coefficients} lists the traces from
which this polynomial can be recomputed by hand.

\paragraph{The deficit on the band.} In the variables $d$ and $\delta$, we
have $-h^3P_\vartheta(k)=\mathcal G_\vartheta(d,\abs\delta)$, where
\[
 \mathcal G_\vartheta(d,x)=-\frac{c_2}\vartheta\,x^2d-\vartheta c_4\frac{x^4}d
 -\vartheta^3c_6\frac{x^6}{d^3}
\]
(see \eqref{eq:cub-G-function}). The leading term is
$2(1-\vartheta)(2-\vartheta)^{-2}\delta^2d$, which is $4\delta^2d/9$ for
$\vartheta=1/2$. The deficit is quadratic in the overlap imbalance and
linear in the gap $q-r$; its minimum on the band is a positive multiple of
$\rho^3$, and it can be larger away from the corner. The scalar factor
$\alpha(r)^2\beta^6\E\chi(r,Y_r)^3$ is bounded below on $I_\beta$ by a constant
$m_\beta>0$, since $\alpha>0$ on $(0,q_\beta]$ and
$\E\chi(r,Y_r)^3\ge\beta^{-3}$ (\cref{cor:sc-mbeta}). For $E=1/100$, the
function $\mathcal G_\vartheta$ is increasing in $d$ and in $\abs\delta$ on the
band, and it attains its minimum at the corner $(\rho,E\rho)$
(\cref{lem:cub-band}). Choosing $d_0$ so small that the remainder in
\eqref{eq:ov-expansion} is absorbed, we obtain, for every fixed
$g'<m_\beta E^2G(\vartheta,E)$,
\begin{equation}\label{eq:ov-band}
 p_{3,n}\bigl(D(r,q,\delta)\bigr)\le\mathcal T(q_0,h,k)\le3P_\beta-g'\rho^3
\end{equation}
for every feasible matrix in the band, all $n$, and all $0<\rho\le d_0$,
uniformly over $r\in J_\beta$ (\cref{prop:cub-band}). Here $G(\vartheta,E)$ is the value of
$E^{-2}\rho^{-3}\mathcal G_\vartheta$ at the corner, and $G(1/2,1/100)>1/3$.
Since the second stationarity identity fails at $t=q_\beta$, where the Parisi
measure has an atom (\cref{rem:sc-upper-atom}), the centers lie in compact
subintervals of $(0,q_\beta)$; no other condition on the Parisi measure near
$r$ is needed, in contrast with \cite[Lemma~7.1]{oai-barriers}.

\section{From pressures to Gibbs probabilities}\label{sec:ov-static}

\paragraph{The obstacle.} The bound \eqref{eq:ov-band} concerns the quenched
pressure $n^{-1}\E\log Z_{3,n}(D)$. However, Gaussian concentration of this
Lipschitz function of the disorder (\cref{rem:inp-lipschitz}) turns a mean
deficit of order $\rho^3$ only into the probability bound $e^{-cn\rho^6}$, and
$n\rho^6$ tends to zero; for $\rho=Rn^{-7/30}$, it is $R^6n^{-2/5}$.

\paragraph{Fractional moments.} To overcome this obstacle, we bound positive
fractional moments of order $s=S\rho$, for a small constant $S>0$. Suppose that
\begin{equation}\label{eq:ov-fractional}
 \frac1{ns}\log\E Z_{3,n}(D)^s\le3P_\beta-g\rho^3 .
\end{equation}
Applying Markov's inequality to $Z_{3,n}(D)^s$, we find that
\[
 \PP\bigl\{n^{-1}\log Z_{3,n}(D)>3P_\beta-\varkappa g\rho^3\bigr\}
 \le e^{-(1-\varkappa)gS\,n\rho^4},
\]
and $n\rho^4$ grows like a power of $n$ (Step~2 of the proof of
\cref{b:lem:static-event}). To prove \eqref{eq:ov-fractional}, we adapt the
matrix comparison of \cite[Proposition~2.3]{oai-barriers} to moments of order
$s$, as in the positive-moment interpolation of Aronow and Lopatto
\cite[Section~2.3]{quantitative-parisi}: the disorder is integrated at the outer
mass $s$, instead of by an expectation. In this way we find that the trial
value of every admissible path whose masses are all at least $s$ bounds
$(ns)^{-1}\log\E Z_{\ell,n}(D)^s+\beta^2\ell^2s/(4n)$
(\cref{b:prop:fractional-matrix}). The interpolation is carried out for the
completed Hamiltonian $H_n+G$ of \cref{sec:model}, whose covariance is exactly
$nR(x,y)^2/2$. The correction $\beta^2\ell^2s/(4n)$ is quadratic in the number
$\ell$ of configurations, because the same variable $G$ enters every
configuration (\cref{lem:model-completion}). Since this correction is
nonnegative, it can be discarded.

\paragraph{The profile $\max\{\alpha,3s\}$.} The masses of the perturbed path
on its initial segment are $\alpha(t)/3$, and $\alpha(t)\to0$ as $t\to0$, so
that the requirement that all masses are at least $s$ fails near the origin.
To overcome this, we replace $\alpha$ by $\max\{\alpha,3s\}$ on the initial
segment. By the trial bound of Aronow and Lopatto
\cite[Lemma~7.2]{quantitative-parisi}, the Parisi functional of this profile
exceeds $P_\beta$ by at most $Cs^5$ (\cref{inp:qp-floor,lem:fr-floor}), which
is much smaller than $\rho^3$. The profile differs from $\alpha$ only on an
interval $[0,Cs]$, far from the window, which starts at $q_0\ge\min I_\beta>0$. We
show that its effect on the response of the initial segment to $V$ is at most
$C\norm{V'}_\infty\norm{\max\{\alpha,3s\}-\alpha}_{L^1}\le C\delta^2s^2/h$
(\cref{lem:fr-prefix-response}). On the band this is at most a constant times
$S^2\rho^3$, and we make it small by the choice of $S$
(see \eqref{b:eq:coarse-floor-error}). Together with \eqref{eq:ov-band}, this
proves \eqref{eq:ov-fractional} for every fixed $g<m_\beta E^2G(\vartheta,E)$,
for all $n$ and all small $\rho$, uniformly over the band
(\cref{lem:fr-local-gap}).

\paragraph{The other regimes.} Every narrow triple falls into one of three
regimes (\cref{lem:st-cover}). In the local regime $q-r\le d_0$, we use
\eqref{eq:ov-fractional}. In the regime $q-r>d_0$ of separated overlaps, with
$\abs{q-\Gamma(q)}$ small, we use the quadratic deficit $c_{\rm sep}\delta^2$ of
\cref{lem:st-remote}, which we derive from
\cite[Section~7, Steps~1--7]{oai-barriers} with fixed parameters; its source is
the decay $\beta^2\E[\chi(q,Y_q)\chi(r,Y_r)]\le1-c(q-r)$ of the mixed
susceptibility \cite[(7.31)]{oai-barriers}. Since $\delta^2$ is of order
$\rho^2$, Gaussian concentration suffices there. In the
remaining pair regime, the two configurations with the larger overlap $q$ have
$\abs{q-\Gamma(q)}>\nu$ for a fixed $\nu>0$, so that the pair deficit
$(\Gamma(q)-q)^2/2$ of \cite[Lemma~4.3]{oai-barriers} (\cref{lem:st-pair}) applies. For
the fixed-scale constraints, we use the three-replica deficit
$c_{\rm fix}\delta^2$ of \cite[(4.20)]{oai-barriers} (\cref{lem:st-fixed}) at
the tolerance $\abs\delta\ge\eta_n$. In the separated-overlap, pair, and
fixed-scale regimes, the mechanisms are those of \cite{oai-barriers}; what
changes are the scales at which the pressure deficits are converted into
probability bounds.

\paragraph{One disorder event.} The disorder event $\mathcal A_n$ of
\cref{b:lem:static-event} requires the lower bound on $F_n$ of
\cref{sec:ov-scales} and upper bounds on all constrained partition functions
of the local, separated-overlap, fixed-scale, and pair families; these are
deterministic families of at most $C(n+1)^3$ matrices. The event
$\mathcal A_n$ fails with probability at most
\[
 C\frac{n^{7/15}}{n^2\rho^6}+C(n+1)^3\bigl(e^{-cn\rho^4}+e^{-cn\eta_n^4}\bigr)
 +C(n+1)e^{-cn}
\]
(see \eqref{b:eq:static-event-failure}), which is $O(n^{-2/15})$ for
$\lambda=7/30$. For every disorder in $\mathcal A_n$,
dividing the numerators by the denominator, we find that
\[
 \pi^{\otimes3}(\mathcal N_n)\le C(n+1)^3e^{-cn\rho^3},\qquad
 \pi^{\otimes3}(\mathcal W_n)\le C(n+1)^3e^{-cn\eta_n^2},
\]
where $\mathcal W_n$ is the set of fixed-scale triples
(see \eqref{b:eq:conditional-static-errors}). We keep two kinds of error apart
(\cref{rem:st-two-levels}). Our estimate for the disorder failure probability
$\PP(\mathcal A_n^c)$ contains a polynomial term, and we use this estimate
once, additively, at the end of the proof. The Gibbs
probabilities at a fixed disorder in $\mathcal A_n$ decay at least as fast as
$e^{-cn^{1-3\lambda}}$, and only these are multiplied by the numbers $e^{O(L)}$
of reference configurations and path positions. This separation replaces the
conversion \cite[Lemma~4.2]{oai-barriers} of pressure bounds into probability
bounds, whose finite-size error $n^{-1/24}$ is far too large at the scale
$\rho^3$.

\section{The dynamical argument}\label{sec:ov-dynamics}

\paragraph{Fixed families of references.} The dynamical argument starts from
two inputs at a fixed disorder. The first is the bound on the Gibbs mass of
overlap neighborhoods: for every $S\subseteq\Sigma_n$, all $x\in S$ outside a
set of Gibbs mass less than $e^{-D_1L}$ have Gibbs mass at least $e^{-D_3L}$ of
configurations $z\in S$ with $\abs{R(x,z)}\ge q_*$. We prove it by the argument
of \cite[Proposition~6.1]{oai-barriers}, at every scale $L\to\infty$ with
$L=o(n)$ (\cref{b:lem:coverage-extension}). The second input is the bound
$\pi^{\otimes3}(\mathcal B_n)\le q_n$ for the exceptional triples
$\mathcal B_n=\mathcal N_n\cup\mathcal B_n(q_*,b_1)\cup\mathcal B_n(b_1,b_2)$ on
the event $\mathcal A_n$, where
$q_n=C(n+1)^3(e^{-cn\rho^3}+e^{-cn\eta_n^2}+e^{-cn})$ satisfies
$e^{CL}q_n\to0$ for every fixed $C$ (\cref{lem:ref-single-event}).

In \cite[Section~8.3]{oai-barriers}, the properties of the lists of references
are proved with probability tending to one jointly over the disorder, the
lists, and the path, with bounds averaged over the disorder. At our scales,
such an averaged bound contains the disorder failure probability, which does
not remain small after multiplication by $e^{O(L)}$. For this reason, we fix
the disorder first. At each
disorder in $\mathcal G_n$, we sample the lists, with
$\lceil e^{(c_s+1)L}\rceil$ independent Gibbs samples at the levels of
position $s$ in a block. The expected Gibbs mass of the configurations at which
some cell contains no suitable entry is at most $2e^{-4L}$. The expected number
of exceptional triples of list entries, and the expected Gibbs mass of the
configurations that form one with two entries, are at most $e^{O(L)}q_n$. We fix
a realization at which the sum of such quantities is at most its mean
(\cref{lem:ref-families}). Outside a set $V$ with $\pi(V)\le3e^{-4L}$, it has
properties (P1)--(P4) of \cref{sec:ind-setting}: every cell supplies a new
reference for each configuration in it, the fixed-scale constraints hold across $q_*$ and across $b_1$, and
no configuration outside $V$ forms a narrow triple with two signed entries.
The family and $V$ depend only on $J$ (\cref{rem:ref-every-path}).

\paragraph{The accumulation of errors.} In
\cite[(8.19)--(8.20)]{oai-barriers}, the overlaps of every retained reference
with the current configuration and with the later references are controlled
within $6j\varepsilon$ after $j$ levels. These errors accumulate, and the final
count of \cite[(8.29)]{oai-barriers} requires $N_{\rm lev}\varepsilon$ to be
much smaller than $\rho$. Since $N_{\rm lev}$ is of order $1/\rho$, this forces
$\varepsilon$ to be much smaller than $\rho^2$. Our choice $\varepsilon=E\rho$,
at which the narrow constraints have a deficit of order $\rho^3$, is far outside
this range.

\paragraph{Next retained references.} To overcome this, we compare each
retained reference $v_i$ with the next retained reference $v_{a_j(i)}$, where
$a_j(i)$ is the smallest retained level larger than $i$ after level $j$
(\cref{sec:ind-algorithm}). We call the overlap $c_j(i)=R(v_i,v_{a_j(i)})$ the
center of $v_i$. When $v_{i+1}$ is selected at the time $k_i$, we have
$R(v_{i+1},x_{k_i})\ge q_*$ and $R(v_i,x_{k_i})\in[r_i-2/n,r_i]$, and the
fixed-scale constraint across $q_*$ implies that
$\abs{c_j(i)-r_i}\le2\eta_n+2/n$. The center changes only when a deletion
removes the next retained reference of $i$. The next retained reference is then
the first reference of the following block, which is never deleted, and the
center moves by at most $2\varepsilon+2\eta_n$. We show by induction that, for
every retained $i<j$,
\[
 \abs{c_j(i)-r_i}\le2\varepsilon+4\eta_n+\frac2n,\qquad
 \abs{R(v_i,x_{k_j})-c_j(i)}<2\varepsilon
\]
(\cref{def:ind-invariant,prop:ind-induction}). The errors are of order
$\varepsilon$ at every level, and they do not grow with the number of levels.

The narrow constraints preserve these windows during the backward search
(\cref{lem:ind-search}). Suppose that some window fails at a time between
$k_j$ and $k_{j-1}$, and let $k$ be the largest such time and $v_i$ a
reference whose window fails at $k$. All windows hold at time $k+1$, and since
one step changes overlaps by at most $2/n$, the overlap $R(v_i,x_k)$ differs
from $c_j(i)$ by an amount in $[2\varepsilon,2\varepsilon+2/n)$. Let
$a=a_j(i)$. Then $R(v_a,x_k)$ is close to $r_a$ or larger: if $a=j$, because the
backward search has not yet reached the crossing of $r_j$, and otherwise
because the window of $v_a$ holds at time $k+1$. Since $r_a-r_i\ge2\rho$ and
all errors are of order $\varepsilon$, we find that $R(v_a,x_k)\ge r+\rho$, where $r$ is the mean of $R(v_i,x_k)$ and
$R(v_i,v_a)=c_j(i)$. The triple $(v_i,x_k,v_a)$ is then narrow, which (P4)
excludes because $x_k\notin V$. In other words, the next retained reference
supplies the large overlap that the narrow constraint needs, and the spacing
$3\rho$ of the threshold intervals makes this overlap large enough.

\paragraph{The count.} At the time $k_*=k_{N_{\rm lev}-1}$, the first
reference $v_{\ell B}$ of each block $\ell$ still has its original next retained
reference $v_{\ell B+1}$, because the first two levels of a block are never
deleted. By its window, we have $\abs{R(v_{\ell B},x_{k_*})-r_{\ell B}}<\Psi_0$ for
$\ell=0,\dots,M-1$ (\cref{cor:ind-witness}). The reference of each level depends
only on the thresholds of the earlier levels (\cref{lem:ind-predictable}); this
is why a selected reference whose crossing fails is kept. Removing the windows
one at a time, from the largest level down, we find that for each fixed time
$k$ the $M$ windows hold with probability at most $\gamma_n^M$ over independent
uniform thresholds (\cref{lem:bp-product}). Summing over $k\le m$, we have the
bound \eqref{eq:ov-count}. If $(m+1)\gamma_n^M<1$, some threshold array admits no such
time, which contradicts the existence of a path that violates
\eqref{eq:ov-obstruction} (\cref{prop:bp-obstruction}). In \cite[(8.29)]{oai-barriers}, the count is a union over
subsets of retained levels; here the windows sit at the deterministic first
levels of the blocks, and we count only $M$ of them. Further, since the
static estimates are uniform over the interval $J_\beta$, we need no random
event that restricts the thresholds, such as \cite[(8.12)]{oai-barriers}.

\paragraph{Quantitative and almost-sure forms.} With the lower bound on $F_n$
from the small-deviation inequality of Paouris and Valettas
\cite[Theorem~2.1 and Remark~2.4(3)]{paouris-valettas-2018} instead of
Chebyshev's inequality, with $\eta_n=\rho_n/1000$, and with
$A<t_0\log(250/11)/(3BR)$, the disorder event fails with probability at most
$Ce^{-cn^{1/15}}$, and on it the initial configurations from which the
distance stays at least $1/2-Ce^{-cn^{7/30}}$ up to the time horizon
$e^{An^{7/30}}$ have Gibbs mass at least $1-Ce^{-cn^{7/30}}$, for all
$n\ge n_1(C_{\rm rate})$ and every single-site generator with rates at most a
fixed $C_{\rm rate}$ that leaves $\pi$ stationary (\cref{thm:barrier-strong}).
By the same theorem, we have the relaxation time bound
$e^{An^{7/30}}/(Cn^{7/30})$ for the reversible ones and, by the
Borel--Cantelli lemma, the almost sure form for every coupling of the
disorders of different sizes.

\section{The upper bound}\label{sec:ov-upper}

We follow the architecture of the proof of
\cite[Theorem~1]{oai-typical-upper}, which proves \cref{thm:upper} for the
saving $\omega=1/40000000$. The proof has four steps. A finite-chain argument
reduces a bound for typical initial configurations to a lower bound on the
stationary flow across every cut whose two sides both have appreciable Gibbs
mass. A random disorder and a random configuration are then constructed
jointly, by Gaussian queries of the disorder matrix that follow the scalar
Parisi diffusion. The law of this pair is compared with the Gibbs joint law in
two ways: through its relative entropy, and through an upper bound on its
likelihood ratio outside a rare event. A Gaussian smoothing of the disorder
reverses the entropy comparison, so that the constructed configuration reaches
every set of appreciable Gibbs mass. Finally, a sequence of such
configurations, built from rotated Gaussian inputs, connects the two sides of
a cut of small flow while avoiding its boundary, which is impossible. In
\cite{oai-typical-upper}, the parameters are fixed numbers, and the exponent
is not optimized.

To reach every $\omega<1/4$, two estimates must be close to sharp: the
relative entropy of the proposal at the longer horizon, and the likelihood
bound at the shorter horizon, together with the probability of its
exceptional event. The first is described in the paragraph on the forward
entropy below, and the second in the paragraph on coarse domination. The other
steps follow \cite{oai-typical-upper}, with the changes of parameters that
these two estimates allow.

\paragraph{Notation.} Let $g\in\R^{n\times n}$ have independent standard
Gaussian entries, with law $\gamma_n$, and let
$W(g)=(g+g^{\mathsf T})/\sqrt{2n}$. With $J_{ij}=(g_{ij}+g_{ji})/\sqrt2$ and
$J'_{ii}=g_{ii}$, the matrix $W(g)$ is the matrix $W$ of
\cref{sec:model-completion}, and $H^c_g(x)=x^{\mathsf T}W(g)x/2$ is the
completed Hamiltonian of the disorder $J(g)$. We write $F^c(g)$ for the
completed log partition function, $\pi_g=\pi^{J(g)}$, $P^g_t=P^{J(g)}_t$, and
\[
 \Pi_n(\dd g,x)=\gamma_n(\dd g)\,\pi_g(x)
\]
for the Gibbs joint law. Its $(J(g),x)$ marginal is the realized-start law
$\PP_{J,X_0\sim\pi^J}$ of \cref{def:model-joint}. For $S\subseteq\Sigma_n$,
the stationary flow across $S$ is
\[
 \mathcal Q_g(S,S^c)=\sum_{x\in S,\,y\notin S}\pi_g(x)\,\mathcal L^{J(g)}(x,y),
\]
where $\mathcal L^{J(g)}(x,x^{i,-x_i})=p_i^{J(g)}(-x_i\mid x)$ is the rate of
a flip at site $i$ (\cref{sec:model-dynamics}). The exponents of the proof
are fixed numbers, written in sans-serif ($\ex a,\ex q,\ex k_{\rm f},\dots$);
they are chosen after $\beta$ and $\omega$, and they are listed in
\cref{tab:ov-upper-exponents}. Constants $C_\beta,c_\beta>0$ depend only on
$\beta$ and on the exponents.

\begin{table}[!t]
\centering
\small
\caption{The exponents of the proof of \cref{thm:upper}, as functions of
$\varepsilon=\varepsilon_\omega=(1/4-\omega)/20$, their values for
$\omega=1/5$, and their roles. They are fixed after $\beta$ and $\omega$, and
before $n\to\infty$; see \cref{u:sec:parameters,u:sec:budget}.}
\label{tab:ov-upper-exponents}
\begin{tabular}{@{}lll>{\raggedright\arraybackslash}p{0.5\textwidth}@{}}
\toprule
Exponent & Value & $\omega=1/5$ & Role\\
\midrule
$\ex a$ & $1/2$ & $1/2$ & mesh $h=n^{-\ex a}$\\
$\ex f$ & $1/2$ & $1/2$ & fine horizon $V_{\ex f}\approx\ex f\log n$\\
$\ex b$ & $1/2-14\varepsilon$ & $93/200$ & coarse horizon $V_{\ex b}\approx\ex b\log n$\\
$\ex k_{\rm f}$ & $1/2-2\varepsilon$ & $99/200$ & forward entropy $n^{1-\ex k_{\rm f}}$\\
$\ex D$ & $1/4-\varepsilon$ & $99/400$ & smoothing $\delta=n^{-\ex D}$\\
$\ex u_{\rm E}$ & $1/4-2\varepsilon$ & $49/200$ & reverse entropy $n^{1-\ex u_{\rm E}}$\\
$\ex d$ & $\varepsilon/2$ & $1/800$ & masses $n^{-\ex d}$ of the cuts\\
$\ex e$ & $1/4-3\varepsilon$ & $97/400$ & crossing probability $e^{-2n^{1-\ex e}}$\\
$\ex w$ & $1/8-3\varepsilon$ & $47/400$ & weighted errors $n^{-\ex w}$\\
$\ex z$ & $1/4-8\varepsilon$ & $23/100$ & accuracy $n^{-\ex z}$ of the row averages\\
$\ex k_{\rm b}$ & $1/4-9\varepsilon$ & $91/400$ & likelihood bound $e^{n^{1-\ex k_{\rm b}}}$\\
$\ex s_{\rm b}$ & $1/4-5\varepsilon$ & $19/80$ & failure $e^{-n^{1-\ex s_{\rm b}}}$ of the coarse proposal\\
$\ex s_{\rm seq}$ & $1/4-4\varepsilon$ & $6/25$ & failure $e^{-n^{1-\ex s_{\rm seq}}}$ of the sequence\\
$\ex r$ & $1/4-15\varepsilon/2$ & $37/160$ & Hamming radius $R_n=\lceil n^{1-\ex r}\rceil$\\
$\ex u_{\rm F}$ & $1/4-39\varepsilon/4$ & $361/1600$ & threshold $n^{1-\ex u_{\rm F}}$ for sums of local fields\\
$\ex q$ & $1/4-10\varepsilon$ & $9/40$ & flow threshold $\phi_n=e^{-n^{1-\ex q}}$\\
$\ex z_{\rm P}$ & $3/10$ & $3/10$ & pressure event $F^c\le nP_\beta+n^{1-\ex z_{\rm P}}$\\
$\ex g_{\rm rot}$ & $3/5$ & $3/5$ & grid of $\ell_n+1\approx e^{n^{\ex g_{\rm rot}}}$ angles\\
\bottomrule
\end{tabular}
\end{table}

\paragraph{From balanced-cut flow to mixing.} By \cref{lem:model-joint},
\cref{thm:upper} states that the Gibbs mass of the configurations $x$ with
$d^{J(g)}(x,e^{n^{1-\omega}})>1/4$ tends to zero in probability over $g$. The
reduction to flows is the finite-chain lemma
\cite[Lemma~8]{oai-typical-upper}, which we prove as \cref{u:lem:pruning}.
Let $0<\epsilon<1/3$ and $\phi>0$, and suppose that
$\mathcal Q_g(S,S^c)>\phi$ for every set $S$ with
$\epsilon\le\pi_g(S)\le1-\epsilon$. Then, for every $t>0$,
\begin{equation}\label{eq:ov-pruning}
 \sum_{x\in\Sigma_n}\pi_g(x)\,d^{J(g)}(x,t)
 \le\frac{n\sqrt{2\log2}}{\phi\sqrt t}+2\epsilon .
\end{equation}
Here $n$ bounds the total jump rate out of a configuration, and $n\log2$
bounds the entropy of $\pi_g$. The proof removes, one at a time, sets of at
most half of the remaining mass whose flow to the rest is less than $\phi$
times their mass. The hypothesis prevents the removed mass from reaching $\epsilon$.
On the remaining set, the flow bound gives a Poincar\'e inequality, by the
argument of Lawler and Sokal \cite[Theorem~3.5]{lawler-sokal-1988}, and the
entropy dissipation of the full chain
\cite[Lemmas~2.5 and~2.7]{diaconis-saloff-coste-1996} bounds the average of
the distances from fixed initial configurations. The trajectory need not
stay in the remaining set. Initial configurations from which the chain is
slow are allowed, and a set of them costs at most its Gibbs mass. For this
reason only cuts of mass at least $\epsilon$ enter, and the conclusion
concerns typical initial configurations; at large $\beta$, the worst-case
mixing time is exponential \cite[Theorem~1.1]{sellke-2025}.

We apply \eqref{eq:ov-pruning} with $\epsilon=n^{-\ex d}$,
$\phi=\phi_n=e^{-n^{1-\ex q}}$, and $t=e^{n^{1-\omega}}$. By Markov's inequality
in the Gibbs draw, we find that the joint probability that
$d^{J(g)}(x,t)>1/4$ is at most
\[
 \gamma_n(\mathcal W_n)+2e^{-n}
 +4n\sqrt{2\log2}\,e^{n^{1-\ex q}-n^{1-\omega}/2}+8n^{-\ex d}
\]
(\cref{u:sec:proof}). Since $\omega<\ex q$, the third term tends to zero.
The term $2e^{-n}$ bounds the probability that
$\norm{W(g)}_{\rm op}>12$ (\cref{u:lem:goe-norm}), and
\[
 \begin{split}
 \mathcal W_n=\bigl\{g:{}&\norm{W(g)}_{\rm op}\le12,\text{ and there exists }S
 \subseteq\Sigma_n\\
 &\text{with }n^{-\ex d}\le\pi_g(S)\le1-n^{-\ex d}\text{ and }
 \mathcal Q_g(S,S^c)\le\phi_n\bigr\}
 \end{split}
\]
is the event that a cut of small flow exists. The rest of the proof shows
that $\gamma_n(\mathcal W_n)\to0$ (\cref{u:prop:witness}).

\paragraph{The proposal.} The flow bound is proved with a random pair $(g,x)$
whose law is close to $\Pi_n$. Its construction is that of
\cite[Section~4]{oai-typical-upper}, which is related to the iterative
construction of solutions of the TAP equations by Bolthausen
\cite{bolthausen-2014} and to the incremental approximate message passing algorithm of
Montanari \cite{montanari-2021}. Let $\alpha_\beta$ be the Parisi distribution
function; the Parisi measure has support $[0,q_\beta]$ (\cref{inp:full-rsb}).
In the variance
time $v=\beta^2t$, the scalar diffusion of \cref{sec:inputs} solves
\[
 \dd Y_v=\dd M_v+\bar\alpha(v)\,m(v,Y_v)\dd v,\qquad Y_0=0,
\]
where $M$ is a standard Brownian motion, $\bar\alpha(v)=\alpha_\beta(v/\beta^2)$,
and $m(v,y)=u_{\alpha_\beta}(v/\beta^2,y)$ is the spatial derivative of the
Parisi solution. The magnetization $m_v=m(v,Y_v)$ is a martingale, with
$\dd m_v=\chi_v\dd M_v$, where $\chi_v=\chi(v,Y_v)$ is the second spatial
derivative. Beyond $v_*=\beta^2q_\beta$, we continue the process with
$\bar\alpha=1$, $m(v,y)=\tanh y$, and $\chi(v,y)={\rm sech}^2y$; the continued
potential solves the same equation (\cref{u:sec:scalar}). We define
\[
 p(v)=\E\chi_v,\qquad r(v)=\E\chi_v^2,\qquad
 \rho(v)=\beta\sqrt{r(v)},\qquad \sigma(v)=\sqrt{1-\rho(v)^2}.
\]
By \cref{cor:inp-identity}, we have $r=\beta^{-2}$ on $[0,v_*]$, so that
$\rho=1$ and $\sigma=0$ there. After $v_*$, the functions $p$ and $r$ decay at
the rate $e^{-v/2}$ up to a factor $\sqrt{1+v}$
(\cref{u:prop:scalar,u:lem:tails}), and at a horizon $V$ of order $\log n$ the
magnetization is close to $\pm1$.

In the proposal, we run one copy of the continued scalar process for each
coordinate, driven by noise extracted from the matrix $W=W(g)$. We cut the time
axis into cells $[t_j,t_{j+1}]$ with
\[
 t_0=0,\qquad h_j=he^{t_j/4},\qquad t_{j+1}=t_j+h_j,\qquad h=n^{-\ex a}.
\]
At the query $j\ge1$, we take the increment of the vector of magnetizations
over the previous cell, normalized as
$\mathcal T_j=(m(t_j)-m(t_{j-1}))/s_j$ with $s_j^2=\int_{t_{j-1}}^{t_j}r$, and
we orthogonalize it against the earlier query directions in the inner product
$n^{-1}\langle u,v\rangle$. This gives a direction $u_j$, and we read the
column $Wu_j$. Its component orthogonal to $u_1,\dots,u_j$, together with an
independent Gaussian completion inside their span, is a vector $e_j$ that is
standard Gaussian given the past. The vector $e_j$ becomes the standardized
increment of the query paths $B_i$ over the next cell $[t_j,t_{j+1}]$, which
is filled with independent Brownian bridges. Each coordinate is driven by
$M_i=\int\rho\dd B_i+\int\sigma\dd D_i$, where the $D_i$ are independent
Brownian motions. Up to $v_*$, the matrix drives the process alone; after
$v_*$, the query noise is mixed with independent noise. By Gaussian orthogonal
invariance, the unread part of $W$ is a compressed Gaussian orthogonal
ensemble, independent of the past. Then the matrix assembled from the answers
and the remaining Gaussian variables has the law of $W(g)$ under $\gamma_n$,
independently of the initial data, and the pairs $(B_i,D_i)$ are independent
pairs of Brownian motions (\cref{u:lem:assembly}). At the horizon $V$, the
spin is linked to the terminal magnetizations by the density
$\prod_i(1+x_im_i(V))$ relative to an independent uniform spin
(\cref{u:def:laws}). The output $Q_V$ is the law of $(g,x)$, where $g$ is
obtained from $W$ by adding an independent antisymmetric Gaussian matrix, so
that $W(g)=W$. Its marginals are
$\gamma_n$ and the uniform law, but $x$ is correlated with $g$.

The construction has two equivalent forms: in one, the matrix is assembled
from the answers to the queries, and in the other, the same recursion reads a
given matrix and given completions (\cref{u:def:laws,u:lem:assembly}). The main
structural fact is that, under the linked law, the rows $(x_i,Y_i,B_i,D_i)$,
$i=1,\dots,n$, are independent, with an explicit one-row law $\nu_V$.
Apart from the Gaussian variables that the recursion never reads, the
statistics of the proposal used below are averages of $n$ independent rows, or
explicit functions of such averages.

We use two horizons on the same mesh: the fine horizon $V_{\ex f}$ and the
coarse horizon $V_{\ex b}$, the first times $t_j$ beyond $\ex f\log n$ and
$\ex b\log n$, with $\ex b<\ex f$ (\cref{u:lem:quad}). The two runs read the
same inputs, so that the coarse run is an initial segment of the fine run. The
growing cells are new compared with \cite{oai-typical-upper}, where the cells
have equal length $n^{-1/16}$. Since $r(t_j)$ decays like $e^{-t_j/2}$, the
lengths $h_j$ grow as the profile decays, and $h_j^2r(t_j)$ stays of order
$h^2$ at most. The number of queries is at most $8n^{\ex a}$ for every horizon
(\cref{u:lem:quad}), and the errors of the mesh add up to $O(h)$, uniformly in
the horizon. Unequal cells require some care in the estimates: $\mathcal T_j$
is an increment over the previous cell, $e_j$ belongs to the next cell, and the
ratio $h_j/h_{j-1}=e^{h_{j-1}/4}$ enters the bounds.

\paragraph{The forward entropy.} The first comparison is
\begin{equation}\label{eq:ov-forward-kl}
 \KL(Q_V\Vert\Pi_n)\le nj(V)-n\log2+\E F^c(g)-\beta\,\E H^c_g(x),
 \qquad j(V)=\frac12\int_0^V\rho(v)^2\varkappa(v)\dd v,
\end{equation}
where $\varkappa(v)=\E[\chi_v^2/(1-m_v^2)]$ and the last expectation is under
the proposal (\cref{u:lem:forward-kl}). The quantity $j(V)$ is the
information per spin that the proposal acquires through the matrix. To see
this, we move the part of the linking density that comes from the independent
noise $D$ into a reference law, as in \cite[Section~4]{oai-typical-upper}, whose $(g,x)$ marginal is $\gamma_n$ times the
uniform law. The logarithm of the remaining density is a sum of $n$
independent rows, each with mean $j(V)$ (\cref{u:lem:reference}). The factor
$\rho^2$ is the share of the matrix in the driving noise. Changing the
reference to $\Pi_n$, whose density is $2^ne^{\beta H^c_g(x)-F^c(g)}$, gives
\eqref{eq:ov-forward-kl}.

The terms of \eqref{eq:ov-forward-kl} are related by the scalar identity
\cite[Lemma~4]{oai-typical-upper}: for every $V\ge v_*$,
\begin{equation}\label{eq:ov-scalar-identity}
 j(V)-\log2+P_\beta-\beta e(V)=\frac{\beta^2}4p(V)^2,
 \qquad e(V)=\beta\int_0^Vr(v)p(v)\dd v
\end{equation}
(\cref{u:prop:scalar}). Here $e(V)$ is the scalar reference for the energy per spin of the proposal,
and the identity balances the information paid through the matrix against the
energy gained. By Guerra's bound and \cref{lem:model-completion}, we have
$\E F^c(g)\le nP_\beta$ (\cref{inp:parisi}). The expected energy satisfies
\[
 \frac1n\E H^c_g(x)\ge e(V)-C_\beta\Bigl[h+\frac{(N+1)(1+V)+\sqrt N}n
 +\frac{(N+1)^2VR_V}{n^2}\Bigr],
\]
where $N$ is the number of queries and $R_V=(\min_{[0,V]}r)^{-1}$
(\cref{u:lem:fine-energy}). Inserting these bounds into
\eqref{eq:ov-forward-kl}, and using $p(V)^2\le C_\beta e^{-V}$, we find that
\[
 \KL(Q_V\Vert\Pi_n)\le C_\beta\Bigl[nh+ne^{-V}+(N+1)(1+V)+\sqrt N
 +\frac{(N+1)^2VR_V}n\Bigr].
\]
At the fine horizon, the five terms have the powers $1-\ex a$, $1-\ex f$,
$\ex a$, $\ex a/2$, and $2\ex a+\ex f/2-1$ of $n$, up to logarithms, and we
obtain $\KL(Q_F\Vert\Pi_n)\le C_\beta n^{1-\ex k_{\rm f}}$ for the fine
output $Q_F$ whenever $\ex a>\ex f/4$, $\ex a+\ex f/2<1$, and
$\ex k_{\rm f}<\min(\ex a,\ex f,1-\ex a)$ (\cref{u:prop:fine-kl}).

This is the first place where a sharp estimate is needed. In
\cite[Proposition~5]{oai-typical-upper}, the relative entropy is at most
$2n^{1-l/2}$ for $0<l\le1/100$; the energy is computed on the event where the
empirical Gram matrix of the queries is close to the identity, and the
information is concentrated separately. Here the expected energy is computed
directly, through an exact conditional identity for the third moments of a
single query (\cref{u:lem:onequery}), and no Gram event enters. The saving
$\ex k_{\rm f}$ can then be taken arbitrarily close to $1/2$, and $1/2$ is the
limit of the estimate: the error $nh$ of the mesh and the number $n^{\ex a}$
of queries balance at $\ex a=1/2$. For $\ex a=\ex f=1/2$, we have
$\KL(Q_F\Vert\Pi_n)\le C_\beta n^{1/2+2\varepsilon_\omega}$, where
$\varepsilon_\omega=(1/4-\omega)/20$.

\paragraph{Coarse domination.} The second comparison is a bound on the
likelihood of the coarse proposal. There exists an event $E_B$ of the
coarse proposal, with $\PP(E_B^c)\le e^{-n^{1-\ex s_{\rm b}}}$, such that the
output restricted to $E_B$ has a kernel $q_B^{\rm good}(g,x)$ with respect to
$\gamma_n$ that satisfies
\begin{equation}\label{eq:ov-coarse-kernel}
 q_B^{\rm good}(g,x)\le e^{-nP_\beta+n^{1-\ex k_{\rm b}}}e^{\beta H^c_g(x)}
 =e^{n^{1-\ex k_{\rm b}}+F^c(g)-nP_\beta}\,\pi_g(x)
\end{equation}
for all $x$ and almost every $g$ (\cref{u:prop:coarse}). When
$F^c(g)\le nP_\beta+n^{1-\ex z_{\rm P}}$, this is a bound by
$e^{o(n^{1-\ex q})}$ times the Gibbs measure once $\ex q<\ex k_{\rm b}$ and
$\ex q<\ex z_{\rm P}$. Since the log density of $\Pi_n$ relative to $\gamma_n$
times the uniform law is $n\log2+\beta H^c_g(x)-F^c(g)$, it suffices to show
that, on $E_B$,
\begin{equation}\label{eq:ov-coarse-target}
 \frac1nI^{(n)}-\log2-\frac\beta nH^c_g(x)\le-P_\beta+n^{-\ex k_{\rm b}},
\end{equation}
where $I^{(n)}=\sum_iI_{V,i}$ is the logarithm of the density of the proposal
relative to the reference law of the paragraph on the forward entropy, at
$V=V_{\ex b}$. Integrating \eqref{eq:ov-coarse-target} against the reference
law, we obtain \eqref{eq:ov-coarse-kernel}; the factor $2^n$ cancels the
uniform spin weight $2^{-n}$.

The obstacle is the accuracy. By \eqref{eq:ov-scalar-identity}, the left
side of \eqref{eq:ov-coarse-target} has mean close to $-P_\beta$, and we must
show that it deviates by at most $n^{-\ex z}$, for an exponent
$\ex z>\ex k_{\rm b}>\ex q>\omega$, outside an event of probability
$e^{-n^{1-\ex s_{\rm b}}}$, with $\ex s_{\rm b}<1/4$. If the information and
the energy are concentrated separately, as in
\cite[Proposition~5]{oai-typical-upper}, each row has variance of order one,
and a deviation $n^{-\ex z}$ has probability of order
$e^{-n^{1-2\ex z}}$. In the parameter system below, the resulting requirement
$2\ex z<\ex s_{\rm b}$, together with $\omega<\ex q<\ex k_{\rm b}<\ex z$ and $\ex s_{\rm b}<1/4$ would
confine $\omega$ below $1/8$. We avoid this loss with three observations.

The first is algebraic. We write $H^c_g(x)/n=E_{\rm qry}+Z$, where $E_{\rm qry}$
is the energy carried by the queried columns and $Z$ collects the diagonal
entries $\langle u_j,Wu_j\rangle/n$ and the unread block. The queried energy is
an explicit function of the empirical Gram and cross matrices of the vectors
$\mathcal T_j$, $e_j$, and $x$, whose population values are known. Its
linearization in the errors of these matrices is itself an average of
independent rows: it equals $n^{-1}\sum_i\zeta_{h,i}$, where
\[
 \zeta_h=\sum_jw_j(A_{t_j}-A_{t_{j-1}})+\sum_js_j(x-m_{t_j})e_j-\frac{e_h}2,
 \qquad A_v=xm_v-\frac{m_v^2}2,
\]
for a row with spin $x$, magnetization $m$, and query endpoints $e_j$. The
deterministic weights are
$b_j=h_j^{-1/2}\int_{t_j}^{t_{j+1}}\rho(v)p(v)\dd v$ and $w_j=b_j/s_j$,
and $e_h=\sum_js_jb_j$ is the discrete scalar energy. With these
normalizations, $\E\zeta_h=e_h$ (\cref{u:lem:telescope}). The remainder of the linearization is quadratic in a
weighted measure $W_{\rm wt}$ of the errors (\cref{u:lem:weighted-energy}).
This gives
\[
 \frac1nI^{(n)}-\frac\beta nH^c_g(x)
 =\frac1n\sum_{i=1}^n\bigl(I_{V,i}-\beta\zeta_{h,i}\bigr)
 +O\bigl(D_*W_{\rm wt}^2\bigr)-\beta Z,
\]
where $D_*=1+\lceil\log_2N\rceil$.

The second observation is a pathwise cancellation between information and
energy (\cref{u:lem:joint}). On a mesh of vanishing size, $\zeta_h$ becomes
$\zeta_\infty=Z_V-e(V)/2$, where
$Z_V=\beta\int_0^Vp\dd A+\int_0^V\sqrt r\,(x-m)\dd B$. For every $V\ge v_*$,
we have exactly
\begin{equation}\label{eq:ov-joint}
 I_V-\beta\zeta_\infty=\log2-P_\beta-\frac{\beta^2}4p(V)^2
 +\frac{\beta^2p(V)}2(x-m_V)^2 .
\end{equation}
In other words, the stochastic integrals of the information and of the
energy cancel along every path. Note that the expectation of
\eqref{eq:ov-joint} is \eqref{eq:ov-scalar-identity}. To prove \eqref{eq:ov-joint}, we use It\^o's
formula for the Parisi solution along the scalar process
(\cref{u:lem:ito}). Before $v_*$, it shows that
$\beta^2pA+xY-U(v,Y)$ has differential $\beta\dd Z_v$, where $U$ is the
potential, and that the driver $B$ equals $M$; at $v_*$, the identity
$\log(1+x\tanh y)=xy-\log\cosh y$ matches the two sides; after $v_*$, the
difference $I_v-\beta Z_v$ moves only through $\beta^2p(v)(x-m_v)^2/2$. The
last term of \eqref{eq:ov-joint} does not have variance of order $h$: its
centered version $T_V$ has variance
$\beta^4p(V)^2\{4p(V)-3r(V)-p(V)^2\}/4\le\beta^4p(V)^3$, which is positive at
every fixed $V$. Its amplitude is at most $2\beta^2p(V)=O_\beta(n^{-\ex b/2})$,
however, and it is absorbed deterministically when $\ex z<\ex b/2$.

The third observation concerns the difference between the mesh and its limit.
The residual $R_h=Z_V-\zeta_h-e_h/2$ has mean $3(e(V)-e_h)/2=O_\beta(h)$, and
its centered moment generating function is at most $e^{C_\beta\theta^2h}$ for
$\abs\theta\le c_\beta/h$, uniformly in the horizon and in the number of cells
(\cref{u:lem:lag}). On the growing mesh, the lag of the magnetization inside a
cell of length $h_j$ contributes $h_j^2r(t_j)$, which is at most of order
$h^2$, and
there are at most $8/h$ cells. Since the rows are independent, we conclude that the average of
$I_{V,i}-\beta\zeta_{h,i}$ exceeds $\log2-P_\beta+C_\beta n^{-\ex z}$ with
probability at most $2e^{-c_\beta n^{1+\ex a-2\ex z}}$ (\cref{u:lem:row}). The
loss is $2\ex z-\ex a<0$, in place of $2\ex z$.

It remains to make $W_{\rm wt}^2$ smaller than $n^{-\ex z}$ and to control
$Z$. We have $W_{\rm wt}\le C_\beta n^{-\ex w}\log^Cn$ except with
probability
\[
 2\exp\bigl\{C_\beta n^{\ex a}-c_\beta n^{1-\max\{2\ex w,\ex w+\ex b/4\}}/\log^Cn\bigr\}
\]
(\cref{u:lem:weighted}). The Gram--Schmidt step is a Cholesky factorization,
and its errors involve the triangular truncation $\Upt$, which keeps the
strict upper triangle and half of the diagonal. As in
\cite[Section~4]{oai-typical-upper}, a dyadic decomposition of the triangle
costs only powers of $D_*$. The quantity $W_{\rm wt}$ measures the errors
through their projections onto dyadic intervals $I$ of query indices, weighted
by the population vectors. These projections are built from martingale
increments over the previous cells, and their variances are bounded by a
polylogarithmic multiple of the total length $d_I=\sum_{j\in I}h_{j-1}$ of
these cells. We divide each projection by $\sqrt{d_I}$ before applying a
variance-sensitive Bernstein inequality. Then the variance parameter is
polylogarithmic, and the inverse profile
$R_V\le C_\beta n^{\ex b/2}\sqrt{\log n}$ enters only the scale of the
exponential tail; this gives the loss $\ex w+\ex b/4$ beside $2\ex w$. The
half of the diagonal is a martingale plus a centered random drift, and both
have variance $O(d_I)$, with no power of $R_V$. At each dyadic depth, the
lengths $d_I$ add up to $t_N$, so that the squared errors, before the
normalization, add up without a
loss in the number of queries. The fixed Gram event
$\norm{\mathcal T^{\mathsf T}\mathcal T/n-I}\le(16D_*^2)^{-1}$, together with
its cross block, fails with probability whose exponent loses $\ex b/2$
(\cref{u:lem:gram}). Given the scalar paths and the spins, $Z$ is a centered
Gaussian variable, since the scalar recursion and the link never read the
diagonal entries or the unread block. Its conditional variance is at most
$C_\beta n^{-1}\{h+\tau^2+p(t_N)^2\}$ when $\norm{c-s}\le\tau$, where
$c_j=n^{-1}\langle u_j,x\rangle$ and $s_j$ is the population value of
$n^{-1}\langle\mathcal T_j,x\rangle$ (\cref{u:lem:weighted-energy}). The cruder bound $1/(2n)$ would
again lose $2\ex z$.

Collecting these estimates, we obtain \eqref{eq:ov-coarse-target} with
$\PP(E_B^c)\le e^{-n^{1-\ex s_{\rm b}}}$ whenever
\begin{gather*}
 \max\{2\ex w,\ex w+\ex b/4,\ex b/2,2\ex z-\ex a,2\ex z-2\ex w,2\ex z-\ex b\}
 <\ex s_{\rm b},\\
 \ex b/4<\ex a<1-\ex s_{\rm b},\qquad
 \ex k_{\rm b}<\ex z<\min(\ex a,\ex b/2,2\ex w).
\end{gather*}
This is the second place where a sharp estimate is needed. The failure bound
holds for the proposal law averaged over the disorder; no bound is asserted
for each fixed disorder.

\paragraph{Smoothing, coverage, and the connecting sequence.} A bound on
$\KL(Q_F\Vert\Pi_n)$ does not imply that the proposal reaches a given set of
appreciable Gibbs mass. To reverse the comparison, we follow
\cite[Section~5]{oai-typical-upper}. Conditionally on the spin $x$, the
disorder under $\Pi_n$ has density proportional to
$e^{-\norm g_F^2/2+\beta H^c_g(x)-F^c(g)}$, and since $F^c$ is convex and
$H^c_g(x)$ is linear in $g$, this law is $1$-strongly log-concave. The
transport inequality of Bogachev, Kolesnikov, and Medvedev
\cite[Corollary~3.10]{bogachev-kolesnikov-medvedev-2005} bounds the squared
Wasserstein distance from it by twice the relative entropy
(\cref{u:lem:transport}). We smooth the disorder by
$g\mapsto\sqrt{1-\delta}\,g+\sqrt\delta\,z$, with $\delta=n^{-\ex D}$ and an
independent standard Gaussian matrix $z$. As in
\cite[Lemma~11]{lugosi-neu-2022}, the transport bound turns the forward
entropy into a reverse entropy for the smoothed laws, at the cost of a factor
$\delta^{-1}$. Let $g_\star,g_\circ$ be independent with law $\gamma_n$, let
$g_{\rm base}=\sqrt{1-\delta}\,g_\star+\sqrt\delta\,g_\circ$, and let
$\nu^{\rm sm}_{g_{\rm base}}$ be the law of the spin of the fine proposal run
on the matrix $\sqrt{1-\delta}\,g_{\rm base}+\sqrt\delta\,z$, with fresh
auxiliary inputs. Then
\[
 \E\KL\bigl(\pi_{g_\star}\Vert\nu^{\rm sm}_{g_{\rm base}}\bigr)
 \le\frac{\beta^2(n-1)}2\delta+\delta^{-1}\KL(Q_F\Vert\Pi_n)
\]
(\cref{u:lem:smoothing}). Both terms are of order $n^{1-\ex D}$ and
$n^{1-\ex k_{\rm f}+\ex D}$, and the relative entropy is at most
$n^{1-\ex u_{\rm E}}$ with probability tending to one when
$\ex u_{\rm E}<\min(\ex D,\ex k_{\rm f}-\ex D)$. On this event, using the
inequality for the binary relative entropy, we find that, simultaneously for
all $S\subseteq\Sigma_n$,
\[
 \pi_{g_\star}(S)\ge n^{-\ex d}\quad\Longrightarrow\quad
 \nu^{\rm sm}_{g_{\rm base}}(S)\ge
 \exp\bigl\{-n^{\ex d}\bigl(n^{1-\ex u_{\rm E}}+\log2\bigr)\bigr\}
\]
(\cref{u:lem:coverage}). The set $S$ may depend on $g_\star$. Two independent
fine spins with law $\nu^{\rm sm}_{g_{\rm base}}$ fall on opposite sides of
every cut with both masses at least $n^{-\ex d}$ with probability
$\varpi_n\ge e^{-2n^{1-\ex e}}$ when $\ex e+\ex d<\ex u_{\rm E}$.

Suppose now that $g_\star\in\mathcal W_n$, and let $S_{g_\star}$ be the first
cut of small flow in a fixed order. The following argument is that of
\cite[Section~6]{oai-typical-upper}. If the two fine spins lie on opposite
sides of $S_{g_\star}$, we join them by a sequence of configurations with
short steps. Every such sequence comes close to the boundary of the cut. An
enlargement of the boundary has small Gibbs mass, and the likelihood bound
keeps the intermediate configurations out of it. We now describe the three
ingredients.

\emph{The excluded set.} Since $\norm{W(g_\star)}_{\rm op}\le12$, every
one-spin energy change is at most $24\sqrt n$, and every flip rate is at least
$e^{-24\beta\sqrt n}/2$. Then the vertex boundary of $S_{g_\star}$ has Gibbs mass
at most $4\phi_ne^{24\beta\sqrt n}$. We enlarge it by the Hamming radius
$R_n=\lceil n^{1-\ex r}\rceil$, and to compare Gibbs weights on the
enlargement, we also exclude configurations $x$ for which
\[
 \sum_i\bigl(\mathbf 1_A(i)-\mathbf 1_B(i)\bigr)
     (W(g_\star)x)_i\ge n^{1-\ex u_{\rm F}}
\]
for some $A,B\subseteq[n]$ with $|A|,|B|\le R_n$.
The expected Gibbs mass of the resulting set
$\mathcal E_n(g_\star)$ is at most
$e^{-n^{1-\ex q}/2}$ when $0<\ex q<\ex u_{\rm F}<\ex r<1$,
$2\ex u_{\rm F}-\ex r<\ex q$, and $\ex q<1/2$ (\cref{u:lem:boundary}). The
bound on the local fields uses the transport inequality for the conditional
disorder law given the spin. The cut is selected from $g_\star$ alone, and we
also restrict to the event $F^c(g_\star)\le nP_\beta+n^{1-\ex z_{\rm P}}$,
which fails with probability at most $e^{-c_\beta n^{1-2\ex z_{\rm P}}}$
(\cref{u:lem:pressure-tail}).

\emph{The sequence.} With independent standard Gaussian matrices $g_1,g_2$,
we set
\[
 g_\theta=\sqrt{1-\delta}\,g_{\rm base}
 +\sqrt\delta\,(\cos\theta\,g_1+\sin\theta\,g_2),\qquad
 0\le\theta\le\pi/2,
\]
and we rotate two independent copies of the other Gaussian inputs of the
proposal in the same way. For each fixed $\theta$, the matrix $g_\theta$ has
law $\gamma_n$, and the rotated inputs have their original law. We run the
coarse proposal at the $\ell_n+1$ points of a grid of spacing of order
$e^{-n^{\ex g_{\rm rot}}}$, and the fine proposal only at the two endpoints
$\theta=0$ and $\theta=\pi/2$. Given $(g_\star,g_\circ)$, the two fine spins
are independent with law $\nu^{\rm sm}_{g_{\rm base}}$. The sequence consists
of the first fine spin, the coarse spins in the order of the grid, and the
second fine spin. On an event whose complement has probability at most
$e^{-n^{1-\ex s_{\rm seq}}}$, the following holds (\cref{u:prop:sequence}):
\begin{itemize}
\item every coarse spin avoids $\mathcal E_n(g_\star)$: by the regression
$g_\theta=(1-\delta)g_\star+\sqrt{\delta(2-\delta)}\,Z_\theta$, with
$Z_\theta$ independent of $g_\star$, the conditional expectation of
$e^{\beta H^c_{g_\theta}(x)}$ given $g_\star$ is at most
$e^{O_\beta(n\delta)}e^{\beta H^c_{g_\star}(x)}$ when
$\norm{W(g_\star)}_{\rm op}\le12$, and
\eqref{eq:ov-coarse-kernel} bounds the probability of hitting the excluded
set by $e^{-n^{1-\ex q}/4}$ when $\ex q<\min(\ex k_{\rm b},\ex z_{\rm P},\ex D)$
(\cref{u:lem:transfer});
\item consecutive coarse magnetizations differ by at most
$e^{-n^{\ex g_{\rm rot}}/2}$ when $\ex a<\ex g_{\rm rot}$, because the
recursion amplifies differences of its inputs by at most
$e^{C_\beta n^{\ex a}\log n}$ on the Gram events at both ends of a grid step
(\cref{u:lem:stability});
\item every spin, coarse or fine, is within squared Euclidean distance $R_n/4$
of its coarse magnetization when $\ex r<\ex b/2$.
The fine and coarse runs share their paths up to $V_{\ex b}$;
the expected squared discrepancy per coordinate is
$p(V_{\ex b})\le C_\beta n^{-\ex b/2}$. The expected total
squared Euclidean distance is $np(V_{\ex b})=o(R_n)$;
independence across rows gives the tail in \cref{u:lem:pinning}.
\end{itemize}
On this event, we have that every step of the sequence has Hamming length at
most $R_n$, and every step contains a coarse spin. The union over the grid costs a factor
$e^{O(n^{\ex g_{\rm rot}})}$, which is absorbed when
$\ex g_{\rm rot}<1-\ex s_{\rm b}$, and the comparison of the failures requires
$\ex r<\ex s_{\rm b}$, $\ex q<\ex s_{\rm b}$, and
$\ex s_{\rm b}<\ex s_{\rm seq}$.

The fine proposal enters only through its relative entropy, which has no
exceptional event, and through the two endpoint spins, which are joined to the
grid by pinning. Every event in the union over the grid concerns the coarse
proposal, so that the sequence needs no Gram event of the fine proposal and
only the failure bound $e^{-n^{1-\ex s_{\rm b}}}$ of the coarse one.

\emph{The contradiction.} If the fine spins lie on opposite sides of
$S_{g_\star}$, some step of the sequence crosses the cut. A shortest path
across this step meets the boundary, so that the coarse endpoint of the step
lies within distance $R_n$ of the boundary, in $\mathcal E_n(g_\star)$. On
the event of the sequence, this is impossible. Given an outer pair
$(g_\star,g_\circ)$ in the event of the coverage bound, with $g_\star$ in
$\mathcal W_n$ and in the pressure event, we conclude that $\varpi_n$ is at most the
conditional probability that the sequence fails. Integrating over the outer
pair, and paying the coverage and pressure exceptions once, we find that
\begin{equation}\label{eq:ov-witness}
 \gamma_n(\mathcal W_n)\le
 C_\beta\bigl(n^{\ex u_{\rm E}-\ex D}+n^{\ex u_{\rm E}-\ex k_{\rm f}+\ex D}\bigr)
 +e^{-c_\beta n^{1-2\ex z_{\rm P}}}
 +e^{2n^{1-\ex e}-n^{1-\ex s_{\rm seq}}}
\end{equation}
(\cref{u:prop:witness}), which tends to zero when $\ex s_{\rm seq}<\ex e$.

\paragraph{The parameters.} The exponents of \cref{tab:ov-upper-exponents}
are affine functions of $\varepsilon_\omega=(1/4-\omega)/20\in(0,1/80)$, and
they satisfy every condition stated above, with strict inequalities, for
every $\omega\in(0,1/4)$ (\cref{u:prop:range}). The complete list of
conditions is given in \cref{u:sec:budget}. Four chains among them limit
the saving:
\begin{gather*}
 \ex k_{\rm f}<\min(\ex a,\ex f,1-\ex a)\le\frac12,\qquad
 \ex e+\ex d<\ex u_{\rm E}<\min(\ex D,\ex k_{\rm f}-\ex D)
 \le\frac{\ex k_{\rm f}}2,\\
 \ex q<\ex s_{\rm b}<\ex s_{\rm seq}<\ex e,\qquad
 \omega<\ex q .
\end{gather*}
They come from the forward entropy, from smoothing and coverage, from the
connecting sequence, and from \eqref{eq:ov-pruning}. Together they give
\[
 \omega<\ex q<\ex s_{\rm b}<\ex s_{\rm seq}<\ex e<\ex u_{\rm E}
 <\frac{\ex k_{\rm f}}2\le\frac14 .
\]
The conditions of coarse domination fit inside the third chain: the family
has $\ex q<\ex k_{\rm b}<\ex z<\ex b/2<\ex s_{\rm b}$, with consecutive gaps
$\varepsilon_\omega$, $\varepsilon_\omega$, $\varepsilon_\omega$, and
$2\varepsilon_\omega$, and a loss $2\ex z$ in place of $2\ex z-\ex a$ would
not fit. For $\omega=1/5$, we have $\varepsilon_\omega=1/400$,
the chain reads
\[
 \frac15<\frac9{40}<\frac{19}{80}<\frac6{25}<\frac{97}{400}<\frac{49}{200}
 <\frac{99}{400},
\]
and \eqref{eq:ov-witness} becomes
$C_\beta n^{-1/400}+e^{-c_\beta n^{2/5}}+\exp\{2n^{303/400}-n^{19/25}\}$.

Both arguments stop at the exponent $1/4$. The proof of
\cref{thm:barrier-730} needs $n\rho^4/\log n\to\infty$ for the overlap scale
$\rho$, and the proof of \cref{thm:upper} needs the chain of inequalities
$\omega<\ex q<\ex s_{\rm b}<\ex s_{\rm seq}<\ex e<\ex u_{\rm E}
<\ex k_{\rm f}/2\le1/4$ of \cref{u:sec:budget}. We do not know whether this
coincidence has a common explanation.


\section{Fluctuations and the mean bias}\label{sec:ov-fluct}

\Cref{f:main} is OpenAI's theorem \cite[Theorem~1.1]{oai-fluctuations}. We
give no new proof of it; our contribution concerning it and
\cref{cor:mean-bias} is described in \cref{sec:formal}. In
\cite{oai-fluctuations}, the theorem is reduced to a two-sided bound on a positive logarithmic moment. Let $F_n^c=F_n+\beta G$ be
the completed log partition function, and let $K_n(t)=\log\E\exp\{t(F_n^c-\E F_n^c)\}$.
For fixed $m\in(0,1/300)$ and $s_n=n^{-1/6+5m}$, it is proved in
\cite[Proposition~7.5]{oai-fluctuations} that
$cn^{20m}\le K_n(s_n)\le Cn^{100m}$ for all sufficiently large $n$
(\cref{f:window}); \cref{sec:fl-route} summarizes this proof with locators.
OpenAI bounds $K_n(s_n)$ by comparison with finite Gaussian hierarchies along
overlap quantiles. The error
of such a comparison is small when its minimizers satisfy a two-sided linear
bound. This bound follows from a one-site cavity estimate
\cite[Proposition~4.1]{oai-fluctuations}, from the asymptotic linearity of the
rescaled overlap quantiles of the minimizers
\cite[Theorem~5.1]{oai-fluctuations}, and from finite propagation through a bounded number of successive comparisons
\cite[Proposition~6.6]{oai-fluctuations}. The lower bound uses a chord
inequality at $\varepsilon=n^{-1/3}$, and the upper bound an iteration in
$\varepsilon$ with a low-field perturbation
\cite[Sections~7.1--7.4]{oai-fluctuations}.

A bound on $K_n$ at one tilt does not control $\Var(F_n)$ by itself, since a
rare deviation can dominate an exponential moment or a variance. The additional
input is that the sum of $F_n^c$ and a random variable with the exponential law
of rate one, independent of $(J,G)$, has a log-concave density
(\cref{f:logconcave}). This observation is
due to Aronow and Lopatto \cite[Proposition~5.1]{quantitative-parisi}, and
\cite[Lemma~7.6]{oai-fluctuations} states it in the present setting. For a
log-concave law with standard deviation $\sigma$, the centered logarithmic
moment is bounded at tilts below $c/\sigma$ and grows at least linearly in
$t\sigma$ at larger tilts $t$, and every interval of length $2r$ has
probability at most $Cr/\sigma$ (\cref{f:one-dim}). We deduce that
$\Var(F_n)=n^{1/3+o(1)}$ and, using the bound on intervals, the lower bound on
$\abs{F_n-\E F_n}$ in \eqref{f:typical} (\cref{sec:fl-logconcave,sec:fl-endgame},
following \cite[Sections~7.5 and~7.6]{oai-fluctuations}).

\Cref{cor:mean-bias} is the lower bound $n^{-5/6-\varepsilon}$ on the mean
bias. It follows from the variance--bias inequality
$\Delta_{n,\beta}\ge c_\beta((\Var(F_n)+\beta^2/2)/n)^{5/4}$ of Aronow and
Lopatto \cite[Lemma~7.4 and Remark~7.14]{quantitative-parisi}
(\cref{prop:bias-variance-deficit}). Inserting
$\Var(F_n)\ge n^{1/3-4\varepsilon/5}$ from \cref{f:main}, we find that
$\Delta_{n,\beta}\ge c_\beta n^{-5/6-\varepsilon}$ for all sufficiently large
$n$. Together with the upper bound $\Delta_{n,\beta}\le C_bn^{-7/10}$ of
\cref{inp:hb-rate}, which determines the exponent of \cref{thm:barrier-730}, this
shows that if $\Delta_{n,\beta}=n^{-a+o(1)}$ for some $a$, then
$7/10\le a\le5/6$. Both deductions are already recorded in
\cite[Section~1.6]{hexagon-bias}.
\part{Preliminaries}\label{part:prelim}
\chapter{The model and the dynamics}\label{sec:model}

This section fixes the model, the dynamics, and the conventions used in the rest of the paper.
\Cref{sec:model-sk} defines the Sherrington--Kirkpatrick model, its Gibbs measure, the overlap, the log partition function $F_n$, the limiting pressure $P_\beta$, and the mean bias $\Delta_{n,\beta}$.
\Cref{sec:model-completion} records exact identities for the Hamiltonian completed by an independent Gaussian variable (\cref{lem:model-completion}). They are used in \cref{sec:inputs} to translate imported results into our normalization, in \cref{sec:fractional} for fractional moments of constrained partition functions, in \cref{part:upper} for the Gaussian-query proposal, and in \cref{sec:fluct,sec:bias}.
\Cref{sec:model-dynamics} defines heat-bath dynamics in discrete and continuous time and proves the elementary facts collected in \cref{lem:model-dynamics}: reversibility, Poissonization, and the contraction of total variation. These facts are used in \cref{sec:barrier-proof} to prove \cref{thm:barrier-730,b:cor:subcritical} and in \cref{part:upper} to prove \cref{thm:upper,cor:upper-discrete}.
\Cref{sec:model-joint} defines the realized-start probability in which \cref{thm:barrier-730,thm:upper} are formulated, and \cref{lem:model-joint} relates statements about this probability to statements about the Gibbs mass of the corresponding initial states. \Cref{sec:model-asymptotics} fixes the asymptotic conventions, and \cref{sec:model-dictionary} compares our normalization with those of the works from which we import results.

The model, the heat-bath dynamics, and the realized-start formulation are those of \cite[Section~1]{oai-barriers}. The facts proved in this section are standard. We include short proofs so that the conventions can be checked directly, including the clock and the order in which the disorder and the initial state are sampled.

\section{The Sherrington--Kirkpatrick model}\label{sec:model-sk}

We consider the zero-field Gaussian Sherrington--Kirkpatrick model, introduced in \cite{sk-original}.
For an integer $n\ge2$, let $\Sigma_n=\{-1,1\}^n$. Its elements, usually denoted $x$, $y$, $z$, or $v$, are called configurations.
The disorder is a family $J=(J_{ij})_{1\le i<j\le n}$ of independent standard Gaussian random variables, and we set $J_{ji}=J_{ij}$ and $J_{ii}=0$. We write $\PP_J$ and $\E_J$ for probability and expectation over $J$. Unsubscripted $\PP$, $\E$, and $\Var$ refer to all random variables under consideration, except those declared fixed. In \cref{sec:model-completion} they average over $(J,G)$, whereas in \cref{lem:model-dynamics} the disorder is held fixed and probabilities average over the chain and the independent clock.
For $\beta>0$ and $x\in\Sigma_n$, we define the Hamiltonian, the partition function, and the log partition function by
\begin{equation}\label{eq:common-model}
 H_n(x)=\frac1{\sqrt n}\sum_{1\le i<j\le n}J_{ij}x_ix_j,\qquad
 Z_n=\sum_{x\in\Sigma_n}e^{\beta H_n(x)},\qquad
 F_n=\log Z_n.
\end{equation}
When the dependence on the disorder or on $\beta$ matters, we write $H_n^J$ and $Z_n=Z_{n,\beta}^J$.
The Gibbs measure is the probability measure on $\Sigma_n$ given by
\begin{equation}\label{b:eq:SK-model}
 \pi(x)=\pi^J(x)=\pi^J_{n,\beta}(x)=\frac{e^{\beta H_n^J(x)}}{Z^J_{n,\beta}},\qquad x\in\Sigma_n.
\end{equation}
For a function $f$ on $\Sigma_n$ and a set $A\subseteq\Sigma_n$, we write $\langle f\rangle=\pi(f)=\sum_xf(x)\pi(x)$ and $\pi(A)=\sum_{x\in A}\pi(x)$.

Three features of \eqref{eq:common-model} are used throughout. First, the inverse temperature multiplies $H_n$ and is not part of it. Second, $Z_n$ is a sum with respect to counting measure on $\Sigma_n$, with no factor $2^{-n}$. Third, $F_n$ is the logarithm of the partition function, not a free energy per spin. In physical terms, the energy is $-H_n$ and the free energy is $-F_n/\beta$; since $\beta$ is fixed, the factor $-1/\beta$ does not affect any exponent in this paper.

The overlap of two configurations $x,y\in\Sigma_n$ is
\[
 R(x,y)=\frac1n\sum_{i=1}^nx_iy_i\in[-1,1].
\]
We have $R(x,x)=1$, $R(-x,y)=-R(x,y)$, and $H_n(-x)=H_n(x)$, the last because $H_n$ is quadratic in the spins. It follows that the Gibbs measure is invariant under the global spin reversal $x\mapsto-x$.
The Hamiltonian is a centered Gaussian process indexed by $\Sigma_n$, with covariance
\begin{equation}\label{eq:common-covariance}
 \E H_n(x)H_n(y)=\frac{nR(x,y)^2-1}{2},\qquad x,y\in\Sigma_n.
\end{equation}
To see this, set $a_i=x_iy_i$. Using that $\E J_{ij}J_{kl}=\mathbf 1\{(i,j)=(k,l)\}$ for $i<j$ and $k<l$, and that $a_i^2=1$, we find
\[
 \E H_n(x)H_n(y)=\frac1n\sum_{i<j}a_ia_j
 =\frac1{2n}\biggl(\Bigl(\sum_{i=1}^na_i\Bigr)^2-\sum_{i=1}^na_i^2\biggr)
 =\frac{n^2R(x,y)^2-n}{2n}.
\]
We have $\Var H_n(x)=(n-1)/2$ for all $x$. The constant $-1/2$ in \eqref{eq:common-covariance} reflects the absence of diagonal terms in $H_n$; it is removed in \cref{sec:model-completion}.

We write
\[
 p_n(\beta)=\frac1n\E F_n
\]
for the expected pressure. It is finite, since $\beta H_n(x_0)\le F_n\le n\log2+\beta\max_xH_n(x)$ for any fixed $x_0\in\Sigma_n$ and each $H_n(x)$ is Gaussian.
In \cite[Theorems~1--2, (17)--(18)]{guerra-toninelli-2002}, Guerra and Toninelli proved that the sequence $n\,p_n(\beta)$ is superadditive in $n$ and that the limit
\[
 P_\beta=\lim_{n\to\infty}p_n(\beta)
\]
exists for every $\beta>0$ and equals the supremum of the sequence.
% CHECK: arXiv:cond-mat/0204280v1, p. 6; their (1)-(3) at h=0 give Z_N(beta,0,J) = our Z_n (their H_N is minus ours) and alpha_N(beta,0) = our p_n(beta).
We call
\begin{equation}\label{eq:model-bias}
 \Delta_{n,\beta}=P_\beta-p_n(\beta)
\end{equation}
the mean bias. We have $\Delta_{n,\beta}\ge0$ for all $n\ge2$, because $P_\beta$ is the supremum of the sequence $p_n(\beta)$; this is also a consequence of Guerra's bound \cite[Theorem~3]{guerra-2003}.
The identification of $P_\beta$ with the minimum of the Parisi functional, due to Guerra \cite{guerra-2003} and Talagrand \cite{parisi-formula}, is stated in \cref{sec:inputs} as \cref{inp:parisi}, where the Parisi functional is also defined.

\section{The completed Hamiltonian}\label{sec:model-completion}

By \eqref{eq:common-covariance}, the covariance of $H_n$ differs from $nR(x,y)^2/2$ by a constant. Several of the results we import, and the Gaussian interpolation of \cref{sec:fractional}, are formulated for a Hamiltonian whose covariance is exactly $nR(x,y)^2/2$. Such a Hamiltonian is obtained by adding a single Gaussian variable, the same for every configuration.
Let $G$ be a centered Gaussian random variable with variance $1/2$, independent of $J$, and set
\begin{equation}\label{eq:model-completion}
 H_n^c(x)=H_n(x)+G,\qquad
 Z_n^c=\sum_{x\in\Sigma_n}e^{\beta H_n^c(x)}=e^{\beta G}Z_n,\qquad
 F_n^c=\log Z_n^c=F_n+\beta G.
\end{equation}
We call $H_n^c$ the completed Hamiltonian and $F_n^c$ the completed log partition function.
In \cite{oai-barriers,quantitative-parisi,hexagon-bias}, the variable $G$ arises from the diagonal of a symmetric Gaussian matrix. Let $J'_{11},\dots,J'_{nn}$ be independent standard Gaussian variables, independent of $J$, and let $W$ be the symmetric $n\times n$ matrix with $W_{ij}=J_{ij}/\sqrt n$ for $i\neq j$ and $W_{ii}=\sqrt{2/n}\,J'_{ii}$. Then
\[
 \frac12\sum_{i,j=1}^nW_{ij}x_ix_j=H_n(x)+\frac1{\sqrt{2n}}\sum_{i=1}^nJ'_{ii},
\]
and the last term has the law of $G$.

\begin{lemma}[Exact identities for the completed Hamiltonian]\label{lem:model-completion}
Fix $n\ge2$ and $\beta>0$. Then the following statements hold.
\begin{enumerate}[label=(\alph*)]
\item For all $x,y\in\Sigma_n$, we have $\E H^c_n(x)H^c_n(y)=nR(x,y)^2/2$.
\item For every realization of $(J,G)$ and every $x\in\Sigma_n$, we have $e^{\beta H_n^c(x)}/Z_n^c=\pi^J(x)$.
\item We have $\E F_n^c=\E F_n$ and $\Var F_n^c=\Var F_n+\beta^2/2$. Further, for every $t\in\R$, both sides of the following identity are finite, and
\begin{equation}\label{eq:model-completion-laplace}
 \log\E\exp\{t(F_n^c-\E F_n^c)\}=\log\E\exp\{t(F_n-\E F_n)\}+\frac{\beta^2t^2}{4}.
\end{equation}
\item Let $\ell\ge1$ be an integer, and let $A\subseteq\Sigma_n^\ell$ be a nonempty set of $\ell$-tuples of configurations. Define
\[
 Z_{\ell,n}(A)=\sum_{(x^1,\dots,x^\ell)\in A}\exp\biggl(\beta\sum_{a=1}^\ell H_n(x^a)\biggr),
\]
and define $Z_{\ell,n}^c(A)$ in the same way, with $H_n^c$ in place of $H_n$. Then $Z^c_{\ell,n}(A)=e^{\beta\ell G}Z_{\ell,n}(A)$ and $\E\log Z^c_{\ell,n}(A)=\E\log Z_{\ell,n}(A)$. For every $s\in\R$, both sides of the following identity are finite, and
\begin{equation}\label{eq:model-completion-moment}
 \E Z_{\ell,n}^c(A)^s=\exp\Bigl(\frac{\beta^2\ell^2s^2}{4}\Bigr)\,\E Z_{\ell,n}(A)^s.
\end{equation}
Equivalently, for every $s\neq0$,
\[
 \frac1{ns}\log\E Z_{\ell,n}^c(A)^s=\frac1{ns}\log\E Z_{\ell,n}(A)^s+\frac{\beta^2\ell^2s}{4n}.
\]
\end{enumerate}
\end{lemma}

\begin{proof}
Part (a) follows from \eqref{eq:common-covariance}, since $G$ is independent of $J$ and $\E G^2=1/2$. For part (b), the factor $e^{\beta G}$ appears in both $e^{\beta H_n^c(x)}$ and $Z_n^c$.

For part (c), recall that $F_n$ is a function of $J$ and that $G$ is independent of $J$. Using $\E G=0$ and $\Var(\beta G)=\beta^2/2$, we have $\E F_n^c=\E F_n$ and $\Var F_n^c=\Var F_n+\beta^2/2$. Next, we check that all exponential moments of $F_n$ are finite. Fix $x_0\in\Sigma_n$. Since $e^{\beta H_n(x_0)}\le Z_n\le2^n\max_xe^{\beta H_n(x)}$, we have
\[
 e^{tF_n}\le2^{nt}\sum_{x\in\Sigma_n}e^{t\beta H_n(x)}\quad(t\ge0),\qquad
 e^{tF_n}\le e^{t\beta H_n(x_0)}\quad(t<0).
\]
As each $H_n(x)$ is Gaussian, this implies that $\E e^{tF_n}<\infty$ for all $t\in\R$. By the independence of $F_n$ and $G$ and the identity $\E e^{t\beta G}=e^{\beta^2t^2/4}$, we have
\[
 \E\exp\{t(F_n^c-\E F_n^c)\}
 =\E\exp\{t(F_n-\E F_n)\}\,\E e^{t\beta G}
 =\E\exp\{t(F_n-\E F_n)\}\,e^{\beta^2t^2/4}.
\]
Taking logarithms gives \eqref{eq:model-completion-laplace}.

For part (d), note that every term in the sum defining $Z_{\ell,n}^c(A)$ contains the factor $\exp(\beta\sum_{a=1}^\ell G)=e^{\beta\ell G}$, which proves $Z^c_{\ell,n}(A)=e^{\beta\ell G}Z_{\ell,n}(A)$. Fix a tuple $(x_0^1,\dots,x_0^\ell)\in A$. The bounds of part (c), with $\sum_aH_n(x_0^a)$ in place of $H_n(x_0)$ and $|A|$ in place of $2^n$, show that $\log Z_{\ell,n}(A)$ is integrable and that $\E Z_{\ell,n}(A)^s<\infty$ for every $s\in\R$, since $\sum_aH_n(x^a)$ is Gaussian for each tuple $(x^1,\dots,x^\ell)$. Taking logarithms and expectations in $Z^c_{\ell,n}(A)=e^{\beta\ell G}Z_{\ell,n}(A)$ and using $\E G=0$, we find $\E\log Z^c_{\ell,n}(A)=\E\log Z_{\ell,n}(A)$. The variable $\beta\ell sG$ is centered Gaussian with variance $\beta^2\ell^2s^2/2$, and it is independent of $Z_{\ell,n}(A)$. We conclude that
\[
 \E Z^c_{\ell,n}(A)^s=\E e^{\beta\ell sG}\,\E Z_{\ell,n}(A)^s=e^{\beta^2\ell^2s^2/4}\,\E Z_{\ell,n}(A)^s,
\]
which is \eqref{eq:model-completion-moment}. Dividing its logarithm by $ns$ gives the last display of the lemma.
\end{proof}

The constrained partition functions of \cref{sec:inputs,sec:fractional,sec:static}, in which the overlaps $R(x^a,x^b)$ of the $\ell$-tuple are prescribed, are of the form $Z_{\ell,n}(A)$. By part (b) of \cref{lem:model-completion} and the identity $Z^c_{\ell,n}(A)=e^{\beta\ell G}Z_{\ell,n}(A)$, Gibbs measures, constrained Gibbs probabilities $Z_{\ell,n}(A)/Z_n^\ell$, and expected logarithms are the same for the two Hamiltonians. Fractional moments are not: the correction $\beta^2\ell^2s/(4n)$ in the last display of the lemma is quadratic in $\ell$, because the same variable $G$ enters every replica, and it is kept explicitly in \cref{sec:fractional}. The identity \eqref{eq:model-completion-laplace} is the form in which the completion enters \cref{sec:fluct}, where the logarithmic-moment bounds of \cite{oai-fluctuations} are stated for $F_n^c$.

\section{Heat-bath dynamics}\label{sec:model-dynamics}

We use single-spin heat-bath dynamics, in the tradition of the stochastic Ising dynamics introduced by Glauber \cite{glauber-1963}. The clock and resampling conventions below are those of \cite[Section~1, pp.~2--3]{oai-barriers}.
For $x\in\Sigma_n$, $i\in\{1,\dots,n\}$, and $\sigma\in\{-1,1\}$, let $x^{i,\sigma}$ be the configuration obtained from $x$ by setting its $i$th spin equal to $\sigma$. The local field at site $i$ and the heat-bath probabilities are
\begin{equation}\label{eq:model-heat-bath}
 h_i(x)=h_i^J(x)=\frac1{\sqrt n}\sum_{j\neq i}J_{ij}x_j,\qquad
 p_i(\sigma\mid x)=p_i^J(\sigma\mid x)=\frac{e^{\beta\sigma h_i(x)}}{2\cosh(\beta h_i(x))}.
\end{equation}
The field $h_i(x)$ does not depend on $x_i$. Since $J_{ji}=J_{ij}$, the terms of $H_n$ that contain $x_i$ add up to $x_ih_i(x)$, and we have
\begin{equation}\label{eq:model-site-split}
 H_n(x)=x_ih_i(x)+H_n^{(i)}(x),\qquad
 H_n^{(i)}(x)=\frac1{\sqrt n}\sum_{\substack{j<k\\ j,k\neq i}}J_{jk}x_jx_k,
\end{equation}
where $H_n^{(i)}(x)$ does not depend on $x_i$. It follows that $p_i(\sigma\mid x)=\pi(x^{i,\sigma})/(\pi(x^{i,1})+\pi(x^{i,-1}))$; that is, $p_i(\cdot\mid x)$ is the conditional law of the $i$th spin under $\pi$, given the other spins.

The discrete heat-bath kernel is
\begin{equation}\label{eq:common-kernel}
 K(x,y)=K^J(x,y)=\frac1n\sum_{i=1}^n\sum_{\sigma\in\{-1,1\}}p_i(\sigma\mid x)\,\mathbf 1\{y=x^{i,\sigma}\},\qquad x,y\in\Sigma_n.
\end{equation}
In one step, a site $i$ is chosen uniformly at random and the $i$th spin is resampled from $p_i(\cdot\mid x)$. The new value may equal the old one, and in this case the update attempt leaves the configuration unchanged. Explicitly, for all $x\in\Sigma_n$ and all sites $i$,
\begin{equation}\label{b:eq:discrete-kernel}
 K(x,x^{i,-x_i})=\frac1n\,p_i(-x_i\mid x),\qquad
 K(x,x)=\frac1n\sum_{i=1}^np_i(x_i\mid x),
\end{equation}
and $K(x,y)=0$ if $x$ and $y$ differ in more than one coordinate. The discrete heat-bath chain $(X_k)_{k\ge0}$ is the Markov chain on $\Sigma_n$ with transition matrix $K$.

In continuous time, we define the generator and the transition probabilities by
\begin{equation}\label{eq:common-semigroup}
 \mathcal L=\mathcal L^J=n(K-I),\qquad P_t=P_t^J=e^{t\mathcal L},\quad t\ge0,
\end{equation}
and we call the Markov process $(X_t)_{t\ge0}$ with transition probabilities $P_t$ the continuous heat-bath process. In this process, each site carries an independent Poisson clock of rate one, and when the clock at site $i$ rings, the $i$th spin of the current configuration $x$ is resampled from $p_i(\cdot\mid x)$. Indeed, the generator of this description acts on functions $f\colon\Sigma_n\to\R$ by
\[
 \mathcal Lf(x)=\sum_{i=1}^n\sum_{\sigma\in\{-1,1\}}p_i(\sigma\mid x)\bigl(f(x^{i,\sigma})-f(x)\bigr)=n\bigl(Kf(x)-f(x)\bigr),
\]
where the second equality follows from \eqref{eq:common-kernel} and $\sum_\sigma p_i(\sigma\mid x)=1$. The total rate of update attempts is $n$, so that one unit of continuous time corresponds to $n$ update attempts on average. In both time conventions, an attempt that leaves the configuration unchanged is counted. We use the letter $t$ for continuous time and $k$ for the number of discrete steps, so that the notation $(X_t)$ or $(X_k)$ indicates which process is meant.

For probability measures $\nu,\zeta$ on $\Sigma_n$, the total variation distance is
\[
 \norm{\nu-\zeta}_\TV=\frac12\sum_{x\in\Sigma_n}|\nu(x)-\zeta(x)|=\max_{A\subseteq\Sigma_n}|\nu(A)-\zeta(A)|,
\]
where the second equality is \cite[Proposition~4.2]{levin-peres-2017}. For $x\in\Sigma_n$, $t\ge0$, and integers $k\ge0$, we define
\begin{equation}\label{eq:model-distances}
 d^J(x,t)=\norm{P^J_t(x,\cdot)-\pi^J}_\TV,\qquad
 d^J_{\rm disc}(x,k)=\norm{(K^J)^k(x,\cdot)-\pi^J}_\TV.
\end{equation}
These are the distances from equilibrium of the laws of the dynamics started at the fixed configuration $x$. No supremum over $x$ is taken.
For the threshold $1/4$, we define the mixing times from $x$ by
\begin{equation}\label{eq:model-mixing-time}
 \begin{aligned}
 T^J(x)&=\inf\{t\ge0:d^J(x,t)\le1/4\},\\
 T^J_{\rm disc}(x)&=\min\{k\ge0:d^J_{\rm disc}(x,k)\le1/4\}.
 \end{aligned}
\end{equation}
Both times are finite. Indeed, every single-site resampling probability is
positive, so $K^n(x,y)>0$ for all $x,y\in\Sigma_n$: one may update each site
once to its value in $y$. The minimum of these finitely many entries is
positive. Writing $K^n$ as a mixture with a positive uniform component and
using contraction shows that $d^J_{\rm disc}(x,k)\to0$; \eqref{eq:model-poisson}
then gives $d^J(x,t)\to0$. By \cref{lem:model-dynamics}(d), the continuous
sublevel set in \eqref{eq:model-mixing-time} is closed and upward closed.
For every $t\ge0$, we have $T^J(x)\le t$ if and only if $d^J(x,t)\le1/4$,
and $T^J(x)>t$ if and only if $d^J(x,t)>1/4$.
The corresponding equivalences hold in discrete time.

In the next lemma, we record classical facts about finite Markov chains for the kernels $K$ and $P_t$; see \cite[Proposition~1.20, Exercise~4.2, and Section~20.1]{levin-peres-2017}.

\begin{lemma}[Elementary properties of heat-bath dynamics]\label{lem:model-dynamics}
Fix $n\ge2$, $\beta>0$, and a realization of $J$. Then the following statements hold.
\begin{enumerate}[label=(\alph*)]
\item The kernel $K$ is reversible with respect to $\pi$, that is, $\pi(x)K(x,y)=\pi(y)K(y,x)$ for all $x,y\in\Sigma_n$. In particular, $\pi K=\pi$.
\item For every $t\ge0$,
\begin{equation}\label{eq:model-poisson}
 P_t=\sum_{k=0}^\infty e^{-nt}\frac{(nt)^k}{k!}\,K^k.
\end{equation}
If $(X_k)_{k\ge0}$ is the discrete heat-bath chain started at $x$ and $(N_t)_{t\ge0}$ is a Poisson process of rate $n$, independent of $(X_k)_{k\ge0}$, then $\PP(X_{N_t}=y)=P_t(x,y)$ for all $y\in\Sigma_n$ and $t\ge0$.
\item For every $t\ge0$, the kernel $P_t$ is reversible with respect to $\pi$, and $\pi P_t=\pi$. In particular, $\sum_x\pi(x)P_t(x,\cdot)=\pi$ and $\sum_x\pi(x)K^k(x,\cdot)=\pi$ for all $t\ge0$ and $k\ge0$.
\item For every transition matrix $Q$ on $\Sigma_n$ and all probability measures $\nu,\zeta$ on $\Sigma_n$, we have $\norm{\nu Q-\zeta Q}_\TV\le\norm{\nu-\zeta}_\TV$. For every $x\in\Sigma_n$, the functions $k\mapsto d^J_{\rm disc}(x,k)$ and $t\mapsto d^J(x,t)$ are nonincreasing, and $t\mapsto d^J(x,t)$ is continuous.
\end{enumerate}
\end{lemma}

\begin{proof}
For part (a), the identity is trivial if $x=y$, and both sides vanish by \eqref{b:eq:discrete-kernel} if $x$ and $y$ differ in more than one coordinate. Suppose that $y=x^{i,-x_i}$ for some $i$. Then $h_i(y)=h_i(x)$ and $H_n^{(i)}(y)=H_n^{(i)}(x)$. Using \eqref{eq:model-heat-bath}, \eqref{eq:model-site-split}, and \eqref{b:eq:discrete-kernel}, we find
\[
 \pi(x)K(x,y)=\frac{e^{\beta H_n^{(i)}(x)}}{nZ_n}\cdot\frac{e^{\beta x_ih_i(x)}e^{\beta y_ih_i(x)}}{2\cosh(\beta h_i(x))}.
\]
The right side is symmetric in $x$ and $y$, which proves the identity. Summing the identity $\pi(x)K(x,y)=\pi(y)K(y,x)$ over $x$ and using $\sum_xK(y,x)=1$, we obtain $\pi K=\pi$.

For part (b), since the matrices $ntK$ and $ntI$ commute, we have $e^{tn(K-I)}=e^{-nt}e^{ntK}$, and expanding $e^{ntK}$ in its power series gives \eqref{eq:model-poisson}. For the second statement, recall that $N_t$ has the Poisson law with mean $nt$ and is independent of $(X_k)_{k\ge0}$. Conditioning on $N_t$, we find
\[
 \PP(X_{N_t}=y)=\sum_{k=0}^\infty\PP(N_t=k)\,K^k(x,y)=\sum_{k=0}^\infty e^{-nt}\frac{(nt)^k}{k!}\,K^k(x,y)=P_t(x,y).
\]

For part (c), we first show by induction on $k$ that $\pi(x)K^k(x,y)=\pi(y)K^k(y,x)$ for all $k\ge0$. The case $k=0$ is trivial. If the identity holds for $k$, then by part (a) we have
\[
 \pi(x)K^{k+1}(x,y)=\sum_z\pi(z)K(z,x)K^k(z,y)=\sum_zK(z,x)\,\pi(y)K^k(y,z)=\pi(y)K^{k+1}(y,x).
\]
By \eqref{eq:model-poisson}, $P_t$ is a convex combination of the matrices $K^k$, with weights that do not depend on $x$ or $y$. This implies that $\pi(x)P_t(x,y)=\pi(y)P_t(y,x)$, and summing over $x$ gives $\pi P_t=\pi$. Summing the identity for $K^k$ over $x$ in the same way, we find $\pi K^k=\pi$.

For part (d), for each $y$ we have $|\nu Q(y)-\zeta Q(y)|\le\sum_x|\nu(x)-\zeta(x)|Q(x,y)$. Summing over $y$ and using $\sum_yQ(x,y)=1$ gives the first claim. Applying it with $Q=K$, $\nu=K^k(x,\cdot)$, and $\zeta=\pi$, and recalling that $\pi K=\pi$, we find
\[
 d^J_{\rm disc}(x,k+1)=\norm{K^k(x,\cdot)K-\pi K}_\TV\le d^J_{\rm disc}(x,k).
\]
Similarly, for $u\ge0$ we have $P_{t+u}=P_tP_u$ and $\pi P_u=\pi$, and the first claim with $Q=P_u$ implies that $d^J(x,t+u)\le d^J(x,t)$. Finally, by \eqref{eq:model-poisson}, every entry of $P_t$ is a continuous function of $t$. As a finite sum of absolute values of such functions, $d^J(x,t)$ is continuous in $t$.
\end{proof}

We can realize the continuous heat-bath process started at $x$ as $(X_{N_t})_{t\ge0}$, with $(X_k)_{k\ge0}$ and $(N_t)_{t\ge0}$ as in part (b) of \cref{lem:model-dynamics}. Indeed, the superposition of the $n$ independent rate-one clocks is a Poisson process of rate $n$. At each of its points, the site that rings is uniform on $\{1,\dots,n\}$ and independent of the past; see \cite[Section~20.1, (20.1)--(20.6)]{levin-peres-2017}.

We use part (d) of \cref{lem:model-dynamics} to turn a lower bound on the distance at one time into lower bounds at all earlier times. If $t\le t'$ and $d^J(x,t')>a$, then $d^J(x,t)>a$, and likewise in discrete time. This is the property used in the proof of \cref{b:cor:subcritical}, and parts (b) and (c) are used in the proof of \cref{thm:barrier-730}. We note that part (c) concerns deterministic times. It says that a chain started from $\pi$ has law $\pi$ at every fixed time. It says nothing about the law of the chain at a random time that depends on the trajectory.

\section{The realized-start experiment}\label{sec:model-joint}

In \cref{thm:barrier-730,thm:upper}, the initial state of the dynamics is a Gibbs sample for the realized disorder, and it is held fixed when the distance from equilibrium is evaluated. We make this precise in the following definition.

\begin{definition}[Realized-start probability]\label{def:model-joint}
Fix $n\ge2$ and $\beta>0$. Let $\PP_{J,X_0\sim\pi^J}$ be the joint law of a pair $(J,X_0)$ obtained by first sampling the disorder $J$ and then sampling $X_0$ from $\pi^J$. That is, for every Borel set $B\subseteq\R^{n(n-1)/2}$ and every $x\in\Sigma_n$,
\[
 \PP_{J,X_0\sim\pi^J}\{J\in B,\ X_0=x\}=\E_J\bigl[\pi^J(x)\,\mathbf 1\{J\in B\}\bigr].
\]
For a deterministic time $t\ge0$ and a threshold $a\in\R$, the realized-start probability is
\begin{equation}\label{eq:common-joint}
 \PP_{J,X_0\sim\pi^J}\{d^J(X_0,t)>a\}
 =\E_J\sum_{x\in\Sigma_n}\pi^J(x)\,\mathbf 1\{d^J(x,t)>a\}
 =\E_J\,\pi^J\{x\in\Sigma_n:d^J(x,t)>a\},
\end{equation}
and the discrete realized-start probability is defined in the same way, with $d^J_{\rm disc}(X_0,k)$ for a deterministic integer $k\ge0$ in place of $d^J(X_0,t)$.
\end{definition}

The first equality in \eqref{eq:common-joint} is the definition of the joint law, and the second is the definition of $\pi^J(\cdot)$ for sets. The quantity $d^J(X_0,t)$ is the distance from $\pi^J$ of the law at time $t$ of the dynamics started at $X_0$. Its randomness comes only from $J$ and $X_0$, since the randomness of the dynamics is integrated in $P_t^J(X_0,\cdot)$.

We evaluate the distance before averaging over the initial state. If we averaged over the initial state first, the law at time $t$ would be $\sum_x\pi^J(x)P^J_t(x,\cdot)=\pi^J$ by part (c) of \cref{lem:model-dynamics}, and its distance from $\pi^J$ would be zero at every time. The realized-start probability measures instead how much Gibbs mass lies on initial states from which the dynamics is still far from equilibrium at time $t$. This is the formulation of \cite[Section~1, following Theorem~1.1, p.~3]{oai-barriers}. It is not a worst-case statement: a lower bound on $\max_xd^J(x,t)$ gives no information about the Gibbs mass of the initial states from which the dynamics mixes slowly.

We note that all quantities in \eqref{eq:common-joint} are measurable. Since $\Sigma_n$ is finite, the Gibbs weights $\pi^J(x)$ and the entries of $K^J$ are continuous functions of $J$, and so are the entries of $(K^J)^k$ and of $P_t^J=e^{tn(K^J-I)}$. It follows that for all $x$, $t$, and $k$, the maps $J\mapsto d^J(x,t)$ and $J\mapsto d^J_{\rm disc}(x,k)$ are continuous, the sets $\{J:d^J(x,t)>a\}$ are open, and $J\mapsto\pi^J\{x:d^J(x,t)>a\}$ is Borel measurable. No measurable selection of initial states or of auxiliary objects is needed to define these probabilities.

\begin{lemma}[Equivalent forms of a realized-start statement]\label{lem:model-joint}
Fix $\beta>0$ and $a\in\R$, and let $(t_n)_{n\ge2}$ be deterministic times. Set
\[
 M_n(J)=\pi^J\{x\in\Sigma_n:d^J(x,t_n)>a\}.
\]
Then the following three statements are equivalent.
\begin{enumerate}[label=(\alph*)]
\item $\PP_{J,X_0\sim\pi^J}\{d^J(X_0,t_n)>a\}\to1$ as $n\to\infty$.
\item $\E_JM_n\to1$ as $n\to\infty$.
\item $M_n\to1$ in probability over the disorder; that is, $\PP_J\{M_n>1-\eps\}\to1$ for every $\eps>0$.
\end{enumerate}
The same holds with $d^J_{\rm disc}(x,k_n)$, for deterministic integers $k_n\ge0$, in place of $d^J(x,t_n)$. If these statements hold both in continuous and in discrete time, then $\PP_{J,X_0\sim\pi^J}\{d^J(X_0,t_n)>a,\ d^J_{\rm disc}(X_0,k_n)>a\}\to1$, and the Gibbs mass of the set of $x$ that satisfy both inequalities tends to one in probability over the disorder.
\end{lemma}

\begin{proof}
By \eqref{eq:common-joint}, statements (a) and (b) are the same. Fix $\eps\in(0,1)$. Since $0\le1-M_n\le1$, by Markov's inequality we have
\[
 \PP_J\{1-M_n\ge\eps\}\le\frac{\E_J(1-M_n)}{\eps},
\]
and splitting the expectation according to whether $1-M_n<\eps$, we also find
\[
 \E_J(1-M_n)\le\eps+\PP_J\{1-M_n\ge\eps\}.
\]
By the first inequality, (b) implies (c). By the second inequality, (c) implies $\limsup_n\E_J(1-M_n)\le\eps$ for every $\eps\in(0,1)$, and hence (b) holds. The argument in discrete time is identical.

For the last statement, let $M'_n$ be the Gibbs mass of $\{x:d^J_{\rm disc}(x,k_n)>a\}$, and let $M''_n$ be the Gibbs mass of the set of $x$ satisfying both inequalities. By a union bound for $\pi^J$, we have $1-M''_n\le(1-M_n)+(1-M'_n)$. Statement (b) holds for $M_n$ and for $M'_n$, and this implies that $\E_J(1-M''_n)\to0$. By the definition of the joint law, $\E_JM''_n$ is the joint probability that $X_0$ satisfies both inequalities, and the equivalence of (b) and (c) for $M''_n$ yields the second assertion.
\end{proof}

The complementary events satisfy the same equivalences. For deterministic
$t_n$ and a fixed $a$, the probability
$\PP_{J,X_0\sim\pi^J}\{d^J(X_0,t_n)\le a\}$ tends to one if and only if
$\PP_{J,X_0\sim\pi^J}\{d^J(X_0,t_n)>a\}$ tends to zero. By
\eqref{eq:common-joint}, this is equivalent to $\E_JM_n\to0$, and hence to
$M_n\to0$ in probability, since $0\le M_n\le1$. Equivalently, the Gibbs
mass of $\{x:d^J(x,t_n)\le a\}$ tends to one in probability over the disorder.
This is the form used in \cref{thm:upper}. The same statements hold in
discrete time.

By \cref{lem:model-joint}, each of the two conclusions of \cref{thm:barrier-730} is equivalent to the convergence to one, in probability over the disorder, of the Gibbs mass of the initial states that satisfy it, and the two conclusions also hold simultaneously. We use the lemma for each fixed threshold; it gives no uniformity in the threshold and no almost sure statement in the disorder.

\section{Asymptotic conventions}\label{sec:model-asymptotics}

The definitions above hold for every $\beta>0$. From now on, unless stated otherwise, $\beta>1$ is fixed before $n\to\infty$, and all limits are taken as $n\to\infty$. We use the following conventions.

\paragraph{Eventual statements.} A statement holds \emph{for all sufficiently large $n$}, or \emph{eventually}, if there exists $n_0$ such that it holds for all $n\ge n_0$. The threshold $n_0$ may depend on $\beta$ and on the fixed parameters named in the statement, such as an accuracy parameter $\eta$ or $\eps$ or a threshold $\tau$. It never depends on the disorder, on a configuration, or on a time.

\paragraph{Constants.} We write $c,C,c_\beta,C_\beta,\dots$ for positive finite constants. They depend only on $\beta$ and on fixed parameters that are listed explicitly, and never on $n$, on the disorder, or on configurations. Within a proof, the value of an unnamed constant may change from line to line. Named constants, such as the coefficient $A_\beta$ of \cref{thm:barrier-730}, keep their meaning throughout. When several parameters are chosen, we say in which order they are fixed. A statement made ``for every fixed $\tau$'' asserts nothing uniform in $\tau$.

\paragraph{Power exponents.} For a positive sequence $(a_n)$ and $r\in\R$, we write $a_n=n^{r+o(1)}$ if, for every fixed $\eta>0$, we have $n^{r-\eta}\le a_n\le n^{r+\eta}$ for all sufficiently large $n$, with a threshold that may depend on $\eta$. For example, $\Var(F_n)=n^{1/3+o(1)}$ means that for every fixed $\eta>0$, eventually $n^{1/3-\eta}\le\Var(F_n)\le n^{1/3+\eta}$.
For nonnegative random variables $U_n$, we write $U_n=n^{r+o_\PP(1)}$ if $\PP\{n^{r-\eta}\le U_n\le n^{r+\eta}\}\to1$ for every fixed $\eta>0$. The notation $|F_n-\E F_n|=n^{1/6+o_\PP(1)}$ means
\begin{equation}\label{eq:common-fluctuation-window}
 \PP\bigl\{n^{1/6-\eta}\le|F_n-\E F_n|\le n^{1/6+\eta}\bigr\}\longrightarrow1\quad\text{for every fixed }\eta>0.
\end{equation}
This is a statement about the typical size of $F_n-\E F_n$. It does not follow from $\Var(F_n)=n^{1/3+o(1)}$, since a variance estimate does not exclude that $|F_n-\E F_n|$ is much smaller than $n^{1/6}$ with probability bounded away from zero.

\section{Normalizations in the cited works}\label{sec:model-dictionary}

The works from which we import results use several normalizations. Three operations relate them to ours, and they act differently.
\begin{enumerate}[label=(\roman*)]
\item Replacing the counting measure on $\Sigma_n$ by the uniform probability measure replaces $Z_n$ by $2^{-n}Z_n$, $F_n$ by $F_n-n\log2$, and $p_n(\beta)$ and $P_\beta$ by $p_n(\beta)-\log2$ and $P_\beta-\log2$. This is a deterministic shift. It changes neither Gibbs measures, nor variances, nor the centered variable $F_n-\E F_n$, nor the mean bias $\Delta_{n,\beta}$.
\item Completing the Hamiltonian as in \cref{sec:model-completion} adds the random variable $\beta G$ to $F_n$. By \cref{lem:model-completion}, it changes neither Gibbs measures nor expected logarithms, but it adds $\beta^2/2$ to the variance, adds $\beta^2t^2/4$ to the logarithm of the Laplace transform of the centered variable, and multiplies the $s$th moment of an $\ell$-replica partition function by $e^{\beta^2\ell^2s^2/4}$.
\item The energy normalization of \cite[Section~7.1]{quantitative-parisi} divides log partition functions and the Parisi functional by $\beta$ and multiplies the cumulative profile by $\beta$. It also rescales the spatial variable of the Parisi equation, which changes spatial derivatives of the Parisi solution by powers of $\beta$.
\end{enumerate}
\Cref{tab:model-dictionary} lists the conventions of the works cited most often. In it, $\Phi_\alpha$ and $\mathcal P_\beta$ are the solution of the Parisi equation and the Parisi functional of \cref{sec:inputs}, with terminal value $\Phi_\alpha(1,z)=\log(2\cosh z)$.

\begin{table}[ht]
\centering
\small
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{>{\raggedright\arraybackslash}p{0.17\textwidth}>{\raggedright\arraybackslash}p{0.36\textwidth}>{\raggedright\arraybackslash}p{0.38\textwidth}}
\toprule
Work & Model and pressure & Completion and other conventions\\
\midrule
OpenAI, barriers \cite[Section~1, (1.1), pp.~2--3; Section~2, (2.1), (2.12)--(2.14)]{oai-barriers}
& As here: $H_n^J$, $Z^J_{n,\beta}=\sum_xe^{\beta H_n^J(x)}$, $\pi^J$, $R$, $p_n(\beta)$; the same $\Phi_\alpha$ and $\mathcal P_\beta$.
& Adds $(2n)^{-1/2}\sum_iJ_{ii}$, with new independent $J_{ii}$, in free-energy calculations only; this variable has the law of $G$. The kernels $K^J$ and $P^J_t$ and the distances $d^J$, $d^J_{\rm disc}$ are as here.\\
\addlinespace
OpenAI, fluctuations \cite[Section~1, p.~1; Section~5.2, p.~48; Section~7, p.~66]{oai-fluctuations}
& Writing $\widetilde H_n$, $\widetilde Z_n$, $\widetilde F_n$ for the $H_n$, $Z_n$, $F_n$ of that work: $g_{ij}=J_{ij}$, $\sigma=x$, $\widetilde H_n=\beta H_n$, $\widetilde Z_n=2^{-n}\sum_\sigma e^{\widetilde H_n(\sigma)}=2^{-n}Z_n$, $\widetilde F_n=F_n-n\log2$.
& $\widetilde F_n^c=\widetilde F_n+D$ with $D\sim N(0,\beta^2/2)$, which has the law of $\beta G$; hence $\widetilde F_n^c-\E\widetilde F_n^c$ has the law of $F_n^c-\E F_n^c$. $T=\beta^2$; the terminal value is $\log\cosh z$.\\
\addlinespace
Aronow and Lopatto \cite[(1.1)--(1.3), p.~4; Section~7.1, pp.~53--54; (7.64), p.~93]{quantitative-parisi}
& $N=n$, $g_{ij}=J_{ij}$, $\sigma=x$; $F_{N,\beta}=F_n$ and $p_{N,\beta}=p_n(\beta)$; $a=\alpha$ and $u_a=\Phi_\alpha$, with the same functional $\mathcal P_\beta$.
& $D_N=\operatorname{Tr}W/2$ has the law of $G$, and $F^c_{N,\beta}=F_n+\beta D_N$. In Section~7: $\gamma=\beta\alpha$, $u^{\rm en}_\gamma(q,x)=\beta^{-1}\Phi_\alpha(q,\beta x)$, the functional of $\gamma$ is $\beta^{-1}\mathcal P_\beta(\alpha)$, and $X_{N,\beta}=F_n/\beta$. The lower bound $\gamma\ge\lambda$ on the cumulative profile is $\alpha\ge\lambda/\beta$; the profile $\max\{\alpha,3s\}$ corresponds to $\lambda=3\beta s$.\\
\addlinespace
Aronow and Lopatto \cite[Section~1.1, p.~1; Section~3.1, p.~33]{hexagon-bias}
& As in \cite{quantitative-parisi}: $F_{N,\beta}=F_n$, $p_{N,\beta}=p_n(\beta)$, and $u_\alpha=\Phi_\alpha$; $\alpha_\beta$ is the distribution function of the Parisi measure.
& $D_N$ and $F^c_{N,\beta}=F_n+\beta D_N$ as in \cite{quantitative-parisi}. The mean bias $P_\beta-p_{N,\beta}$ is $\Delta_{n,\beta}$.\\
\bottomrule
\end{tabular}
\caption{Normalizations of the works cited most often, compared with \eqref{eq:common-model} and \eqref{eq:model-completion}. In the second and third columns, the symbol of the cited work appears on the left of each identity. In \cite{oai-barriers,quantitative-parisi,hexagon-bias}, $P_\beta$ is defined as the infimum of $\mathcal P_\beta$; it equals our $P_\beta$ by \cref{inp:parisi}.}
\label{tab:model-dictionary}
\end{table}

OpenAI's upper-bound paper uses the same normalization
\cite[Section~2]{oai-typical-upper}. Its completion matrix $W(G)$ is the
matrix $W$ of \cref{sec:model-completion}; its $\widehat H_G$, $F(G)$, and
$\pi_G$ are our $H_n^c$, $F_n^c$, and $\pi^J$, respectively. The diagonal
contributes only the common Gaussian variable $G$ of
\eqref{eq:model-completion}, so the heat-bath kernels also agree by
\eqref{eq:model-heat-bath}. The primitive matrix representation used in
\cref{part:upper} is given in \cref{u:sec:primitive}.

We record the consequences for the results of this paper. The barrier preprint \cite{oai-barriers} uses our normalization, and its results apply without translation. The fluctuation preprint \cite{oai-fluctuations} places $\beta$ inside the Hamiltonian and uses the uniform prior. Since $\widetilde F_n-\E\widetilde F_n=F_n-\E F_n$ in the notation of \cref{tab:model-dictionary}, its fluctuation theorem has the same meaning in our normalization, and its logarithmic-moment bounds concern the variable $F_n^c-\E F_n^c$, whose Laplace transform is related to that of $F_n-\E F_n$ by \eqref{eq:model-completion-laplace}. The works \cite{quantitative-parisi,hexagon-bias} use our normalization of the model. The variance--bias inequality of \cite[Remark~7.14, (7.68)]{quantitative-parisi} and the deduction in \cite[Section~1.6]{hexagon-bias} are stated directly for $\Var F_n$ and $\Delta_{n,\beta}$. The additive term $\beta^2/2$ that appears there is the variance of $\beta G$ from part (c) of \cref{lem:model-completion}, not the effect of a change of prior. Neither the completion nor the choice of prior changes any exponent in \cref{f:main} or \cref{cor:mean-bias}.
\chapter{Gaussian hierarchies and imported results}\label{sec:inputs}

This section fixes the scalar and matrix objects used in
\cref{part:static,part:dynamics}, and it states, once, the results from the
literature on which the proof of \cref{thm:barrier-730} relies. In
\cref{sec:inp-steps} we define the Gaussian integration step and record its
elementary properties. In \cref{b:sec:scalar} we define the Parisi functional,
the scalar diffusion and the response $\Gamma$, and we state the Parisi formula,
the uniqueness of its minimizer, the scalar estimates and stationarity
identities of OpenAI's barrier preprint \cite{oai-barriers}, and the fact that the support
of the Parisi measure is an interval. Together these imply that $\Gamma(t)=t$ on
$[0,q_\beta]$ (\cref{cor:inp-identity}). \Cref{sec:inp-finite} states the
finite-size estimates of \cite{quantitative-parisi,hexagon-bias}. In
\cref{sec:inp-hierarchies} we define Gaussian hierarchies, their trial values
and the constrained partition functions, we prove the block factorization of
the unperturbed three-replica path (\cref{lem:inp-blocks}), and we state the
matrix comparison and two Gaussian tools of \cite{oai-barriers}.
\Cref{sec:inp-three} states the two- and three-replica pressure bounds of
\cite{oai-barriers}, and \cref{sec:inp-coverage} its energy and rotation
estimates and its bound on the masses of overlap neighborhoods.

We use the scalar objects from \cref{sec:scalar} on. The hierarchies and the
matrix comparison enter \cref{sec:flow,sec:cubic,sec:fractional,sec:static},
the finite-size estimates enter
\cref{sec:fractional,sec:static,sec:references,sec:barrier-proof},
and the bounds on the masses of overlap neighborhoods enter
\cref{sec:references}. We prove
\cref{lem:inp-steps,lem:inp-blocks,cor:inp-identity} and the elementary facts
recorded in the remarks. Every theorem and proposition of this section is
quoted, without proof, from the source named in its heading, or is a consequence
of a proof given there; in the latter case we say so and explain how it is
obtained (\cref{inp:ob-remote}). A note after each result lists the sections
in which it is used.

\paragraph{Imported results and their normalization.}
Apart from standard tools of probability, such as Gaussian integration by
parts \cite[Section~A.4, (A.17)]{talagrand-2011}, It\^o's formula
\cite[Theorem~3.3.6]{karatzas-shreve-1991}, Gronwall's inequality
\cite[Section~A.13]{talagrand-2011}, and the inequalities of Markov and
Chebyshev (apply \cite[(A.7)]{talagrand-2011} to a nonnegative variable and to
a squared centered variable, respectively), the
results stated in this section with a citation are the only mathematical inputs
from the literature in the proof of \cref{thm:barrier-730}. The
inputs used only in \cref{part:fluct} are stated in \cref{sec:fluct,sec:bias}.
The inputs to \cref{part:upper} are listed in \cref{u:sec:scalar}.
The barrier preprint \cite{oai-barriers} uses our normalization
(\cref{tab:model-dictionary}), and it defines constrained pressures as
$n^{-1}\E\log$ of unnormalized constrained sums. Every statement of
\cite{oai-barriers} quoted below holds after the changes of letters listed in
\cref{tab:inp-ob}, apart from the explicitly described consequences of proofs
in \cref{inp:ob-remote}. For each of the other sources, we state, where it is
quoted, only the change of normalization that the quoted result needs;
\cref{sec:model-dictionary} compares all of them with ours.

\begin{table}[ht]
\centering
\small
\caption{Notation of \cite{oai-barriers} and of this paper.}\label{tab:inp-ob}
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.25\textwidth}>{\raggedright\arraybackslash}p{0.33\textwidth}>{\raggedright\arraybackslash}p{0.34\textwidth}@{}}
\toprule
Object & In \cite{oai-barriers} & Here\\
\midrule
number of replicas & $d$ & $\ell$\\
constrained pressure & $F_{d,n}(D)$ & $p_{\ell,n}(D)$ (\cref{def:inp-constrained})\\
hierarchy masses & $x_{-1}=0$, $x_k=1$, and $x_{j-1}\le x_j$; the $j$th field is
  integrated at mass $x_{j-1}$ & $0\le m_1\le\dots\le m_N\le1$; the $j$th
  increment is integrated at mass $m_j$ (\cref{def:inp-hierarchy})\\
squared norm, penalty & $|U|^2$, $\int x\,\dd|Q|^2$ & $\norm{U}_F^2$,
  $\int m\,\dd\norm{Q}_F^2$ \eqref{eq:inp-penalty}\\
minimizer & a minimizer $\mu$ & $\mu_\beta$, unique by \cref{inp:unique}\\
scalar diffusion & $X_s$ & $Y_t$\\
scalar quantities & $A(s)$, $d_*(s)$ & $\int_s^1\alpha(t)\dd t$,
  $s\alpha(s)+\int_s^1\alpha(t)\dd t$\\
large-overlap cutoff in \cite[Lemma~4.1 and (4.20)]{oai-barriers} & $h$ & $b_*$\\
scales in \cite[Sections~6--7]{oai-barriers} & $L=n^{1/10000}$, $\rho=L^{-1}$
  & $L_{\rm OB}$, $\rho_{\rm OB}$\\
\bottomrule
\end{tabular}
\end{table}

\paragraph{Conventions.}
Throughout, we fix $\beta>0$. Every statement that involves the
Parisi measure $\mu_\beta$, its support $[0,q_\beta]$, or the response $\Gamma$ of
the minimizer assumes $\beta>1$. Constants $C,c$ and thresholds $n_0$ depend only
on $\beta$ and on the parameters named in the statement where they appear.

\section{Gaussian integration steps}\label{sec:inp-steps}

In this section, $\ell\ge1$ is an integer. We write vectors of $\R^\ell$ in bold,
$\mathbf z\cdot\mathbf y$ and $\abs{\mathbf z}$ for the Euclidean inner product
and norm, and $\mathbf g$ for a standard Gaussian vector in $\R^\ell$. The notation
$\Sigma\succeq0$ means that the symmetric $\ell\times\ell$ matrix $\Sigma$ is
positive semidefinite, and $\Sigma^{1/2}$ denotes its positive semidefinite
square root. A function $f\colon\R^\ell\to\R$ has \emph{linear growth} if it is
measurable and there exists $C<\infty$ such that $\abs{f(\mathbf z)}\le
C(1+\abs{\mathbf z})$ for all $\mathbf z$.

\begin{definition}[Gaussian integration step]\label{def:inp-step}
Let $m\in[0,1]$, let $\Sigma\succeq0$ be an $\ell\times\ell$ matrix, and let $f$
have linear growth. The \emph{Gaussian integration step} at mass $m$ with
covariance $\Sigma$ is
\begin{equation}\label{eq:inp-step}
 T_{m,\Sigma}f(\mathbf z)=
 \begin{cases}
  m^{-1}\log\E\exp\bigl(mf(\mathbf z+\Sigma^{1/2}\mathbf g)\bigr), & m>0,\\
  \E f(\mathbf z+\Sigma^{1/2}\mathbf g), & m=0.
 \end{cases}
\end{equation}
The \emph{tilted Gaussian kernel} is the probability kernel
\begin{equation}\label{eq:inp-kernel}
 \mathcal K_{m,\Sigma,f}(\mathbf z,\dd\mathbf y)
 =\exp\bigl\{m\bigl[f(\mathbf y)-T_{m,\Sigma}f(\mathbf z)\bigr]\bigr\}\,
  N(\mathbf z,\Sigma)(\dd\mathbf y),
\end{equation}
where $N(\mathbf z,\Sigma)$ is the law of $\mathbf z+\Sigma^{1/2}\mathbf g$. For
$m=0$ the exponential factor equals one.
\end{definition}

Note that $T_{m,\Sigma}f$ is the normalization that makes
$\mathcal K_{m,\Sigma,f}(\mathbf z,\cdot)$ a probability measure. A singular
$\Sigma$ is allowed. In that case, $N(\mathbf z,\Sigma)$ is carried by $\mathbf
z+\operatorname{range}\Sigma$, and no inverse of $\Sigma$ is used. A rank-one
covariance $\Sigma=\mathbf v\mathbf v^{\mathsf T}$ integrates a single Gaussian
variable in the direction $\mathbf v$, and $T_{m,0}f=f$.

\begin{lemma}[Elementary properties of the integration step]\label{lem:inp-steps}
Let $m\in[0,1]$, let $\Sigma,\Sigma'\succeq0$ be $\ell\times\ell$ matrices, and
let $f,f'$ have linear growth.
\begin{enumerate}[label=\textup{(\alph*)}]
\item The function $T_{m,\Sigma}f$ has linear growth. If $f\le f'$, then
 $T_{m,\Sigma}f\le T_{m,\Sigma}f'$, and $T_{m,\Sigma}(f+c)=T_{m,\Sigma}f+c$ for
 every $c\in\R$.
\item $T_{m,\Sigma}T_{m,\Sigma'}f=T_{m,\Sigma+\Sigma'}f$.
\item Let $\ell=\ell_1+\ell_2$, write $\mathbf z=(\mathbf z_1,\mathbf z_2)$
 accordingly, and suppose that $f(\mathbf z)=f_1(\mathbf z_1)+f_2(\mathbf z_2)$,
 with $f_1,f_2$ of linear growth,
 and that $\Sigma$ is block diagonal with blocks $\Sigma_1,\Sigma_2$. Then
 $T_{m,\Sigma}f(\mathbf z)=T_{m,\Sigma_1}f_1(\mathbf z_1)+T_{m,\Sigma_2}f_2(\mathbf z_2)$.
\item For every integer $k\ge1$, we have $T_{m/k,\Sigma}(kf)=k\,T_{m,\Sigma}f$.
\item The value $T_{m,\Sigma}f(\mathbf z)$ depends only on the restriction of
 $f$ to $\mathbf z+\operatorname{range}\Sigma$.
\end{enumerate}
\end{lemma}

\begin{proof}
For (a), suppose that $\abs{f(\mathbf z)}\le C(1+\abs{\mathbf z})$, and set
$M=C\norm{\Sigma^{1/2}}\abs{\mathbf g}$, where $\norm{\cdot}$ is the operator
norm. Then $\abs{f(\mathbf z+\Sigma^{1/2}\mathbf g)}\le C(1+\abs{\mathbf
z})+M$. For $m>0$, by Jensen's inequality we have $T_{m,\Sigma}f(\mathbf z)\ge
\E f(\mathbf z+\Sigma^{1/2}\mathbf g)\ge -C(1+\abs{\mathbf z})-\E M$. By H\"older's
inequality, the map $m\mapsto m^{-1}\log\E e^{mM}$ is nondecreasing on $(0,1]$,
which implies that $T_{m,\Sigma}f(\mathbf z)\le C(1+\abs{\mathbf z})+\log\E
e^{M}$. The case $m=0$ is the same with expectations, and the monotonicity and
the translation identity are immediate from \eqref{eq:inp-step}.

For (b) with $m>0$, we have $\exp(mT_{m,\Sigma'}f(\mathbf y))=\E\exp(mf(\mathbf
y+\Sigma'^{1/2}\mathbf g'))$ for an independent standard Gaussian vector $\mathbf
g'$. Applying $T_{m,\Sigma}$, we find
\[
 \exp\bigl(mT_{m,\Sigma}T_{m,\Sigma'}f(\mathbf z)\bigr)
 =\E\exp\bigl(mf(\mathbf z+\Sigma^{1/2}\mathbf g+\Sigma'^{1/2}\mathbf g')\bigr),
\]
and $\Sigma^{1/2}\mathbf g+\Sigma'^{1/2}\mathbf g'$ is centered Gaussian with
covariance $\Sigma+\Sigma'$. The case $m=0$ is the same computation without
exponentials. For (c), if $\Sigma$ is block diagonal, then the two blocks of
$\Sigma^{1/2}\mathbf g$ are independent, and the expectation of
$e^{mf_1}e^{mf_2}$ factorizes (for $m=0$, the expectation of $f_1+f_2$ splits).
Part (d) follows from \eqref{eq:inp-step}, since $(k/m)\log\E e^{(m/k)kf}
=k\,m^{-1}\log\E e^{mf}$ for $m>0$. Part (e) holds because $\mathbf
z+\Sigma^{1/2}\mathbf g\in\mathbf z+\operatorname{range}\Sigma$ almost surely.
\end{proof}
\usedin{\cref{lem:inp-blocks} below, and \cref{sec:flow,sec:cubic,sec:fractional},
in the computations with Gaussian hierarchies.}

\section{The Parisi functional and the scalar diffusion}\label{b:sec:scalar}

A \emph{distribution function} is a right-continuous nondecreasing function
$\alpha\colon[0,1]\to[0,1]$ such that $\alpha(1)=1$. Equivalently,
$\alpha(t)=\mu([0,t])$ for a probability measure $\mu$ on $[0,1]$. We pass
freely between $\alpha$ and $\mu$. We say that $\alpha$ is a \emph{step}
distribution function if there exist $0=t_0<t_1<\dots<t_p=1$ such that $\alpha$
is constant on each interval $[t_{i-1},t_i)$. We call such points
\emph{breakpoints} of $\alpha$, and $m_i=\alpha(t_{i-1})$ denotes the value of
$\alpha$ on $[t_{i-1},t_i)$.

For a step distribution function $\alpha$, we define
$\Phi_\alpha\colon[0,1]\times\R\to\R$ by $\Phi_\alpha(1,z)=\log(2\cosh z)$ and,
for $i=p,p-1,\dots,1$ and $t_{i-1}\le t<t_i$,
\begin{equation}\label{eq:inp-scalar-recursion}
 \Phi_\alpha(t,\cdot)=T_{m_i,\beta^2(t_i-t)}\Phi_\alpha(t_i,\cdot),
\end{equation}
with the scalar integration step of \cref{def:inp-step} ($\ell=1$). On each
interval $(t_{i-1},t_i)$, the function $e^{m_i\Phi_\alpha}$ (or $\Phi_\alpha$
itself, if $m_i=0$) solves the backward heat equation
$-\partial_t\psi=(\beta^2/2)\partial_{zz}\psi$. It follows that $\Phi_\alpha$ is
smooth there and solves the Parisi equation
\begin{equation}\label{b:eq:Parisi-PDE}
 -\partial_t\Phi_\alpha(t,z)=\frac{\beta^2}{2}
 \Bigl(\partial_{zz}\Phi_\alpha(t,z)+\alpha(t)\bigl(\partial_z\Phi_\alpha(t,z)\bigr)^2\Bigr),
 \qquad \Phi_\alpha(1,z)=\log(2\cosh z).
\end{equation}
For a general distribution function $\alpha$, we define $\Phi_\alpha$ as the
uniform limit of $\Phi_{\alpha_k}$ along any sequence of step distribution
functions $\alpha_k$ such that $\int_0^1\abs{\alpha_k-\alpha}\dd t\to0$. By
\cref{inp:ob-scalar} below, the limit exists and does not depend on the
sequence, and \eqref{b:eq:Parisi-PDE} holds in time-integrated form, with the
time derivative existing for almost every $t$. The \emph{Parisi functional} is
\begin{equation}\label{b:eq:Parisi-functional}
 \mathcal P_\beta(\alpha)=\Phi_\alpha(0,0)-\frac{\beta^2}{2}\int_0^1t\alpha(t)\dd t.
\end{equation}
We write $u_\alpha=\partial_z\Phi_\alpha$ and $\chi_\alpha=\partial_{zz}\Phi_\alpha$,
and we define the scalar diffusion and the response by
\begin{equation}\label{b:eq:scalar-diffusion}
 \dd Y_t=\beta\dd B_t+\beta^2\alpha(t)u_\alpha(t,Y_t)\dd t,\qquad Y_0=0,\qquad
 \Gamma_\alpha(t)=\E u_\alpha(t,Y_t)^2,
\end{equation}
where $B$ is a standard Brownian motion. These are the scalar conventions of
\cite[Section~3]{oai-barriers}, where the diffusion is denoted $X$.

\begin{proposition}[{Regularity, stability, and scalar evolution \cite[Lemma~3.1]{oai-barriers}}]\label{inp:ob-scalar}
Let $\beta>0$. There exists $C_\beta<\infty$ such that the following holds for
all distribution functions $\alpha$ and $\widetilde\alpha$.
\begin{enumerate}[label=\textup{(\roman*)}]
\item The limit defining $\Phi_\alpha$ exists and does not depend on the
 approximating sequence. The functions $\Phi_\alpha$, $u_\alpha$ and
 $\chi_\alpha$ are continuous on $[0,1]\times\R$, $\Phi_\alpha(t,\cdot)$ is
 even, and
 \[
  \abs{u_\alpha}\le1,\qquad 0<\chi_\alpha\le1,\qquad
  \abs{\chi_\alpha(t,z)-\chi_\alpha(t,z')}\le C_\beta\abs{z-z'}.
 \]
\item We have
 \[
  \sup_{t,z}\Bigl(\abs{\Phi_\alpha-\Phi_{\widetilde\alpha}}
  +\abs{u_\alpha-u_{\widetilde\alpha}}+\abs{\chi_\alpha-\chi_{\widetilde\alpha}}\Bigr)
  \le C_\beta\int_0^1\abs{\alpha(t)-\widetilde\alpha(t)}\dd t.
 \]
\item The equation \eqref{b:eq:scalar-diffusion} has a unique strong solution. If
 $\alpha$ is a step distribution function with breakpoints $t_0<\dots<t_p$, then
 $(Y_{t_i})_{0\le i\le p}$ is a Markov chain whose transition from $t_{i-1}$ to
 $t_i$ is the tilted Gaussian kernel
 $\mathcal K_{m_i,\beta^2(t_i-t_{i-1}),\Phi_\alpha(t_i,\cdot)}$.
\item For $0\le s\le t\le1$,
 \begin{align*}
  u_\alpha(t,Y_t)-u_\alpha(s,Y_s)&=\beta\int_s^t\chi_\alpha(r,Y_r)\dd B_r,\\
  \E\bigl[\chi_\alpha(t,Y_t)\bigm| Y_s\bigr]&=\chi_\alpha(s,Y_s)
   -\beta^2\int_s^t\alpha(r)\E\bigl[\chi_\alpha(r,Y_r)^2\bigm| Y_s\bigr]\dd r.
 \end{align*}
 Moreover, $\Gamma_\alpha$ is continuously differentiable on $[0,1]$, with
 one-sided derivatives at the endpoints, and
 $\Gamma_\alpha(0)=0$, $\Gamma_\alpha'(t)=\beta^2\E\chi_\alpha(t,Y_t)^2$.
\item Let $0\le a\le1$ and $z\in\R$, and let $Y^{a,z}$ solve
 $\dd Y^{a,z}_t=\beta\dd B_t+\beta^2\alpha(t)u_\alpha(t,Y^{a,z}_t)\dd t$ for
 $a\le t\le1$, with $Y^{a,z}_a=z$. This equation has a unique strong solution,
 and the two identities in \textup{(iv)} hold for $Y^{a,z}$ and $a\le s\le t\le1$.
 If $\alpha$ is a step distribution function, then the transition of $Y^{a,z}$
 across each interval of constancy of $\alpha$ after $a$, with $a$ added to the
 breakpoints, is the tilted Gaussian kernel of \textup{(iii)}. The function
 $\Gamma^{a,z}_\alpha(t)=\E u_\alpha(t,Y^{a,z}_t)^2$ is continuously
 differentiable on $[a,1]$, with $\Gamma^{a,z}_\alpha(a)=u_\alpha(a,z)^2$ and
 $(\Gamma^{a,z}_\alpha)'(t)=\beta^2\E\chi_\alpha(t,Y^{a,z}_t)^2$.
\end{enumerate}
\end{proposition}

We note that, in \cite{oai-barriers}, part (i) collects (3.2) and the evenness of
$\Phi_\alpha$ (so $u_\alpha(t,0)=0$), part (ii) is (3.3), part (iii) is the
statement about (3.4), and part (iv) contains (3.5), (3.7), and (3.8). Taking
expectations in the second identity of (iv), we obtain the unconditional form
\cite[(3.6)]{oai-barriers}. The last sentence of \cite[Lemma~3.1]{oai-barriers}
extends the well-posedness and the identities (3.5)--(3.6) to a diffusion started
at any deterministic time and position. In part (v), the conditional identity
then follows from the Markov property. The statement about the transition
kernels follows from the identification of the kernels on each interval of
constant mass in the proof of \cite[Lemma~3.1]{oai-barriers}, which starts from
an arbitrary position. The statement about $\Gamma^{a,z}_\alpha$ follows from the
martingale identity, It\^o's isometry, and the continuity of $\chi_\alpha$ and
$Y^{a,z}$. The initial value
$\Gamma_\alpha(0)=0$ in (iv) uses the start $Y_0=0$. The constant $C_\beta$
does not depend on $\alpha$.
The proof in \cite{oai-barriers} also bounds $\abs{\partial_z^3\Phi_\alpha}$ by
$2e^{3\beta^2}$ for step distribution functions, and it proves continuity in
$\beta$. We prove bounds on all positive spatial derivatives, for every
distribution function, in \cref{sec:scalar}. For the step approximation and the
spatial regularity of the Parisi evolution, \cite{oai-barriers} credits
Auffinger and Chen \cite[Propositions~1--2]{auffinger-chen-2015}.
\usedin{\cref{sec:scalar}, for all scalar estimates, and
\cref{sec:flow,sec:cubic,sec:fractional,sec:static}, for the stability under a
change of the distribution function and for the passage from step
approximations to the minimizer, and \cref{part:upper}, for the scalar
process and its stochastic identities.}

\begin{theorem}[{Guerra's bound and the Parisi formula \cite[Theorem~3]{guerra-2003}, \cite[Theorems~1--2]{guerra-toninelli-2002},
\cite[Theorem~1.1]{parisi-formula}}]\label{inp:parisi}
Let $\beta>0$. For every $n\ge2$ and every distribution function $\alpha$, we
have $p_n(\beta)\le\mathcal P_\beta(\alpha)$. The limit
$P_\beta=\lim_{n\to\infty}p_n(\beta)$ exists, and
$P_\beta=\inf_\alpha\mathcal P_\beta(\alpha)$, where the infimum is over all
distribution functions. In particular, $\Delta_{n,\beta}=P_\beta-p_n(\beta)\ge0$
for every $n\ge2$.
\end{theorem}

The bound $p_n(\beta)\le\mathcal P_\beta(\alpha)$ is Guerra's
replica-symmetry-breaking bound \cite[Theorem~3]{guerra-2003}, whose proof first
treats step distribution functions and extends to the full class by
\cite[Theorem~1]{guerra-2003}. In our normalization it is
\cite[(2.14)]{oai-barriers}, where it is deduced from \cref{inp:ob-matrix} below
with $\ell=1$ (see \cref{lem:inp-blocks}(a)) and extended to all $\alpha$ by
\cref{inp:ob-scalar}(ii). The existence of the limit was proved by Guerra and
Toninelli \cite[Theorems~1--2]{guerra-toninelli-2002} by an interpolation that
shows superadditivity. The identification of the limit is Talagrand's Parisi
formula \cite[Theorem~1.1, (1.14)]{parisi-formula}, applied at zero external
field with counting measure and covariance function $\xi(q)=\beta^2q^2/2$.
% CHECK: Talagrand (2006) Thm 1.1, eq. (1.14), p. 223; Guerra (2003) Thm 3
% (arXiv cond-mat/0205123v1, p. 7); Guerra--Toninelli (2002) Thms 1--2, eqs. (17)--(18).
\usedin{\cref{sec:model}, for the identification of $P_\beta$ with the
infimum of $\mathcal P_\beta$, and \cref{sec:cubic,sec:fractional,sec:static},
where $P_\beta$ is the baseline of every pressure comparison, and
\cref{part:upper}, for the finite-volume pressure bound.}

\begin{theorem}[{Uniqueness of the Parisi measure \cite[Theorem~2 and Corollary~1]{auffinger-chen-uniqueness-2015}}]\label{inp:unique}
Let $\beta>0$. The functional $\mathcal P_\beta$ has exactly one minimizer among
distribution functions.
\end{theorem}

A minimizer exists by an elementary compactness argument. The probability
measures on $[0,1]$ form a weakly compact set, weak convergence implies $L^1$ convergence of the
distribution functions, and $\mathcal P_\beta$ is continuous in $L^1$ by
\cref{inp:ob-scalar}(ii), as in \cite[proof of Proposition~3.2]{oai-barriers}.
The functional of Auffinger and Chen
\cite[(7)]{auffinger-chen-uniqueness-2015} is $\mathcal P^{\rm AC}_{\xi,h}(\mu)
=\Phi^{\rm AC}_\mu(0,h)$, where $\Phi^{\rm AC}_\mu$ solves the Parisi equation
with coefficient $\xi''$ and terminal value $\log\cosh$. For
$\xi(q)=\beta^2q^2/2$ and $h=0$, we have $\xi''=\beta^2$, and
\cref{rem:inp-logcosh} gives
\[
 \mathcal P_\beta(\alpha)=\log2+\mathcal P^{\rm AC}_{\xi,0}(\mu)
 -\frac{\beta^2}{2}\int_0^1t\alpha(t)\dd t,
\]
which is their full variational objective \cite[(8)]{auffinger-chen-uniqueness-2015}.
They prove that $\mathcal P^{\rm AC}_{\xi,0}$ is strictly convex
\cite[Theorem~2]{auffinger-chen-uniqueness-2015}. Subtracting the linear
functional displayed above preserves strict convexity, and
\cite[Corollary~1]{auffinger-chen-uniqueness-2015} gives uniqueness of the
minimizer. We denote the minimizer by $\alpha_\beta$ and the corresponding
probability measure, the \emph{Parisi measure}, by $\mu_\beta$. By uniqueness,
we may speak of the scalar data of the minimizer; the results of
\cite{oai-barriers} quoted below are stated there for an arbitrary minimizer.
\usedin{every section of \cref{part:static,part:dynamics,part:upper}.}

\paragraph{Standing notation.}
Let $\beta>1$. From now on we write $\alpha=\alpha_\beta$ and $\mu=\mu_\beta$
when no confusion can arise, and $\Phi$, $u$, $\chi$, $Y$, and $\Gamma$ denote
$\Phi_\alpha$, $u_\alpha$, $\chi_\alpha$, the diffusion
\eqref{b:eq:scalar-diffusion} of $\alpha$, and $\Gamma_\alpha$. We also write
$w=\partial_z^3\Phi$; this derivative exists and is bounded by the estimates
of \cref{sec:scalar}. Some sections of \cref{part:static} abbreviate
$b=\beta^2$. In this notation, \eqref{b:eq:Parisi-PDE} and
\eqref{b:eq:scalar-diffusion} read
\begin{equation}\label{b:eq:os-scalar}
\begin{gathered}
 \partial_t\Phi=-\frac b2\bigl(\partial_{zz}\Phi+\alpha(\partial_z\Phi)^2\bigr),\qquad
 \Phi(1,z)=\log(2\cosh z),\\
 \dd Y_t=b\alpha(t)u(t,Y_t)\dd t+\sqrt b\dd B_t,\qquad Y_0=0.
\end{gathered}
\end{equation}

\begin{proposition}[{Support conditions and the mean of $\chi$ \cite[Proposition~3.2]{oai-barriers}}]\label{inp:ob-stationarity}
Let $\beta>0$, and let $\alpha$ be a minimizer of $\mathcal P_\beta$, with
probability measure $\mu$, diffusion $Y$ and response $\Gamma$.
\begin{enumerate}[label=\textup{(\roman*)}]
\item We have $0\in\supp\mu$, and $\supp\mu\subset[0,1-\eps_0]$ for some
 $\eps_0>0$.
\item At every $q\in\supp\mu\cap(0,1)$, we have $\Gamma(q)=q$ and
 $\Gamma'(q)\le1$. Also $\Gamma(0)=0$ and $\Gamma'(0)\le1$.
\item For every $s\in[0,1]$,
 \begin{equation}\label{eq:inp-stationarity-identity}
  \E\bigl[\chi(s,Y_s)+\alpha(s)u(s,Y_s)^2\bigr]=s\alpha(s)+\int_s^1\alpha(t)\dd t,
 \end{equation}
 or equivalently
 \begin{equation}\label{eq:inp-stationarity}
  \E\chi(s,Y_s)=\int_s^1\alpha(t)\dd t+\alpha(s)\bigl(s-\Gamma(s)\bigr).
 \end{equation}
 In particular, $\chi(0,0)=\int_0^1\alpha(t)\dd t$ and
 $\beta\int_0^1\alpha(t)\dd t\le1$. Moreover, $\alpha(s)>0$ for every $s\in(0,1]$.
\end{enumerate}
\end{proposition}

In \cite{oai-barriers}, the right side of \eqref{eq:inp-stationarity-identity}
is written $d_*(s)=1-\int_{(s,1]}t\,\mu(\dd t)$, which equals
$s\alpha(s)+\int_s^1\alpha(t)\dd t$ by Fubini's theorem, and
$\int_s^1\alpha(t)\dd t$ is written $A(s)$. The identities
\eqref{eq:inp-stationarity-identity} and \eqref{eq:inp-stationarity} are
\cite[(3.14)--(3.15)]{oai-barriers}, where they are called stationarity
identities. They hold at every $s$, including atoms of $\mu$. In this paper,
the name \emph{stationarity identities} refers instead to the two identities
\eqref{b:eq:contacts} on $(0,q_\beta)$ described after \cref{cor:inp-identity}. The support conditions in (ii) come from the first variation
\cite[(3.9)]{oai-barriers}. As credited in \cite[Section~3]{oai-barriers}, the
first-variation characterization appears in Chen
\cite[Theorem~2 and Proposition~1]{chen-2017} and in Jagannath and Tobasco
\cite[Proposition~1.1 and Corollary~3.10]{jagannath-tobasco-2017}. Part (ii) does
not describe the support of $\mu$; for $\beta>1$, \cref{inp:full-rsb}
describes it.
\usedin{\cref{cor:inp-identity}, \cref{sec:scalar}, and \cref{sec:static},
where \eqref{eq:inp-stationarity} bounds $\E\chi(q,Y_q)-\int_q^1\alpha(t)\dd t$
at overlaps $q$ with $\abs{q-\Gamma(q)}$ small.}

\begin{theorem}[{Full replica symmetry breaking \cite[Theorem~1.1]{full-rsb}}]\label{inp:full-rsb}
For every $\beta>1$, there exists $q_\beta\in(0,1)$ with the following
properties. We have $\supp\mu_\beta=[0,q_\beta]$. The restriction of $\mu_\beta$
to $[0,q_\beta)$ has a nonnegative density with respect to Lebesgue measure that
is infinitely differentiable on $(0,q_\beta)$ and whose derivatives of every
order extend continuously to $0$. Finally, $\mu_\beta(\{q_\beta\})\in(0,1)$.
\end{theorem}

\cite{full-rsb} works with the Hamiltonian $\beta(2n)^{-1/2}\sum_{i,j=1}^n g_{ij}x_ix_j$,
with $(g_{ij})$ independent standard Gaussians, which equals $\beta H_n$ plus an
independent centered Gaussian of variance $\beta^2/2$ common to all
configurations, that is, $\beta H_n^c$ with $H_n^c$ the completed Hamiltonian of
\cref{sec:model-completion}. The Parisi
functional there is $\log2+u_\mu(0,0)-(\beta^2/2)\int_0^1s\mu([0,s])\dd s$, where
$u_\mu$ solves the Parisi equation with terminal value $\log\cosh x$. By
\cref{rem:inp-logcosh}, this functional equals $\mathcal P_\beta$, and $\mu_\beta$
is the same measure in both papers. We cite version~3 of \cite{full-rsb}. Its
version~4 (October 2026) states the same support conclusion, with a density
that is smooth up to $q_\beta$, and adds a quantitative bound on the atom. We
use only the statement above.
% CHECK: arXiv:2607.11756v3, Theorem 1.1, eqs. (1.4)--(1.5), p. 2; normalization (1.1)--(1.3).
\usedin{\cref{cor:inp-identity}, and through it the stationarity identities of
\cref{sec:scalar} and the choice of the intervals of overlaps in
\cref{sec:cubic,sec:static,sec:references}, and \cref{part:upper}, for the
continuation of the scalar process beyond the support.}

\begin{corollary}[The interval on which $\Gamma(t)=t$]\label{cor:inp-identity}
Let $\beta>1$. Then
\begin{equation}\label{b:eq:identity-interval}
 \Gamma(t)=t\qquad(0\le t\le q_\beta),
\end{equation}
and $\beta^2\E\chi(t,Y_t)^2=1$ for $0<t<q_\beta$. Moreover, $\alpha_\beta(0)=0$,
$\alpha_\beta(t)>0$ for $0<t\le1$, $\alpha_\beta(t)=1$ for $t\ge q_\beta$, and
$\alpha_\beta$ is infinitely differentiable on $[0,q_\beta)$, with one-sided
derivatives at $0$.
\end{corollary}

\begin{proof}
By \cref{inp:full-rsb}, we have $\supp\mu_\beta=[0,q_\beta]$ with $q_\beta<1$.
Then $\supp\mu_\beta\cap(0,1)=(0,q_\beta]$, and \cref{inp:ob-stationarity}(ii)
implies that $\Gamma(q)=q$ for $0<q\le q_\beta$. Also $\Gamma(0)=0$ by
\cref{inp:ob-scalar}(iv), which proves \eqref{b:eq:identity-interval}. Recall
that $\Gamma$ is continuously differentiable with
$\Gamma'(t)=\beta^2\E\chi(t,Y_t)^2$. Differentiating
\eqref{b:eq:identity-interval} on $(0,q_\beta)$, we obtain the second claim. For
$t<q_\beta$, \cref{inp:full-rsb} implies that $\alpha_\beta(t)=\mu_\beta([0,t])$ is
the integral from $0$ to $t$ of a density that is smooth on $[0,q_\beta)$; in
particular $\alpha_\beta(0)=0$. For $t\ge q_\beta$ we have $\alpha_\beta(t)=1$.
Finally, $\alpha_\beta(t)>0$ for $t>0$ because $0\in\supp\mu_\beta$.
\end{proof}

In \cref{sec:scalar}, we prove that the identity $\beta^2\E\chi(t,Y_t)^2=1$ and
its companion $\E w(t,Y_t)^2=2\alpha(t)\E\chi(t,Y_t)^3$ hold on $(0,q_\beta)$
with the regularity needed later. These are the stationarity identities
\eqref{b:eq:contacts}.
\usedin{\cref{sec:scalar}, for the stationarity identities, and
\cref{sec:cubic,sec:static}, for the local comparisons, which take place in
$(0,q_\beta)$, and \cref{part:upper}, for the pre-support scalar identities.}

\begin{remark}[Terminal value $\log\cosh$]\label{rem:inp-logcosh}
Some of the sources quoted here and in \cref{part:fluct}, among them
OpenAI's fluctuation preprint \cite{oai-fluctuations} and \cite{full-rsb}, use
the terminal value $\log\cosh z$ in \eqref{b:eq:Parisi-PDE}. Since
$\log(2\cosh z)=\log2+\log\cosh z$, the two solutions differ by the constant
$\log2$ (by \cref{lem:inp-steps}(a) for step distribution functions, and by
approximation in general). They have the same spatial derivatives, the same
diffusion \eqref{b:eq:scalar-diffusion} and the same response $\Gamma$. Adding
$\log2$ to the functional with terminal value $\log\cosh$ recovers
\eqref{b:eq:Parisi-functional}, and the minimizer does not change. The papers
\cite{quantitative-parisi,hexagon-bias} use exactly
\eqref{b:eq:Parisi-PDE} and \eqref{b:eq:Parisi-functional}.
\end{remark}

\section{Finite-size estimates}\label{sec:inp-finite}

We use three estimates of Aronow and Lopatto on the convergence in the Parisi
formula at a fixed temperature $\beta>1$: an upper bound on the variance of
$F_n$, an upper bound on the mean bias $\Delta_{n,\beta}=P_\beta-p_n(\beta)$,
and a bound on the increase of the Parisi functional when a lower bound is
imposed on the distribution function.

\begin{theorem}[{Upper bound on the variance \cite[Theorem~1.4, (1.11)]{quantitative-parisi}}]\label{inp:qp-rate}
For every fixed $\beta>1$, there exist $C_v<\infty$ and $n_0(\beta)$ such that,
for every $n\ge n_0(\beta)$,
\begin{equation}\label{b:eq:external-rate}
 \Var(F_n)\le C_vn^{7/15}.
\end{equation}
\end{theorem}

The normalization of \cite[(1.1)--(1.2)]{quantitative-parisi} is ours
(\cref{tab:model-dictionary}). The theorem in \cite{quantitative-parisi} also
contains the lower bound $\Var(F_n)\ge c_\beta n^{4/15}$ and the bounds
$c_\beta n^{-11/12}\le P_\beta-p_n(\beta)\le C_\beta n^{-2/3}$ on the mean bias,
which we do not use. The upper bound on the mean bias that we use is
\cref{inp:hb-rate}.
% CHECK: arXiv:2609.09103v1 (v1, 8 Sep 2026), Theorem 1.4, eq. (1.11), p. 5; normalization (1.1), p. 4.
\usedin{\cref{sec:static}, for the lower bound on $Z_n$ in the disorder event
of \cref{b:lem:static-event}; and \cref{sec:bp-strong}, for the same lower
bound in \cref{lem:bp-strong-event}.}

\begin{theorem}[{Upper bound on the mean bias with exponent $7/10$ \cite[Theorem~1.1]{hexagon-bias}}]\label{inp:hb-rate}
For every fixed $\beta>1$, there exists $C_b<\infty$ such that
\[
 0\le P_\beta-p_n(\beta)\le C_b n^{-7/10}\qquad(n\ge2).
\]
\end{theorem}

The normalization of \cite[Section~1.1]{hexagon-bias} is ours, and the lower
bound $0\le P_\beta-p_n(\beta)$ is \cref{inp:parisi}.
% CHECK: hexagon:2610.00177v1, Theorem 1.1, p. 2; model and functional in Section 1.1, (1.1).
\usedin{\cref{sec:static}, for the lower bound on $Z_n$ in the disorder event
of \cref{b:lem:static-event}, together with \cref{inp:qp-rate};
\cref{sec:references}, where it is the input \eqref{eq:ref-mean-input} for
\cref{thm:barrier-730}, and its constant enters the choice of the scale
coefficient in \eqref{b:dyn:R-condition}; and \cref{sec:barrier-proof},
through these choices.}

\begin{proposition}[{The profile $\max\{\alpha,3s\}$ \cite[Lemma~7.2, (7.3), and
Proposition~7.12]{quantitative-parisi}}]\label{inp:qp-floor}
Fix $\beta>1$ and let $\alpha=\alpha_\beta$. There exist constants $T_*>0$,
$0<c_1\le c_2<\infty$, $s_0\in(0,1/3]$, and $C<\infty$, depending only on
$\beta$, such that the following holds.
\begin{enumerate}[label=\textup{(\roman*)}]
\item For $0\le t\le T_*$, we have $c_1t\le\alpha(t)\le c_2t$.
\item For $0<s\le s_0$, the function $\max\{\alpha,3s\}$ is a distribution
 function, and
 \begin{equation}\label{b:eq:external-floor}
  0\le\mathcal P_\beta(\max\{\alpha,3s\})-P_\beta\le Cs^5.
 \end{equation}
\end{enumerate}
\end{proposition}

\begin{remark}[Translation and consequences of \cref{inp:qp-floor}]\label{rem:inp-floor}
In \cite[Section~7.1]{quantitative-parisi}, the scalar problem is rescaled by
$\gamma=\beta\alpha$ and $x_{\rm old}=\beta x$, with rescaled functional
$\mathcal P^{\rm en}_\beta(\gamma)=\beta^{-1}\mathcal P_\beta(\gamma/\beta)$.
By Lemma~7.2 there, applied with temperature threshold $\beta_0=\beta$ to the
rescaled minimizer $\gamma_\beta=\beta\alpha_\beta$, we have
$kq\le\gamma_\beta(q)\le C_0q$ on $[0,T_*]$, which is (i) with $c_1=k/\beta$
and $c_2=C_0/\beta$. For a lower bound $a>0$ on the rescaled profile
(denoted $\lambda$ there), its estimate (7.3) bounds the value of the particular
trial profile $\max\{a,\gamma_\beta\}$ by $\mathcal
P^{\rm en}_\beta(\gamma_\beta)+C_1a^5$ for $0<a\le a_*$, where $a_*>0$ depends
only on $\beta$ (denoted $\lambda_*$ there). With $a=3\beta s$, we have
$\max\{a,\gamma_\beta\}=\beta\max\{3s,\alpha\}$, and we find that
\[
 \mathcal P_\beta(\max\{\alpha,3s\})
 =\beta\,\mathcal P^{\rm en}_\beta(\max\{3\beta s,\gamma_\beta\})
 \le P_\beta+3^5C_1\beta^6s^5\qquad(3\beta s\le a_*),
\]
which is (ii). The same estimate, in the original variables, is
\cite[(7.65)--(7.66)]{quantitative-parisi} with imposed lower bound $3s$. The bound
\eqref{b:eq:external-floor} concerns this particular profile, not only the
minimum over profiles with an atom at zero. For the minimizer of that
constrained problem, Pei \cite[Theorem~1(a) and Proposition~2.3(c)]{pei-fifth-order-2026}
obtains sharp fifth-order asymptotics, which we do not use.

Part (i) localizes the modification. If in addition $3s\le c_1T_*$, then
$\max\{\alpha,3s\}=\alpha$ on $[3s/c_1,1]$, since $\alpha(t)\ge c_1t\ge3s$ for
$3s/c_1\le t\le T_*$ and $\alpha(t)\ge\alpha(T_*)\ge c_1T_*\ge3s$ for $t>T_*$.
On $[0,3s/c_1)$ we have $0\le\max\{\alpha,3s\}-\alpha\le3s$. We conclude
that $\int_0^1\abs{\max\{\alpha,3s\}-\alpha}\dd t\le9s^2/c_1$. The lower bound
$3s$ ensures that the mass $\max\{\alpha,3s\}/3$ of a common block of three
coordinates is at least $s$ (\cref{lem:inp-blocks}(b)), as required in
\cref{sec:fractional}.
\end{remark}
% CHECK: arXiv:2609.09103v1, Lemma 7.2 and (7.3), p. 60; Section 7.1 rescaling, pp. 53--54;
% Proposition 7.12, (7.65)--(7.66), pp. 93--94.
\usedin{\cref{sec:fractional}, for the profile used in the positive fractional
comparison.}

\section{Gaussian hierarchies and constrained pressures}\label{sec:inp-hierarchies}

We now define the finite Gaussian hierarchies that serve as trials in the
pressure comparisons of \cref{part:static}. Their scalar case is the recursion
\eqref{eq:inp-scalar-recursion}. For $\ell\times\ell$ matrices $U,V$ we write
$U\mathbin{:}V=\sum_{a,b}U_{ab}V_{ab}$ and $\norm{U}_F^2=U\mathbin{:}U$.

\begin{definition}[Gaussian hierarchies]\label{def:inp-hierarchy}
Fix $\ell\ge1$.
\begin{enumerate}[label=\textup{(\alph*)}]
\item An \emph{admissible path} of length $N\ge1$ is a pair $(Q,m)$, where
 $Q=(Q_0,\dots,Q_N)$ are symmetric $\ell\times\ell$ matrices with $Q_0=0$ and
 $\Delta Q_j=Q_j-Q_{j-1}\succeq0$ for $1\le j\le N$, and $m=(m_1,\dots,m_N)$
 satisfies $0\le m_1\le m_2\le\dots\le m_N\le1$. We call $Q_N$ the
 \emph{endpoint} of the path and $m_j$ the \emph{mass} of the $j$th increment.
\item For a terminal function $f$ of linear growth, the \emph{partial values} are
 $\Psi_N=f$ and
 \begin{equation}\label{eq:inp-hierarchy-recursion}
  \Psi_{j-1}=T_{m_j,\beta^2\Delta Q_j}\Psi_j\qquad(j=N,N-1,\dots,1),
 \end{equation}
 and we set $\Psi(Q,m;f)=\Psi_0$. For a symmetric $\ell\times\ell$ matrix
 $\lambda$, we define
 \[
  f_\lambda(\mathbf z)=\log\sum_{\epsilon\in\{-1,1\}^\ell}
  \exp\bigl(\mathbf z\cdot\epsilon+\epsilon^{\mathsf T}\lambda\epsilon\bigr),
  \qquad
  \Psi(Q,m,\lambda)=\Psi(Q,m;f_\lambda)(0).
 \]
 In particular, $f_0(\mathbf z)=\sum_{a=1}^\ell\log(2\cosh z_a)$.
\item The \emph{path law} of $(Q,m,\lambda)$ is the law of
 $(\mathbf z_1,\dots,\mathbf z_N,\epsilon)$ defined as follows. Let $\mathbf
 z_0=0$. Given $\mathbf z_0,\dots,\mathbf z_{j-1}$, the vector $\mathbf z_j$ has
 law $\mathcal K_{m_j,\beta^2\Delta Q_j,\Psi_j}(\mathbf z_{j-1},\cdot)$, with
 the partial values of $f=f_\lambda$. Given $\mathbf z_1,\dots,\mathbf z_N$,
 the vector $\epsilon\in\{-1,1\}^\ell$ equals $\epsilon'$ with probability
 $\exp(\mathbf z_N\cdot\epsilon'+\epsilon'^{\mathsf T}\lambda\epsilon'-f_\lambda(\mathbf z_N))$.
 For $0\le l\le N$, the \emph{pair law} sharing the initial segment through level $l$ is
 the law of two such sequences that coincide in $(\mathbf z_1,\dots,\mathbf z_l)$
 and whose continuations, including the terminal draws, are conditionally
 independent given $(\mathbf z_1,\dots,\mathbf z_l)$, each with the
 transitions above. We write $\E_l^{(2)}$ for the corresponding expectation.
\end{enumerate}
\end{definition}

Note that the $j$th increment is integrated at mass $m_j$, that mass zero means
an ordinary Gaussian expectation, and that the terminal Gibbs draw of $\epsilon$
plays the role of a level of mass one. Equivalently, one draws independent increments $\mathbf
z_j-\mathbf z_{j-1}$ with law $N(0,\beta^2\Delta Q_j)$ and tilts the $j$th by
the density $\exp\{m_j[\Psi_j(\mathbf z_j)-\Psi_{j-1}(\mathbf z_{j-1})]\}$
given the past. This is the law of \cite[(2.5)--(2.6)]{oai-barriers}, in which
the continuations of a common initial segment are conditionally independent. By
\cref{lem:inp-steps}(a), every partial value has linear growth, and all
quantities above are finite. The \emph{penalty} of an admissible path is
\begin{equation}\label{eq:inp-penalty}
 \int m\dd\norm{Q}_F^2=\sum_{j=1}^Nm_j\bigl(\norm{Q_j}_F^2-\norm{Q_{j-1}}_F^2\bigr).
\end{equation}
By \cref{lem:inp-steps}(b), merging two consecutive increments with the same
mass, or inserting a zero increment whose mass keeps the masses ordered,
changes neither $\Psi(Q,m,\lambda)$ nor the penalty. \Cref{sec:fractional}
uses a variant of \cref{def:inp-hierarchy} in which all masses are at least a
positive number and the disorder is also integrated at positive mass.

\begin{definition}[Trial values]\label{def:inp-trial}
Let $D$ and $\lambda$ be symmetric $\ell\times\ell$ matrices, and let $(Q,m)$ be
an admissible path. The \emph{trial value} of $(Q,m,\lambda)$ for $D$ is
\begin{equation}\label{eq:inp-trial}
 \mathcal T_D(Q,m,\lambda)=\Psi(Q,m,\lambda)-\lambda\mathbin{:}D
 +\frac{\beta^2}{4}\Bigl(\norm{D-Q_N}_F^2-\int m\dd\norm{Q}_F^2\Bigr).
\end{equation}
If $(Q^{(k)},m^{(k)})$ is a sequence of admissible paths along which the
endpoints, the values $\Psi(Q^{(k)},m^{(k)},\lambda)$, and the penalties
converge, we also call the limit of $\mathcal T_D(Q^{(k)},m^{(k)},\lambda)$ a
trial value for $D$.
\end{definition}

When the endpoint equals $D$ and $\lambda=0$, the trial value is
$\Psi(Q,m,0)-(\beta^2/4)\int m\dd\norm{Q}_F^2$. This is the form of the trial
values in \cref{sec:cubic}. The term $(\beta^2/4)\norm{D-Q_N}_F^2$ is the cost
of an endpoint that differs from the prescribed overlap matrix, and it is used in
\cref{sec:static}.

\begin{definition}[Constrained partition functions]\label{def:inp-constrained}
Let $\ell\ge1$ and $n\ge2$. The \emph{overlap matrix} of
$x^1,\dots,x^\ell\in\Sigma_n$ is $(R(x^a,x^b))_{a,b=1}^\ell$. A symmetric
$\ell\times\ell$ matrix $D$ is \emph{feasible} if it is the overlap matrix of at
least one $\ell$-tuple. For feasible $D$, we define
\[
 Z_{\ell,n}(D)=\sum_{\substack{x^1,\dots,x^\ell\in\Sigma_n\\
 R(x^a,x^b)=D_{ab}\ \forall a,b}}
 \exp\Bigl(\beta\sum_{a=1}^\ell H_n(x^a)\Bigr),\qquad
 p_{\ell,n}(D)=\frac1n\E\log Z_{\ell,n}(D).
\]
\end{definition}

Note that $Z_{\ell,n}(D)/Z_n^\ell$ is the probability that $\ell$ independent samples
from $\pi^J$ have overlap matrix $D$. Since this probability is at most one, we have
$p_{\ell,n}(D)\le\ell p_n(\beta)$. Also $p_{1,n}(1)=p_n(\beta)$. A feasible matrix has unit diagonal
and off-diagonal entries in $\{-1,-1+2/n,\dots,1\}$, and there are at most
$(n+1)^{\ell(\ell-1)/2}$ feasible $\ell\times\ell$ matrices. For two and three
replicas we use the matrices
\begin{equation}\label{eq:inp-triple}
 D_2(q)=\begin{pmatrix}1&q\\q&1\end{pmatrix},\qquad
 D(r,q,\delta)=\begin{pmatrix}1&r+\delta&r-\delta\\r+\delta&1&q\\r-\delta&q&1\end{pmatrix},
\end{equation}
the latter in the coordinate order $(v,x,y)$. If $v,x,y\in\Sigma_n$ have overlap
matrix $D(r,q,\delta)$, then $q=R(x,y)$, $r=(R(v,x)+R(v,y))/2$ and
$\delta=(R(v,x)-R(v,y))/2$.

\begin{remark}[Spin flips and permutations]\label{rem:inp-orientation}
Let $\varpi$ be a permutation of $\{1,\dots,\ell\}$ and let
$\epsilon\in\{-1,1\}^\ell$. For a symmetric $D$, set
$(\mathsf O D)_{ab}=\epsilon_a\epsilon_bD_{\varpi(a)\varpi(b)}$. If $(x^b)_{b\le\ell}$ has
overlap matrix $D$, then $(\epsilon_ax^{\varpi(a)})_{a\le\ell}$ has overlap matrix
$\mathsf OD$,
and this map is a bijection between the two sets of tuples. Since
$H_n(-x)=H_n(x)$, it preserves $\sum_aH_n(x^a)$. We conclude that $\mathsf OD$ is
feasible if and only if $D$ is, and that $Z_{\ell,n}(\mathsf OD)=Z_{\ell,n}(D)$ for every
realization of the disorder. In particular, $Z_{2,n}(D_2(-q))=Z_{2,n}(D_2(q))$,
reversing $v$ gives $Z_{3,n}(D(-r,q,-\delta))=Z_{3,n}(D(r,q,\delta))$, and
exchanging $x$ and $y$ gives $Z_{3,n}(D(r,q,-\delta))=Z_{3,n}(D(r,q,\delta))$.
\end{remark}

The next two results, from \cite[Section~2]{oai-barriers}, are the tools used
to compare constrained pressures with trial values. The first is stated for
a hierarchy with a general terminal function, which is the form used in the
interpolation proofs.

\begin{proposition}[{Differentiation of a hierarchy \cite[Lemma~2.2]{oai-barriers}}]\label{inp:ob-hierarchy-derivative}
Let $\mathcal S$ be a nonempty finite set, let $\omega$ be a random element (the
quenched randomness), let $\mathbf g_1,\dots,\mathbf g_N$ be standard Gaussian
vectors, independent of each other and of $\omega$, and let $0\le m_1\le\dots\le
m_N\le1$. For $s\in\mathcal S$, let
\[
 U(s)=a(s,\omega)+\sum_{i=1}^N\mathbf a_i(s)\cdot\mathbf g_i,
 \qquad\E\max_{s\in\mathcal S}\abs{a(s,\omega)}<\infty,
\]
with deterministic vectors $\mathbf a_i(s)$. Set $\Psi_N=\log\sum_{s\in\mathcal S}e^{U(s)}$, and for $j=N,\dots,1$ set
$\Psi_{j-1}=m_j^{-1}\log\E_je^{m_j\Psi_j}$, or $\Psi_{j-1}=\E_j\Psi_j$ if
$m_j=0$, where $\E_j$ integrates $\mathbf g_j$. Under the path law, given
$(\omega,\mathbf g_1,\dots,\mathbf g_{j-1})$ the law of $\mathbf g_j$ has
density $\exp\{m_j(\Psi_j-\Psi_{j-1})\}$ with respect to its Gaussian law, and
the terminal configuration $S$ has the Gibbs law proportional to $e^{U(S)}$.
For $0\le l\le N$, let $\E_l^{(2)}$ be the expectation under the pair law of
$(S^1,S^2)$ sharing $(\omega,\mathbf g_1,\dots,\mathbf g_l)$. Let $m_0=0$,
$m_{N+1}=1$, and $\Delta m_l=m_{l+1}-m_l$ for $0\le l\le N$. Let $e$ be a fixed vector-valued function on $\mathcal S$, and suppose that
$U$ depends on a parameter $v>0$ in one of the following two ways: either
$\mathbf a_j(s)=\sqrt v\,e(s)$
for some $1\le j\le N$, or $j=0$, $\omega=(\omega',\mathbf g)$ with independent
components, $\mathbf g$ a standard Gaussian vector, and
$a(s,\omega)=a_*(s,\omega')+\sqrt v\,\mathbf g\cdot e(s)$ with
$\E\max_s\abs{a_*(s,\omega')}<\infty$. In both cases $v$ occurs only in the
displayed coefficient. Then, with all other parameters fixed, for $v>0$,
\[
 \frac{\partial}{\partial v}\E\Psi_0
 =\frac12\Bigl(\E\norm{e(S)}^2-\sum_{l=j}^N\Delta m_l\,\E_l^{(2)}e(S^1)\cdot e(S^2)\Bigr).
\]
The identity extends to $v=0$ by integration and continuity.
\end{proposition}

In \cite{oai-barriers}, the quenched root is level $0$ and the $j$th field is
integrated at mass $x_{j-1}$; with $x_{j-1}=m_j$, the weights
$x_l-x_{l-1}$ of \cite{oai-barriers} are our $\Delta m_l$. Under $\E_N^{(2)}$ the two
copies share all fields and only the terminal draws are independent. The
hypotheses on $U$ are those of the applications in \cite{oai-barriers}: the
terminal exponent is affine in the Gaussian variables, and $\mathcal S$ is
finite, so all coefficients are bounded. Gaussian exponential integrability and
the integrability of the root term then make every recursion finite and justify
the differentiation, as in \cite[proof of Lemma~2.2]{oai-barriers}.
\usedin{\cref{sec:fractional}, in the proof of the positive fractional matrix
comparison \cref{b:prop:fractional-matrix}, where the first level has positive
mass.}

\begin{proposition}[{Matrix interpolation bound \cite[Proposition~2.3]{oai-barriers}}]\label{inp:ob-matrix}
Let $n\ge2$ and $\ell\ge1$, let $D$ be a feasible $\ell\times\ell$ matrix, let
$(Q,m)$ be an admissible path of $\ell\times\ell$ matrices, and let $\lambda$ be
a symmetric $\ell\times\ell$ matrix. Then
\begin{equation}\label{eq:inp-matrix}
 p_{\ell,n}(D)\le\mathcal T_D(Q,m,\lambda)
 =\Psi(Q,m,\lambda)-\lambda\mathbin{:}D
 +\frac{\beta^2}{4}\Bigl(\norm{D-Q_N}_F^2-\int m\dd\norm{Q}_F^2\Bigr).
\end{equation}
For fixed $D$ and $\lambda$, the bound passes to every trial value for $D$ in the
sense of \cref{def:inp-trial}.
\end{proposition}

This is \cite[(2.11)]{oai-barriers}, with the constrained pressure
\cite[(2.9)]{oai-barriers} and the path integral \cite[(2.10)]{oai-barriers}. In
\cite{oai-barriers} the path is indexed from $Q_{-1}=0$; inserting the quenched
level $Q_0=0$ allowed there and setting $x_{j-1}=m_j$ identifies the two
conventions, including the penalty. The endpoint $Q_N$ is arbitrary, subject only
to positive semidefinite increments. Its diagonal need not equal
one, and the cost of a mismatch with $D$ is
$(\beta^2/4)\norm{D-Q_N}_F^2$. The proposition belongs to Guerra's interpolation
framework \cite{guerra-2003}. As \cite[Section~2]{oai-barriers} explains,
Talagrand's bounds for coupled replicas
\cite[Propositions~3.4 and~4.4]{talagrand-2007} already retain a cost for an
endpoint that differs from the prescribed overlaps, and Panchenko's vector-spin
formulation \cite{panchenko-2018} proves the asymptotic sharpness of such
bounds. The bound is quenched, and the positive fractional version needed in
\cref{sec:fractional} is proved there.
\usedin{\cref{sec:cubic}, for the trial values of the perturbed three-replica
path, with zero mismatch; \cref{sec:static}, for the comparison at separated
overlaps, with mismatch cost; and \cref{sec:fractional}, which adapts it to
positive fractional moments.}

\begin{lemma}[Block factorization]\label{lem:inp-blocks}
Let $\alpha$ be a step distribution function, and let $0=t_0<t_1<\dots<t_p=1$
contain the points where $\alpha$ jumps. Write $m_i=\alpha(t_{i-1})$ and
$\delta_i=t_i-t_{i-1}$.
\begin{enumerate}[label=\textup{(\alph*)}]
\item \textup{(One coordinate.)} The path with $\ell=1$, $Q_i=t_i$ and masses
 $m_i$ ($1\le i\le p$) is admissible. Its partial values for $f_0$ are
 $\Psi_i=\Phi_\alpha(t_i,\cdot)$, its trial value for $D=1$ and $\lambda=0$ is
 $\mathcal P_\beta(\alpha)$, and under its path law $(\mathbf z_1,\dots,\mathbf
 z_p)$ has the law of $(Y_{t_1},\dots,Y_{t_p})$, where $Y$ is the diffusion
 \eqref{b:eq:scalar-diffusion} of $\alpha$.
\item \textup{(Three coordinates.)} Let $0\le r\le q\le1$ belong to
 $\{t_0,\dots,t_p\}$. In the coordinate order $(v,x,y)$, let
 $\mathbf e=(1,1,1)$ and let $\mathbf e_v,\mathbf e_x,\mathbf e_y$ be the unit
 vectors. We use the following increments of $Q$, with their masses:
 \begin{itemize}
 \item common increments $\delta_i\mathbf e\mathbf e^{\mathsf T}$ at mass
  $m_i/3$, for $t_i\le r$;
 \item $v$-increments $\delta_i\mathbf e_v\mathbf e_v^{\mathsf T}$ at mass $m_i$,
  and $xy$-increments $\delta_i(\mathbf e_x+\mathbf e_y)(\mathbf e_x+\mathbf
  e_y)^{\mathsf T}$ at mass $m_i/2$, for $r<t_i\le q$;
 \item independent increments $\delta_i\mathrm{Id}$ at mass $m_i$, for $t_i>q$.
 \end{itemize}
 Let $(Q,m)$ take the common increments in increasing order of $i$, then the
 $v$- and $xy$-increments in any order that keeps each of these two lists in
 increasing order of $i$ and has nondecreasing masses, and then the
 independent increments in increasing order of $i$. Then $(Q,m)$ is admissible,
 $Q_N=D(r,q,0)$,
 \[
  \Psi(Q,m,0)=3\Phi_\alpha(0,0),\qquad
  \int m\dd\norm{Q}_F^2=6\int_0^1t\alpha(t)\dd t,
 \]
 and $\mathcal T_{D(r,q,0)}(Q,m,0)=3\mathcal P_\beta(\alpha)$.
\item \textup{(Two coordinates.)} Let $q\in\{t_0,\dots,t_p\}$. The path with
 increments $\delta_i(\mathbf e_1+\mathbf e_2)(\mathbf e_1+\mathbf
 e_2)^{\mathsf T}$ at mass $m_i/2$ for $t_i\le q$, followed by $\delta_iI$ at
 mass $m_i$ for $t_i>q$, is admissible, its endpoint is $D_2(q)$, and its trial
 value for $D_2(q)$ and $\lambda=0$ is $2\mathcal P_\beta(\alpha)$.
\item \textup{(Path law in (b).)} For $t_i\le q$, let $Y^v_{t_i}$ be the
 $v$-coordinate of the cumulative field $\mathbf z$ after the common
 increments and the $v$-increments with index at most $i$, and let $Y^x_{t_i}$
 be its $x$-coordinate after the common increments and the $xy$-increments with
 index at most $i$. Under the path law of $(Q,m,0)$, each of
 $(Y^v_{t_i})_{t_i\le q}$ and $(Y^x_{t_i})_{t_i\le q}$ has the law of
 $(Y_{t_i})_{t_i\le q}$, the two coincide for $t_i\le r$, and conditionally on
 $(Y^v_{t_i})_{t_i\le r}$ the families $(Y^v_{t_i})_{r<t_i\le q}$ and
 $(Y^x_{t_i})_{r<t_i\le q}$ are independent, each with the conditional law of
 $(Y_{t_i})_{r<t_i\le q}$ given $Y_r=Y^v_r$.
\end{enumerate}
\end{lemma}

Part (b) with step approximations of $\alpha_\beta$ is the base path of
\cite[Section~4.2]{oai-barriers} and \cite[Section~7, Step~1]{oai-barriers},
and its value is \cite[(7.7)]{oai-barriers}. Part (d) is
\cite[(4.6)]{oai-barriers}. The comparisons of \cref{part:static} perturb this
path near $r$ (\cref{sec:cubic,sec:static}) or replace $\alpha$ by
$\max\{\alpha,3s\}$ (\cref{sec:fractional}).

\begin{proof}
We prove (a) first and use it for the other parts. In (b), we call the common
increments, the $v$- and $xy$-increments, and the independent increments the
first, the middle and the last stage. The proof of (b) has three steps:
admissibility and the endpoint, the value, and the penalty.

\paragraph{Part (a).} Since $\alpha$ is nondecreasing and $m_p\le1$, the path is
admissible. By \eqref{eq:inp-scalar-recursion} at $t=t_{i-1}$, we have
$\Phi_\alpha(t_{i-1},\cdot)=T_{m_i,\beta^2\delta_i}\Phi_\alpha(t_i,\cdot)$, and
backward induction from $\Psi_p=\log(2\cosh\cdot)=\Phi_\alpha(1,\cdot)$ shows that
$\Psi_i=\Phi_\alpha(t_i,\cdot)$ for every $i$. Note that the penalty is
$\sum_im_i(t_i^2-t_{i-1}^2)=2\int_0^1t\alpha(t)\dd t$ and that the endpoint is
$1=D$. Then the trial value is $\Phi_\alpha(0,0)-(\beta^2/2)\int_0^1t\alpha(t)\dd t
=\mathcal P_\beta(\alpha)$. Finally, the transition of $\mathbf z_i$ is
$\mathcal K_{m_i,\beta^2\delta_i,\Phi_\alpha(t_i,\cdot)}$, and
\cref{inp:ob-scalar}(iii) states that this is the transition of $Y$ from
$t_{i-1}$ to $t_i$.

\paragraph{Admissibility and the endpoint in part (b).} Every increment is positive
semidefinite. We check that the masses are nondecreasing. For $t_i\le r$ we
have $t_{i-1}<r$, so that the common masses satisfy $m_i/3\le\alpha(r)/3$. Since
$r$ and $q$ are breakpoints, we see that the first $v$- and $xy$-masses are
$\alpha(r)$ and $\alpha(r)/2$, that the first independent mass is $\alpha(q)$,
and that every $v$- and $xy$-mass is at most $\alpha(q)$. Two nondecreasing
lists can always be merged into one nondecreasing list that keeps the order of
each, and this shows that an order as in (b) exists. The masses of $(Q,m)$ are
then nondecreasing and at most $m_p\le1$. Summing the increments, we obtain
\[
 Q_N=r\,\mathbf e\mathbf e^{\mathsf T}+(q-r)\bigl(\mathbf e_v\mathbf e_v^{\mathsf T}
 +(\mathbf e_x+\mathbf e_y)(\mathbf e_x+\mathbf e_y)^{\mathsf T}\bigr)+(1-q)\mathrm{Id},
\]
which has diagonal entries $1$, entries $r$ in the positions $(v,x)$ and $(v,y)$,
and entry $q$ in the position $(x,y)$. This is the matrix $D(r,q,0)$.

\paragraph{The value in part (b).} We integrate backward. The independent increments
are block diagonal with three one-dimensional blocks, and $f_0$ is a sum of
functions of single coordinates. By \cref{lem:inp-steps}(c) and part (a), after
the independent increments the partial value is $\sum_{a\in\{v,x,y\}}\Phi_\alpha(q,z_a)$.
Let $V=\{\mathbf z\in\R^3:z_x=z_y\}$. Note that the common, $v$-, and
$xy$-increments have ranges in $V$ and that the path starts at $0\in V$. By
\cref{lem:inp-steps}(e), at points of $V$ these steps depend only on
restrictions to $V$, which we write as
functions of $(z_v,z)$, where $z=z_x=z_y$. On $V$, a $v$-increment adds a scalar
Gaussian of variance $\beta^2\delta_i$ to $z_v$ and leaves $z$ fixed, and an
$xy$-increment adds a scalar Gaussian of variance $\beta^2\delta_i$ to $z$ and
leaves $z_v$ fixed. We claim that, during the middle stage, the restriction to
$V$ of the partial value has the form
\begin{equation}\label{eq:inp-middle-form}
 \Phi_\alpha(t_v,z_v)+2\Phi_\alpha(t_{xy},z),
\end{equation}
where $t_v$ (respectively $t_{xy}$) is the largest $t_i\le q$ whose $v$-increment
(respectively $xy$-increment) has not yet been integrated, and $t_v=r$
(respectively $t_{xy}=r$) if there is none. At the end of the middle stage this is
the restriction of $\sum_{a\in\{v,x,y\}}\Phi_\alpha(q,z_a)$, with $t_v=t_{xy}=q$. If the next step
backward is the $v$-increment with $t_i=t_v$, then \cref{lem:inp-steps}(c) and
\eqref{eq:inp-scalar-recursion} turn \eqref{eq:inp-middle-form} into
$\Phi_\alpha(t_{i-1},z_v)+2\Phi_\alpha(t_{xy},z)$. If it is the $xy$-increment with
$t_i=t_{xy}$, then \cref{lem:inp-steps}(c) and (d), with $k=2$, give
$T_{m_i/2,\beta^2\delta_i}\bigl(2\Phi_\alpha(t_{xy},\cdot)\bigr)=2T_{m_i,\beta^2\delta_i}\Phi_\alpha(t_{xy},\cdot)
=2\Phi_\alpha(t_{i-1},\cdot)$. This proves the claim. We conclude that, at the start
of the middle stage, the restriction to $V$ is
$\Phi_\alpha(r,z_v)+2\Phi_\alpha(r,z)$. The
common increments have range spanned by $\mathbf e$. On this line, where
$z_v=z_x=z_y=z$, the function equals $3\Phi_\alpha(r,z)$, and
\cref{lem:inp-steps}(d), with $k=3$, gives
$T_{m_i/3,\beta^2\delta_i}\bigl(3\Phi_\alpha(t_i,\cdot)\bigr)=3\Phi_\alpha(t_{i-1},\cdot)$ at each
common increment. After all common increments, the value at $0$ is
$3\Phi_\alpha(0,0)$.

\paragraph{The penalty in part (b).} During the first stage we have $Q=t\,\mathbf e\mathbf
e^{\mathsf T}$ and $\norm{Q}_F^2=9t^2$, so that this stage contributes
$\sum_{t_i\le r}(m_i/3)\cdot9(t_i^2-t_{i-1}^2)$. During the middle stage, if
the $v$-clock is $t_v$ and the $xy$-clock is $t_{xy}$, then $\norm{Q}_F^2=t_v^2+4t_{xy}^2+4r^2$.
A $v$-increment at mass $m_i$ contributes $m_i(t_i^2-t_{i-1}^2)$, and an
$xy$-increment at mass $m_i/2$ contributes $(m_i/2)\cdot4(t_i^2-t_{i-1}^2)$.
During the last stage only the diagonal changes, and each increment contributes
$3m_i(t_i^2-t_{i-1}^2)$. Every index $i$ contributes
$3m_i(t_i^2-t_{i-1}^2)$ in total, through one increment in the first and last
stages and through two increments in the middle stage. The penalty is
$3\sum_{i=1}^pm_i(t_i^2-t_{i-1}^2)=6\int_0^1t\alpha(t)\dd t$. Since the endpoint
is $D(r,q,0)$, the trial value is
$3\Phi_\alpha(0,0)-(3\beta^2/2)\int_0^1t\alpha(t)\dd t=3\mathcal P_\beta(\alpha)$.

\paragraph{Part (c).} The proof is the same, with the $v$-coordinate removed and
the common stage replaced by the $xy$-increments on $[0,q]$.

\paragraph{Part (d).} Under the path law, the $j$th increment has density
$\exp\{m_j[\Psi_j(\mathbf z_j)-\Psi_{j-1}(\mathbf z_{j-1})]\}$ with respect to its
Gaussian law. By the computation of the value, we see that for a $v$-increment with
index $i$ the exponent is $m_i[\Phi_\alpha(t_i,Y^v_{t_i})-\Phi_\alpha(t_{i-1},Y^v_{t_{i-1}})]$:
the term $2\Phi_\alpha(t_{xy},z)$ of \eqref{eq:inp-middle-form} is the same before and
after the step and cancels. For an $xy$-increment at mass $m_i/2$ the exponent is
$(m_i/2)\cdot2[\Phi_\alpha(t_i,Y^x_{t_i})-\Phi_\alpha(t_{i-1},Y^x_{t_{i-1}})]$,
and for a common increment at mass $m_i/3$ it is
$(m_i/3)\cdot3[\Phi_\alpha(t_i,z)-\Phi_\alpha(t_{i-1},z')]$ along the diagonal.
In each case the moving coordinate makes the scalar transition
$\mathcal K_{m_i,\beta^2\delta_i,\Phi_\alpha(t_i,\cdot)}$, which depends only on its own
current value, and the increments of different branches are driven by
independent Gaussian variables. This proves (d), in view of part (a).
\end{proof}

When $\alpha$ is not a step function, for instance $\alpha=\alpha_\beta$, we
use \cref{lem:inp-blocks} with step distribution functions $\alpha_k\to\alpha$
in $L^1$ whose breakpoints contain $r$ and $q$. By \cref{inp:ob-scalar}(ii),
$\mathcal P_\beta(\alpha_k)\to\mathcal P_\beta(\alpha)$, and the last sentence
of \cref{inp:ob-matrix} applies to the limit. The path-law statement (d) also
passes to the limit, in the following sense. Let $Y^{(k)}$ and $Y$ be the scalar
diffusions of $\alpha_k$ and $\alpha$, driven by the same Brownian motion, and
let $u_k=u_{\alpha_k}$. Since $\abs{u_k}\le1$ and $u$ is $1$-Lipschitz in space
($0<\chi\le1$), by Gronwall's inequality we have
\[
 \sup_{t\le1}\abs{Y^{(k)}_t-Y_t}
 \le\beta^2e^{\beta^2}\bigl(\norm{\alpha_k-\alpha}_{L^1}+\norm{u_k-u}_\infty\bigr),
\]
which tends to zero by \cref{inp:ob-scalar}(ii). For the two branches, we use the
same Brownian path up to $r$ and independent Brownian continuations after $r$.
The branch laws of (d) for $\alpha_k$, evaluated at any fixed finite set of
common breakpoints, therefore converge to the corresponding laws of two
diffusions of $\alpha$ that coincide up to $r$ and are conditionally independent
after $r$ given their common value at $r$.
\usedin{\cref{sec:cubic,sec:fractional,sec:static}, for the value and the law
of the unperturbed path.}

\begin{proposition}[{Gaussian tools \cite[Lemma~2.1]{oai-barriers}}]\label{inp:ob-concentration}
Let $\mathbf g$ be a standard Gaussian vector in $\R^M$, and let
$f\colon\R^M\to\R$ be $L$-Lipschitz. Then, for $a>0$ and $\theta\in\R$,
\begin{equation}\label{eq:inp-concentration}
 \PP\bigl(\abs{f(\mathbf g)-\E f(\mathbf g)}\ge a\bigr)\le2\exp\Bigl(-\frac{a^2}{2L^2}\Bigr),
 \qquad
 \log\E\exp\bigl\{\theta\bigl(f(\mathbf g)-\E f(\mathbf g)\bigr)\bigr\}\le\frac{\theta^2L^2}{2}.
\end{equation}
If a probability density on $\R^M$ is proportional to $e^{-V}$, where $V$ is twice
differentiable and $\nabla^2V\succeq I$, then $\Var(f)\le\E\norm{\nabla f}^2$
under this density for every continuously differentiable $f$ with finite right
side. In particular, the variance of a linear functional $\mathbf a\cdot\mathbf
X$ of a random vector $\mathbf X$ with this density is at most $\abs{\mathbf a}^2$.
\end{proposition}

The bounds \eqref{eq:inp-concentration} are the Gaussian concentration
inequality of Tsirelson, Ibragimov, and Sudakov \cite{tsirelson-ibragimov-sudakov-1976},
in the logarithmic-moment and tail forms of
\cite[Section~5.4, Theorems~5.5--5.6]{boucheron-lugosi-massart-2013},
and the variance bound is the case $\nabla^2V\succeq I$ of the Brascamp--Lieb
inequality \cite[Theorem~4.1]{brascamp-lieb-1976}. The proof in
\cite{oai-barriers} covers \eqref{eq:inp-concentration} and the case of linear
$f$, under an upper bound on $\nabla^2V$ that holds in its applications. The
linear case enters this paper only through \cref{inp:ob-energy-tail}.
\usedin{\cref{sec:static}, for the concentration of logarithms of constrained
partition functions, and \cref{sec:references}, in the proof of
\cref{b:lem:coverage-extension}.}

\begin{remark}[Lipschitz constants of logarithms of constrained partition functions]\label{rem:inp-lipschitz}
The Gaussian vector in applications of \eqref{eq:inp-concentration} is
$J=(J_{ij})_{i<j}\in\R^{n(n-1)/2}$. Let $\mathcal S\subset\Sigma_n^\ell$ be
nonempty and $F_{\mathcal S}(J)=\log\sum_{(x^a)\in\mathcal
S}\exp(\beta\sum_{a=1}^\ell H_n^J(x^a))$. Then $\partial F_{\mathcal S}/\partial
J_{ij}=\beta n^{-1/2}\langle\sum_ax^a_ix^a_j\rangle_{\mathcal S}$, where
$\langle\cdot\rangle_{\mathcal S}$ is the Gibbs average on $\mathcal S$. Each
derivative has absolute value at most $\beta\ell/\sqrt n$. This implies that
$F_{\mathcal S}$ is Lipschitz with constant $\beta\ell\sqrt{(n-1)/2}$; this is the bound used in
\cite[proof of Lemma~4.2]{oai-barriers}. It applies to $\log Z_{\ell,n}(D)$ and,
with $\ell=1$ and $\mathcal S=\Sigma_n$, to $F_n$.
\end{remark}

\section{Two- and three-replica pressure bounds}\label{sec:inp-three}

In this section, we assume that $\beta>1$, we write $\alpha=\alpha_\beta$, and we let $Y$
and $\Gamma$ be the diffusion and response of $\alpha_\beta$. The three results below are proved in
\cite[Sections~4 and~7]{oai-barriers} from \cref{inp:ob-matrix}, applied to
perturbations of the paths of \cref{lem:inp-blocks}, and from the scalar
identities of \cref{inp:ob-scalar,inp:ob-stationarity}. We import their pressure
bounds, not the probability bounds that \cite{oai-barriers} derives from them at
its own scales. We convert them into probability bounds at our scales in
\cref{sec:static}.

\begin{proposition}[{Pair pressure bound \cite[Lemma~4.3, (4.4)]{oai-barriers}}]\label{inp:ob-pair-gap}
For every $n\ge2$ and every $q\in[0,1]$ such that $D_2(q)$ is feasible,
\[
 p_{2,n}(D_2(q))\le2P_\beta-\frac12\bigl(\Gamma(q)-q\bigr)^2.
\]
\end{proposition}

By \cref{rem:inp-orientation}, the same bound holds for $q\in[-1,0]$ with
$\Gamma(\abs q)-\abs q$ in place of $\Gamma(q)-q$. The proof in
\cite{oai-barriers} uses the path of \cref{lem:inp-blocks}(c) and the terminal
multiplier $\lambda_{12}=\lambda_{21}=L/2$ with $L=q-\Gamma(q)$. The lemma in
\cite{oai-barriers} also contains a probability bound, which we do not use, for
the overlaps $R$ at which $\abs{\Gamma(\abs{R})-\abs{R}}$ exceeds a fixed
positive tolerance.
\usedin{\cref{sec:static}, for the pair overlaps at which $\abs{q-\Gamma(q)}$ is
not small and for the overlaps near one.}

\begin{proposition}[{Signed fixed-scale pressure bound \cite[(4.20)]{oai-barriers}}]\label{inp:ob-fixed-gap}
For every $b_*\in(0,1)$, there exist $r_0,\delta_0,\nu,c_0>0$, depending only on
$\beta$ and $b_*$, with the following property. For every $n\ge2$ and every
feasible $D(r,q,\delta)$ with
\[
 0\le r\le r_0,\qquad\abs\delta\le\delta_0,\qquad b_*\le q\le1,\qquad
 \abs{q-\Gamma(q)}\le\nu,
\]
we have
\[
 p_{3,n}(D(r,q,\delta))\le3P_\beta-c_0\delta^2.
\]
\end{proposition}

In \cite{oai-barriers} the cutoff $b_*$ is called $h$ and the tolerance $\nu$ is
called $\tau$, and (4.20) is proved in
\cite[Sections~4.2--4.4]{oai-barriers} for nonempty constraints. No condition
$r\ge\abs\delta$ is imposed, and the two small overlaps $r\pm\delta$ may have
opposite signs. Reversing $v$ (\cref{rem:inp-orientation}), we obtain the same
bound for $-r_0\le r\le0$. With the conditions on $q$, the bound holds whenever
$\abs{r+\delta},\abs{r-\delta}\le\min\{r_0,\delta_0\}$, since then $\abs r$ and
$\abs\delta$ are at most this minimum. The perturbation in the proof acts on a
fixed interval $[s_1,s_2]\subset(0,b_*)$, and the gain comes from the strict
inequality $\beta^2(\int_{b_*}^1\alpha(t)\dd t)(\int_0^1\alpha(t)\dd t)<1$,
which follows from \cref{inp:ob-stationarity}(iii). The paper \cite{oai-barriers} converts (4.20)
into the probability bound of its Lemma~4.1, with tolerance $2n^{-1/100}$, using
its Lemma~4.2 and its upper bound $Cn^{-1/24}$ on the mean bias. We use neither of these
lemmas.
% CHECK: OB (4.20), printed p. 30.
\usedin{\cref{sec:static}, for the fixed-scale bound \eqref{b:eq:fixed-pressure-gap},
and through it for the signed overlap constraints of
\cref{sec:references,sec:induction}.}

\begin{proposition}[{Narrow-annulus comparison \cite[Section~7, Steps~1--8, (7.30)--(7.31)]{oai-barriers}}]\label{inp:ob-remote}
Let $t_0>0$ satisfy $3t_0\le1$ and $\Gamma(t)=t$ for $t_0/2<t<3t_0$. Let
$\rho_{\rm OB}=n^{-1/10000}$, $h=\rho_{\rm OB}^8$, $\eps=\rho_{\rm OB}^{20}$,
$\theta=\rho_{\rm OB}^2$, and $W=(2(1-\theta)^2+\theta^2)/h$. There exist $C,c>0$
and $n_0$, depending only on $\beta$ and $t_0$, with the following properties.
\begin{enumerate}[label=\textup{(\roman*)}]
\item For $n\ge n_0$ and every feasible $D(r,q,\delta)$ with $t_0/2<r<3t_0$,
 $\mu_\beta([r-h,r+h])\le\rho_{\rm OB}^4$, $q\ge r+\rho_{\rm OB}$ and
 $\eps\le\abs\delta\le2\eps$,
 \begin{equation}\label{eq:inp-remote}
 \begin{split}
  p_{3,n}(D(r,q,\delta))\le{}&3P_\beta
  +\delta^2\Bigl\{\beta^2W\Bigl(\E\chi(q,Y_q)-\int_q^1\alpha(t)\dd t\Bigr)\\
  &+2(1-\theta)\theta\beta^2\alpha(r)\bigl(\beta^2\E[\chi(q,Y_q)\chi(r,Y_r)]-1\bigr)\\
  &+C(\rho_{\rm OB}^4+\theta^2)\Bigr\}
  +C\bigl(\abs\delta^3h^{-3/2}+\delta^4h^{-2}\bigr).
 \end{split}
 \end{equation}
\item For every $t_0/2<r<3t_0$ and $r\le q\le1$,
 \begin{equation}\label{eq:inp-susceptibility-loss}
  \beta^2\E[\chi(q,Y_q)\chi(r,Y_r)]
  =1-\beta^4\int_r^q\alpha(t)\E[\chi(t,Y_t)^2\chi(r,Y_r)]\dd t
  \le1-c(q-r).
 \end{equation}
\end{enumerate}
\end{proposition}

By \cref{cor:inp-identity}, the hypothesis on $t_0$ holds for every $t_0\in(0,q_\beta/3]$,
and finitely many intervals $(t_0/2,3t_0)$ with such $t_0$ cover any compact
subset of $(0,q_\beta)$. The estimate \eqref{eq:inp-remote} is
\cite[(7.30)]{oai-barriers}, obtained there from the step version
\cite[(7.29)]{oai-barriers} by passing to $\alpha_\beta$ at fixed $n,r,q,\delta$.
It is proved by applying \cref{inp:ob-matrix} with $\lambda=0$ to the path of
\cref{lem:inp-blocks}(b), perturbed as follows. With the notation of \cref{lem:inp-blocks}(d), for step
approximations whose breakpoints contain $r-h$, $r$, $r+h$, and $q$, let
\[
 \mathsf Y_0=\frac{(Y^x_r-Y^x_{r-h})-(Y^x_{r+h}-Y^x_r)}{h},\qquad
 \mathsf Y_1=\frac{Y^v_{r+h}-Y^v_r}{h},\qquad
 \mathsf Y=(1-\theta)\mathsf Y_0+\theta\mathsf Y_1,
\]
as linear combinations of the Gaussian increments \cite[(7.8)]{oai-barriers}.
The $x$ field receives $+\delta$ times, and the $y$ field $-\delta$ times, the
contribution of each increment to $\mathsf Y$. The endpoint
becomes $D(r,q,\delta)+\delta^2W(\mathbf e_x-\mathbf e_y)(\mathbf e_x-\mathbf
e_y)^{\mathsf T}$, and the mismatch cost $\beta^2\delta^4W^2$
\cite[(7.13)]{oai-barriers} is part of the last error term. The coefficient $W$
is denoted $K$ in \cite{oai-barriers}. The identity in
\eqref{eq:inp-susceptibility-loss} follows from \cref{inp:ob-scalar}(iv) and
$\Gamma'(r)=1$, and the inequality is \cite[(7.31)]{oai-barriers}.

\Cref{inp:ob-remote} is a consequence of the proof in
\cite[Section~7]{oai-barriers} rather than a verbatim statement there, and we
record how it is obtained. The setup of \cite[Section~7]{oai-barriers} also
imposes $3t_0<q_*$, with $q_*$ the constant of \cref{inp:ob-coverage}, and,
before Step~1, the restriction $\abs{q-\Gamma(q)}\le\rho_{\rm OB}^{13}$. The
Gaussian calculations through (7.30) use neither restriction. Their geometric
requirements are $0<r-h$ and $r+h<q\le1$, which hold under the hypotheses of
(i) for all sufficiently large $n$, and the pair restriction is needed only to
obtain (7.32). In \cite[Section~7]{oai-barriers}, (7.31) is stated for the
pairs $(r,q)$ of Lemma~7.1, but its proof uses only $t_0/2<r<3t_0$ and
$r\le q\le1$. Indeed, let $M$ be such that the event
$\{\beta\sup_{t\le1}\abs{B_t}+\beta^2\le M\}$ has probability $p_M>0$; on it
$\abs{Y_t}\le M$ for all $t\le1$. With $c_M=\min_{0\le t\le1,\,\abs z\le M}
\chi(t,z)>0$, we have $\E[\chi(t,Y_t)^2\chi(r,Y_r)]\ge p_Mc_M^3$, and
$\alpha(t)\ge\alpha(t_0/2)>0$ for $t\ge r$. We may take
$c=\beta^4\alpha(t_0/2)p_Mc_M^3$. After imposing the additional restriction
$\abs{q-\Gamma(q)}\le\rho_{\rm OB}^{13}$, \cite{oai-barriers} deduces from
\eqref{eq:inp-remote} and \eqref{eq:inp-susceptibility-loss} the deficit
$c\delta^2\rho_{\rm OB}^3$ \cite[(7.32)]{oai-barriers} and the probability bound
of its Lemma~7.1. We use neither.
% CHECK: OB Section 7, (7.29)--(7.31), printed pp. 42--43.
\usedin{\cref{sec:static}, for the comparison at separated overlaps in
\eqref{b:eq:remote-pressure-gap} and \eqref{b:eq:remote-tent-estimate}.}

\begin{remark}[Where the scales enter the proof of \eqref{eq:inp-remote}]\label{rem:inp-remote-scales}
\Cref{sec:static} needs a version of \eqref{eq:inp-remote} with $h$ and $\theta$
fixed, and with the oscillation $\alpha(r+h)-\alpha(r-h)$ in place of
$\rho_{\rm OB}^4$. This version is not stated in \cite{oai-barriers}, and we
derive it in \cref{sec:static} by following Steps~1--7 of
\cite[Section~7]{oai-barriers}. In those steps, the specific scales enter only
at the following places.
\begin{enumerate}[label=\textup{(\alph*)}]
\item Step~1 uses $0<r-h$ and $r+h<q$, so that the triangular function on
 $[r-h,r+h]$ that defines the triangular covariance perturbation vanishes at
 $0$ and at $q$.
\item The errors $O(\rho_{\rm OB}^4)$ in \cite[(7.14), (7.15), (7.20), (7.23)--(7.28)]{oai-barriers}
 come from the oscillation bound $\alpha(r+h)-\alpha(r-h)\le\rho_{\rm OB}^4$
 \cite[(7.6)]{oai-barriers}, which follows from the hypothesis on
 $\mu_\beta([r-h,r+h])$. Each of them is at most a constant times
 $\alpha(r+h)-\alpha(r-h)$. The ramp error is $O(1)$ and is multiplied by
 $\theta^2$.
\item Step~4 uses $\delta^2/h\to0$ only to bound
 $\E[\mathsf Y^4e^{\delta^2\mathsf Y^2}]$ through Gaussian exponential moments, since
 $\mathsf Y$ is a centered Gaussian of variance $\beta^2W$ plus a term bounded by
 a constant depending on $\beta$; a bound $\delta^2/h\le c_\beta$ with $c_\beta>0$
 small, together with $h\le1$, suffices.
\item Once the geometric requirements in (a) and the moment restriction in
 (c) are satisfied, the quantitative lower bound $q-r\ge\rho_{\rm OB}$, the
 restriction $\abs{q-\Gamma(q)}\le\rho_{\rm OB}^{13}$, and the relations
 between the powers of $\rho_{\rm OB}$ that define $h$, $\eps$ and $\theta$ are
 used only in Steps~8 and~9.
\end{enumerate}
At fixed $h$ and $\theta$, the last error term of \eqref{eq:inp-remote} is
absorbed only for small amplitudes, so the fixed-parameter version in
\cref{sec:static} holds under a condition $\abs\delta\le\delta_0$. It must be
derived from Steps~1--7 with the oscillation of $\alpha$ kept explicit; it does
not follow by applying \cref{inp:ob-remote}(i) at fixed $h$ and $\theta$.
\end{remark}

\section{Energy, disorder rotation, and overlap-neighborhood masses}\label{sec:inp-coverage}

In this section, we assume that $\beta>1$ and that $J,\widetilde J$ are independent
standard Gaussian vectors in $\R^{n(n-1)/2}$, indexed by pairs $i<j$. We write $Z_n^J$, $\pi^J$, and $H_n^J$
to indicate the disorder. The three results below, from \cite[Sections~5--6]{oai-barriers},
use the off-diagonal Hamiltonian without the auxiliary diagonal variables.

\begin{proposition}[{Upper energy tail \cite[Lemma~5.1]{oai-barriers}}]\label{inp:ob-energy-tail}
There exist constants $e_1\in(0,\beta/2)$ and $c_E>0$ such that, for all
sufficiently large $n$,
\[
 \PP_J\Bigl(\pi^J\{x\in\Sigma_n:H_n^J(x)>ne_1\}>e^{-c_En}\Bigr)\le e^{-c_En}.
\]
\end{proposition}

The proof in \cite{oai-barriers} combines the strict bound
$n^{-1}\E\pi^J(H_n^J)\le e_0<\beta/2$ for all sufficiently large $n$ of
\cite[Lemma~3.3]{oai-barriers}, which
uses $\beta>1$, with the linear case of the variance bound in
\cref{inp:ob-concentration} for the law of $J$ given a Gibbs sample.
\usedin{\cref{sec:references}, in the proof of \cref{b:lem:coverage-extension}.}

\begin{proposition}[{Disorder rotation \cite[Lemma~5.2 and (5.5)]{oai-barriers}}]\label{inp:ob-rotation}
For $\theta\in(0,\pi/2)$, let $J_\theta=\cos\theta\,J+\sin\theta\,\widetilde J$. For every
$n\ge2$, every $s\in\R$ and every $u>0$,
\[
 \E\exp\bigl\{s\bigl(\log Z_n^{J_\theta}-\log Z_n^J\bigr)\bigr\}
 \le\exp\Bigl\{\frac{s^2\theta^2\beta^2(n-1)}4\Bigr\}
\]
and
\[
 \PP\bigl(\log Z_n^{J_\theta}-\log Z_n^J\ge u\bigr)
 \le\exp\Bigl\{-\frac{u^2}{\theta^2\beta^2(n-1)}\Bigr\}.
\]
If $0<t<1/2$ and $J'=\sqrt{1-t}\,J+\sqrt t\,\widetilde J$, then for every $u>0$,
\[
 \PP\bigl(\log Z_n^{J'}-\log Z_n^J\ge u\bigr)\le\exp\Bigl(-\frac{u^2}{2\beta^2nt}\Bigr).
\]
\end{proposition}

The first two bounds are \cite[(5.3)--(5.4)]{oai-barriers}. The last is
\cite[(5.5)]{oai-barriers}, which follows from the second with
$\theta=\arcsin\sqrt t$, since $\theta^2\le2t$ for $0<t<1/2$. All three bounds
are uniform in $\theta$ and $t$. They also apply to rotations whose
size depends on $n$, and they concern the joint law of $(J,\widetilde J)$.
\usedin{\cref{sec:references}, in the proof of \cref{b:lem:coverage-extension}.}

\begin{proposition}[{Overlap-neighborhood mass bound \cite[Proposition~6.1]{oai-barriers}}]\label{inp:ob-coverage}
Let $L_{\rm OB}=n^{1/10000}$. There exists $q_*\in(0,1)$, depending only on
$\beta$, with the following property. For every fixed $D_1>0$ there exist
$D_3=D_3(\beta,D_1)>0$ and events $\mathcal C_n(D_1)$, measurable with respect to
$J$, such that $\PP_J(\mathcal C_n(D_1))\to1$ and, on $\mathcal C_n(D_1)$, for
every $S\subset\Sigma_n$,
\[
 \pi^J\Bigl(\Bigl\{x\in S:\pi^J\{z\in S:\abs{R(x,z)}\ge q_*\}<e^{-D_3L_{\rm OB}}\Bigr\}\Bigr)
 <e^{-D_1L_{\rm OB}}.
\]
\end{proposition}

Note that both masses in the display are masses under $\pi^J$ itself, not under
$\pi^J$ conditioned on $S$. The proposition, titled ``Stretched-exponential coverage'' in
\cite{oai-barriers}, is stated there at the scale $L_{\rm OB}$ only. Its proof uses this scale only through $L_{\rm
OB}\to\infty$ and $L_{\rm OB}/n\to0$, and $q_*$ is chosen before $D_1$ from the
constant $e_1$ of \cref{inp:ob-energy-tail}. \Cref{b:lem:coverage-extension} in
\cref{sec:references} states and proves the corresponding bound for every
deterministic sequence $L_n\to\infty$ with $L_n=o(n)$, by the argument of
\cite[Section~6]{oai-barriers}, using
\cref{inp:ob-energy-tail,inp:ob-rotation,inp:ob-concentration}.
\usedin{\cref{sec:references}, as the statement extended in
\cref{b:lem:coverage-extension}.}
\part{Static estimates for the lower bound}\label{part:static}
\chapter{The scalar Parisi problem on the support}\label{sec:scalar}

In this section, we collect the properties of the scalar Parisi equation used
in the barrier argument. Recall from \cref{sec:inputs} the solution
$\Phi_\alpha$ of the Parisi equation for a distribution function $\alpha$,
its spatial derivatives $u$, $\chi$, and $w$, the scalar diffusion $Y_t$,
and the function $\Gamma(t)=\E u(t,Y_t)^2$. We first prove two statements
that hold for every distribution function. In \cref{lem:sc-derivatives}, we
bound each positive spatial derivative of $\Phi_\alpha$ by a constant that
depends only on $\beta$ and on the order of the derivative. In
\cref{lem:sc-ito}, we record the stochastic differential equations that these
derivatives satisfy along $Y$. We then specialize to the Parisi distribution
function $\alpha_\beta$ on the interval $(0,q_\beta)$, where $\Gamma(t)=t$.
In \cref{b:lem:os-scalar-jets}, we prove the two stationarity identities
$\beta^2\E\chi(t,Y_t)^2=1$ and $\E w(t,Y_t)^2=2\alpha(t)\E\chi(t,Y_t)^3$,
the formula $\alpha=\E w^2/(2\E\chi^3)$, smoothness and a Lipschitz bound for
$\alpha$, and bounds on mixed derivatives of $\Phi$ on compact subintervals.
The positive constant $m_\beta$ of \eqref{b:eq:main-analytic-constants} is
defined in \cref{cor:sc-mbeta}.

We use three external inputs: \cref{inp:ob-scalar}, on the regularity and
stability of the scalar equation; \cref{inp:full-rsb}, which states that the
support of $\mu_\beta$ is $[0,q_\beta]$; and the identity
\eqref{b:eq:identity-interval}, which states that $\Gamma(t)=t$ for
$0<t<q_\beta$. The derivative bounds are used in
\cref{sec:flow,sec:cubic,sec:fractional}. The stationarity identities enter
the coefficient computation of \cref{sec:cubic}, and the constant $m_\beta$
enters the cubic pressure deficit there and, through it, the choice of
parameters in \cref{sec:fractional,sec:references,sec:barrier-proof}.

The statements of this section are known, some of them in a slightly
different form. We include proofs because the later sections need them with
explicit uniformity. For each fixed distribution function, bounds on all
positive spatial derivatives, and their convergence when the profile is
approximated, were proved by Auffinger and Chen
\cite[Propositions~1--2]{auffinger-chen-2015} (see also
\cite[Proposition~2.4]{full-rsb}). The bounds of order at most three, the
martingale identity \eqref{b:eq:u-martingale}, and the expectation of
\eqref{b:eq:chi-evolution} are proved in \cite[Lemma~3.1]{oai-barriers}. Our
proof of \cref{lem:sc-derivatives} extends the Feynman--Kac argument given
there to every order. The identities \eqref{b:eq:os-jet-generator} with
$j\le3$ and the first two identities of \eqref{eq:sc-moments} appear in
\cite[Lemma~2.5, (2.12)--(2.16)]{full-rsb}; the formulas for $\Gamma'$
given there are stochastic versions of the differentiation formulas of
\cite[Proposition~3]{auffinger-chen-2015}. The identity $\Gamma(q)=q$ at
points of the support goes back to Talagrand
\cite{talagrand-2006-parisi-measures} (see
\cite[Theorem~5 and its proof]{auffinger-chen-2015}); it is part of the
characterization of Parisi measures by Chen \cite[Proposition~1]{chen-2017}
and by Jagannath and Tobasco \cite[Proposition~1.1]{jagannath-tobasco-2017},
and we use it in the form proved in \cite[Proposition~3.2]{oai-barriers}.
Since $\supp\mu_\beta=[0,q_\beta]$ \cite[Theorem~1.1]{full-rsb}, the identity holds on
all of $(0,q_\beta)$. In the proof
of \cite[Theorem~2(ii)]{auffinger-chen-2015}, Auffinger and Chen derive the
quotient formula for $\alpha$ and use an induction to prove its smoothness.
The same quotient argument, including the passage from almost every point
to every point, was carried out near $t=0$ in
\cite[proof of Lemma~7.2, Step~3, (7.8)--(7.12)]{quantitative-parisi}, in a
rescaled normalization. The smoothness of $\alpha_\beta$ on
$(0,q_\beta)$ is also part of \cref{inp:full-rsb} (see
\cref{cor:inp-identity}). The use of the second stationarity identity in the
covariance expansion of \cref{sec:cubic} is new.
% CHECK: Auffinger--Chen arXiv:1303.3573v1 read: Propositions 1--3 (Sections 2--3),
% Theorem 5 (Section 3; its proof credits Talagrand, Parisi measures, Prop. 3.2),
% Theorem 2(ii) with proof in Section 4.2, quotient (40); Lopatto arXiv:2607.11756v3, Proposition 2.4 and
% Lemma 2.5, (2.12)--(2.16), pp. 5--7; Aronow--Lopatto arXiv:2609.09103v1,
% proof of Lemma 7.2, Step 3, (7.8)--(7.12), pp. 63--64.

\paragraph{Notation.} Throughout this section, $\beta>1$ is fixed and
$b=\beta^2$. We use the conventions of \cref{b:sec:scalar}. A distribution
function is a nondecreasing, right-continuous function
$\alpha\colon[0,1]\to[0,1]$ with $\alpha(1)=1$; equivalently,
$\alpha(t)=\mu([0,t])$ for a probability measure $\mu$ on $[0,1]$. Let
$\alpha$ be a step distribution function, with breakpoints
$0=t_0<t_1<\dots<t_p=1$ and $\alpha=m_i$ on $[t_{i-1},t_i)$. Then
$\Phi_\alpha(1,z)=\log(2\cosh z)$, and the Gaussian recursion
\eqref{eq:inp-scalar-recursion} reads
\begin{equation}\label{eq:sc-recursion}
 \Phi_\alpha(t,z)=\frac1{m_i}\log\E\exp\Bigl(m_i\,
   \Phi_\alpha\bigl(t_i,z+\beta\sqrt{t_i-t}\,Z\bigr)\Bigr),
 \qquad t_{i-1}\le t<t_i,
\end{equation}
where $Z$ is a standard Gaussian variable, and where the right side is read
as $\E\Phi_\alpha(t_i,z+\beta\sqrt{t_i-t}\,Z)$ when $m_i=0$. For a
general distribution function $\alpha$, the function $\Phi_\alpha$ is the
uniform limit of $\Phi_{\alpha_N}$ along any sequence of step distribution
functions with $\norm{\alpha_N-\alpha}_{L^1}\to0$ (\cref{inp:ob-scalar}\,(i)). We
write
\[
 f_0=\Phi_\alpha,\qquad f_j=f_j^\alpha=\partial_z^j\Phi_\alpha\quad(j\ge1),
\]
so that $f_1=u$, $f_2=\chi$, and $f_3=w$. For $(t,z)\in[0,1]\times\R$, let
$(Y^{t,z}_r)_{t\le r\le1}$ be the solution of
\[
 \dd Y^{t,z}_r=\beta\dd B_r+b\,\alpha(r)\,u(r,Y^{t,z}_r)\dd r,\qquad
 Y^{t,z}_t=z,
\]
where $B$ is a standard Brownian motion. This equation has a unique strong
solution (\cref{inp:ob-scalar}\,(iii) and (v)), and the scalar diffusion of
\cref{sec:inputs} is $Y=Y^{0,0}$. We write $\norm{\cdot}_\infty$ for the
supremum over $[0,1]\times\R$, or over $\R$ for a function of $z$ alone.
From \cref{inp:ob-scalar}, we use the bounds $\abs u\le1$ and $0<\chi\le1$
(the proof of \cite[Lemma~3.1]{oai-barriers} gives the sharper bound
$\chi\le1-u^2$, which we do not need), the joint continuity of $\Phi_\alpha$,
$u$, and $\chi$, the well-posedness of the scalar diffusion, and the
stability estimate in part (ii) there,
\[
 \sup_{t,z}\bigl(\abs{\Phi_\alpha-\Phi_{\tilde\alpha}}
   +\abs{u_\alpha-u_{\tilde\alpha}}+\abs{\chi_\alpha-\chi_{\tilde\alpha}}\bigr)
 \le C_\beta\norm{\alpha-\tilde\alpha}_{L^1},
\]
which holds for all distribution functions $\alpha$ and $\tilde\alpha$.

\section{Bounds on spatial derivatives}\label{sec:sc-derivatives}

The derivative bounds below are uniform over all distribution functions. We
need this uniformity because the later sections apply the bounds to step
approximations and to profiles other than $\alpha_\beta$, such as
$\max\{\alpha_\beta,3s\}$ with $0\le s\le1/3$. The constants are defined by $C_1=C_2=1$ and, for
$j\ge3$,
\begin{equation}\label{eq:sc-constants}
 C_j=e^{jb}\Bigl(\norm{\partial_z^j\log(2\cosh z)}_\infty
   +\frac b2\sum_{a=2}^{j-2}\binom ja C_{a+1}C_{j+1-a}\Bigr).
\end{equation}
The sum is empty for $j=3$, and it involves only $C_3,\dots,C_{j-1}$. Every
derivative of positive order of $\log(2\cosh z)$ is a polynomial in
$\tanh z$ and is bounded.

\begin{lemma}[Bounds on spatial derivatives]\label{lem:sc-derivatives}
Let $\alpha$ be a distribution function. For all $j\ge1$, the derivative
$f_j=\partial_z^j\Phi_\alpha$ exists and is continuous on $[0,1]\times\R$,
and
\begin{equation}\label{eq:sc-derivative-bounds}
 \sup_{(t,z)\in[0,1]\times\R}\abs{\partial_z^j\Phi_\alpha(t,z)}\le C_j.
\end{equation}
Moreover, the following hold.
\begin{enumerate}[label=\textup{(\roman*)}]
\item If $\alpha_N$ are distribution functions such that
  $\norm{\alpha_N-\alpha}_{L^1}\to0$, then
  $\norm{\partial_z^j\Phi_{\alpha_N}-\partial_z^j\Phi_\alpha}_\infty\to0$ for
  all $j\ge1$.
\item For all $j\ge0$, $t\in[0,1]$, and $z\in\R$,
  \begin{equation}\label{eq:sc-integrated-pde}
   f_j(t,z)=\partial_z^j\log(2\cosh z)
     +\frac b2\int_t^1\bigl(f_{j+2}+\alpha\,\partial_z^j(u^2)\bigr)(r,z)\dd r,
  \end{equation}
  where
  $\partial_z^j(u^2)=\sum_{a=0}^j\binom ja f_{a+1}f_{j+1-a}$.
\end{enumerate}
\end{lemma}

The bounds concern derivatives of positive order only; the function
$\Phi_\alpha$ itself grows linearly in $z$. By \eqref{eq:sc-integrated-pde},
the Parisi equation and its spatial derivatives hold for almost every $t$,
and at every $t$ at which $\alpha$ is continuous.

\begin{proof}
We first recall why, for a fixed step distribution function, all spatial
derivatives exist and are bounded, with bounds that may depend on the steps.
We then write the equation for $f_j$ as a linear equation and use its
Feynman--Kac representation (\cite[Theorem~5.7.6]{karatzas-shreve-1991}, after
an exponential rescaling for a bounded potential) to obtain bounds independent
of the steps, extending the third-derivative argument in
\cite[proof of Lemma~3.1]{oai-barriers}.
Finally, we pass to general distribution functions by a forward difference
argument and prove \eqref{eq:sc-integrated-pde}.

\paragraph{Step 1: a fixed step distribution function.}
Let $\alpha$ be a step distribution function as in \eqref{eq:sc-recursion},
fix $i$, and let $\phi=\Phi_\alpha(t_i,\cdot)$. Suppose that $\phi$ is
smooth, that all its derivatives of positive order are bounded, and that
$\abs{\phi(z)}\le c(1+\abs z)$ for some $c$. These properties hold for
$\phi(z)=\log(2\cosh z)$, which is the case $i=p$. For
$t_{i-1}\le t\le t_i$, the function $\Phi_\alpha(t,\cdot)$ is $1/m_i$ times
the logarithm of the convolution of $e^{m_i\phi}$ with the centered Gaussian
density of variance $b(t_i-t)$, or the convolution of $\phi$ itself when
$m_i=0$. The linear growth of $\phi$ justifies differentiation under the
integral. We find that, for $j\ge1$, $f_j(t,z)$ is a polynomial, with
coefficients depending only on $j$ and $m_i$, in expectations of products of
$\phi'(z+\sigma Z),\dots,\phi^{(j)}(z+\sigma Z)$ under the probability
measure with density proportional to $e^{m_i\phi(z+\sigma Z)}$, where
$\sigma=\beta\sqrt{t_i-t}$. In particular, $f_j$ is bounded and
continuous on $[t_{i-1},t_i]\times\R$. For $t<t_i$, the Gaussian
convolution is smooth in $(t,z)$, and $e^{m_i\Phi_\alpha}$ (or
$\Phi_\alpha$ itself, when $m_i=0$) solves the backward heat equation
$-\partial_tv=(b/2)\partial_{zz}v$. It follows that
$\Phi_\alpha$ is smooth on $[t_{i-1},t_i)\times\R$, where it satisfies
\[
 -\partial_t\Phi_\alpha=\frac b2\bigl(\partial_{zz}\Phi_\alpha
   +m_i(\partial_z\Phi_\alpha)^2\bigr).
\]
The function $\Phi_\alpha(t_{i-1},\cdot)$ has the three properties assumed for
$\phi$ (it grows at most linearly because $\abs u\le1$). Proceeding backward
through the steps, we obtain these properties on every strip; this argument
is recorded in the proof of \cite[Lemma~3.1]{oai-barriers}. The bounds it gives depend on the number and
the location of the steps, and the next two steps remove this dependence.

\paragraph{Step 2: the differentiated equation.}
Differentiating the equation $j$ times in $z$ on each strip
$[t_{i-1},t_i)\times\R$, we find that
\begin{equation}\label{eq:sc-differentiated}
 -\partial_tf_j=\frac b2\bigl(f_{j+2}+\alpha\,\partial_z^j(u^2)\bigr)
 \qquad(j\ge0),
\end{equation}
with $\partial_z^j(u^2)$ as in \eqref{eq:sc-integrated-pde}. Let $j\ge3$. In
the sum defining $\partial_z^j(u^2)$, the terms with $a\in\{0,j\}$ add up to
$2uf_{j+1}$, and the terms with $a\in\{1,j-1\}$ add up to $2j\chi f_j$.
Then \eqref{eq:sc-differentiated} becomes
\begin{equation}\label{eq:sc-linear}
 -\partial_tf_j=\frac b2\partial_{zz}f_j+b\alpha u\,\partial_zf_j
   +jb\alpha\chi f_j+\mathcal Q_j,\qquad
 \mathcal Q_j=\frac{b\alpha}2\sum_{a=2}^{j-2}\binom ja f_{a+1}f_{j+1-a}.
\end{equation}
The sum defining $\mathcal Q_j$ is empty for $j=3$, and its factors are derivatives of
orders between $3$ and $j-1$. We regard \eqref{eq:sc-linear} as a linear
equation for $f_j$, with coefficients $u$ and $\chi$ and source term $\mathcal Q_j$.

\paragraph{Step 3: bounds independent of the steps.}
We prove \eqref{eq:sc-derivative-bounds} for step distribution functions by
induction on $j$. For $j\in\{1,2\}$, it is the bound $\abs u\le1$,
$0<\chi\le1$ of \cref{inp:ob-scalar}. Let $j\ge3$, and suppose that
$\norm{f_l}_\infty\le C_l$ for $1\le l\le j-1$. Since $0\le\alpha\le1$, we
have
\[
 \norm{\mathcal Q_j}_\infty\le\frac b2\sum_{a=2}^{j-2}\binom ja C_{a+1}C_{j+1-a}.
\]
Fix $(t,z)\in[0,1)\times\R$ and let $Y_r=Y^{t,z}_r$. We abbreviate
\[
 \Xi_r=jb\int_t^r\alpha(r')\chi(r',Y_{r'})\dd r',\qquad t\le r\le1,
\]
so that $0\le\Xi_r\le jb$. By \eqref{eq:sc-linear}, the drift of
$f_j(r,Y_r)$ in It\^o's formula is
\[
 \partial_rf_j+b\alpha u\,\partial_zf_j+\frac b2\partial_{zz}f_j
 =-jb\alpha\chi f_j-\mathcal Q_j,
\]
evaluated at $(r,Y_r)$. On each interval $[t_{i-1},t_i)\cap[t,1]$, we then
have
\[
 \dd\bigl(e^{\Xi_r}f_j(r,Y_r)\bigr)
 =-e^{\Xi_r}\mathcal Q_j(r,Y_r)\dd r
   +\beta e^{\Xi_r}f_{j+1}(r,Y_r)\dd B_r.
\]
Recall from Step~1 that the functions $f_j$ and $f_{j+1}$ are bounded and
continuous on $[0,1]\times\R$. The process
\[
 e^{\Xi_r}f_j(r,Y_r)
   +\int_t^re^{\Xi_{r'}}\mathcal Q_j(r',Y_{r'})\dd r',\qquad t\le r\le1,
\]
is therefore continuous, and it is a martingale. Comparing its expectations
at $r=t$ and $r=1$, we obtain
\[
 f_j(t,z)=\E\Bigl[e^{\Xi_1}\,\partial_z^j\log(2\cosh z)\big|_{z=Y_1}
   +\int_t^1e^{\Xi_r}\mathcal Q_j(r,Y_r)\dd r\Bigr].
\]
Since $0\le\Xi_r\le jb$ for $t\le r\le1$, we have
\[
 \abs{f_j(t,z)}\le e^{jb}\bigl(\norm{\partial_z^j\log(2\cosh z)}_\infty
   +\norm{\mathcal Q_j}_\infty\bigr)\le C_j.
\]
The same bound holds at $t=1$, which completes the induction. The
elapsed time is at most one, and the only property of $\alpha$ we used is
$0\le\alpha\le1$, so the constants depend neither on the number
nor on the location of the steps.

\paragraph{Step 4: general distribution functions.}
Let $\alpha$ be a distribution function, and let $\alpha_N$ be step
distribution functions such that $\norm{\alpha_N-\alpha}_{L^1}\to0$; for
instance, $\alpha_N(t)=\alpha((k+1)/N)$ for $k/N\le t<(k+1)/N$ and
$\alpha_N(1)=1$, which converges to $\alpha$ pointwise by right continuity.
We write $f_j^N=f_j^{\alpha_N}$. The stability estimate of
\cref{inp:ob-scalar} implies that $\Phi_{\alpha_N}$, $f_1^N$, and $f_2^N$
converge uniformly to $\Phi_\alpha$, $u$, and $\chi$. Let $j\ge3$, and suppose that
$f_{j-1}^N$ converges uniformly to $f_{j-1}=\partial_z^{j-1}\Phi_\alpha$,
which is continuous. For $\zeta>0$, Taylor's theorem and Step~3 imply
that
\[
 \Bigl|f_j^N(t,z)-\frac{f_{j-1}^N(t,z+\zeta)-f_{j-1}^N(t,z)}\zeta\Bigr|
 \le\frac{C_{j+1}\zeta}2,
\]
and that
\[
 \norm{f_j^N-f_j^M}_\infty
 \le\frac2\zeta\norm{f_{j-1}^N-f_{j-1}^M}_\infty+C_{j+1}\zeta.
\]
Given $\eps>0$, we first choose $\zeta$ with $C_{j+1}\zeta<\eps/2$, and then
$N_0$
such that the first term is less than $\eps/2$ for $N,M\ge N_0$. Then
$f_j^N$ converges uniformly to a continuous function $\widehat f_j$ with
$\norm{\widehat f_j}_\infty\le C_j$. Letting $N\to\infty$ in
$f_{j-1}^N(t,z)=f_{j-1}^N(t,0)+\int_0^zf_j^N(t,y)\dd y$, we find that
$\partial_zf_{j-1}$ exists and equals $\widehat f_j$. By induction on $j$, this proves
the existence and continuity of $f_j$, the bound
\eqref{eq:sc-derivative-bounds}, and the uniform convergence
$f_j^N\to f_j$ for all $j\ge1$. Now that \eqref{eq:sc-derivative-bounds}
holds for all distribution functions, the same argument applies to any
sequence of distribution functions $\alpha_N$ with
$\norm{\alpha_N-\alpha}_{L^1}\to0$, step functions or not. This proves (i).

\paragraph{Step 5: the integrated equation.}
Let $\alpha$ be a step distribution function and $j\ge0$. For fixed $z$, the
function $t\mapsto f_j(t,z)$ is continuous on $[0,1]$ by Step~1, and
continuously differentiable on each interval $[t_{i-1},t_i)$, with derivative
given by \eqref{eq:sc-differentiated}. Integrating over the steps, we obtain
\eqref{eq:sc-integrated-pde}. For a general $\alpha$, we choose $\alpha_N$ as
in Step~4 and write $u_N=f_1^N$. Then $\Phi_{\alpha_N}\to\Phi_\alpha$ and
$f_l^N\to f_l$ uniformly for all $l\ge1$, and
\[
 \Bigl|\int_t^1\bigl(\alpha_N\partial_z^j(u_N^2)-\alpha\,\partial_z^j(u^2)\bigr)
   (r,z)\dd r\Bigr|
 \le\norm{\alpha_N-\alpha}_{L^1}\norm{\partial_z^j(u_N^2)}_\infty
   +\norm{\partial_z^j(u_N^2)-\partial_z^j(u^2)}_\infty.
\]
The right side tends to zero by \eqref{eq:sc-derivative-bounds} and the
uniform convergence. Letting $N\to\infty$ in \eqref{eq:sc-integrated-pde}
for $\alpha_N$ yields \eqref{eq:sc-integrated-pde} for $\alpha$, which
completes the proof.
\end{proof}

\section{Stochastic identities along the scalar diffusion}
\label{sec:sc-ito}

We now record the equations satisfied by the spatial derivatives of
$\Phi_\alpha$ along the scalar diffusion. As usual, a differential identity
such as \eqref{b:eq:os-jet-generator} below means that the corresponding
integrated identity holds almost surely, simultaneously for all
$0\le s\le t\le1$. For a polynomial $\Theta$ in the variables
$x_1,\dots,x_k$, we define the polynomials in $x_1,\dots,x_{k+1}$
\begin{equation}\label{eq:sc-generator}
 \Theta^0=\frac b2\sum_{i,l=1}^kx_{i+1}x_{l+1}\,\partial_i\partial_l\Theta,
 \qquad
 \Theta^1=-b\sum_{i=1}^k\partial_i\Theta
   \sum_{a=1}^{i-1}\binom{i-1}ax_{a+1}x_{i+1-a},
\end{equation}
where $\partial_i$ is the derivative in $x_i$. We write
$\Theta_t=\Theta(f_1(t,Y_t),\dots,f_k(t,Y_t))$, and $\Theta^0_t$ and
$\Theta^1_t$ for the analogous expressions with $k+1$ arguments.

\begin{lemma}[Stochastic identities along the scalar diffusion]
\label{lem:sc-ito}
Let $\alpha$ be a distribution function, and let $Y=Y^{0,0}$. Write
$u_t=u(t,Y_t)$, $\chi_t=\chi(t,Y_t)$, and $w_t=w(t,Y_t)$.
\begin{enumerate}[label=\textup{(\roman*)}]
\item For all $j\ge1$,
  \begin{equation}\label{b:eq:os-jet-generator}
   \dd f_j(t,Y_t)=\beta f_{j+1}(t,Y_t)\dd B_t
     -b\,\alpha(t)\sum_{a=1}^{j-1}\binom{j-1}a
       f_{a+1}(t,Y_t)f_{j+1-a}(t,Y_t)\dd t.
  \end{equation}
  In particular,
  \begin{align}
   \dd u_t&=\beta\chi_t\dd B_t,\label{b:eq:u-martingale}\\
   \dd\chi_t&=\beta w_t\dd B_t-b\,\alpha(t)\chi_t^2\dd t.
     \label{b:eq:chi-evolution}
  \end{align}
\item Let $\Theta$ be a polynomial in $x_1,\dots,x_k$. Then
  $t\mapsto\E\Theta_t$ is Lipschitz on $[0,1]$, and for
  $0\le s\le t\le1$,
  \begin{equation}\label{eq:sc-moment-identity}
   \E\Theta_t-\E\Theta_s
   =\int_s^t\bigl(\E\Theta^0_r+\alpha(r)\E\Theta^1_r\bigr)\dd r.
  \end{equation}
\item For $0\le s\le t\le1$,
  \begin{equation}\label{eq:sc-moments}
  \begin{aligned}
   \Gamma(t)-\Gamma(s)&=b\int_s^t\E\chi_r^2\dd r,\\
   \E\chi_t^2-\E\chi_s^2
     &=b\int_s^t\bigl(\E w_r^2-2\alpha(r)\E\chi_r^3\bigr)\dd r,\\
   \E w_t^2-\E w_s^2
     &=b\int_s^t\E\bigl(f_4(r,Y_r)^2-6\alpha(r)\chi_rw_r^2\bigr)\dd r,\\
   \E\chi_t^3-\E\chi_s^3
     &=3b\int_s^t\E\bigl(\chi_rw_r^2-\alpha(r)\chi_r^4\bigr)\dd r.
  \end{aligned}
  \end{equation}
  In particular, $\Gamma$ is continuously differentiable with
  $\Gamma'(t)=b\E\chi_t^2$. The function $t\mapsto\E\chi_t^2$ has a right
  derivative at every $t\in[0,1)$, and a derivative at every point of
  continuity of $\alpha$, given by
  \begin{equation}\label{b:eq:contact-derivative}
   \frac{\dd}{\dd t}\E\chi_t^2=b\bigl(\E w_t^2-2\alpha(t)\E\chi_t^3\bigr).
  \end{equation}
\end{enumerate}
\end{lemma}

Parts (i) and (ii) hold, with the same proof, for the diffusion $Y^{t,z}$
started from any $(t,z)\in[0,1]\times\R$, on the time interval $[t,1]$.

\begin{proof}
We prove (i) first for step distribution functions, for which the equation
holds classically between the breakpoints, and then pass to the limit along
a coupling of the diffusions. Parts (ii) and (iii) then follow from It\^o's
formula \cite[Theorem~3.3.6]{karatzas-shreve-1991}.

\paragraph{Step 1: step distribution functions.}
Let $\alpha$ be a step distribution function. By Step~1 of the proof of
\cref{lem:sc-derivatives}, each $f_j$ is smooth on the strips
$[t_{i-1},t_i)\times\R$, bounded, and continuous on $[0,1]\times\R$. On each
strip, It\^o's formula shows that the martingale part of $f_j(t,Y_t)$ is
$\beta f_{j+1}(t,Y_t)\dd B_t$. Using \eqref{eq:sc-differentiated}, we find
that its drift is
\[
 \partial_tf_j+b\alpha u f_{j+1}+\frac b2f_{j+2}
 =-\frac{b\alpha}2\sum_{a=0}^j\binom ja f_{a+1}f_{j+1-a}+b\alpha uf_{j+1}
 =-\frac{b\alpha}2\sum_{a=1}^{j-1}\binom ja f_{a+1}f_{j+1-a},
\]
evaluated at $(t,Y_t)$. In the second equality, we used that the terms with
$a\in\{0,j\}$ equal $2uf_{j+1}$. By Pascal's rule
$\binom ja=\binom{j-1}a+\binom{j-1}{a-1}$ and the substitution
$a\mapsto j-a$ in the sum containing $\binom{j-1}{a-1}$, we have
\[
 \sum_{a=1}^{j-1}\binom ja f_{a+1}f_{j+1-a}
 =2\sum_{a=1}^{j-1}\binom{j-1}a f_{a+1}f_{j+1-a}.
\]
This proves \eqref{b:eq:os-jet-generator} on each interval $[t_{i-1},t_i)$.
The process $f_j(t,Y_t)$ is continuous in $t$, and the integrated identities
on the successive intervals add up to \eqref{b:eq:os-jet-generator} on
$[0,1]$.

\paragraph{Step 2: general distribution functions.}
Let $\alpha$ be a distribution function, let $\alpha_N$ be step distribution
functions such that $\norm{\alpha_N-\alpha}_{L^1}\to0$, and let $Y^N$ be the
scalar diffusion of $\alpha_N$, driven by the same Brownian motion $B$ as
$Y$. Let $u_N=f_1^{\alpha_N}$. Recall that $\abs{u_N}\le1$, and that $u$ is
$1$-Lipschitz in $z$ because $\abs\chi\le1$. Then
\[
 \abs{Y^N_t-Y_t}\le b\int_0^t\abs{\alpha_N(r)-\alpha(r)}\dd r
   +b\norm{u_N-u}_\infty+b\int_0^t\abs{Y^N_r-Y_r}\dd r ,
\]
and Gronwall's inequality yields
\[
 \sup_{0\le t\le1}\abs{Y^N_t-Y_t}
 \le be^b\bigl(\norm{\alpha_N-\alpha}_{L^1}+\norm{u_N-u}_\infty\bigr)
\]
for every realization of $B$. The right side tends to zero by the stability
estimate of \cref{inp:ob-scalar}. Using \cref{lem:sc-derivatives}, we find
that, for each $l\ge1$,
\[
 \abs{f_l^{\alpha_N}(t,Y^N_t)-f_l(t,Y_t)}
 \le\norm{f_l^{\alpha_N}-f_l}_\infty+C_{l+1}\abs{Y^N_t-Y_t},
\]
which tends to zero uniformly in $t$ and in the realization. We now let
$N\to\infty$ in \eqref{b:eq:os-jet-generator} for $\alpha_N$, in its
integrated form between fixed times $s\le t$. The terms $f_j(t,Y_t)$ and
$f_j(s,Y_s)$ converge uniformly. The drift integrals also converge
uniformly, because their integrands are bounded by
\cref{lem:sc-derivatives} and converge uniformly, and
$\norm{\alpha_N-\alpha}_{L^1}\to0$. By It\^o's isometry, the stochastic
integrals converge in $L^2$. We conclude that \eqref{b:eq:os-jet-generator}
holds almost surely for each fixed $s\le t$. Since both sides are continuous
in $(s,t)$, it holds almost surely for all $s\le t$ simultaneously. The cases
$j=1$ and $j=2$ are \eqref{b:eq:u-martingale} and
\eqref{b:eq:chi-evolution}.

\paragraph{Step 3: polynomial moments.}
It follows from (i) that the processes $f_1(t,Y_t),\dots,f_{k+1}(t,Y_t)$ are
continuous semimartingales with
$\dd\langle f_i(\cdot,Y),f_l(\cdot,Y)\rangle_t
 =b\,f_{i+1}(t,Y_t)f_{l+1}(t,Y_t)\dd t$. By It\^o's formula, we have
\[
 \dd\Theta_t=\bigl(\Theta^0_t+\alpha(t)\Theta^1_t\bigr)\dd t
   +\beta\sum_{i=1}^k\partial_i\Theta\bigl(f_1(t,Y_t),\dots,f_k(t,Y_t)\bigr)
     f_{i+1}(t,Y_t)\dd B_t.
\]
Here the second-order term of It\^o's formula produces $\Theta^0_t$, and the
drifts in \eqref{b:eq:os-jet-generator} produce $\alpha(t)\Theta^1_t$. By
\cref{lem:sc-derivatives}, the integrand of the stochastic integral is
bounded. Taking expectations, we obtain \eqref{eq:sc-moment-identity}. The
integrand in \eqref{eq:sc-moment-identity} is bounded by the supremum of
$\abs{\Theta^0}+\abs{\Theta^1}$ over the box $\prod_{i\le k+1}[-C_i,C_i]$,
which proves the Lipschitz bound.

\paragraph{Step 4: four moments.}
For $\Theta=x_1^2$, $x_2^2$, $x_3^2$, and $x_2^3$, the definition
\eqref{eq:sc-generator} gives
\[
\begin{aligned}
 (x_1^2)^0&=bx_2^2,&(x_1^2)^1&=0,\\
 (x_2^2)^0&=bx_3^2,&(x_2^2)^1&=-2bx_2^3,\\
 (x_3^2)^0&=bx_4^2,&(x_3^2)^1&=-6bx_2x_3^2,\\
 (x_2^3)^0&=3bx_2x_3^2,&(x_2^3)^1&=-3bx_2^4.
\end{aligned}
\]
For example, for $\Theta=x_3^2$ we have $\Theta^0=(b/2)\cdot2x_4^2$ and
$\Theta^1=-b\cdot2x_3\bigl(\binom21x_2x_3+\binom22x_3x_2\bigr)$. Inserting
these into \eqref{eq:sc-moment-identity}, we obtain \eqref{eq:sc-moments}.
The integrand $\E\chi_r^2$ in the first identity is continuous in $r$ by
(ii), and the first identity implies that $\Gamma$ is continuously
differentiable with $\Gamma'=b\E\chi^2$. In the second identity, the
functions $r\mapsto\E w_r^2$ and $r\mapsto\E\chi_r^3$ are continuous by (ii),
and $\alpha$ is right-continuous. The integrand is therefore right-continuous,
and it is continuous at every point of continuity of $\alpha$. The
fundamental theorem of calculus now gives \eqref{b:eq:contact-derivative}.
\end{proof}

\section{The Parisi distribution function on \texorpdfstring{$(0,q_\beta)$}{(0,q)}}
\label{sec:sc-support}

From now on, $\alpha=\alpha_\beta$ is the Parisi distribution function, which
is unique by \cref{inp:unique}, $\mu_\beta$ is the Parisi measure, $Y$ is the
associated scalar diffusion, and $u_t$, $\chi_t$, $w_t$ are as in
\cref{lem:sc-ito}. By \cref{inp:full-rsb}, $\supp\mu_\beta=[0,q_\beta]$ with
$0<q_\beta<1$, and $\mu_\beta$ has an atom at $q_\beta$. We set
\begin{equation}\label{eq:sc-lipschitz-constant}
 K_\beta=\frac{\beta^5}2\bigl(C_4^2+12C_3^2+6\bigr),
\end{equation}
with $C_3$ and $C_4$ as in \eqref{eq:sc-constants}.

\begin{lemma}[The Parisi distribution function on the support]
\label{b:lem:os-scalar-jets}
The following hold.
\begin{enumerate}[label=\textup{(\roman*)}]
\item For all $t\in(0,q_\beta)$,
  % NOTE: second label via the kernel \label saved by amsmath (amsmath forbids
  % two \label's in one display); cleveref also records b:eq:os-contacts@cref.
  \begin{equation}\label{b:eq:contacts}\csname ltx@label\endcsname{b:eq:os-contacts}
   \beta^2\E\chi(t,Y_t)^2=1,\qquad
   \E w(t,Y_t)^2=2\alpha(t)\E\chi(t,Y_t)^3.
  \end{equation}
\item For all $t\in(0,q_\beta)$, we have $\E\chi_t^3\ge\beta^{-3}$ and
  \begin{equation}\label{eq:sc-quotient}
   \alpha(t)=\frac{\E w_t^2}{2\E\chi_t^3}.
  \end{equation}
\item We have $\alpha(t)>0$ for all $t\in(0,1]$. For
  $0<s\le t<q_\beta$,
  \[
   0\le\alpha(t)-\alpha(s)=\mu_\beta((s,t])\le K_\beta(t-s).
  \]
  In particular, $\mu_\beta$ has no atoms in $(0,q_\beta)$.
\item The function $\alpha$ is infinitely differentiable on $(0,q_\beta)$.
  For every polynomial $\Theta$, the function $t\mapsto\E\Theta_t$ of
  \cref{lem:sc-ito} is infinitely differentiable on $(0,q_\beta)$. In
  particular, $t\mapsto\E\chi_t^3$ and
  $t\mapsto\alpha(t)^2\beta^6\E\chi_t^3$ are infinitely differentiable on
  $(0,q_\beta)$.
\item The function $\Phi_\alpha$ is infinitely differentiable on
  $(0,q_\beta)\times\R$. For every compact interval $I\subset(0,q_\beta)$
  and all integers $i,j\ge0$ with $i+j\ge1$, there exists
  $C(\beta,I,i,j)<\infty$ such that
  \[
   \sup_{t\in I}\abs{\alpha^{(i)}(t)}
   +\sup_{(t,z)\in I\times\R}\abs{\partial_t^i\partial_z^j\Phi_\alpha(t,z)}
   \le C(\beta,I,i,j).
  \]
\end{enumerate}
\end{lemma}

The proof of (i) uses the identity $\Gamma(t)=t$, which is a property of
$\alpha_\beta$ itself. The step distribution functions used in the proof of
\cref{lem:sc-ito} do not satisfy it, and this is why we established the
identities of \cref{lem:sc-ito} for every distribution function first.

\begin{proof}
We first obtain the two identities in (i): the first at every point, from
$\Gamma(t)=t$, and the second at almost every point, from the constancy of
$\E\chi_t^2$. Since $\alpha$ is right-continuous, the second identity then
holds at every point, which gives the quotient formula
\eqref{eq:sc-quotient}. Using the quotient formula and the Lipschitz bounds of
\cref{lem:sc-ito}, we show that $\alpha$ is Lipschitz, and an induction on the
order of differentiability gives its smoothness. The
bounds on mixed derivatives then follow from the differentiated Parisi
equation.

\paragraph{Step 1: the first identity.}
The first identity is also stated in \cref{cor:inp-identity}; we recall the
argument. By \eqref{b:eq:identity-interval}, $\Gamma(t)=t$ for
$0<t<q_\beta$. By \cref{lem:sc-ito}\,(iii), $\Gamma$ is continuously
differentiable with $\Gamma'(t)=b\E\chi_t^2$. Differentiating, we find that
$b\E\chi_t^2=\Gamma'(t)=1$ for all $t\in(0,q_\beta)$, which is the first
identity in \eqref{b:eq:contacts}.

\paragraph{Step 2: the second identity almost everywhere.}
By Step~1, $\E\chi_t^2=\beta^{-2}$ is constant on $(0,q_\beta)$. The
second identity in \eqref{eq:sc-moments} implies that, for $0<s<t<q_\beta$,
\[
 0=\E\chi_t^2-\E\chi_s^2
 =b\int_s^t\bigl(\E w_r^2-2\alpha(r)\E\chi_r^3\bigr)\dd r.
\]
The integral of the bounded function
$r\mapsto\E w_r^2-2\alpha(r)\E\chi_r^3$ vanishes over every subinterval of
$(0,q_\beta)$. Lebesgue's differentiation theorem then implies that
$\E w_r^2=2\alpha(r)\E\chi_r^3$ for almost every $r\in(0,q_\beta)$.

\paragraph{Step 3: the second identity at every point, and the quotient.}
Let $t\in(0,q_\beta)$. By Step~2, there exist $r_n\downarrow t$ such that
$\E w_{r_n}^2=2\alpha(r_n)\E\chi_{r_n}^3$ for all $n$. Recall from \cref{lem:sc-ito}\,(ii) that the
functions $r\mapsto\E w_r^2$ and $r\mapsto\E\chi_r^3$ are continuous, and
that $\alpha$ is right-continuous. Letting $n\to\infty$, we find that
$\E w_t^2=2\alpha(t)\E\chi_t^3$. This proves (i). Since $\chi_t>0$ and
$x\mapsto x^{3/2}$ is convex on $[0,\infty)$, Jensen's inequality and Step~1
imply that
\[
 \E\chi_t^3=\E\bigl[(\chi_t^2)^{3/2}\bigr]\ge\bigl(\E\chi_t^2\bigr)^{3/2}
 =\beta^{-3}.
\]
Dividing the second identity in \eqref{b:eq:contacts} by $2\E\chi_t^3$, we
obtain \eqref{eq:sc-quotient}, which proves (ii).

\paragraph{Step 4: positivity and the Lipschitz bound.}
Since $0\in\supp\mu_\beta$, every interval $[0,t]$ with $t>0$ has
positive $\mu_\beta$-mass, so $\alpha(t)>0$ for $t\in(0,1]$. By
\cref{lem:sc-derivatives}, we have $0<\chi\le1$, $\abs w\le C_3$, and
$\abs{f_4}\le C_4$. Inserting these bounds into the last two identities in
\eqref{eq:sc-moments}, we find that, for all $s,t\in[0,1]$,
\begin{equation}\label{eq:sc-moment-lipschitz}
 \abs{\E w_t^2-\E w_s^2}\le b\bigl(C_4^2+6C_3^2\bigr)\abs{t-s},\qquad
 \abs{\E\chi_t^3-\E\chi_s^3}\le3b\bigl(C_3^2+1\bigr)\abs{t-s}.
\end{equation}
Let $0<s\le t<q_\beta$. By \eqref{eq:sc-quotient} at $t$ and the identity
$\E w_s^2=2\alpha(s)\E\chi_s^3$ at $s$, we have
\[
 \alpha(t)-\alpha(s)
 =\frac{\E w_t^2-\E w_s^2}{2\E\chi_t^3}
   -\alpha(s)\,\frac{\E\chi_t^3-\E\chi_s^3}{\E\chi_t^3}.
\]
Using \eqref{eq:sc-moment-lipschitz}, $\E\chi_t^3\ge\beta^{-3}$, and
$0\le\alpha(s)\le1$, we conclude that
\[
 \abs{\alpha(t)-\alpha(s)}
 \le\beta^3\Bigl(\frac{b}{2}\bigl(C_4^2+6C_3^2\bigr)
   +3b\bigl(C_3^2+1\bigr)\Bigr)(t-s)=K_\beta(t-s).
\]
Since $\alpha$ is nondecreasing and $\alpha(t)-\alpha(s)=\mu_\beta((s,t])$,
this proves (iii); the continuity of $\alpha$ on $(0,q_\beta)$ excludes
atoms there.

\paragraph{Step 5: smoothness of $\alpha$ and of the moments.}
Let $I_0=(0,q_\beta)$. We prove by induction on $i\ge0$ that
$\alpha\in C^i(I_0)$ and that $t\mapsto\E\Theta_t$ belongs to $C^{i+1}(I_0)$
for every polynomial $\Theta$. By Step~4, $\alpha$ is continuous on $I_0$.
Then the integrand $r\mapsto\E\Theta^0_r+\alpha(r)\E\Theta^1_r$ in
\eqref{eq:sc-moment-identity} is continuous on $I_0$, and $\E\Theta_t$ is
continuously differentiable on $I_0$ by the fundamental theorem of calculus,
with
\begin{equation}\label{eq:sc-moment-derivative}
 \frac{\dd}{\dd t}\E\Theta_t=\E\Theta^0_t+\alpha(t)\E\Theta^1_t
 \qquad(t\in I_0).
\end{equation}
This proves the claim for $i=0$. Suppose that it holds for some $i\ge0$.
Since the moments $\E w_t^2$ and $\E\chi_t^3$ belong to $C^{i+1}(I_0)$ and
$\E\chi_t^3\ge\beta^{-3}$, the formula \eqref{eq:sc-quotient} implies that
$\alpha\in C^{i+1}(I_0)$. By the induction hypothesis, the moments
$\E\Theta^0_t$ and $\E\Theta^1_t$ belong to $C^{i+1}(I_0)$. Then the right
side of \eqref{eq:sc-moment-derivative} belongs to $C^{i+1}(I_0)$, and we
obtain $\E\Theta_t\in C^{i+2}(I_0)$. This completes the induction and proves
(iv).

\paragraph{Step 6: mixed derivatives.}
Let $j\ge0$. By \eqref{eq:sc-integrated-pde} and the continuity of $\alpha$
on $I_0$, the function $t\mapsto f_j(t,z)$ is continuously differentiable on
$I_0$ for every $z$, and
\begin{equation}\label{eq:sc-time-derivative}
 \partial_tf_j=-\frac b2\bigl(f_{j+2}+\alpha\,\partial_z^j(u^2)\bigr)
 \qquad\text{on }I_0\times\R.
\end{equation}
The right side is a polynomial in $\alpha$ and $f_1,\dots,f_{j+2}$, and it is
continuous on $I_0\times\R$. We claim that, for every $i\ge1$, the
derivative $\partial_t^if_j$ exists on $I_0\times\R$ and is a polynomial,
with coefficients depending only on $b$, $i$, and $j$, in
$\alpha,\alpha',\dots,\alpha^{(i-1)}$ (evaluated at $t$) and
$f_1,\dots,f_{j+2i}$ (evaluated at $(t,z)$). For $i=1$, this is
\eqref{eq:sc-time-derivative}. If it holds for $i$, we differentiate the
polynomial in $t$, using (iv) for the derivatives of $\alpha$ and
\eqref{eq:sc-time-derivative} for the derivatives of $f_1,\dots,f_{j+2i}$,
which gives the claim for $i+1$. Let $I\subset I_0$ be compact. On
$I\times\R$, the functions $\alpha,\alpha',\dots,\alpha^{(i)}$ are bounded
because they are continuous on $I$, and we have $\abs{f_l}\le C_l$ by
\cref{lem:sc-derivatives}. We obtain the bound in (v), with a constant
depending on $\beta$, $I$, $i$, and $j$. Finally, the right side of \eqref{eq:sc-time-derivative} for $j+1$
is the $z$-derivative of the right side for $j$, because
$\partial_z\partial_z^j(u^2)=\partial_z^{j+1}(u^2)$. This shows that
$\partial_z\partial_tf_j=\partial_tf_{j+1}$, and by Schwarz's theorem and
induction on $i$, we have $\partial_z\partial_t^if_j=\partial_t^if_{j+1}$.
Every partial derivative of $\Phi_\alpha$, taken in any order, is therefore
one of the continuous functions $\partial_t^i\partial_z^j\Phi_\alpha$, and
$\Phi_\alpha$ is infinitely differentiable on $I_0\times\R$. This completes
the proof.
\end{proof}

\begin{remark}\label{rem:sc-upper-atom}
The second identity in \eqref{b:eq:contacts} does not extend to
$t=q_\beta$. Indeed, $t\mapsto\E w_t^2$ and $t\mapsto\E\chi_t^3$ are
continuous on $[0,1]$, and letting $t\uparrow q_\beta$ in
\eqref{b:eq:contacts}, we find that
$\E w_{q_\beta}^2=2\alpha(q_\beta-)\E\chi_{q_\beta}^3$. However,
$\E\chi_{q_\beta}^3>0$, and
$\alpha(q_\beta-)=1-\mu_\beta(\{q_\beta\})<1=\alpha(q_\beta)$ because
$\mu_\beta$ has an atom at $q_\beta$. The barrier argument uses
\eqref{b:eq:contacts} and \cref{b:lem:os-scalar-jets} only on compact
subintervals of $(0,q_\beta)$.
\end{remark}

We conclude with the positive constant that enters the cubic pressure
deficit. The quantity bounded below in \eqref{b:eq:main-analytic-constants} is
the scalar factor of the cubic term in the expansion of
\cref{b:prop:os-expansion}.

\begin{corollary}[The constant $m_\beta$]\label{cor:sc-mbeta}
Let $I_\beta\subset(0,q_\beta)$ be a nonempty compact interval, and set
$m_\beta=\beta^3\alpha(\min I_\beta)^2$. Then $m_\beta>0$, and
\begin{equation}\label{b:eq:main-analytic-constants}
 m_\beta\le\inf_{r\in I_\beta}\alpha(r)^2\beta^6\E\chi(r,Y_r)^3.
\end{equation}
\end{corollary}

\begin{proof}
By \cref{b:lem:os-scalar-jets}\,(iii), we have $\alpha(\min I_\beta)>0$, so
$m_\beta>0$. Let $r\in I_\beta$. Since $\alpha$ is nondecreasing, we have
$\alpha(r)\ge\alpha(\min I_\beta)$, and \cref{b:lem:os-scalar-jets}\,(ii)
gives $\E\chi(r,Y_r)^3\ge\beta^{-3}$. Multiplying these bounds, we find that
$\alpha(r)^2\beta^6\E\chi(r,Y_r)^3\ge\beta^3\alpha(\min I_\beta)^2=m_\beta$.
\end{proof}

The constant $m_\beta$ depends only on $\beta$ and $I_\beta$. In the proof of
\cref{thm:barrier-730}, the interval $I_\beta$ is fixed in \cref{sec:references},
before any of the small parameters of the local comparison is chosen.
\chapter{Short-time estimates for Gaussian hierarchies}\label{sec:flow}

In this section, we prove the analytic estimates on Gaussian hierarchies that
the three-replica covariance perturbation of \cref{sec:cubic} requires. Near the
window where the covariance path is perturbed, the one-site recursion of that
comparison is a composition of short semilinear parabolic flows on $\R^3$, one
for each of the fixed number of stages of the perturbed path. We must differentiate its value
four times in the width parameter $h$ of the window and twice in the
parameter $k=\delta/h$, uniformly in the starting field. Several covariance
matrices of the path are singular, and the stage of the terminal correction has
a duration proportional to $hk^2$, which vanishes at $k=0$. We state and prove
the estimates for a general finite composition of such flows on $\R^\ell$.

The section has four main results. \Cref{lem:flow-step-limit} identifies the
limit of finite Gaussian hierarchies (\cref{def:inp-hierarchy}) along
refinements with the solution of a flow equation, with an error controlled by
the $L^1$ distance between the masses. \Cref{prop:flow-uniform} bounds all mixed
derivatives of the flow in the parameters $(h,k)$ and in the field, uniformly in
the field, including the endpoint $h=0$ and stages of zero duration, and
\cref{lem:flow-penalty} gives the corresponding bounds for the penalty obtained
by weighting the squared covariance by the mass. \Cref{b:lem:os-prefix} and \cref{cor:flow-prefix}
control the response of a Gaussian hierarchy to a small bounded change of its
terminal function, with an error that does not depend on the number of levels.
In \cref{sec:cubic}, we use the first three results to obtain \cref{b:lem:os-flow}, and
\cref{cor:flow-prefix} is applied to the part of the hierarchy that precedes the
window, in the proof of \cref{b:prop:os-expansion}. From earlier sections, we use
the integration step and the hierarchies of \cref{sec:inputs}
(\cref{def:inp-step,lem:inp-steps,def:inp-hierarchy}). The regularity and
stability of the scalar Parisi equation (\cref{inp:ob-scalar}) enter only the
proof of \cref{cor:flow-prefix}. The remaining tools are the Gaussian
interpolation formula, It\^o's formula, and Gronwall's inequality
\cite[Section~A.13]{talagrand-2011}.

For the scalar Parisi equation, the approximation by step profiles and the
bounds on spatial derivatives that are uniform in the profile are due to
Auffinger and Chen \cite[Propositions~1--2]{auffinger-chen-2015}. The existence,
uniqueness, and spatial smoothness of weak solutions were proved by Jagannath and
Tobasco \cite[Theorem~1]{jagannath-tobasco-2016}. In our
normalization, these facts and the Lipschitz dependence on the profile are
stated in \cref{inp:ob-scalar}, which is \cite[Lemma~3.1]{oai-barriers}. Bovier
and Klimovsky derived the matrix form of the Parisi equation for vector spins
\cite[Section~6, Proposition~6.1 and (6.5)]{bovier-klimovsky-2009}. The
arguments below use the Feynman--Kac representation for the linear equations
obtained by differentiating the flow. We give the estimates for a vector
equation whose diffusion matrices may be singular and whose coefficients depend
on parameters, and we give in detail the bookkeeping of mixed derivatives in the
parameters, which the later sections need.

\paragraph{Notation.}
Throughout this section, $\ell\ge1$ is an integer ($\ell=3$ in \cref{sec:cubic},
one coordinate for each replica), $b>0$ ($b=\beta^2$ in all applications), and
$x\in\R^\ell$ denotes the field variable, with coordinates indexed by
$i,i'\in\{1,\dots,\ell\}$. We write $\langle\cdot,\cdot\rangle$ and
$\abs{\cdot}$ for the Euclidean inner product and norm, $\abs{\Sigma}$ for the
operator norm of a matrix, $U\mathbin{:}V=\sum_{i,i'}U_{ii'}V_{ii'}$ and
$\norm{U}_F^2=U\mathbin{:}U$, and $\Sigma\succeq0$ when $\Sigma$ is symmetric
and positive semidefinite; $\Sigma^{1/2}$ is its positive semidefinite square
root, and $\mathbf g$ is a standard Gaussian vector in $\R^\ell$. For a multi-index
$\iota\in\N^\ell$, $D^\iota$ is the corresponding derivative in $x$ and
$\abs\iota$ its order; $\nabla$ and $D^2$ are the gradient and the Hessian in
$x$. For $\varphi\colon\R^\ell\to\R$ and an integer $r\ge1$, we set
$\norm{D^r\varphi}_\infty=\max_{\abs\iota=r}\sup_x\abs{D^\iota\varphi(x)}$, and
for $2\le r\le\infty$ we let $\mathcal C^r$ be the set of
$\varphi\in C^r(\R^\ell)$ with $\norm{D^{r'}\varphi}_\infty<\infty$ for all finite
$1\le r'\le r$. A function in $\mathcal C^r$ grows at most linearly and need
not itself be bounded. Parameter derivatives are taken in two real variables
$(h,k)$: for $\gamma=(\gamma_h,\gamma_k)\in\N^2$, we write
$\partial^\gamma=\partial_h^{\gamma_h}\partial_k^{\gamma_k}$,
$\abs\gamma=\gamma_h+\gamma_k$, $\gamma'\le\gamma$ for the componentwise order,
$\binom{\gamma}{\gamma'}=\binom{\gamma_h}{\gamma_h'}\binom{\gamma_k}{\gamma_k'}$,
and $\gamma!=\gamma_h!\gamma_k!$. We say that the mixed derivative
$\partial^\gamma D^\iota$ has \emph{weight} $\abs\iota+2\abs\gamma$. One parameter
derivative counts as two spatial derivatives, because a parameter derivative of
a covariance coefficient multiplies a second derivative in $x$.

\section{Flows and the main estimates}\label{sec:flow-statements}

\begin{definition}[Stage data and flows]\label{def:flow}
Let $N\ge1$. A \emph{stage datum} consists of, for each $j\in\{1,\dots,N\}$, a
matrix $B_j\succeq0$, a duration $\lambda_j\ge0$, and a measurable mass function
$m_j\colon[0,1]\to[0,1]$. Stage $j$ occupies the interval $[j-1,j]$ of the
variable $\tau\in[0,N]$. For a terminal function $f\in\mathcal C^2$, the
\emph{flow equation} is
\begin{equation}\label{eq:flow-equation}
 -\partial_\tau F=\frac{b\lambda_j}{2}\Bigl\{B_j\mathbin{:}D^2F
 +m_j(\tau-j+1)\bigl\langle\nabla F,B_j\nabla F\bigr\rangle\Bigr\}
 \quad\text{on }(j-1,j)\times\R^\ell,
\end{equation}
for $j=1,\dots,N$, with $F(N,\cdot)=f$ and $F$ continuous on
$[0,N]\times\R^\ell$. We call $F(0,\cdot)$ the \emph{output} of the flow.
\end{definition}

We solve the equation backward, from $\tau=N$ to $\tau=0$, as the Parisi
equation \eqref{b:eq:Parisi-PDE} is solved backward from time one. Stage~$1$ is
the outermost stage, nearest to the root of the hierarchy, and stage~$N$ is the
innermost. On stage $j$ the Gaussian field has covariance $b\lambda_jB_j$ per unit
of $\tau$; in the notation of \cref{def:inp-hierarchy}, the overlap covariance
path increases by $\lambda_jB_j$ over stage $j$. For example, the Parisi equation
for a distribution function $\alpha$ on an interval $[t_1,t_2]$ is the case
$\ell=1$, $N=1$, $B_1=1$, $\lambda_1=t_2-t_1$, and
$m_1(\tau)=\alpha(t_1+\tau(t_2-t_1))$, with $b=\beta^2$.

\paragraph{Step flows.}
We first define solutions for masses that are step functions. A \emph{step
flow} is the flow of a stage datum in which, for each $j$, there exists a partition
$0=\tau_{j,0}<\tau_{j,1}<\dots<\tau_{j,M_j}=1$ such that $m_j=\mu_{j,i}$ on
$[\tau_{j,i-1},\tau_{j,i})$. With the integration step of
\cref{def:inp-step}, we set $F(N,\cdot)=f$ and
\begin{equation}\label{eq:flow-step-solution}
 F(\tau,\cdot)=T_{\mu_{j,i},\,(j-1+\tau_{j,i}-\tau)b\lambda_jB_j}
 F(j-1+\tau_{j,i},\cdot)
 \quad\text{for } j-1+\tau_{j,i-1}\le\tau\le j-1+\tau_{j,i},
\end{equation}
proceeding from the piece $(N,M_N)$ to the piece $(1,1)$. The formulas for two
consecutive pieces agree at their common endpoint, because
$T_{m,0}\varphi=\varphi$. The output $F(0,x)$ is the value at $x$ of a finite Gaussian hierarchy:
its levels, listed from the root, are the pieces $(j,i)$ in lexicographic order,
and the level $(j,i)$ has covariance $b\lambda_jB_j(\tau_{j,i}-\tau_{j,i-1})$ and
is integrated at mass $\mu_{j,i}$. By \cref{lem:flow-cole-hopf} below, $F$ is
continuous and solves \eqref{eq:flow-equation} on each piece, where the mass is
constant.

If the masses $m_j$ are continuous and $\mathfrak p=(\tau_{j,i})$ is a family of
partitions, the \emph{sampled masses} are $m_j^{\mathfrak p}=m_j(\tau_{j,i-1})$
on $[\tau_{j,i-1},\tau_{j,i})$, and the \emph{mesh} $\abs{\mathfrak p}$ is the
largest length $\tau_{j,i}-\tau_{j,i-1}$. If each $m_j$ is nondecreasing and
$m_j(1)\le m_{j+1}(0)$ for $j<N$, then the sampled masses are nondecreasing from
the root, as in an admissible path of \cref{def:inp-hierarchy}.

\begin{lemma}[Step approximation]\label{lem:flow-step-limit}
Fix a stage datum with matrices $B_j$ and durations $\lambda_j$, and a terminal
function $f\in\mathcal C^2$. Let $L_f=\sup_x\abs{\nabla f(x)}$.
\begin{enumerate}[label=\textup{(\alph*)}]
\item Let $F$ and $\widetilde F$ be the step flows with step masses $m_j$ and
 $\widetilde m_j$. Then $\sup_{\tau,x}\abs{\nabla F(\tau,x)}\le L_f$ and
 \begin{equation}\label{eq:flow-L1}
  \sup_{\tau,x}\abs{F(\tau,x)-\widetilde F(\tau,x)}
  \le\frac{bL_f^2}{2}\sum_{j=1}^N\lambda_j\abs{B_j}
  \int_0^1\abs{m_j(\tau)-\widetilde m_j(\tau)}\dd\tau.
 \end{equation}
 If $F'$ is the step flow with the masses $m_j$ and a terminal function
 $f'\in\mathcal C^2$ such that $f-f'$ is bounded, then
 $\sup_{\tau,x}\abs{F-F'}\le\sup_x\abs{f-f'}$.
\item Let $m_j$ be measurable. For every sequence of step masses $m_j^{(l)}$ with
 $\int_0^1\abs{m_j^{(l)}-m_j}\dd\tau\to0$ for each $j$, the step flows converge
 uniformly on $[0,N]\times\R^\ell$ to a limit $F$ that does not depend on the
 sequence. We call $F$ the \emph{flow} of the stage datum with terminal
 function $f$. The three estimates of \textup{(a)} hold for these flows.
\item Let $1\le N'<N$. Let $F_{\rm in}$ be the flow of stages $N'+1,\dots,N$
 with terminal function $f$, and suppose that $f_{\rm in}=F_{\rm in}(N',\cdot)$
 belongs to $\mathcal C^2$. Then $F$ coincides on $[0,N']\times\R^\ell$ with the
 flow of stages $1,\dots,N'$ with terminal function $f_{\rm in}$.
\item Suppose that $f\in\mathcal C^\infty$ and
 $2\ell b\norm{D^2f}_\infty\sum_j\lambda_j\abs{B_j}\le1$. Then the flow $F$ of
 \textup{(b)} has continuous and bounded spatial derivatives of every order
 $r\ge1$, with $\norm{D^2F(\tau,\cdot)}_\infty\le2\norm{D^2f}_\infty$, and, for
 the step flows $F^{(l)}$ of \textup{(b)} and every $\iota\ne0$,
 \begin{equation}\label{eq:flow-L1-derivatives}
  \sup_{\tau,x}\abs{D^\iota F^{(l)}(\tau,x)-D^\iota F(\tau,x)}
  \le C\sum_{j=1}^N\lambda_j\int_0^1\abs{m^{(l)}_j-m_j}\dd\tau,
 \end{equation}
 where $C$ depends only on $\ell$, $b$, $\abs\iota$, $\max_j\abs{B_j}$,
 $\sum_j\lambda_j\abs{B_j}$, and $\norm{D^{r}f}_\infty$ for $1\le r\le\abs\iota+1$.
 If moreover each $m_j$ is continuous on $[0,1]$, then $F$ is continuously
 differentiable in $\tau$ on each closed stage $[j-1,j]$, with one-sided
 derivatives at its endpoints, and solves \eqref{eq:flow-equation}. It is the
 unique such solution with bounded first and second spatial derivatives that is
 of class $C^{1,2}$ on each closed stage.
\end{enumerate}
\end{lemma}

In words, the hierarchy values converge along refinements of the masses, at a
rate given by the $L^1$ distance between the masses, and the limit is the
solution of the flow equation; for short flows the same holds for every spatial
derivative. Part (c) says that the limiting flows compose stage by stage. Parts
(a)--(c) hold for stages of any duration. The constant in \eqref{eq:flow-L1}
involves only the gradient of $f$. To identify the gradient of the limit in (b),
we also use the bound $K_f+L_f^2$ of \cref{lem:flow-spatial}(a) on the Hessians
of the step flows, where $K_f=\sup_x\abs{D^2f(x)}$. This bound does not require
short stages either. Part (d) is stated only for short flows, which are the only
ones we need.

\paragraph{Parameter-dependent data.}
We now let the data depend on two parameters. Fix $N\ge1$, $h_1>0$, and
$k_0>0$, and let $\mathcal U=[0,h_1]\times[-k_0,k_0]$. A function on $\mathcal U$
is called \emph{smooth} if all its partial derivatives exist and are continuous
on $\mathcal U$; at boundary points, the derivatives are one-sided. In
particular, derivatives in $h$ at $h=0$ are right derivatives. We assume the
following.
\begin{enumerate}[label=\textup{(H\arabic*)}]
\item\label{H1} For each $j$, the map $(h,k)\mapsto B_j(h,k)$ is smooth on
 $\mathcal U$, with values in the positive semidefinite matrices.
\item\label{H2} For each $j$, $\lambda_j(h,k)=h\kappa_j(h,k)$, where
 $\kappa_j\colon\mathcal U\to[0,\infty)$ is smooth. The function $\kappa_j$
 may vanish.
\item\label{H3} For each $j$, $m_j\colon[0,1]\times\mathcal U\to[0,1]$ is
 continuous, and all its derivatives in $(h,k)$ exist and are continuous on
 $[0,1]\times\mathcal U$.
\item\label{H4} The terminal function $f\colon\mathcal U\times\R^\ell\to\R$ is
 smooth in the same sense, and $\partial^\gamma D^\iota f$ is bounded on $\mathcal U\times\R^\ell$
 for every $(\gamma,\iota)\ne(0,0)$.
\item\label{H5} The number $h_0\in(0,h_1]$ satisfies
 $2\ell b\,h_0\Lambda_*\norm{D^2f}_\infty\le1$, where
 $\Lambda_*=\max_{\mathcal U}\sum_j\kappa_j\abs{B_j}$ and
 $\norm{D^2f}_\infty=\sup_{\mathcal U\times\R^\ell}\max_{i,i'}\abs{\partial_{ii'}f}$.
\end{enumerate}
Let $\mathcal U_0=[0,h_0]\times[-k_0,k_0]$. For $(h,k)\in\mathcal U_0$, let
$F(\tau,h,k,x)$ be the flow, in the sense of \cref{lem:flow-step-limit}, of the
stage datum $(B_j(h,k),\lambda_j(h,k),m_j(\cdot,h,k))_{j\le N}$ with terminal
function $f(h,k,\cdot)$, and let $\mathcal B(h,k,x)=F(0,h,k,x)$ be its output.

\begin{proposition}[Uniform bounds for short flows]\label{prop:flow-uniform}
Assume \ref{H1}--\ref{H5}.
\begin{enumerate}[label=\textup{(\roman*)}]
\item For each $(h,k)\in\mathcal U_0$, the function $F(\cdot,h,k,\cdot)$ is
 continuously differentiable in $\tau$ on each closed stage, infinitely
 differentiable in $x$, and it is the unique solution of \eqref{eq:flow-equation}
 with bounded first and second spatial derivatives that is of class $C^{1,2}$ on
 each closed stage.
\item All partial derivatives $\partial^\gamma D^\iota F$ exist and are
 continuous on $[0,N]\times\mathcal U_0\times\R^\ell$. For every
 $(\gamma,\iota)\ne(0,0)$, there exists $C$ such that
 \[
  \sup_{[0,N]\times\mathcal U_0\times\R^\ell}\abs{\partial^\gamma D^\iota F}\le C,
 \]
 where $C$ depends only on $\ell$, $N$, $b$, $h_1$, $\gamma$, and $\abs\iota$,
 on the suprema over $\mathcal U$ of $\abs{\partial^{\gamma'}(\kappa_jB_j)}$ and
 over $[0,1]\times\mathcal U$ of $\abs{\partial^{\gamma'}(\kappa_jm_jB_j)}$, for
 $\gamma'\le\gamma$, and on the suprema over $\mathcal U\times\R^\ell$ of
 $\abs{\partial^{\gamma'}D^{\iota'}f}$ for $\gamma'\le\gamma$ and
 $0<\abs{\iota'}+2\abs{\gamma'}\le\abs\iota+2\abs\gamma$.
 Moreover, $\abs{F(\tau,h,k,x)}\le C_{\rm grow}(1+\abs x)$, where $C_{\rm grow}$
 depends only on $\ell$, $N$, $b$, $h_1$, the supremum over $\mathcal U$ of
 $\abs{\kappa_jB_j}$, the suprema over $\mathcal U\times\R^\ell$ of
 $\abs{\nabla f}$ and $\abs{D^2f}$, and $\sup_{\mathcal U}\abs{f(h,k,0)}$.
\item For $\gamma\ne0$, the derivative $\partial^\gamma F$ is continuously
 differentiable in $\tau$ on each closed stage, and it solves the equation
 obtained by applying $\partial^\gamma$ to \eqref{eq:flow-equation}, with
 terminal value $\partial^\gamma f$.
\item Let $\mathfrak p_l$ be partitions with $\abs{\mathfrak p_l}\to0$, and let
 $F^{\mathfrak p_l}$ be the step flows with the sampled masses $m_j^{\mathfrak p_l}(\cdot,h,k)$.
 Then $F^{\mathfrak p_l}\to F$ uniformly on $[0,N]\times\mathcal U_0\times\R^\ell$, and
 $\partial^\gamma D^\iota F^{\mathfrak p_l}\to\partial^\gamma D^\iota F$ uniformly on compact
 subsets of $[0,N]\times\mathcal U_0\times\R^\ell$, for every $(\gamma,\iota)$.
\end{enumerate}
\end{proposition}

To bound a derivative $\partial^\gamma D^\iota\mathcal B$, we use only
derivatives of the terminal function of weight at most $\abs\iota+2\abs\gamma$
and $(h,k)$-derivatives of order at most $\gamma$ of the coefficients. In
\cref{sec:cubic}, we need the derivatives $\partial_h^{\gamma_h}\partial_k^{\gamma_k}\mathcal B$ with
$\gamma_h\le4$ and $\gamma_k\le2$ together with a few spatial derivatives; with at
most eight spatial derivatives they have weight at most $20$, and their bounds
use $(h,k)$-derivatives of the masses and of the terminal function of order at
most six. The value $F$ itself grows at most linearly in $x$ and is
not estimated, except through its derivatives and through bounded differences
such as $F(\tau,h,k,x)-F(\tau,h',k',x)$.

\begin{remark}[The coefficients at $h=0$]\label{rem:flow-first-order}
Since $\lambda_j(0,k)=0$, we have $F(\tau,0,k,\cdot)=f(0,k,\cdot)$ for all
$\tau$, and hence $\partial_k^{\gamma_k}\mathcal B(0,k,\cdot)=\partial_k^{\gamma_k}f(0,k,\cdot)$ for
every $\gamma_k\ge0$.
Writing $\bar m_j(k)=\int_0^1m_j(\tau,0,k)\dd\tau$ and evaluating $f$, $B_j$ and
$\kappa_j$ at $(0,k)$, we have
\begin{equation}\label{eq:flow-first-order}
 \partial_h\mathcal B(0,k,x)=\partial_hf(0,k,x)+\frac b2\sum_{j=1}^N\kappa_j
 \Bigl\{B_j\mathbin{:}D^2f(0,k,x)
 +\bar m_j\bigl\langle\nabla f(0,k,x),B_j\nabla f(0,k,x)\bigr\rangle\Bigr\}.
\end{equation}
Indeed, at $h=0$ the equation of \cref{prop:flow-uniform}(iii) for
$\gamma=(1,0)$ has no terms involving $\partial_hF$, since the coefficients of
\eqref{eq:flow-equation} vanish, and its source is the derivative in $h$ of the
right side of \eqref{eq:flow-equation} at $F=f(0,k,\cdot)$. Integrating it over
$\tau\in[0,N]$, we obtain \eqref{eq:flow-first-order}.
\end{remark}

\begin{remark}[Singular matrices, zero durations, and regularization]
\label{rem:flow-regularization}
The hypotheses \ref{H1}--\ref{H5} and the constants in \cref{prop:flow-uniform}
involve no lower bound on $B_j$ or on $\kappa_j$. In the proof, every Gaussian
integral is parametrized by its covariance matrix
$t\,bh\kappa_j(h,k)B_j(h,k)$, which is a smooth function of $(h,k)$ with values
in the positive semidefinite matrices, including where it vanishes, and
\cref{lem:flow-gauss} differentiates Gaussian integrals as functions of the
covariance. No square root of a covariance or of a duration is differentiated.
For instance, a stage with $\kappa_j(h,k)=ck^2$ has zero duration at $k=0$, and
its contribution is differentiable in $k$ there through the polynomial $hk^2$.
One may also replace each $B_j$ by $B_j+\eps\mathrm{Id}$ with $0<\eps\le1$. Let
$K_*=\max_{\mathcal U}\sum_j\kappa_j$, and decrease $h_0$, if necessary, so that
$2\ell b\,h_0(\Lambda_*+K_*)\norm{D^2f}_\infty\le1$. The regularized data then
satisfy \ref{H1}--\ref{H5}, with upper bounds uniform in $\eps$. For parameters
with $h\kappa_j(h,k)>0$, the diffusion matrix of stage $j$ is positive definite.
Stages with $h\kappa_j(h,k)=0$, in particular all stages at $h=0$, still have zero
diffusion, and the regularized equations are not uniformly parabolic on
$\mathcal U_0$. The
regularized flows satisfy the bounds of \cref{prop:flow-uniform} uniformly in
$\eps$, and they converge to $F$ as $\eps\to0$, together with all derivatives
$\partial^\gamma D^\iota$, uniformly on compact sets. Indeed, the difference
between the regularized flow and $F$ satisfies the linear equation in the proof
of \cref{lem:flow-step-limit}(a), with $B_j+\eps\mathrm{Id}$ in place of $B_j$ and with
equal masses. Its source is
$(b\lambda_j\eps/2)\{\mathrm{Id}\mathbin{:}D^2F+m_j\abs{\nabla F}^2\}$, which is bounded
by $C\eps$. Then \cref{lem:flow-linear} shows that the values
converge, and the compactness argument in the proof of
\cref{prop:flow-uniform}(iv) identifies the limits of all derivatives. We do not
use the regularization.
\end{remark}

\begin{remark}[Further parameters and restriction to a line]
\label{rem:flow-uniformity}
If the data depend on a further parameter in which no derivatives are needed
(in \cref{sec:cubic}, the left endpoint of the window), and the suprema listed in
\cref{prop:flow-uniform}(ii) and the number $\Lambda_*\norm{D^2f}_\infty$ are
bounded uniformly in it, then the conclusions hold uniformly in that parameter;
for the linear growth bound, $\sup_{\mathcal U}\abs{f(h,k,0)}$ must also be
bounded uniformly in that parameter.
The statements and the proofs of this section hold without change if $k$ is a
vector of parameters ranging over a product of compact intervals, with
multi-indices in place of $\gamma_k$; a further parameter in which derivatives
are needed can be included in this way. For a vector $e\in\R^\ell$ and $z\in\R$,
the restriction $z\mapsto\mathcal B(h,k,ze)$
satisfies
\[
 \partial^\gamma\partial_z^r\bigl[\mathcal B(h,k,ze)\bigr]
 =\sum_{\abs\iota=r}\frac{r!}{\iota!}\,e^\iota\,\partial^\gamma D^\iota\mathcal B(h,k,ze),
\]
where $\iota!=\prod_i\iota_i!$ and $e^\iota=\prod_ie_i^{\iota_i}$. Its derivatives
obey the bounds of \cref{prop:flow-uniform}(ii), multiplied by
$(\sum_i\abs{e_i})^r$.
\end{remark}

The last estimate of this section concerns the penalty. For a smooth function
$Q_\star\colon\mathcal U\to\R^{\ell\times\ell}$ with symmetric values, let
\[
 Q(\tau)=Q_\star+\sum_{j'<j}\lambda_{j'}B_{j'}+(\tau-j+1)\lambda_jB_j
 \qquad(j-1\le\tau\le j),
\]
where all functions are evaluated at $(h,k)$. This is the overlap covariance
path through the stages, started at $Q_\star$. We set
\begin{equation}\label{eq:flow-penalty}
 \Pi(h,k)=\sum_{j=1}^N\int_{j-1}^jm_j(\tau-j+1)\,\partial_\tau\norm{Q(\tau)}_F^2\dd\tau
 =2\sum_{j=1}^N\lambda_j\int_0^1m_j(\tau)\,Q(j-1+\tau)\mathbin{:}B_j\dd\tau.
\end{equation}
This is the contribution of the stages to the penalty $\int m\dd\norm{Q}_F^2$
of \eqref{eq:inp-penalty}. For partitions $\mathfrak p$, the corresponding
contribution of the step hierarchy with the sampled masses is
\[
 \Pi^{\mathfrak p}(h,k)=\sum_{j=1}^N\sum_{i=1}^{M_j}m_j(\tau_{j,i-1})
 \Bigl(\norm{Q(j-1+\tau_{j,i})}_F^2-\norm{Q(j-1+\tau_{j,i-1})}_F^2\Bigr).
\]

\begin{lemma}[The penalty]\label{lem:flow-penalty}
Assume \ref{H1}--\ref{H3}. Then $\Pi$ is smooth on $\mathcal U$, and for every
$\gamma$ we have $\sup_{\mathcal U}\abs{\partial^\gamma\Pi}\le C$, where $C$
depends only on $\ell$, $N$, $h_1$, $\gamma$, and the suprema over $\gamma'\le\gamma$
of $\abs{\partial^{\gamma'}Q_\star}$, $\abs{\partial^{\gamma'}(\kappa_jB_j)}$ and
$\abs{\partial^{\gamma'}(\kappa_jm_jB_j)}$. Moreover
$\sup_{\mathcal U}\abs{\partial^\gamma(\Pi^{\mathfrak p}-\Pi)}\to0$ as
$\abs{\mathfrak p}\to0$, for every $\gamma$.
\end{lemma}

\begin{proof}
Since $m_j^{\mathfrak p}$ is constant on each piece, we have
$\Pi^{\mathfrak p}=2\sum_j\lambda_j\int_0^1m_j^{\mathfrak p}(\tau)\,Q(j-1+\tau)\mathbin{:}B_j\dd\tau$,
which is \eqref{eq:flow-penalty} with $m_j$ replaced by $m_j^{\mathfrak p}$. The
integrand on stage $j$ is $Q(j-1+\tau)\mathbin{:}(h\kappa_jm_jB_j)$, where
$Q(j-1+\tau)$ is a polynomial in $\tau$, $Q_\star$, and the matrices
$h\kappa_{j'}B_{j'}$, and the $(h,k)$-derivatives of $m_j^{\mathfrak p}$ are those
of $m_j$ evaluated at the left endpoints of the pieces. Differentiating under the integral sign, we
find that $\Pi$ and $\Pi^{\mathfrak p}$ are smooth with the stated bounds. By
\ref{H3}, every $(h,k)$-derivative of $m_j$ is uniformly continuous on the
compact set $[0,1]\times\mathcal U$, and we have
$\partial^{\gamma'}m_j^{\mathfrak p}\to\partial^{\gamma'}m_j$ uniformly as
$\abs{\mathfrak p}\to0$. This implies the last statement.
\end{proof}

\section{Three tools}\label{sec:flow-tools}

With the first tool, we differentiate a Gaussian integral with respect to its
covariance, including at singular covariances.

\begin{lemma}[Gaussian integrals as functions of the covariance]\label{lem:flow-gauss}
Let $\Omega$ be a convex subset of a Euclidean space. Let $\mathcal F\colon\Omega\times\R^\ell\to\R$ be
continuous and twice continuously differentiable in its second variable $y$,
with $D^2_y\mathcal F$ continuous on $\Omega\times\R^\ell$, and suppose that there exists
$c<\infty$ such that
\begin{equation}\label{eq:flow-growth}
 \abs{\mathcal F(\omega,y)}+\abs{\nabla_y\mathcal F(\omega,y)}+\abs{D^2_y\mathcal F(\omega,y)}
 \le c\,e^{c\abs y}\qquad(\omega\in\Omega,\ y\in\R^\ell).
\end{equation}
\begin{enumerate}[label=\textup{(\alph*)}]
\item The map $(\omega,\Sigma)\mapsto\E\mathcal F(\omega,\Sigma^{1/2}\mathbf g)$ is continuous
 on $\Omega\times\{\Sigma\succeq0\}$. This holds for every continuous $\mathcal F$ with
 $\abs{\mathcal F}\le ce^{c\abs y}$.
\item For $\omega\in\Omega$ and $\Sigma,\Sigma'\succeq0$, with
 $\Sigma_t=\Sigma+t(\Sigma'-\Sigma)$,
 \[
  \E\mathcal F(\omega,\Sigma'^{1/2}\mathbf g)-\E\mathcal F(\omega,\Sigma^{1/2}\mathbf g)
  =\frac12\int_0^1(\Sigma'-\Sigma)\mathbin{:}
  \E D^2_y\mathcal F(\omega,\Sigma_t^{1/2}\mathbf g)\dd t.
 \]
\item Let $\omega\mapsto\Sigma(\omega)\succeq0$ be continuously differentiable on
 $\Omega$, and suppose that $\partial_{\omega_a}\mathcal F$ exists, is continuous, and
 satisfies $\abs{\partial_{\omega_a}\mathcal F}\le ce^{c\abs y}$. Then the function
 $\omega\mapsto\E\mathcal F(\omega,\Sigma(\omega)^{1/2}\mathbf g)$ has a continuous partial
 derivative in $\omega_a$, one-sided at boundary points of $\Omega$, and
 \begin{equation}\label{eq:flow-gauss-derivative}
  \partial_{\omega_a}\E\mathcal F(\omega,\Sigma(\omega)^{1/2}\mathbf g)
  =\E\partial_{\omega_a}\mathcal F(\omega,\Sigma(\omega)^{1/2}\mathbf g)
  +\frac12\,\partial_{\omega_a}\Sigma(\omega)\mathbin{:}
  \E D^2_y\mathcal F(\omega,\Sigma(\omega)^{1/2}\mathbf g).
 \end{equation}
\end{enumerate}
\end{lemma}

\begin{proof}
For (a), recall that the positive square root is continuous on the positive
semidefinite matrices. Indeed, if the spectra of $\Sigma$ and $\Sigma'$ lie in
$[0,c']$, then $\abs{\Sigma^{1/2}-\Sigma'^{1/2}}\le2\sup_{[0,c']}\abs{\sqrt s-P(s)}
+\abs{P(\Sigma)-P(\Sigma')}$ for every polynomial $P$, and $\sqrt s$ is a uniform
limit of polynomials on $[0,c']$. If $(\omega_n,\Sigma_n)\to(\omega,\Sigma)$, then
$\mathcal F(\omega_n,\Sigma_n^{1/2}\mathbf g)\to\mathcal F(\omega,\Sigma^{1/2}\mathbf g)$ almost surely, and
these random variables are dominated by $c\exp(cc'^{1/2}\abs{\mathbf g})$, where $c'$ bounds
the spectra of all $\Sigma_n$. By dominated convergence, the expectations
converge, which proves (a).

For (b), let $\mathbf g'$ be a standard Gaussian vector independent of $\mathbf g$, and let
$\zeta(t)=\sqrt t\,\Sigma'^{1/2}\mathbf g'+\sqrt{1-t}\,\Sigma^{1/2}\mathbf g$. The vector $\zeta(t)$ is
centered Gaussian with covariance $\Sigma_t$, so it has the law of
$\Sigma_t^{1/2}\mathbf g$. By the Gaussian interpolation formula
\cite[Lemma~1.3.1]{talagrand-2011}, for $0<t<1$,
\[
 \frac{\dd}{\dd t}\E\mathcal F(\omega,\zeta(t))
 =\frac12(\Sigma'-\Sigma)\mathbin{:}\E D_y^2\mathcal F(\omega,\zeta(t)).
\]
That lemma is stated for infinitely differentiable functions whose derivatives
grow more slowly than $e^{a\abs y^2}$ for every $a>0$. For $\mathcal F(\omega,\cdot)$
as in \eqref{eq:flow-growth}, we apply it to the convolution of $\mathcal F(\omega,\cdot)$
with a smooth compactly supported mollifier, integrate the identity between two
times in $(0,1)$, and let the mollifier shrink; both sides converge by dominated
convergence. The right side of the displayed identity is continuous on $[0,1]$ by
(a), applied to $D^2_y\mathcal F$. Integrating over $t\in[0,1]$, we obtain (b).

For (c), fix $\omega$ and write $\Sigma=\Sigma(\omega)$ and
$\Sigma_\eta=\Sigma(\omega+\eta e_a)$ for $\omega+\eta e_a\in\Omega$, where
$e_a$ is the $a$th coordinate vector. Then
\begin{multline*}
 \E\mathcal F(\omega+\eta e_a,\Sigma_\eta^{1/2}\mathbf g)-\E\mathcal F(\omega,\Sigma^{1/2}\mathbf g)
 =\int_0^\eta\E\partial_{\omega_a}\mathcal F(\omega+\eta'e_a,\Sigma_\eta^{1/2}\mathbf g)\dd\eta'\\
 +\frac12\int_0^1(\Sigma_\eta-\Sigma)\mathbin{:}
 \E D_y^2\mathcal F\bigl(\omega,(\Sigma+t(\Sigma_\eta-\Sigma))^{1/2}\mathbf g\bigr)\dd t,
\end{multline*}
where we used the fundamental theorem of calculus and Fubini's theorem for the
first term and (b) for the second; the segment from $\Sigma$ to $\Sigma_\eta$
consists of positive semidefinite matrices. We divide by $\eta$ and let
$\eta\to0$. By (a), applied to $\partial_{\omega_a}\mathcal F$ and to the entries of
$D^2_y\mathcal F$, the two integrands converge uniformly to their values at $\eta=0$,
and we obtain \eqref{eq:flow-gauss-derivative}. The right side of
\eqref{eq:flow-gauss-derivative} is continuous by (a).
\end{proof}

The second tool collects the properties of one integration step
$T_{m,\Sigma}$ of \cref{def:inp-step} that we need. Recall that monotonicity, the
translation identity, and the semigroup property are \cref{lem:inp-steps}(a)
and (b). For constant mass, the exponential change of variables that turns
the flow into the heat equation is the Cole--Hopf transformation; its scalar
Parisi form is recorded in \cite[Section~2, (10)--(12)]{auffinger-chen-2015}.
The next lemma gives the corresponding finite-dimensional Gaussian step.

\begin{lemma}[One integration step]\label{lem:flow-cole-hopf}
Let $\Sigma\succeq0$, let $m\in[0,1]$, and let $\varphi\in\mathcal C^2$.
\begin{enumerate}[label=\textup{(\alph*)}]
\item We have $T_{m,\Sigma}\varphi\in\mathcal C^2$ and
 $\sup_x\abs{\nabla T_{m,\Sigma}\varphi(x)}\le\sup_x\abs{\nabla\varphi(x)}$. If
 $\varphi\in\mathcal C^r$, then $T_{m,\Sigma}\varphi\in\mathcal C^r$.
\item The function $\psi(t,x)=T_{m,t\Sigma}\varphi(x)$, $t\ge0$, is continuous and
 continuously differentiable in $t$, with a right derivative at $t=0$; its first
 and second derivatives in $x$ are continuous in $(t,x)$; and
 \begin{equation}\label{eq:flow-one-step-pde}
  \partial_t\psi=\frac12\Bigl\{\Sigma\mathbin{:}D^2\psi
  +m\bigl\langle\nabla\psi,\Sigma\nabla\psi\bigr\rangle\Bigr\}.
 \end{equation}
\item Let $\Omega$ be a compact convex subset of a Euclidean space, and let $\Sigma(\omega)\succeq0$,
 $m(\omega)\in[0,1]$, and $\varphi(\omega,x)$ be smooth functions of
 $\omega\in\Omega$ and of $(\omega,x)\in\Omega\times\R^\ell$, such that
 $\partial^\gamma_\omega D^\iota\varphi$ is bounded for every $(\gamma,\iota)\ne(0,0)$.
 Then $\psi(t,\omega,x)=T_{m(\omega),t\Sigma(\omega)}\varphi(\omega,\cdot)(x)$ is
 infinitely differentiable in $(t,\omega,x)\in[0,1]\times\Omega\times\R^\ell$
 (one-sided at the boundary), the function $\psi-\varphi$ is bounded, and every
 derivative $\partial^\gamma_\omega D^\iota\psi$ with $(\gamma,\iota)\ne(0,0)$ is
 bounded on $[0,1]\times\Omega\times\R^\ell$.
\end{enumerate}
\end{lemma}

\begin{proof}
Let $L_\varphi=\sup_x\abs{\nabla\varphi(x)}$, and, for fixed $x$, let
$\mathcal Y_x=\varphi(x+\Sigma^{1/2}\mathbf g)-\varphi(x)$, so that $\abs{\mathcal Y_x}\le L_\varphi\abs{\Sigma^{1/2}\mathbf g}$.
For $m>0$, we have
$T_{m,\Sigma}\varphi(x)=\varphi(x)+m^{-1}\log\E e^{m\mathcal Y_x}$.

For (a), since the derivatives of $x\mapsto e^{m\varphi(x+y)-m\varphi(x)}$ of
order at most two are bounded by $C(1+\abs y)^2e^{L_\varphi\abs y}$, we may differentiate
under the expectation. We obtain
\begin{equation}\label{eq:flow-tilted-gradient}
 \nabla T_{m,\Sigma}\varphi(x)
 =\frac{\E\bigl[\nabla\varphi(x+\Sigma^{1/2}\mathbf g)\,e^{m\mathcal Y_x}\bigr]}{\E e^{m\mathcal Y_x}}.
\end{equation}
The right side is an average of values of $\nabla\varphi$ with respect to a
probability measure, so its norm is at most $L_\varphi$.
Differentiating further, we find that $D^\iota T_{m,\Sigma}\varphi(x)$, for
$1\le\abs\iota\le r$, is a polynomial, with coefficients depending only on $\iota$
and $m$, in the averages
$\E[P(x+\Sigma^{1/2}\mathbf g)e^{m\mathcal Y_x}]/\E e^{m\mathcal Y_x}$ of products $P$ of the derivatives
$D^{\iota'}\varphi$ with $1\le\abs{\iota'}\le\abs\iota$. These derivatives are
bounded, and so are the averages. The case $m=0$ is the same without the
exponential weights.

For (b) and $m>0$, let $U(t,x)=\E\exp\{m\varphi(x+(t\Sigma)^{1/2}\mathbf g)\}$. The
function $y\mapsto e^{m\varphi(x+y)}$ and its first two derivatives are bounded
by $Ce^{m\abs{\varphi(x)}+L_\varphi\abs y}$, locally uniformly in $x$. By
\cref{lem:flow-gauss}(a) and (b), with the covariances $t\Sigma$ and
$t'\Sigma$, the function $U$ is continuous, continuously differentiable in $t$,
and $\partial_tU=(1/2)\Sigma\mathbin{:}D^2U$. By \cref{lem:flow-gauss}(a),
$\nabla U$ and $D^2U$ are continuous in $(t,x)$. Writing $\psi=m^{-1}\log U$, we
have $\nabla U=mU\nabla\psi$ and $D^2U=mU\{D^2\psi+m\nabla\psi\nabla\psi^{\mathsf T}\}$,
and \eqref{eq:flow-one-step-pde} follows. For $m=0$, we apply
\cref{lem:flow-gauss}(b) to $\varphi$ directly.

For (c), we write $m=m(\omega)$, $\Sigma=\Sigma(\omega)$, and
$\mathcal Y=\varphi(\omega,x+(t\Sigma)^{1/2}\mathbf g)-\varphi(\omega,x)$. For every $m\ge0$,
\begin{equation}\label{eq:flow-ch-integral}
 \psi(t,\omega,x)-\varphi(\omega,x)
 =\int_0^1\frac{\E\bigl[\mathcal Y\,e^{\rho m\mathcal Y}\bigr]}{\E e^{\rho m\mathcal Y}}\dd\rho.
\end{equation}
Indeed, for $m>0$ the integrand is the derivative of
$\rho\mapsto m^{-1}\log\E e^{\rho m\mathcal Y}$, and for $m=0$ both sides equal $\E \mathcal Y$.
We write $\widehat\varphi(\omega,x,y)=\varphi(\omega,x+y)-\varphi(\omega,x)$. The
denominator is the Gaussian integral $\E\mathcal F(\omega,x,\rho;(t\Sigma)^{1/2}\mathbf g)$ of
the function
\[
 \mathcal F(\omega,x,\rho;y)=\exp\bigl\{\rho\,m(\omega)\,\widehat\varphi(\omega,x,y)\bigr\},
\]
and the numerator is the Gaussian integral of $\widehat\varphi\,\mathcal F$. Every
derivative of $\mathcal F$ and of $\widehat\varphi\,\mathcal F$ in $(\omega,x,\rho,y)$ is a
polynomial in
$\varphi(\omega,x+y)-\varphi(\omega,x)$, in the derivatives of $m$, and in the
derivatives $\partial^{\gamma'}_\omega D^{\iota'}\varphi$ with
$(\gamma',\iota')\ne(0,0)$ evaluated at $(\omega,x+y)$ and $(\omega,x)$, times
the exponential factor. Since $\abs{\varphi(\omega,x+y)-\varphi(\omega,x)}\le
L_\varphi\abs y$ with $L_\varphi=\sup\abs{\nabla_x\varphi}$, all these derivatives are bounded by
$C(1+\abs y)^{C}e^{L_\varphi\abs y}$, uniformly in $(t,\omega,x,\rho)$. The covariance
$(t,\omega)\mapsto t\Sigma(\omega)$ is smooth with positive semidefinite values.
Applying \cref{lem:flow-gauss}(c) repeatedly, with the parameter
$(t,\omega,x,\rho)$ in the convex set $[0,1]\times\Omega\times\R^\ell\times[0,1]$,
we find that the numerator and the denominator are infinitely differentiable,
and that each of their derivatives is a Gaussian integral of a function with the
same bound; all these derivatives are bounded. The denominator is
at least $\E e^{-L_\varphi\abs{(t\Sigma)^{1/2}\mathbf g}}$, which is bounded below by a positive
constant uniformly in $t\le1$ and $\omega\in\Omega$, since $\Omega$ is compact.
Then the right side of \eqref{eq:flow-ch-integral} is infinitely
differentiable, with all its derivatives and the function itself bounded. The
derivatives of $\varphi$ of order at least one are bounded by hypothesis, and
this proves (c).
\end{proof}

The third tool is a bound for linear equations of the type
\eqref{eq:flow-equation}. Its proof uses the stochastic representation of the
solution and needs no lower bound on the diffusion matrix. If the
diffusion matrix is degenerate, the representing diffusion moves in fewer
directions.

\begin{lemma}[A linear estimate]\label{lem:flow-linear}
Let $T_*>0$ and suppose that $[0,T_*]$ is the union of finitely many closed
intervals with disjoint interiors. Let $\mathcal A(t)\succeq0$ be constant on each
of them. Let $\phi\colon[0,T_*]\times\R^\ell\to\R^\ell$ and
$\Upsilon\colon[0,T_*]\times\R^\ell\to\R$ be bounded and measurable, with $\phi$
Lipschitz in $x$ uniformly in $t$, and with $t\mapsto\sup_x\abs{\Upsilon(t,x)}$
measurable. Let $\psi\colon[0,T_*]\times\R^\ell\to\R$ be continuous, with
$\abs{\psi(t,x)}+\abs{\nabla\psi(t,x)}\le c(1+\abs x)$, and suppose that on each
of the intervals $\psi$ coincides with a function of class $C^{1,2}$ and satisfies
\[
 -\partial_t\psi=\mathcal A(t)\mathbin{:}D^2\psi+\langle\phi,\nabla\psi\rangle+\Upsilon
\]
in its interior. Then, for every $t\in[0,T_*]$,
\begin{equation}\label{eq:flow-linear}
 \sup_x\abs{\psi(t,x)}\le\sup_x\abs{\psi(T_*,x)}
 +\int_t^{T_*}\sup_x\abs{\Upsilon(t',x)}\dd t'.
\end{equation}
\end{lemma}

\begin{proof}
Fix $t$ and $x$, let $W$ be a standard Brownian motion in $\R^\ell$, and let
$\mathcal X$ solve
\[
 \mathcal X_{t'}=x+\int_t^{t'}\phi(t'',\mathcal X_{t''})\dd t''
 +\int_t^{t'}\bigl(2\mathcal A(t'')\bigr)^{1/2}\dd W_{t''},
 \qquad t\le t'\le T_*.
\]
Since the noise term does not depend on $\mathcal X$ and $\phi$ is bounded and Lipschitz
in space, this equation has a unique continuous adapted solution, obtained
pathwise by Picard iteration and Gronwall's inequality. On each of the finitely
many intervals, It\^o's formula \cite[Theorem~3.3.6]{karatzas-shreve-1991} applies
to the $C^{1,2}$ function that coincides with $\psi$ there. Adding the results and
using the equation for $\psi$, we obtain
\[
 \psi(T_*,\mathcal X_{T_*})-\psi(t,x)
 =-\int_t^{T_*}\Upsilon(t',\mathcal X_{t'})\dd t'
 +\int_t^{T_*}\bigl\langle\nabla\psi(t',\mathcal X_{t'}),
 \bigl(2\mathcal A(t')\bigr)^{1/2}\dd W_{t'}\bigr\rangle.
\]
We have $\E\sup_{t'}\abs{\mathcal X_{t'}}^2<\infty$ because $\phi$ is bounded, and the
stochastic integral has mean zero because $\nabla\psi$ has linear growth. Taking expectations, we
find
\[
 \psi(t,x)=\E\psi(T_*,\mathcal X_{T_*})+\E\int_t^{T_*}\Upsilon(t',\mathcal X_{t'})\dd t',
\]
and \eqref{eq:flow-linear} follows. This is the Feynman--Kac representation
\cite[Theorem~5.7.6]{karatzas-shreve-1991}, which we derived directly because
$\psi$ is of class $C^{1,2}$ only on each of the intervals.
\end{proof}

\section{Spatial derivatives and the step limit}\label{sec:flow-spatial}

We first bound the spatial derivatives of step flows. The constants depend only
on the durations and the matrices, not on the masses or the partitions.

\begin{lemma}[Spatial bounds for step flows]\label{lem:flow-spatial}
Let $F$ be a step flow with terminal function $f$.
\begin{enumerate}[label=\textup{(\alph*)}]
\item If $f\in\mathcal C^2$, then, for every $\tau$,
 \[
  \sup_x\abs{\nabla F(\tau,x)}\le L_f,\qquad
  \sup_x\abs{D^2F(\tau,x)}\le K_f+L_f^2,
 \]
 where $L_f=\sup_x\abs{\nabla f(x)}$, $K_f=\sup_x\abs{D^2f(x)}$, and $\abs{\cdot}$
 is the operator norm. This holds for stages of any duration and any masses in
 $[0,1]$.
\item If $f\in\mathcal C^\infty$ and
 $2\ell b\norm{D^2f}_\infty\sum_j\lambda_j\abs{B_j}\le1$, then
 $\norm{D^2F(\tau,\cdot)}_\infty\le2\norm{D^2f}_\infty$ for every $\tau$.
\item Under the hypotheses of \textup{(b)}, for every $r\ge3$ there exists $C$,
 depending only on $\ell$, $b$, $r$, $\sum_j\lambda_j\abs{B_j}$, and
 $\norm{D^{r'}f}_\infty$ for $2\le r'\le r$, such that
 $\norm{D^rF(\tau,\cdot)}_\infty\le C$ for every $\tau$.
\end{enumerate}
\end{lemma}

\begin{proof}
For (a), we use the weighted martingale-variance decomposition in the proof of
\cite[Lemma~3.1]{oai-barriers}, with a general vector terminal function in place
of the scalar terminal function $\log(2\cosh z)$.
The gradient bound in (a) follows from \cref{lem:flow-cole-hopf}(a), applied to
each step of \eqref{eq:flow-step-solution}. For the Hessian, fix $(\tau,x)$ and a
unit vector $v$. By \eqref{eq:flow-step-solution}, $F(\tau,x)$ is the root value
of a finite hierarchy, whose levels are the pieces after $\tau$; if $\tau$ lies
inside a piece, the first level is the remaining part of that piece. We list the
levels from the root as $1,\dots,M$, with covariances $\Sigma_j$ and masses
$\mu_j\in[0,1]$, and let $\Psi_M=f$ and $\Psi_{j-1}=T_{\mu_j,\Sigma_j}\Psi_j$, so
that $\Psi_0=F(\tau,\cdot)$. Each $\Psi_j$ belongs to $\mathcal C^2$ by
\cref{lem:flow-cole-hopf}(a). Let $z_0=x$, and given $z_0,\dots,z_{j-1}$, let $z_j$
have the law $\mathcal K_{\mu_j,\Sigma_j,\Psi_j}(z_{j-1},\cdot)$ of
\eqref{eq:inp-kernel}. The weights in \eqref{eq:flow-tilted-gradient} are the
densities of this kernel, and therefore
$\partial_v\Psi_{j-1}(z)=\E[\partial_v\Psi_j(z_j)\mid z_{j-1}=z]$.
Differentiating \eqref{eq:flow-tilted-gradient} once more in the direction $v$,
under the expectation as in the proof of \cref{lem:flow-cole-hopf}(a), we find
that
\[
 \partial_v^2\Psi_{j-1}(z)=\E\bigl[\partial_v^2\Psi_j(z_j)\bigm|z_{j-1}=z\bigr]
 +\mu_j\Var\bigl(\partial_v\Psi_j(z_j)\bigm|z_{j-1}=z\bigr).
\]
The process $\mathcal M_j=\partial_v\Psi_j(z_j)$ is a bounded martingale, by the
Markov property, and the conditional variance in the display, evaluated at
$z=z_{j-1}$, has expectation $\E(\mathcal M_j-\mathcal M_{j-1})^2$. Iterating the
display from $j=1$ to $j=M$, we obtain
\[
 \partial_v^2F(\tau,x)=\E\,\partial_v^2f(z_M)
 +\sum_{j=1}^M\mu_j\E\bigl(\mathcal M_j-\mathcal M_{j-1}\bigr)^2.
\]
Since $0\le\mu_j\le1$ and the increments of a martingale are orthogonal, the sum
lies between $0$ and $\sum_j\E(\mathcal M_j-\mathcal M_{j-1})^2=\Var(\mathcal M_M)\le
L_f^2$. We obtain $-K_f\le\partial_v^2F(\tau,x)\le K_f+L_f^2$ for every unit vector
$v$, and since $D^2F(\tau,x)$ is symmetric, this is the Hessian bound. The bound
does not depend on $M$, on the covariances, or on the order of the masses. This
is the argument of the proof of \cref{b:lem:os-prefix} below, with a derivative
in the field in place of a derivative in the terminal perturbation.

For (b) and (c), we note that, by \cref{lem:flow-cole-hopf}(c) with a one-point
parameter set, $F(\tau,\cdot)\in\mathcal C^\infty$ for every $\tau$, all spatial
derivatives of $F$ are continuous, and on each closed piece they are continuously
differentiable in $\tau$ and satisfy the equations obtained by differentiating
\eqref{eq:flow-equation} in $x$. On each piece, each spatial derivative of
$F$ is bounded. For $r\ge1$, let
$\Theta_r(\tau)=\max_{\abs\iota=r}\sup_x\abs{D^\iota F(\tau,x)}$. It is bounded on
$[0,N]$ and lower semicontinuous, hence measurable. On stage $j$ we write
$\lambda(\tau)\abs{B(\tau)}=\lambda_j\abs{B_j}$.

Differentiating \eqref{eq:flow-equation} in $x_i$ and $x_{i'}$, we find that
$\psi=\partial_{ii'}F$ satisfies, on stage $j$,
\[
 -\partial_\tau\psi=\frac{b\lambda_j}{2}B_j\mathbin{:}D^2\psi
 +b\lambda_jm_j\bigl\langle B_j\nabla F,\nabla\psi\bigr\rangle
 +b\lambda_jm_j\bigl\langle\nabla\partial_iF,B_j\nabla\partial_{i'}F\bigr\rangle.
\]
We apply \cref{lem:flow-linear} with $\mathcal A=(b\lambda_j/2)B_j$, with the drift
$\phi=b\lambda_jm_jB_j\nabla F$, which is bounded by (a) and Lipschitz in $x$
since $D^2F$ is bounded on each piece, and with the last term as $\Upsilon$.
Using $0\le m_j\le1$ and $\abs{\nabla\partial_iF}\le\sqrt\ell\,\Theta_2$, we find
that $\abs\Upsilon\le\ell b\lambda_j\abs{B_j}\Theta_2^2$, and \cref{lem:flow-linear}
yields
\[
 \Theta_2(\tau)\le \Xi(\tau),\qquad
 \Xi(\tau)=\norm{D^2f}_\infty+\ell b\int_\tau^N\lambda(\tau')\abs{B(\tau')}\,
 \Theta_2(\tau')^2\dd\tau'.
\]
If $\norm{D^2f}_\infty=0$, then $\Xi(\tau)\le C\int_\tau^N\Xi(\tau')\dd\tau'$, and
Gronwall's inequality implies that $\Xi=0$. Otherwise $\Xi$ is positive and absolutely
continuous, and $-\Xi'=\ell b\lambda\abs B\Theta_2^2\le\ell b\lambda\abs B\,\Xi^2$
almost everywhere. Then $(1/\Xi)'\le\ell b\lambda\abs B$, and integrating from
$\tau$ to $N$, we obtain
\[
 \frac1{\Xi(\tau)}\ge\frac1{\norm{D^2f}_\infty}-\ell b\sum_j\lambda_j\abs{B_j}
 \ge\frac1{2\norm{D^2f}_\infty},
\]
where we used the hypothesis of (b) in the last step. This proves (b).

For (c), let $\abs\iota=r\ge3$. By the Leibniz rule,
\begin{multline*}
 -\partial_\tau D^\iota F=\frac{b\lambda_j}{2}B_j\mathbin{:}D^2D^\iota F
 +b\lambda_jm_j\bigl\langle B_j\nabla F,\nabla D^\iota F\bigr\rangle\\
 +\frac{b\lambda_jm_j}{2}\sum_{0<\iota'<\iota}\binom{\iota}{\iota'}
 \bigl\langle\nabla D^{\iota'}F,B_j\nabla D^{\iota-\iota'}F\bigr\rangle,
\end{multline*}
where $0<\iota'<\iota$ means $\iota'\le\iota$ componentwise with
$\iota'\notin\{0,\iota\}$. The two factors in the last sum have orders
$\abs{\iota'}+1$ and $r-\abs{\iota'}+1$, both between $2$ and $r$. In the terms
with $\abs{\iota'}\in\{1,r-1\}$, one factor has order $r$ and the other order $2$;
the remaining terms involve only orders between $3$ and $r-1$. By
\cref{lem:flow-linear}, applied as before with the last sum as the source, we
have
\[
 \Theta_r(\tau)\le\norm{D^rf}_\infty+C_{\ell,r}b\int_\tau^N\lambda(\tau')
 \abs{B(\tau')}\Bigl\{\Theta_2\Theta_r+\sum_{q=3}^{r-1}\Theta_q\Theta_{r+2-q}\Bigr\}
 (\tau')\dd\tau'.
\]
By (b) and induction on $r$, the sum over $q$ is bounded. Gronwall's inequality
then bounds $\Theta_r$, with a constant of the stated type. This proves (c).
\end{proof}

\begin{proof}[Proof of \cref{lem:flow-step-limit}]
We begin with (a). The gradient bound is \cref{lem:flow-spatial}(a). By
\cref{lem:inp-steps}(a), each integration step is monotone and commutes with
the addition of constants, which implies that it does not increase the supremum
of the difference of two functions. Composing the steps, we find that
$\sup\abs{F-F'}\le\sup\abs{f-f'}$. For \eqref{eq:flow-L1}, we refine the partitions
so that both $m_j$ and $\widetilde m_j$ are constant on each piece; by
\cref{lem:inp-steps}(b) this does not change $F$ or $\widetilde F$. Because $B_j$
is symmetric, we have
\[
 m_j\langle\nabla F,B_j\nabla F\rangle-\widetilde m_j\langle\nabla\widetilde F,
 B_j\nabla\widetilde F\rangle
 =m_j\bigl\langle B_j(\nabla F+\nabla\widetilde F),\nabla(F-\widetilde F)\bigr\rangle
 +(m_j-\widetilde m_j)\langle\nabla\widetilde F,B_j\nabla\widetilde F\rangle.
\]
Then $\mathcal D=F-\widetilde F$ satisfies, on stage $j$,
\[
 -\partial_\tau \mathcal D=\frac{b\lambda_j}{2}B_j\mathbin{:}D^2\mathcal D
 +\frac{b\lambda_jm_j}{2}\bigl\langle B_j(\nabla F+\nabla\widetilde F),\nabla \mathcal D\bigr\rangle
 +\frac{b\lambda_j}{2}(m_j-\widetilde m_j)\langle\nabla\widetilde F,B_j\nabla\widetilde F\rangle,
\]
with $\mathcal D(N,\cdot)=0$. By \cref{lem:flow-cole-hopf}(a) and (b), $\mathcal D$ is of class
$C^{1,2}$ on each piece, $\nabla \mathcal D$ is bounded, and the drift is bounded and
Lipschitz in $x$, since the Hessians of $F$ and $\widetilde F$ are bounded on each
piece. The source is bounded by $(b/2)\lambda_j\abs{B_j}\abs{m_j-\widetilde m_j}L_f^2$,
by the gradient bound. Then \cref{lem:flow-linear} implies \eqref{eq:flow-L1}.

For (b), by \eqref{eq:flow-L1}, we see that the step flows $F^{(l)}$ of any
sequence with $\int_0^1\abs{m_j^{(l)}-m_j}\to0$ form a Cauchy sequence for
uniform convergence. Interlacing two such sequences gives another one, and we
conclude that the limit does not depend on the sequence. The estimate
\eqref{eq:flow-L1} and the bound on $\sup\abs{F-F'}$ pass to uniform limits. For the
gradient, let $e_{l,l'}=\sup_{\tau,x}\abs{F^{(l)}-F^{(l')}}$, let $v$ be a unit
vector, and let $a>0$. By Taylor's formula and the Hessian bound of
\cref{lem:flow-spatial}(a), which holds for every $l$, the difference quotient
$(F^{(l)}(\tau,x+av)-F^{(l)}(\tau,x))/a$ differs from $\partial_vF^{(l)}(\tau,x)$ by
at most $(K_f+L_f^2)a/2$. Comparing the difference quotients of $F^{(l)}$ and
$F^{(l')}$, we find that
\[
 \sup_{\tau,x}\abs{\partial_vF^{(l)}(\tau,x)-\partial_vF^{(l')}(\tau,x)}
 \le\frac{2e_{l,l'}}a+(K_f+L_f^2)a.
\]
Choosing $a=e_{l,l'}^{1/2}$, we see that the gradients of the step flows are
uniformly Cauchy. Their uniform limit is continuous, and by the fundamental
theorem of calculus it is the gradient of $F$, which therefore satisfies
$\abs{\nabla F}\le L_f$.

For (c), let $F^{(l)}$ be step flows of stages $1,\dots,N$ with masses
$m_j^{(l)}$ as in (b), and let $f_{\rm in}^{(l)}=F^{(l)}(N',\cdot)$, which is the
output of the step flow of stages $N'+1,\dots,N$. Let $\bar F^{(l)}$ and $\bar F$ be the
step flow of stages $1,\dots,N'$ with masses $m_j^{(l)}$ and terminal function
$f_{\rm in}$, and its limit. On $[0,N']$, the function $F^{(l)}$ is the step flow
of stages $1,\dots,N'$ with terminal function $f^{(l)}_{\rm in}$. By part (a), we have
\[
 \sup\abs{F^{(l)}-\bar F}\le\sup\abs{f^{(l)}_{\rm in}-f_{\rm in}}+\sup\abs{\bar F^{(l)}-\bar F}.
\]
Both terms tend to zero, by (b) for the stages $N'+1,\dots,N$ and for the stages
$1,\dots,N'$. This proves (c).

For (d), we first note that the hypothesis of \cref{lem:flow-spatial}(b) does
not involve the masses and holds for every step flow. Using
\cref{lem:flow-spatial}, we see that all spatial derivatives of the step flows are
bounded by constants of the type stated there, uniformly in $l$. Let $l,l'$ be
two indices, refine the partitions so that $m_j^{(l)}$ and $m_j^{(l')}$ are
constant on each piece, and let $\mathcal D=F^{(l)}-F^{(l')}$. Then $\mathcal D$
satisfies the equation in the proof of (a), with $F=F^{(l)}$,
$\widetilde F=F^{(l')}$, and the masses $m_j^{(l)}$, $m_j^{(l')}$ in place of
$m_j$, $\widetilde m_j$. For $\iota\ne0$, we apply $D^\iota$ to that equation and
use the Leibniz rule. We find that
\[
 -\partial_\tau D^\iota\mathcal D=\frac{b\lambda_j}{2}B_j\mathbin{:}D^2D^\iota\mathcal D
 +\frac{b\lambda_jm^{(l)}_j}{2}\bigl\langle B_j(\nabla F^{(l)}+\nabla F^{(l')}),
 \nabla D^\iota\mathcal D\bigr\rangle+\Upsilon_\iota,
\]
where $\Upsilon_\iota$ is the sum of the terms
$(b\lambda_jm^{(l)}_j/2)\binom\iota{\iota'}\langle B_jD^{\iota'}\nabla(F^{(l)}+F^{(l')}),
\nabla D^{\iota-\iota'}\mathcal D\rangle$ with $0\ne\iota'\le\iota$ and of
$(b\lambda_j/2)(m^{(l)}_j-m^{(l')}_j)D^\iota\langle\nabla F^{(l')},B_j\nabla F^{(l')}\rangle$.
The terms with $\abs{\iota'}=1$ are linear in the derivatives of $\mathcal D$ of
order $\abs\iota$, with coefficients bounded by $C\lambda_j\abs{B_j}$; the terms
with $\abs{\iota'}\ge2$ involve derivatives of $\mathcal D$ of order less than
$\abs\iota$; and the last term is bounded by
$C\lambda_j\abs{B_j}\abs{m^{(l)}_j-m^{(l')}_j}$, with $C$ depending on the
derivatives of $F^{(l')}$ of order at most $\abs\iota+1$. By
\cref{lem:flow-linear}, induction on $\abs\iota$, and Gronwall's inequality, we
have
\[
 \sup_{\tau,x}\abs{D^\iota\mathcal D(\tau,x)}
 \le C\sum_j\lambda_j\int_0^1\abs{m^{(l)}_j-m^{(l')}_j}\dd\tau.
\]
It follows that every spatial derivative of the step flows converges uniformly,
and the limit $F$ has continuous spatial derivatives of all orders, which are the
limits of those of the step flows. Letting $l'\to\infty$ in the last
display gives \eqref{eq:flow-L1-derivatives}, and the Hessian bound follows from
\cref{lem:flow-spatial}(b).

Now let each $m_j$ be continuous, and choose the step flows with the sampled
masses $m_j^{\mathfrak p_l}$ for partitions with $\abs{\mathfrak p_l}\to0$; then
$m_j^{\mathfrak p_l}\to m_j$ pointwise and in $L^1$. The step flows are
continuous in $\tau$ and continuously differentiable on each piece, and, for
$j-1\le\tau\le\tau_1\le j$, they satisfy
\begin{multline*}
 F^{(l)}(\tau,x)=F^{(l)}(\tau_1,x)\\
 +\int_\tau^{\tau_1}\frac{b\lambda_j}{2}
 \Bigl\{B_j\mathbin{:}D^2F^{(l)}+m_j^{\mathfrak p_l}(\tau'-j+1)
 \bigl\langle\nabla F^{(l)},B_j\nabla F^{(l)}\bigr\rangle\Bigr\}(\tau',x)\dd\tau'.
\end{multline*}
By bounded convergence, the same identity holds for $F$ and $m_j$. Its integrand
is continuous in $\tau'$, since $\nabla F$ and $D^2F$ are uniform limits of
continuous functions and $m_j$ is continuous. This shows that $F$ is
continuously differentiable in $\tau$ on $[j-1,j]$ and solves
\eqref{eq:flow-equation}. For uniqueness, let $F'$ be another solution with the
stated properties. The difference $\mathcal D=F-F'$ satisfies the equation in
the proof of (a), with $\widetilde F=F'$, $\widetilde m_j=m_j$, and
$\mathcal D(N,\cdot)=0$. Its drift is bounded and Lipschitz in $x$, since the first
and second derivatives of $F$ and $F'$ are bounded, and $\mathcal D$ and
$\nabla\mathcal D$ have linear growth.
Then \cref{lem:flow-linear}, with $\Upsilon=0$, implies that $\mathcal D=0$.
\end{proof}

\section{Parameter derivatives}\label{sec:flow-parameters}

We now prove \cref{prop:flow-uniform}. Throughout this section we assume
\ref{H1}--\ref{H5}, and on stage $j$ we abbreviate
\[
 \mathcal A_j=\frac{bh}{2}\kappa_jB_j,\qquad
 \mathcal V_j=\frac{bh}{2}\kappa_jm_jB_j,
\]
so that \eqref{eq:flow-equation} reads
$-\partial_\tau F=\mathcal A_j\mathbin{:}D^2F+\langle\nabla F,\mathcal V_j\nabla F\rangle$.
Both matrices are symmetric and depend on $(h,k)$, and $\mathcal V_j$ also depends
on $\tau$ through the mass. Their derivatives
$\partial^{\gamma'}\mathcal A_j=(b/2)\partial^{\gamma'}(h\kappa_jB_j)$ and
$\partial^{\gamma'}\mathcal V_j$ are bounded by the suprema listed in
\cref{prop:flow-uniform}(ii). Note that $\partial_h\mathcal A_j$ does not vanish
at $h=0$; these derivatives are bounded but not small. The proof has two
parts. In \cref{lem:flow-step-bounds}, we prove the bounds for the step flows
with sampled masses, with constants that do not depend on the partitions. We
then pass to the limit.

\begin{lemma}[Uniform bounds for step flows]\label{lem:flow-step-bounds}
Let $\mathfrak p$ be partitions, and let $F^{\mathfrak p}(\tau,h,k,x)$ be the step
flow with the sampled masses $m_j^{\mathfrak p}(\cdot,h,k)$ and terminal function
$f(h,k,\cdot)$, for $(h,k)\in\mathcal U_0$. Then $F^{\mathfrak p}$ is infinitely
differentiable in $(h,k,x)$, its derivatives $\partial^\gamma D^\iota F^{\mathfrak p}$
are continuous in $(\tau,h,k,x)$ and, on each closed piece, continuously
differentiable in $\tau$, with $\partial_\tau$ commuting with $\partial^\gamma
D^\iota$. For every $(\gamma,\iota)\ne(0,0)$, the function
$\partial^\gamma D^\iota F^{\mathfrak p}$ is bounded on $[0,N]\times\mathcal
U_0\times\R^\ell$ by a constant $C$ of the type stated in
\cref{prop:flow-uniform}(ii), which does not depend on $\mathfrak p$.
\end{lemma}

\begin{proof}
On the piece $(j,i)$, the step \eqref{eq:flow-step-solution} has mass
$m_j(\tau_{j,i-1},h,k)$ and covariance $t\,bh\kappa_j(h,k)B_j(h,k)$, with
$t=j-1+\tau_{j,i}-\tau\in[0,1]$. Both are smooth functions of $(h,k)$ on the
compact convex set $\mathcal U_0$, and the covariance is positive semidefinite
for $h\ge0$. Applying \cref{lem:flow-cole-hopf}(c) to the pieces one after
another, from the innermost, we obtain the regularity statements. The
commutation of $\partial_\tau$ with $\partial^\gamma D^\iota$ holds because
$F^{\mathfrak p}$ is infinitely differentiable in $(\tau,h,k,x)$ on each closed
piece. It remains to prove the bounds, which we do by induction. We drop the
superscript $\mathfrak p$ and write $F_\gamma=\partial^\gamma F$ and
\[
 \Theta_{\gamma,r}(\tau)=\max_{\abs\iota=r}\sup_{x}
 \abs{D^\iota F_\gamma(\tau,h,k,x)},
\]
for fixed $(h,k)\in\mathcal U_0$; all bounds below are uniform in $(h,k)$.

\paragraph{Step 1: spatial derivatives.} For $\gamma=0$ and $r\ge1$, the bound
on $\Theta_{0,r}$ is \cref{lem:flow-spatial}, which applies since
$2\ell b\norm{D^2f}_\infty\sum_j\lambda_j\abs{B_j}\le2\ell b\,h_0\Lambda_*
\norm{D^2f}_\infty\le1$ by \ref{H5}. We have $\Theta_{0,2}\le2\norm{D^2f}_\infty$.

\paragraph{Step 2: the differentiated equations.} Let $\gamma\ne0$. Applying
$\partial^\gamma$ to \eqref{eq:flow-equation} and separating the terms that
contain $F_\gamma$, we find that, on each piece of stage $j$,
\begin{equation}\label{eq:flow-Fgamma}
 -\partial_\tau F_\gamma=\mathcal A_j\mathbin{:}D^2F_\gamma
 +2\langle\mathcal V_j\nabla F,\nabla F_\gamma\rangle+\Upsilon_\gamma,
\end{equation}
with
\begin{equation}\label{eq:flow-Ugamma}
 \Upsilon_\gamma=\sum_{0\ne\gamma'\le\gamma}\binom{\gamma}{\gamma'}
 \partial^{\gamma'}\!\mathcal A_j\mathbin{:}D^2F_{\gamma-\gamma'}
 +\sum_{\substack{\gamma'+\gamma''+\gamma'''=\gamma\\ \gamma''\ne\gamma,\ \gamma'''\ne\gamma}}
 \frac{\gamma!}{\gamma'!\,\gamma''!\,\gamma'''!}
 \bigl\langle\nabla F_{\gamma''},\partial^{\gamma'}\!\mathcal V_j\nabla F_{\gamma'''}\bigr\rangle,
\end{equation}
where the second sum runs over the triples of multi-indices with
$\gamma'+\gamma''+\gamma'''=\gamma$ other than $(0,\gamma,0)$ and $(0,0,\gamma)$; since
$\gamma\ne0$, these are the triples with $\gamma''\ne\gamma$ and $\gamma'''\ne\gamma$.
Every $F_{\gamma''}$, $F_{\gamma'''}$, and $F_{\gamma-\gamma'}$ appearing in
$\Upsilon_\gamma$ has parameter order strictly less than $\abs\gamma$, and
$F_0=F$ enters only through $\nabla F$ and $D^2F$. Applying $D^\iota$ with
$\abs\iota=r$ to \eqref{eq:flow-Fgamma}, we obtain
\begin{multline}\label{eq:flow-Fgamma-iota}
 -\partial_\tau D^\iota F_\gamma=\mathcal A_j\mathbin{:}D^2D^\iota F_\gamma
 +2\langle\mathcal V_j\nabla F,\nabla D^\iota F_\gamma\rangle\\
 +2\sum_{0\ne\iota'\le\iota}\binom{\iota}{\iota'}
 \bigl\langle\mathcal V_j\nabla D^{\iota'}F,\nabla D^{\iota-\iota'}F_\gamma\bigr\rangle
 +D^\iota\Upsilon_\gamma,
\end{multline}
with terminal value $D^\iota F_\gamma(N,\cdot)=\partial^\gamma D^\iota f$.

\paragraph{Step 3: derivative bounds by weight.} We prove the bound on
$\Theta_{\gamma,r}$ by induction on $\abs\gamma$, and, for fixed $\gamma$, by
induction on $r=0,1,2,\dots$. \Cref{tab:flow-weights} lists the terms of
\eqref{eq:flow-Fgamma-iota} and the derivatives of $F$ that they contain. Let
$\varpi=r+2\abs\gamma$ be the weight of $D^\iota F_\gamma$. We read off from the
table that every derivative $\partial^{\gamma^\circ}D^{\iota^\circ}F$ other than
the unknown $D^\iota F_\gamma$ has $\gamma^\circ\le\gamma$ and weight
$\abs{\iota^\circ}+2\abs{\gamma^\circ}\le\varpi$. Moreover, either
$\abs{\gamma^\circ}<\abs\gamma$, or $\gamma^\circ=\gamma$ and $\abs{\iota^\circ}\le r$.
The terms with $\gamma^\circ=\gamma$ and $\abs{\iota^\circ}=r$ are those in the
second row. They are linear in the derivatives of order $r$ of $F_\gamma$, with
coefficients bounded by $2\ell r\abs{\mathcal V_j}\Theta_{0,2}$. In the fourth
row, a parameter derivative is paid for by two spatial derivatives. Indeed, when
$\partial^{\gamma'}$ falls on the coefficient $\mathcal A_j$ of $D^2$, the
derivative $F_{\gamma-\gamma'}$ appears with $r+2$ spatial derivatives, and its
weight $r+2+2(\abs\gamma-\abs{\gamma'})$ is at most $\varpi$ because
$\abs{\gamma'}\ge1$.

\begin{table}[ht]
\centering
\small
\caption{The terms of \eqref{eq:flow-Fgamma-iota}, for $\abs\iota=r$ and
$\varpi=r+2\abs\gamma$. The second column lists the derivatives of $F$ that each term
contains, and the third says how they are controlled.}\label{tab:flow-weights}
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.33\textwidth}>{\raggedright\arraybackslash}p{0.36\textwidth}>{\raggedright\arraybackslash}p{0.23\textwidth}@{}}
\toprule
Term & Derivatives of $F$ & Control\\
\midrule
$\mathcal A_j\mathbin{:}D^2D^\iota F_\gamma$,
$2\langle\mathcal V_j\nabla F,\nabla D^\iota F_\gamma\rangle$
 & the unknown $D^\iota F_\gamma$ & operator of \cref{lem:flow-linear}\\
$\langle\mathcal V_j\nabla D^{\iota'}F,\nabla D^{\iota-\iota'}F_\gamma\rangle$,
$\abs{\iota'}=1$
 & $D^2F$; derivatives of order $r$ of $F_\gamma$ (weight $\varpi$)
 & Gronwall's inequality\\
the same, $\abs{\iota'}\ge2$
 & $D^{\abs{\iota'}+1}F$ (weight $\le r+1$); derivatives of order
 $r-\abs{\iota'}+1<r$ of $F_\gamma$ & induction on $r$\\
$\partial^{\gamma'}\mathcal A_j\mathbin{:}D^2D^\iota F_{\gamma-\gamma'}$,
$\gamma'\ne0$
 & order $r+2$ of $F_{\gamma-\gamma'}$, weight $\le\varpi$ & induction on $\abs\gamma$\\
$\langle D^{\iota_1}\nabla F_{\gamma''},\partial^{\gamma'}\mathcal V_j
D^{\iota_2}\nabla F_{\gamma'''}\rangle$, $\iota_1+\iota_2=\iota$
 & orders $\abs{\iota_1}+1$, $\abs{\iota_2}+1\le r+1$ of $F_{\gamma''}$,
 $F_{\gamma'''}$ with $\abs{\gamma''},\abs{\gamma'''}<\abs\gamma$; weights $<\varpi$
 & induction on $\abs\gamma$\\
\bottomrule
\end{tabular}
\end{table}

In the last row, $\gamma''=0$ or $\gamma'''=0$ is allowed; the corresponding
factor is a spatial derivative of $F$ of order at most $r+1\le\varpi-1$, bounded in
Step~1. By the induction hypothesis and Step~1, we conclude that all terms of
\eqref{eq:flow-Fgamma-iota} other than those in the first two rows are bounded by
a constant $c_0$. The coefficients in the second row are bounded by
$c_1=2\ell r\max_j\sup\abs{\mathcal V_j}\cdot2\norm{D^2f}_\infty$.

\paragraph{Step 4: the estimate.} We apply \cref{lem:flow-linear} to each
$\psi=D^\iota F_\gamma$ with $\abs\iota=r$, with $\mathcal A=\mathcal A_j$ on stage
$j$, the drift $\phi=2\mathcal V_j\nabla F$, and the source formed by all other
terms of \eqref{eq:flow-Fgamma-iota}. The drift is bounded by Step~1, and it is
Lipschitz in $x$ since $D^2F$ is bounded. By the regularity statement, $\psi$,
$\nabla\psi$, and the source are bounded and continuous on each piece. Taking the
maximum over $\abs\iota=r$, we find
\[
 \Theta_{\gamma,r}(\tau)\le\max_{\abs\iota=r}\sup_x\abs{\partial^\gamma D^\iota f}
 +\int_\tau^N\bigl\{c_1\Theta_{\gamma,r}(\tau')+c_0\bigr\}\dd\tau',
\]
and by Gronwall's inequality we have
$\Theta_{\gamma,r}\le(\max_{\abs\iota=r}\sup\abs{\partial^\gamma D^\iota f}+Nc_0)e^{Nc_1}$.
For $r=0$ the second row is absent and $c_1=0$. This completes the induction.

The constant obtained for $(\gamma,r)$ involves the bounds for the pairs
$(\gamma^\circ,r^\circ)$ listed in \cref{tab:flow-weights}, the bounds on
$\partial^{\gamma'}\mathcal A_j$ and $\partial^{\gamma'}\mathcal V_j$ for
$\gamma'\le\gamma$, and the terminal values. Following the induction back, it
depends only on $\ell$, $N$, $b$, $h_1$, $\gamma$, $r$, and the suprema of
$\abs{\partial^{\gamma'}(\kappa_jB_j)}$, $\abs{\partial^{\gamma'}(\kappa_jm_jB_j)}$, and
$\abs{\partial^{\gamma'}D^{\iota'}f}$ over $\gamma'\le\gamma$ and
$0<\abs{\iota'}+2\abs{\gamma'}\le\varpi$, as claimed. None of them depends on
$\mathfrak p$, since the sampled masses have the same values and
$(h,k)$-derivatives as $m_j$ at the sampling points.
\end{proof}

\begin{proof}[Proof of \cref{prop:flow-uniform}]
Let $\mathfrak p_l$ be partitions with $\abs{\mathfrak p_l}\to0$, and let $F^{\mathfrak p_l}$
be the step flows with the sampled masses. The proof has three steps: convergence
of the values, convergence of the derivatives, and the equations.

\paragraph{Step 1: values.} By \ref{H3}, $m_j$ is uniformly continuous on
$[0,1]\times\mathcal U$, and we have
$\varepsilon_l=\sup_{\mathcal U}\sum_j\int_0^1\abs{m_j^{\mathfrak p_l}-m_j}\dd\tau\to0$.
Then \cref{lem:flow-step-limit}(a) and (b), applied at each $(h,k)$ with
$\sum_j\lambda_j\abs{B_j}\le h_0\Lambda_*$, give
\[
 \sup_{[0,N]\times\mathcal U_0\times\R^\ell}\abs{F^{\mathfrak p_l}-F}
 \le\frac b2\sup\abs{\nabla_xf}^2\,h_0\Lambda_*\,\varepsilon_l\longrightarrow0.
\]

\paragraph{Step 2: derivatives.} Fix $(\gamma,\iota)\ne(0,0)$. By
\cref{lem:flow-step-bounds}, the functions $\partial^\gamma D^\iota F^{\mathfrak p_l}$ are
bounded uniformly in $l$, and so are their first derivatives in $x$ and in
$(h,k)$, whose weights are larger by one and by two, respectively. Their derivative in
$\tau$ is given on each piece by \eqref{eq:flow-Fgamma-iota}, and it is also bounded
uniformly in $l$, since it involves derivatives of weight at most
$\abs\iota+2\abs\gamma+2$. Since $\partial^\gamma D^\iota F^{\mathfrak p_l}$ is also continuous in
$\tau$, the family $\{\partial^\gamma D^\iota F^{\mathfrak p_l}\}_l$ is uniformly bounded and uniformly
Lipschitz on $[0,N]\times\mathcal U_0\times\R^\ell$. By the Arzel\`a--Ascoli
theorem and a diagonal argument over the compact sets
$[0,N]\times\mathcal U_0\times\{\abs x\le R\}$, every subsequence has a further
subsequence along which, for all $(\gamma,\iota)$ at once,
$\partial^\gamma D^\iota F^{\mathfrak p_l}$ converges uniformly on compact sets to a continuous
limit $L_{\gamma,\iota}$. We claim that $L_{\gamma,\iota}=\partial^\gamma
D^\iota F$, so that the limits do not depend on the subsequence and the
whole sequence converges. Indeed, $L_{0,0}=F$ by Step~1. If $\varphi_l\to\varphi$ and
$\partial_a\varphi_l\to\psi$ uniformly on compact sets, where $\partial_a$ is a
derivative in $h$, in $k$, or in a coordinate of $x$, then letting
$l\to\infty$ in $\varphi_l(y+\eta e_a)-\varphi_l(y)=\int_0^\eta\partial_a\varphi_l(y+\eta'e_a)\dd\eta'$
shows that $\partial_a\varphi=\psi$; at boundary points of $\mathcal U_0$, such as $h=0$,
we take $\eta$ of the sign that keeps $y+\eta e_a$ in the domain. Induction on $\abs\gamma+\abs\iota$
proves the claim. This proves (iv), the existence and continuity of all
derivatives in (ii), and the bounds in (ii), which pass to the limit. For the
linear growth, note that $\abs{F(\tau,h,k,x)-F(\tau,h,k,0)}\le\sup\abs{\nabla_xf}
\abs x$, that $\abs{F(\tau,h,k,0)-f(h,k,0)}\le N\sup\abs{\partial_\tau F}$ is
bounded by \eqref{eq:flow-equation} and the bounds on $\nabla F$ and $D^2F$, and
that $\sup_{\mathcal U}\abs{f(h,k,0)}$ is finite, since $f(\cdot,\cdot,0)$ is
continuous on the compact set $\mathcal U$. The constant obtained depends on this
supremum in addition to the derivative bounds.

\paragraph{Step 3: the equations.} Part (i) is \cref{lem:flow-step-limit}(d),
whose hypothesis holds at each $(h,k)\in\mathcal U_0$ by \ref{H5}. For (iii), on
stage $j$ and for $j-1\le\tau\le\tau_1\le j$, the step flows satisfy
\[
 F^{\mathfrak p_l}_\gamma(\tau)=F^{\mathfrak p_l}_\gamma(\tau_1)+\int_\tau^{\tau_1}
 \bigl\{\mathcal A_j\mathbin{:}D^2F^{\mathfrak p_l}_\gamma
 +2\langle\mathcal V_j^{\mathfrak p_l}\nabla F^{\mathfrak p_l},\nabla F^{\mathfrak p_l}_\gamma\rangle
 +\Upsilon^{\mathfrak p_l}_\gamma\bigr\}(\tau')\dd\tau',
\]
where $F^{\mathfrak p_l}_\gamma=\partial^\gamma F^{\mathfrak p_l}$ and where $\mathcal V_j^{\mathfrak p_l}$ and
$\Upsilon^{\mathfrak p_l}_\gamma$ are formed with the sampled masses. By \ref{H3}, the
$(h,k)$-derivatives of $m_j^{\mathfrak p_l}$ converge pointwise to those of $m_j$,
and by (iv) the derivatives of $F^{\mathfrak p_l}$ converge pointwise. All terms are bounded,
and by bounded convergence the same identity holds for $F_\gamma=\partial^\gamma F$,
with the masses $m_j$. Since its integrand is continuous in $\tau'$, the function
$F_\gamma$ is continuously differentiable in $\tau$ on $[j-1,j]$ and satisfies
\eqref{eq:flow-Fgamma}, which is the equation obtained by applying $\partial^\gamma$
to \eqref{eq:flow-equation}. This proves (iii).
\end{proof}

\section{Response to a change of the terminal function}\label{sec:flow-response}

We now compare a hierarchy with terminal function $f$ and the same hierarchy with
terminal function $f+V$, for a bounded function $V$. The first-order change is
the expectation of $V$ under the path law of the unperturbed hierarchy, and the
error is quadratic in $\norm V_\infty$, with a constant that does not depend on
the number of levels. For a hierarchy with $M$ levels, let $\Sigma_1,\dots,\Sigma_M
\succeq0$ be the covariances and $m_1,\dots,m_M\in[0,1]$ the masses, listed from
the root; the masses need not be ordered. For a function $f$ of linear growth, the
partial values are $\Psi^f_M=f$ and $\Psi^f_{j-1}=T_{m_j,\Sigma_j}\Psi^f_j$ for
$j=M,\dots,1$, as in \eqref{eq:inp-hierarchy-recursion}, and we write
$\mathcal Hf=\Psi_0^f$. For $z_0\in\R^\ell$, we let $\E_f$ denote expectation for
the Markov chain $z_0,z_1,\dots,z_M$ in which $z_j$ has law
$\mathcal K_{m_j,\Sigma_j,\Psi^f_j}(z_{j-1},\cdot)$ given $z_0,\dots,z_{j-1}$.
This is the path law of \cref{def:inp-hierarchy}(c), started at $z_0$ and without
the terminal draw of spins.

\begin{lemma}[Response of a hierarchy to its terminal function]\label{b:lem:os-prefix}
Let $f$ have linear growth, and let $V\colon\R^\ell\to\R$ be bounded and
measurable. Then, for every $z_0\in\R^\ell$,
\[
 \bigl\lvert\mathcal H(f+V)(z_0)-\mathcal Hf(z_0)-\E_f V(z_M)\bigr\rvert
 \le\frac12\norm V_\infty^2.
\]
The bound does not depend on the number $M$ of levels or on the covariances.
\end{lemma}

The argument is the martingale argument that the proof of
\cite[Lemma~3.1]{oai-barriers} uses to show that $\chi\le1-u^2$, and the
differentiation of a hierarchy behind \cref{inp:ob-hierarchy-derivative}, namely
\cite[Lemma~2.2]{oai-barriers}; we apply them to the terminal function instead
of the field.

\begin{proof}
For $0\le t\le1$, let $\Psi^t_j=\Psi^{f+tV}_j$ be the partial values with
terminal function $f+tV$. Using the monotonicity and the translation identity of
\cref{lem:inp-steps}(a), we have $\abs{\Psi_j^t-\Psi_j^0}\le t\norm V_\infty$, and
all integrals below are finite. Let $\E^t_j$ and $\Var_j^t$ denote the expectation and
the variance under the kernel $\mathcal K_{m_j,\Sigma_j,\Psi^t_j}(z,\cdot)$ for a
fixed starting point $z$. We claim that $t\mapsto\Psi^t_{j-1}(z)$ is twice
continuously differentiable, with
\begin{equation}\label{eq:flow-response-recursion}
 \partial_t\Psi^t_{j-1}(z)=\E^t_j\bigl[\partial_t\Psi^t_j\bigr],\qquad
 \partial_t^2\Psi^t_{j-1}(z)=\E^t_j\bigl[\partial_t^2\Psi^t_j\bigr]
 +m_j\Var^t_j\bigl(\partial_t\Psi^t_j\bigr),
\end{equation}
where the expectation and the variance act on the function $y\mapsto\partial_t\Psi^t_j(y)$
or $\partial^2_t\Psi^t_j(y)$. Indeed, let $\mathcal Z_t$ be random variables, twice
differentiable in $t$ with $\abs{\partial_t\mathcal Z_t}+\abs{\partial^2_t\mathcal Z_t}$ bounded and
$\E e^{\abs{\mathcal Z_0}}<\infty$, and let $\mathfrak a(t)=m^{-1}\log\E e^{m\mathcal Z_t}$ for some
$m\in(0,1]$. Differentiation under the expectation gives
$\mathfrak a'=\E[\partial_t\mathcal Z_t\,e^{m\mathcal Z_t}]/\E e^{m\mathcal Z_t}$ and
\[
 \mathfrak a''=\frac{\E[\partial_t^2\mathcal Z_t\,e^{m\mathcal Z_t}]}{\E e^{m\mathcal Z_t}}
 +m\Bigl(\frac{\E[(\partial_t\mathcal Z_t)^2\,e^{m\mathcal Z_t}]}{\E e^{m\mathcal Z_t}}
 -\Bigl(\frac{\E[\partial_t\mathcal Z_t\,e^{m\mathcal Z_t}]}{\E e^{m\mathcal Z_t}}\Bigr)^2\Bigr).
\]
For $m=0$ and $\mathfrak a(t)=\E\mathcal Z_t$, the same formulas hold without the variance term. We
apply this with $\mathcal Z_t=\Psi^t_j(z+\Sigma_j^{1/2}\mathbf g)$ and $m=m_j$; the weights
$e^{m_j\mathcal Z_t}/\E e^{m_j\mathcal Z_t}$ are the densities of the kernel
$\mathcal K_{m_j,\Sigma_j,\Psi^t_j}(z,\cdot)$. Starting from $\partial_t\Psi^t_M=V$
and $\partial^2_t\Psi^t_M=0$, induction on $M-j$ shows that the derivatives are
bounded and proves \eqref{eq:flow-response-recursion}.

Now let $z_0,\dots,z_M$ be the chain of the lemma for the terminal function $f+tV$,
and let $\mathcal M_j=\partial_t\Psi^t_j(z_j)$. By the first identity in
\eqref{eq:flow-response-recursion} and the Markov property, we have
$\E^t[\mathcal M_j\mid z_0,\dots,z_{j-1}]=\mathcal M_{j-1}$. This implies that
$(\mathcal M_j)$ is a martingale, with $\mathcal M_M=V(z_M)$ and
$\mathcal M_0=\partial_t\mathcal H(f+tV)(z_0)$. Iterating the second identity in
\eqref{eq:flow-response-recursion} from $\partial^2_t\Psi^t_M=0$, we obtain
\[
 \partial_t^2\mathcal H(f+tV)(z_0)=\sum_{j=1}^Mm_j\,\E^t\bigl(\mathcal M_j-\mathcal M_{j-1}\bigr)^2.
\]
Since $0\le m_j\le1$ and the increments of a martingale are orthogonal,
\[
 0\le\partial_t^2\mathcal H(f+tV)(z_0)\le\sum_{j=1}^M\E^t\bigl(\mathcal M_j-\mathcal M_{j-1}\bigr)^2
 =\E^tV(z_M)^2-\bigl(\E^tV(z_M)\bigr)^2\le\norm V_\infty^2.
\]
At $t=0$, we have $\partial_t\mathcal H(f+tV)(z_0)=\mathcal M_0=\E_fV(z_M)$. By
Taylor's formula with integral remainder,
\[
 \mathcal H(f+V)(z_0)-\mathcal Hf(z_0)-\E_fV(z_M)
 =\int_0^1(1-t)\,\partial_t^2\mathcal H(f+tV)(z_0)\dd t,
\]
which lies in $[0,\norm V_\infty^2/2]$. This completes the proof.
\end{proof}

We next apply \cref{b:lem:os-prefix} to the scalar hierarchy that integrates a
block of $\mathfrak n$ identical coordinates over a time interval $[0,t_\star]$, and pass to
a general distribution function. For a distribution function $\alpha$, an
integer $\mathfrak n\ge1$, a time $t_\star\in(0,1]$, and $f\in\mathcal C^2(\R)$, let
$\mathcal H^\alpha_{t_\star,\mathfrak n}f$ denote the value at $z=0$ of the flow of
\cref{lem:flow-step-limit}(b) with $\ell=1$, $N=1$, $B_1=1$, $\lambda_1=t_\star$,
mass $m_1(\tau)=\alpha(t_\star\tau)/\mathfrak n$, and terminal function $f$. If
$\alpha_n$ are step distribution functions with $\int_0^1\abs{\alpha_n-\alpha}\dd t\to0$,
then $\mathcal H^\alpha_{t_\star,\mathfrak n}f$ is the limit of the values at $z=0$ of the
step flows with masses $\alpha_n(t_\star\tau)/\mathfrak n$. In the scalar time
$t=t_\star\tau$, these step flows solve
$-\partial_tF=(b/2)\{\partial_{zz}F+(\alpha_n(t)/\mathfrak n)(\partial_zF)^2\}$ on each
interval where $\alpha_n$ is constant, with $F(t_\star,\cdot)=f$. For a general
$\alpha$, we do not assert that the limiting flow is differentiable in time at
the jumps of $\alpha$. Recall the function $\Phi=\Phi_\alpha$ and the diffusion
$Y$ of \eqref{b:eq:scalar-diffusion}.

\begin{corollary}[Response of the common part of a hierarchy]\label{cor:flow-prefix}
Let $\alpha$ be a distribution function, let $\mathfrak n\ge1$ be an integer, and let
$t_\star\in(0,1]$. Then $\mathcal H^\alpha_{t_\star,\mathfrak n}(\mathfrak n\Phi_\alpha(t_\star,\cdot))
=\mathfrak n\Phi_\alpha(0,0)$. Moreover, let $V\in C^2(\R)$ with $V$, $V'$, and $V''$ bounded,
and suppose that $\norm V_\infty\le\eps$. Then
\[
 \bigl\lvert\mathcal H^\alpha_{t_\star,\mathfrak n}\bigl(\mathfrak n\Phi_\alpha(t_\star,\cdot)+V\bigr)
 -\mathfrak n\Phi_\alpha(0,0)-\E V(Y_{t_\star})\bigr\rvert\le\frac{\eps^2}{2}.
\]
\end{corollary}

In \cref{sec:cubic}, we apply the corollary with $\mathfrak n=3$, with $t_\star$ the left
endpoint of the window, and with $V$ the difference between the restriction of
the local value to the diagonal and $3\Phi_\alpha(t_\star,\cdot)$, for which
$\eps=Ch^2k^2$. The initial segment of the hierarchy then contributes $\E V(Y_{t_\star})+O(h^4k^4)$. Note that
the law of $Y_{t_\star}$ does not depend on the perturbation.

\begin{proof}
Let $\alpha_n$ be step distribution functions with $t_\star$ among their breakpoints
and $\int_0^1\abs{\alpha_n-\alpha}\dd t\to0$; for instance, let $\alpha_n$ be
equal to $\alpha(t_{n,i-1})$ on $[t_{n,i-1},t_{n,i})$, where the $t_{n,i}$ are the
points $i2^{-n}$ together with $t_\star$. Let $\Phi_n=\Phi_{\alpha_n}$, let $Y^n$ be
the diffusion of $\alpha_n$, driven by the same Brownian motion as $Y$, and let
$\mathcal H_n$ denote the step flow with masses $\alpha_n(t_\star\tau)/\mathfrak n$, that is,
the finite hierarchy whose levels are the intervals between consecutive
breakpoints of $\alpha_n$ in $[0,t_\star]$.

\paragraph{Step 1: the step hierarchies.} By \eqref{eq:inp-scalar-recursion} and
\cref{lem:inp-steps}(d), the partial values of $\mathcal H_n$ with terminal
function $\mathfrak n\Phi_n(t_\star,\cdot)$ are $\mathfrak n\Phi_n(t_{n,i},\cdot)$, and in particular
$\mathcal H_n(\mathfrak n\Phi_n(t_\star,\cdot))(0)=\mathfrak n\Phi_n(0,0)$. The level from
$t_{n,i-1}$ to $t_{n,i}$ has mass $\alpha_n(t_{n,i-1})/\mathfrak n$ and partial value
$\mathfrak n\Phi_n(t_{n,i},\cdot)$. Its kernel has the density
$\exp\{\alpha_n(t_{n,i-1})(\Phi_n(t_{n,i},y)-\Phi_n(t_{n,i-1},z))\}$ with respect to
$N(z,b(t_{n,i}-t_{n,i-1}))$. This is the kernel of the scalar recursion of
$\alpha_n$ on the same interval.
By \cref{inp:ob-scalar}(iii), the path law of this hierarchy, started at $0$, is
the law of $(Y^n_{t_{n,i}})_{t_{n,i}\le t_\star}$. Then \cref{b:lem:os-prefix}, with
$f=\mathfrak n\Phi_n(t_\star,\cdot)$ and $z_0=0$, gives
\begin{equation}\label{eq:flow-prefix-step}
 \bigl\lvert\mathcal H_n\bigl(\mathfrak n\Phi_n(t_\star,\cdot)+V\bigr)(0)-\mathfrak n\Phi_n(0,0)
 -\E V(Y^n_{t_\star})\bigr\rvert\le\frac{\eps^2}{2}.
\end{equation}

\paragraph{Step 2: the limit.} By \cref{inp:ob-scalar}(i), $\abs{\partial_z\Phi_n}\le1$
and $0<\partial_{zz}\Phi_n\le1$, and the same holds for $\Phi_\alpha$; hence the
terminal functions below belong to $\mathcal C^2(\R)$, with gradients bounded by
$\mathfrak n+\norm{V'}_\infty$. Using \cref{lem:flow-step-limit}(a), we find that
\[
 \abs{\mathcal H_n(\mathfrak n\Phi_n(t_\star,\cdot)+V)-\mathcal H_n(\mathfrak n\Phi_\alpha(t_\star,\cdot)+V)}
 \le\mathfrak n\sup_z\abs{\Phi_n(t_\star,z)-\Phi_\alpha(t_\star,z)},
\]
which tends to zero by \cref{inp:ob-scalar}(ii). By \cref{lem:flow-step-limit}(b),
$\mathcal H_n(\mathfrak n\Phi_\alpha(t_\star,\cdot)+V)(0)\to\mathcal H^\alpha_{t_\star,\mathfrak n}
(\mathfrak n\Phi_\alpha(t_\star,\cdot)+V)$, since
$\int_0^1\abs{\alpha_n(t_\star\tau)-\alpha(t_\star\tau)}\dd\tau\to0$. Also,
\cref{inp:ob-scalar}(ii) implies that $\Phi_n(0,0)\to\Phi_\alpha(0,0)$. Finally, we write
$u_n=\partial_z\Phi_n$ and $u=\partial_z\Phi_\alpha$. Since $\abs{u_n}\le1$,
$\alpha\le1$, and $\abs{\partial_zu}\le1$, we have
\[
 \abs{\alpha_n(t)u_n(t,y')-\alpha(t)u(t,y)}
 \le\abs{\alpha_n(t)-\alpha(t)}+\sup_{t,z}\abs{u_n-u}+\abs{y'-y},
\]
and the equations for $Y^n$ and $Y$ give, for $t\le t_\star$,
\[
 \abs{Y^n_t-Y_t}\le b\int_0^1\abs{\alpha_n-\alpha}\dd t'+b\sup\abs{u_n-u}
 +b\int_0^t\abs{Y^n_{t'}-Y_{t'}}\dd t'.
\]
By Gronwall's inequality and \cref{inp:ob-scalar}(ii), $Y^n_{t_\star}\to Y_{t_\star}$
almost surely, and $\E V(Y^n_{t_\star})\to\E V(Y_{t_\star})$ by bounded convergence.
Taking $n\to\infty$ in \eqref{eq:flow-prefix-step} proves the second claim. The
first claim follows in the same way from the identity
$\mathcal H_n(\mathfrak n\Phi_n(t_\star,\cdot))(0)=\mathfrak n\Phi_n(0,0)$.
\end{proof}

\begin{remark}[A common block acts on the diagonal]\label{rem:flow-diagonal}
Let $e\in\R^\ell$, and suppose that stage $j$ of a stage datum has $B_j=ee^{\mathsf T}$.
Then the Gaussian increments of that stage are of the form $\zeta e$ with a
centered Gaussian variable $\zeta$, and a point of the line $\{ze:z\in\R\}$ stays
on it. It follows that, for $F\in\mathcal C^2(\R^\ell)$, the flow of this stage with
terminal function $F$, evaluated at $ze$, is the scalar flow with $B_j=1$, the
same $\lambda_j$ and $m_j$, and terminal function $z\mapsto F(ze)$, evaluated at
$z$. This holds for step flows by \cref{lem:inp-steps}(e) and passes to the limit
by \cref{lem:flow-step-limit}(b). In \cref{sec:cubic}, the part of the hierarchy
before the window has $B=ee^{\mathsf T}$ with $e=(1,1,1)$ and masses $\alpha/3$,
and its value is $\mathcal H^\alpha_{t_\star,3}$ applied to the restriction of the
local value to the diagonal.
\end{remark}
\chapter{A three-replica covariance perturbation}\label{sec:cubic}

In this section, we construct a trial path for the pressure of three
configurations whose overlaps are prescribed near a level
$r\in(0,q_\beta)$, and we show that its trial value lies below $3P_\beta$ by
an amount of third order in the width parameter of the perturbation. The prescribed
overlap matrix is $D(r,q,\delta)$ of \eqref{eq:inp-triple}, with $q-r$ and
$\delta$ small. There are two main results. \Cref{b:prop:os-expansion}
expands the trial value to third order with an explicit polynomial
coefficient. \Cref{prop:cub-band} turns this expansion into a
deficit of order $\rho^3$, uniformly over the matrices with
$\rho\le q-r\le d_0$ and $E\rho\le\abs\delta\le2E\rho$. The deficit is used in
\cref{sec:fractional}, in the proof of \cref{lem:fr-local-gap}, and through it
in the static estimate of \cref{sec:static}.

We use the following earlier results. From \cref{sec:inputs}, we use the
integration steps of \cref{def:inp-step} and their properties
(\cref{lem:inp-steps}), the hierarchies, penalties and trial values of
\cref{def:inp-hierarchy,def:inp-trial} and \eqref{eq:inp-penalty}, block
factorization (\cref{lem:inp-blocks}), and the matrix interpolation bound
\cref{inp:ob-matrix}. From \cref{sec:scalar}, we use the bounds on spatial
derivatives of \cref{lem:sc-derivatives} and the constant
$m_\beta$ of \cref{cor:sc-mbeta}. We also use the regularity of $\alpha$ and
$\Phi$ on $(0,q_\beta)$ and the stationarity identities
\eqref{b:eq:os-contacts}, both from \cref{b:lem:os-scalar-jets}. From \cref{sec:flow}, we use the uniform
estimates for short compositions of Gaussian flows (\cref{prop:flow-uniform}),
the identification of these flows with limits of hierarchies
(\cref{lem:flow-step-limit}), and the response of a hierarchy to a small
change of its terminal function (\cref{b:lem:os-prefix,cor:flow-prefix}).

\paragraph{Relation to \cite[Section~7]{oai-barriers}.}
The construction refines the narrow-annulus comparison of OpenAI's barrier
paper \cite[Section~7, Steps~1--8]{oai-barriers}. We keep three of its
ingredients. The first is the three-coordinate hierarchy in which the masses
are divided by the sizes of the common blocks
\cite[Section~4.2 and Section~7, Step~1, (7.7)]{oai-barriers}; OpenAI attributes
it to Talagrand's three-replica construction
\cite[discussion after Proposition~4.4]{talagrand-2007}. The second is the
perturbation that adds a Gaussian linear combination of field increments to
the field of $x$, subtracts it from the field of $y$, and leaves $v$
unchanged \cite[Section~7, Step~2, (7.8)--(7.12)]{oai-barriers}. The third
is the source of the loss: the identity $\beta^2\E\chi(r,Y_r)^2=1$, which
follows from stationarity, together with the evolution of the susceptibility
$\chi$ along the scalar diffusion \cite[Section~7, Step~8, (7.31)]{oai-barriers}. In
\cite{oai-barriers}, the window has width $\rho^8$ and the imbalance $\delta$ is
of order $\rho^{20}$. The endpoint of the perturbed path differs from the
prescribed matrix, and the mismatch is paid through the term
$(\beta^2/4)\norm{D-Q_N}_F^2$ of \cref{inp:ob-matrix}
\cite[(7.12)--(7.13)]{oai-barriers}. The resulting deficit is of order $\rho^{43}$
\cite[(7.32)]{oai-barriers}.

Three parts of the present construction are new. First, the window has width
parameter $h=(q-r)/\vartheta$ for a fixed ratio $\vartheta$. The Gaussian contrast has
fixed weights and extends past the time at which $x$ and $y$ separate, and a
terminal correction of the covariance removes the excess covariance in the
antisymmetric direction $\mathbf e_x-\mathbf e_y$ exactly. The path is positive semidefinite at every stage, its
endpoint is $D(r,q,\delta)$, and no mismatch is paid (\cref{sec:cub-path}).
Second, we expand the trial value to third order in $h$. At second order the
first stationarity identity cancels the penalty, and at third order we use the
second stationarity identity to combine two coefficient polynomials into one
(\cref{sec:cub-coefficients,sec:cub-profile,sec:cub-expansion}). The identities
themselves are known; their sources are recalled in \cref{sec:scalar}. Third,
the resulting polynomial gives a deficit of order $\rho^3$ on a band where
$q-r$ and $\abs\delta$ are both comparable to $\rho$ (\cref{sec:cub-band}).

\paragraph{Organization.}
\Cref{sec:cub-path} defines the perturbed path and its trial value and reduces
the trial value to a boundary perturbation $V$ on the diagonal. \Cref{sec:cub-derivatives} records
the derivative bounds for the local part of the path. In
\cref{sec:cub-coefficients} we compute the Taylor coefficients of $V$ in the
width parameter $h$ when the Parisi distribution function is replaced by a constant on
the window, using covariance matrices that are not positive semidefinite as
formal bookkeeping devices. In \cref{sec:cub-profile} we compute the
contribution of the variation of $\alpha$ on the window and prove the
pointwise expansion \cref{prop:cub-pointwise}. \Cref{sec:cub-expansion} proves
\cref{b:prop:os-expansion}, and \cref{sec:cub-band} proves the uniform deficit
\cref{prop:cub-band}.

\paragraph{Standing notation.}
Throughout this section, $\beta>1$ is fixed and $b=\beta^2$. We write
$\alpha=\alpha_\beta$, $\Phi=\Phi_\alpha$, $u=\partial_z\Phi$,
$\chi=\partial_z^2\Phi$ and $w=\partial_z^3\Phi$, and $Y$ is the scalar
diffusion of \eqref{b:eq:os-scalar}. Vectors of $\R^3$ are written in bold,
in the coordinate order $(v,x,y)$. We write $\mathbf e=(1,1,1)^{\mathsf T}$,
$\mathbf e_v,\mathbf e_x,\mathbf e_y$ for the unit vectors,
\[
 L_2=(\mathbf e_x-\mathbf e_y)(\mathbf e_x-\mathbf e_y)^{\mathsf T},
\]
and $\mathbf z=(z_v,z_x,z_y)$ for a point of $\R^3$. A real number $z$
denotes the point $z\mathbf e$ of the diagonal. As in
\cref{sec:inp-hierarchies}, $U\mathbin:U'=\sum_{i,l}U_{il}U'_{il}$ and
$\norm U_F^2=U\mathbin:U$. For a symmetric $3\times3$ matrix $B$ and a function
$F$ on $\R^3$, we write $B\mathbin:D^2F=\sum_{i,l}B_{il}\partial_i\partial_lF$
and $\nabla F\cdot B\nabla F$. We also set
$\operatorname{cub}(U)=\sum_{i,l}U_{il}^3$.

We fix a ratio $\vartheta\in(0,1/2]$, a bound $k_0>0$ for the relative
imbalance $k$ below, and a compact interval $K\subset(0,q_\beta)$ of left
endpoints. With $L_\vartheta$ as in \eqref{b:eq:os-parameters} below, we set
$\tau_{\max}=2+4k_0^2L_\vartheta$, and we fix $h_1\in(0,1]$ such that
\[
 K'=\bigl[\min K,\ \max K+\tau_{\max}h_1\bigr]\subset(0,q_\beta).
\]
By \cref{b:lem:os-scalar-jets}, $\alpha$ is infinitely differentiable on
$K'$, the number $\alpha_*=\min_{K'}\alpha$ is positive, $\alpha$ is
Lipschitz on $K'$ with constant $K_\beta$, and every derivative
$\partial_t^i\partial_z^j\Phi$ with $i+j\ge1$ is bounded on $K'\times\R$. In
this section, $C$ denotes a constant that depends only on $\beta$,
$\vartheta$, $k_0$ and $K$ (and therefore on $h_1$ and $K'$); its value may
change from line to line. We make no claim of uniformity as
$\vartheta\downarrow0$. The symbols of the construction are collected in
\cref{tab:cub-symbols}.

\begin{table}[ht]
\centering
\small
\begin{tabular}{@{}lp{0.45\textwidth}l@{}}
\toprule
Symbol & Meaning & Defined in\\
\midrule
$\vartheta$ & ratio $(q-r)/h$, fixed in $(0,1/2]$ & \eqref{b:eq:os-parameters}\\
$q_0$, $h$, $k$ & left endpoint, width parameter, relative imbalance & \eqref{b:eq:os-parameters}\\
$r$, $q$, $d$, $\delta$ & $q_0+h$, $q_0+(1+\vartheta)h$, $\vartheta h$, $hk$ & \eqref{b:eq:os-parameters}\\
$\omega_0$, $\omega_1$, $L_\vartheta$ & weights of the contrast and its variance & \eqref{b:eq:os-parameters}\\
$\kappa_j$, $\mathfrak n_j$, $B_j$, $B_j^\circ$ & lengths, block sizes and rates of the stages & \cref{tab:cub-stages}\\
$\tau_T$, $T$ & $2+4k^2L_\vartheta$ and $q_0+h\tau_T$ & \eqref{eq:cub-T}\\
$\mathcal T$, $\mathcal B$, $V$, $\Delta\Pi$ & trial value, local block, boundary perturbation, change of the penalty & \cref{sec:cub-trial}\\
$\Lambda_0$, $\Lambda_1$, $\Lambda_2$ & coefficient matrices at constant mass & \eqref{eq:cub-Lambda}\\
$C_\chi$, $C_w$, $N_\vartheta$ & coefficient polynomials & \eqref{b:eq:os-R}, \eqref{b:eq:os-W}, \eqref{eq:cub-N}\\
$P_\vartheta$, $c_j$, $M$ & cubic polynomial, its coefficients, scalar factor & \cref{sec:cub-expansion}\\
\bottomrule
\end{tabular}
\caption{Notation of \cref{sec:cubic}.}\label{tab:cub-symbols}
\end{table}

\section{The perturbed covariance path}\label{sec:cub-path}

\subsection{Parameters and stages}

For $q_0\in(0,1)$, $h\ge0$ and $k\in\R$, we set
\begin{equation}\label{b:eq:os-parameters}
\begin{gathered}
 r=q_0+h,\qquad q=q_0+(1+\vartheta)h,\qquad d=q-r=\vartheta h,
 \qquad \delta=hk,\\
 \omega_0=\frac1{2-\vartheta},\qquad \omega_1=\frac{1-\vartheta}{2-\vartheta},
 \qquad L_\vartheta=\omega_0^2(3-\vartheta)+\omega_1^2
 =\frac{\vartheta^2-3\vartheta+4}{(2-\vartheta)^2}.
\end{gathered}
\end{equation}
Note that $\omega_0+\omega_1=1$ and $\omega_1=(1-\vartheta)\omega_0$. The number $k$
is the imbalance relative to the width. We define the vectors
\[
\begin{gathered}
 \mathbf c_0=\begin{pmatrix}1\\1+\omega_0k\\1-\omega_0k\end{pmatrix},\qquad
 \mathbf c_1=\begin{pmatrix}0\\1-\omega_0k\\1+\omega_0k\end{pmatrix},\qquad
 \mathbf c_v=\begin{pmatrix}1\\\omega_1k\\-\omega_1k\end{pmatrix},\\
 \mathbf c_x=\begin{pmatrix}0\\1-\omega_0k\\\omega_0k\end{pmatrix},\qquad
 \mathbf c_y=\begin{pmatrix}0\\-\omega_0k\\1+\omega_0k\end{pmatrix},
\end{gathered}
\]
the rates
\begin{equation}\label{eq:cub-rates}
\begin{gathered}
 B_1=\mathbf c_0\mathbf c_0^{\mathsf T},\quad
 B_2=\mathbf c_1\mathbf c_1^{\mathsf T},\quad
 B_3=\mathbf c_v\mathbf c_v^{\mathsf T},\quad
 B_4=\mathbf c_v\mathbf c_v^{\mathsf T}+\mathbf c_x\mathbf c_x^{\mathsf T}
   +\mathbf c_y\mathbf c_y^{\mathsf T},\quad
 B_5=\mathrm{Id}-\tfrac14L_2,\\
 B_1^\circ=\mathbf e\mathbf e^{\mathsf T},\quad
 B_2^\circ=(\mathbf e_x+\mathbf e_y)(\mathbf e_x+\mathbf e_y)^{\mathsf T},\quad
 B_3^\circ=\mathbf e_v\mathbf e_v^{\mathsf T},\quad
 B_4^\circ=B_5^\circ=\mathrm{Id},
\end{gathered}
\end{equation}
and the lengths $\kappa_1=1$, $\kappa_2=\kappa_3=\vartheta$,
$\kappa_4=1-\vartheta$ and $\kappa_5=4k^2L_\vartheta$. Note that
$B_j=B_j^\circ$ for $j\le4$ when $k=0$, and that $\kappa_5=0$ when $k=0$. We
also set
\begin{equation}\label{eq:cub-T}
 \tau_T=\kappa_1+\kappa_2+\kappa_4+\kappa_5=2+4k^2L_\vartheta,\qquad T=q_0+h\tau_T.
\end{equation}
The length $\kappa_3$ does not enter $\tau_T$, because stages $2$ and $3$ below
have the same scalar interval; $h\tau_T$ is the length of the window $[q_0,T]$.

The \emph{local window} consists of the five stages of \cref{tab:cub-stages},
listed from the root to the terminal time; their Gaussian integrations are
applied in the reverse order, from stage $5$ to stage $1$. Stage $j$ has scalar interval
$I_j=[q_0+ho_j,\,q_0+h(o_j+\kappa_j)]$, with offsets $o_1=0$,
$o_2=o_3=1$, $o_4=1+\vartheta$ and $o_5=2$, and its covariance increment over
a scalar time $\dd t$ in $I_j$ is $B_j\dd t$ (before multiplication by $b$),
integrated at mass $\alpha(t)/\mathfrak n_j$. Stages $2$ and $3$ have the same scalar
interval $[r,q]$; they are different blocks of the hierarchy, and stage $2$
precedes stage $3$ along the path, so that the integration of stage $3$ is
applied first. In the \emph{baseline} window, the rates
$B_j$ are replaced by $B_j^\circ$ and everything else is unchanged; in
particular, the baseline has the same stage $5$, of length $\kappa_5$, with
the rate $\mathrm{Id}$.

\begin{table}[ht]
\centering
\begin{tabular}{@{}clcccc@{}}
\toprule
$j$ & stage & scalar interval $I_j$ & length $\kappa_j$ & mass & rate\\
\midrule
1 & common & $[q_0,r]$ & $1$ & $\alpha(t)/3$ & $B_1$\\
2 & shared & $[r,q]$ & $\vartheta$ & $\alpha(t)/2$ & $B_2$\\
3 & singleton, first part & $[r,q]$ & $\vartheta$ & $\alpha(t)$ & $B_3$\\
4 & singleton, second part & $[q,q_0+2h]$ & $1-\vartheta$ & $\alpha(t)$ & $B_4$\\
5 & terminal correction & $[q_0+2h,T]$ & $4k^2L_\vartheta$ & $\alpha(t)$ & $B_5$\\
\bottomrule
\end{tabular}
\caption{The five stages of the local window. The length is the length of
$I_j$ divided by $h$, and the mass of stage $j$ is $\alpha(t)/\mathfrak n_j$ with block
sizes $\mathfrak n_1=3$, $\mathfrak n_2=2$, $\mathfrak n_3=\mathfrak n_4=\mathfrak n_5=1$.}\label{tab:cub-stages}
\end{table}

The \emph{perturbed path} consists of an initial segment on $[0,q_0]$, on
which the three coordinates share every increment, with rate
$\mathbf e\mathbf e^{\mathsf T}$ and mass $\alpha(t)/3$; the local window;
and a final segment on $[T,1]$ with rate $\mathrm{Id}$ and mass $\alpha(t)$. The
\emph{baseline path} is the same with the baseline window. It is the path of
\cref{lem:inp-blocks}(b), with the stages split at the scalar times $q_0$,
$q_0+2h$ and $T$.

\paragraph{The contrast.} The rates have the following meaning. In this
description we assume $h>0$; at $h=0$ the window has length zero, and all its
covariance increments vanish. Let $g_1$,
$g_2$, $g_v$, $g_x$ and $g_y$ be independent centered Gaussian variables with
variances $bh$, $b\vartheta h$, $bh$, $b(1-\vartheta)h$ and
$b(1-\vartheta)h$. In the baseline window, these are the field increments: the
three coordinates share $g_1$ on $[q_0,r]$, the coordinates $x$ and $y$ share
$g_2$ on $[r,q]$, the coordinate $v$ receives $g_v$ on $[r,q_0+2h]$ (split
between stages $3$ and $4$), and $x$ and $y$ receive $g_x$ and $g_y$ on
$[q,q_0+2h]$. We define the contrast
\begin{equation}\label{eq:cub-contrast}
 \Xi=\frac1h\bigl\{\omega_0(g_1-g_2-g_x-g_y)+\omega_1g_v\bigr\}.
\end{equation}
The perturbation adds $\delta\,\Xi$ to the field of $x$, subtracts it from the
field of $y$, and leaves the field of $v$ unchanged. Collecting the
coefficients of the five Gaussian variables, we find that the perturbed field
vector of the window is
\[
 \mathbf c_0g_1+\mathbf c_1g_2+\mathbf c_vg_v+\mathbf c_xg_x+\mathbf c_yg_y.
\]
This explains the rates $B_1,\dots,B_4$. By independence,
\begin{equation}\label{eq:cub-contrast-moments}
\begin{gathered}
 \Var\Xi=\frac bh\bigl\{\omega_0^2\bigl(1+\vartheta+2(1-\vartheta)\bigr)+\omega_1^2\bigr\}
 =\frac{bL_\vartheta}h,\\
 \Cov(\Xi,g_1+g_2+g_x)=b\,\omega_0\bigl(1-\vartheta-(1-\vartheta)\bigr)=0,\\
 \Cov(\Xi,g_1+g_v)=b(\omega_0+\omega_1)=b,
\end{gathered}
\end{equation}
and the same holds with $g_y$ in place of $g_x$. These identities imply that
the perturbation changes the covariance of the window fields, divided by $b$, by
\[
 \delta\bigl(\mathbf e_v(\mathbf e_x-\mathbf e_y)^{\mathsf T}
 +(\mathbf e_x-\mathbf e_y)\mathbf e_v^{\mathsf T}\bigr)
 +\frac{\delta^2L_\vartheta}h\,L_2.
\]
The first term changes the overlaps of $v$ with $x$ and with $y$ by $+\delta$
and $-\delta$, respectively. The second term is an excess covariance in the antisymmetric direction
$\mathbf e_x-\mathbf e_y$.
We have $\delta^2L_\vartheta/h=hk^2L_\vartheta=h\kappa_5/4$. Stage $5$ uses the
rate $\mathrm{Id}-L_2/4$ instead of $\mathrm{Id}$ during a time $h\kappa_5$, and it removes this
excess exactly. We call stage $5$ the terminal correction.

\begin{lemma}[Identities of the window]\label{lem:cub-identities}
For every $k\in\R$, the following hold.
\begin{enumerate}[label=\textup{(\alph*)}]
\item $\mathbf e\cdot B_j\mathbf e=\mathbf e\cdot B_j^\circ\mathbf e$ for
 $1\le j\le5$; these numbers are $9$, $4$, $1$, $3$ and $3$.
\item $\sum_j\kappa_jB_j=\sum_j\kappa_jB_j^\circ
 +k\bigl(\mathbf e_v(\mathbf e_x-\mathbf e_y)^{\mathsf T}
 +(\mathbf e_x-\mathbf e_y)\mathbf e_v^{\mathsf T}\bigr)$. In particular, both
 sums have all diagonal entries equal to $\tau_T$.
\item $\sum_j\mathfrak n_j^{-1}\kappa_jB_j\mathbf e
 =\sum_j\mathfrak n_j^{-1}\kappa_jB_j^\circ\mathbf e=\tau_T\mathbf e$.
\item Let $\mathsf S$ be the permutation matrix that exchanges the coordinates
 $x$ and $y$, and write $B_j(k)$ for the dependence on $k$. Then
 $\mathsf SB_j(k)\mathsf S=B_j(-k)$ and $\mathsf SB_j^\circ\mathsf S=B_j^\circ$ for
 every $j$.
\end{enumerate}
\end{lemma}

\begin{proof}
For (a), we have $\mathbf e\cdot\mathbf c_0=3$, $\mathbf e\cdot\mathbf c_1=2$,
$\mathbf e\cdot\mathbf c_v=\mathbf e\cdot\mathbf c_x=\mathbf e\cdot\mathbf c_y=1$
and $L_2\mathbf e=0$, and the same numbers for the baseline vectors
$\mathbf e$, $\mathbf e_x+\mathbf e_y$ and $\mathbf e_v$. For (b), note that
$h\sum_{j\le4}\kappa_jB_j$ is the covariance of the perturbed window fields
divided by $b$. The computation above gives
\[
 h\sum_{j\le4}\kappa_jB_j=h\sum_{j\le4}\kappa_jB_j^\circ
 +\delta\bigl(\mathbf e_v(\mathbf e_x-\mathbf e_y)^{\mathsf T}
 +(\mathbf e_x-\mathbf e_y)\mathbf e_v^{\mathsf T}\bigr)+hk^2L_\vartheta L_2,
\]
while $h\kappa_5B_5=h\kappa_5B_5^\circ-hk^2L_\vartheta L_2$. Adding and
dividing by $h$ gives (b). The diagonal of the baseline sum is
$\kappa_1+\kappa_2+\kappa_4+\kappa_5=\tau_T$ in the coordinates $x,y$ and
$\kappa_1+\kappa_3+\kappa_4+\kappa_5=\tau_T$ in the coordinate $v$. For (c),
we have $B_1\mathbf e=3\mathbf c_0$, $B_2\mathbf e=2\mathbf c_1$,
$B_3\mathbf e=\mathbf c_v$, $B_4\mathbf e=\mathbf c_v+\mathbf c_x+\mathbf c_y$
and $B_5\mathbf e=\mathbf e$, so that
\[
 \sum_j\mathfrak n_j^{-1}\kappa_jB_j\mathbf e
 =\mathbf c_0+\vartheta\mathbf c_1+\mathbf c_v
 +(1-\vartheta)(\mathbf c_x+\mathbf c_y)+\kappa_5\mathbf e.
\]
Its $v$ entry is $2+\kappa_5$. Its $x$ entry is
\[
\begin{aligned}
 &(1+\omega_0k)+\vartheta(1-\omega_0k)+\omega_1k+(1-\vartheta)(1-2\omega_0k)
 +\kappa_5\\
 &\qquad=2+\kappa_5+k\bigl(\omega_1-(1-\vartheta)\omega_0\bigr)=\tau_T,
\end{aligned}
\]
and the $y$ entry is the same with $k$ replaced by $-k$. For the baseline we
find $\mathbf e+\vartheta(\mathbf e_x+\mathbf e_y)+\vartheta\mathbf e_v
+(1-\vartheta)\mathbf e+\kappa_5\mathbf e=\tau_T\mathbf e$. Finally, (d)
holds because $\mathsf S\mathbf c_0(k)=\mathbf c_0(-k)$, $\mathsf S\mathbf
c_1(k)=\mathbf c_1(-k)$, $\mathsf S\mathbf c_v(k)=\mathbf c_v(-k)$, $\mathsf
S\mathbf c_x(k)=\mathbf c_y(-k)$, $\mathsf S\mathbf c_y(k)=\mathbf c_x(-k)$ and
$\mathsf SL_2\mathsf S=L_2$.
\end{proof}

Part (b) says that the window shifts the overlaps of $v$ with $x$ and $y$ by
$\pm\delta$ and changes nothing else in the total covariance. Part (c) is the
identity behind the cancellation of the linear part of the terminal function
in \cref{sec:cub-coefficients,sec:cub-profile}.

\begin{lemma}[Admissibility and endpoint]\label{lem:cub-path}
Let $q_0\in K$, $0\le h\le h_1$ and $\abs k\le k_0$, and suppose that
$\vartheta K_\beta h\le\alpha_*$. Then every increment of the perturbed path is
positive semidefinite, and its masses are nondecreasing along the path. Its
endpoint is
\begin{equation}\label{b:eq:os-target}
 D_\delta=D(r,q,\delta)=\begin{pmatrix}
 1&r+\delta&r-\delta\\r+\delta&1&q\\r-\delta&q&1
 \end{pmatrix}.
\end{equation}
The same holds for the baseline path, whose endpoint is $D(r,q,0)$.
\end{lemma}

\begin{proof}
The rates $B_1,\dots,B_4$ and the baseline rates are sums of matrices of the
form $\mathbf c\mathbf c^{\mathsf T}$. The matrix $L_2$ has the eigenvalue
$\abs{\mathbf e_x-\mathbf e_y}^2=2$ on $\mathbf e_x-\mathbf e_y$ and the
eigenvalue $0$ on the orthogonal complement, and $\mathrm{Id}-L_2/4$ has eigenvalues
$1,1,1/2$. This proves that the increments are positive semidefinite.

The masses are nondecreasing within each stage because $\alpha$ is
nondecreasing. At the junctions, they are $\alpha(q_0)/3$ on both sides of
$q_0$, then $\alpha(r)/3\le\alpha(r)/2$, then $\alpha(q)/2\le\alpha(r)$, and
they are continuous at the remaining junctions. The inequality
$\alpha(q)\le2\alpha(r)$ holds because $[q_0,T]\subset K'$, so that
\[
 \alpha(q)-\alpha(r)\le K_\beta(q-r)=\vartheta K_\beta h\le\alpha_*\le\alpha(r).
\]
All masses are at most $1$.

For the endpoint, the baseline path ends at
$q_0\mathbf e\mathbf e^{\mathsf T}+h\sum_j\kappa_jB_j^\circ+(1-T)\mathrm{Id}$. We have
$h\kappa_1=h$, $h\kappa_2=h\kappa_3=\vartheta h$ and
$h(\kappa_4+\kappa_5)+1-T=1-q$, and this endpoint equals
\[
 r\,\mathbf e\mathbf e^{\mathsf T}
 +\vartheta h\bigl((\mathbf e_x+\mathbf e_y)(\mathbf e_x+\mathbf e_y)^{\mathsf T}
 +\mathbf e_v\mathbf e_v^{\mathsf T}\bigr)+(1-q)\mathrm{Id}=D(r,q,0).
\]
By \cref{lem:cub-identities}(b), the endpoint of the perturbed path is
$D(r,q,0)+hk\bigl(\mathbf e_v(\mathbf e_x-\mathbf e_y)^{\mathsf T}
+(\mathbf e_x-\mathbf e_y)\mathbf e_v^{\mathsf T}\bigr)=D_\delta$.
\end{proof}

\subsection{The trial value}\label{sec:cub-trial}

Let $\Lambda_*=\max_{\abs k\le k_0}\sum_j\kappa_j\abs{B_j}$, where $\abs{B_j}$
is the operator norm, and fix $h_0\in(0,h_1]$ such that
\begin{equation}\label{eq:cub-h0}
 \vartheta K_\beta h_0\le\alpha_*,\qquad 6bh_0\Lambda_*\le1.
\end{equation}
In the rest of this section we assume that $q_0\in K$, $0\le h\le h_0$ and
$\abs k\le k_0$, so that \cref{lem:cub-path} applies. We now define the trial
value and express it through the local window.

For $N\ge1$, let $(Q^{(N)},m^{(N)})$ be the admissible path obtained from the
perturbed path by dividing $[0,q_0]$, each scalar interval $I_j$ and $[T,1]$
into $N$ intervals of equal length, keeping the covariance increments, and
giving each increment the mass of the path at the left endpoint of its scalar
interval. The proof of \cref{lem:cub-path} shows that these masses are
nondecreasing, and the endpoint is $D_\delta$. We define
\begin{equation}\label{b:eq:os-trial}
 \mathcal T(q_0,h,k)=\lim_{N\to\infty}\Bigl(\Psi\bigl(Q^{(N)},m^{(N)},0\bigr)
 -\frac b4\int m^{(N)}\dd\norm{Q^{(N)}}_F^2\Bigr).
\end{equation}
The expression in parentheses is the trial value
$\mathcal T_{D_\delta}(Q^{(N)},m^{(N)},0)$ of \cref{def:inp-trial}, whose
mismatch term $\norm{D_\delta-Q_N}_F^2$ vanishes because the endpoint is
$D_\delta$; by \cref{lem:cub-trial} below,
$\mathcal T(q_0,h,k)$ is a trial value for $D_\delta$ in the sense of that
definition. We write $\mathcal T^\circ(q_0,h,k)$ for the same limit for the
baseline path.

To describe the limit, we use the flows of \cref{sec:flow}. The \emph{local
block} is the flow, in the sense of \cref{def:flow,lem:flow-step-limit}, of the
stage datum with $\ell=3$, $N=5$, matrices $B_1,\dots,B_5$, durations
$h\kappa_1,\dots,h\kappa_5$, and masses
$m_j(\tau)=\mathfrak n_j^{-1}\alpha\bigl(q_0+h(o_j+\kappa_j\tau)\bigr)$ for
$\tau\in[0,1]$. Here $\tau=0$ is the end of $I_j$ closest to the root of
the hierarchy, and $\tau=1$ the end closest to the terminal time. The flow
$F$ solves
\begin{equation}\label{b:eq:os-vector-flow}
 -\partial_\tau F=\frac{bh\kappa_j}2\Bigl\{B_j\mathbin:D^2F
 +m_j(\tau-j+1)\,\nabla F\cdot B_j\nabla F\Bigr\}\quad\text{on }(j-1,j)\times\R^3,
 \qquad j=1,\dots,5,
\end{equation}
with $F(5,\mathbf z)=\sum_{i\in\{v,x,y\}}\Phi(T,z_i)$, and we set
\[
 \mathcal B(q_0,h,k,\mathbf z)=F(0,\mathbf z).
\]
We define $\mathcal B^\circ$ in the same way with the rates $B_j^\circ$. For
masses that are constant on the pieces of a partition, the flow is the
composition of the integration steps of \cref{def:inp-step}
(see \eqref{eq:flow-step-solution}), that is, the value of a finite hierarchy; for
the masses above it is the limit of these values (\cref{lem:flow-step-limit}(b)).

For the initial segment, let $f\colon\R\to\R$ have bounded derivatives of orders
one and two. We write $\mathcal H(f)=\mathcal H^\alpha_{q_0,3}f$, in the notation
of \cref{cor:flow-prefix}: this is the value at $(0,0)$ of the solution
$\Phi^\alpha_f$ of
\[
 -\partial_t\Phi^\alpha_f=\frac b2\Bigl(\partial_{zz}\Phi^\alpha_f+\frac{\alpha(t)}3
 (\partial_z\Phi^\alpha_f)^2\Bigr)\quad\text{on }[0,q_0]\times\R,\qquad
 \Phi^\alpha_f(q_0,\cdot)=f,
\]
in the sense of \cref{lem:flow-step-limit}. It is the one-site recursion of
three coordinates that share every field increment on $[0,q_0]$, acting on a
function of their common value (\cref{rem:flow-diagonal}), and it is the
operator $\mathcal H_\alpha$ of \cref{sec:fr-prefix} on the interval $[0,q_0]$.
By the first claim of \cref{cor:flow-prefix},
\begin{equation}\label{eq:cub-prefix-scalar}
 \mathcal H\bigl(3\Phi(q_0,\cdot)\bigr)=3\Phi(0,0).
\end{equation}

Finally, the penalties of the two paths are the limits
$\Pi(q_0,h,k)$ and $\Pi^\circ$ of $\int m^{(N)}\dd\norm{Q^{(N)}}_F^2$, and
we write $\Delta\Pi=\Pi-\Pi^\circ$ and
\begin{equation}\label{eq:cub-V}
 V(q_0,h,k,z)=\mathcal B(q_0,h,k,z\mathbf e)-3\Phi(q_0,z)\qquad(z\in\R).
\end{equation}

\begin{lemma}[The trial value]\label{lem:cub-trial}
The limit \eqref{b:eq:os-trial} exists, and the following hold.
\begin{enumerate}[label=\textup{(\alph*)}]
\item For every $n\ge2$ such that $D_\delta$ is feasible, we have
 $p_{3,n}(D_\delta)\le\mathcal T(q_0,h,k)$.
\item $\mathcal T(q_0,h,k)=\mathcal H\bigl(\mathcal B(q_0,h,k,\cdot\,\mathbf e)\bigr)
 -\frac b4\Pi(q_0,h,k)$, and the same holds for the baseline path with
 $\mathcal B^\circ$ and $\Pi^\circ$.
\item $\mathcal B^\circ(q_0,h,k,z\mathbf e)=3\Phi(q_0,z)$ for all $z\in\R$,
 $\Pi^\circ=6\int_0^1t\alpha(t)\dd t$, and $\mathcal T^\circ(q_0,h,k)=3P_\beta$.
\item We have
 \begin{equation}\label{eq:cub-decomposition}
  \mathcal T(q_0,h,k)-3P_\beta
  =\mathcal H\bigl(3\Phi(q_0,\cdot)+V\bigr)-\mathcal H\bigl(3\Phi(q_0,\cdot)\bigr)
  -\frac b4\Delta\Pi.
 \end{equation}
\item Let $\widetilde Q_0=0$ and $\widetilde Q_j=\sum_{i\le j}\kappa_iB_i$, and
 define $\widetilde Q_j^\circ$ in the same way with the rates $B_i^\circ$. Then
 \begin{equation}\label{eq:cub-penalty}
  \Delta\Pi=\sum_{j=1}^5\frac{2h^2\kappa_j}{\mathfrak n_j}\int_0^1
  \alpha\bigl(q_0+h(o_j+\kappa_j\tau)\bigr)
  \Bigl[(\widetilde Q_{j-1}+\kappa_j\tau B_j)\mathbin:B_j
  -(\widetilde Q_{j-1}^\circ+\kappa_j\tau B_j^\circ)\mathbin:B_j^\circ\Bigr]\dd\tau .
 \end{equation}
\end{enumerate}
\end{lemma}

\begin{proof}
We first identify the values of the finite paths with step flows, then pass to
the limit, and then treat the baseline and the penalties.

\paragraph{Step 1: the finite paths as step flows.} We consider the stage datum
with seven stages: the initial segment, with matrix $\mathbf e\mathbf
e^{\mathsf T}$, duration $q_0$ and mass $\alpha(q_0\tau)/3$; the five stages
of the window; and the final segment, with matrix $\mathrm{Id}$, duration $1-T$ and mass
$\alpha(T+(1-T)\tau)$. Its terminal function is
$\mathbf z\mapsto\sum_i\log(2\cosh z_i)$. The masses of $(Q^{(N)},m^{(N)})$ are the sampled
masses of this datum for the partition of each stage into $N$ pieces of equal
length. By the remark after \eqref{eq:flow-step-solution},
$\Psi(Q^{(N)},m^{(N)},0)$ is the output at $\mathbf z=0$ of the corresponding
step flow.

\paragraph{Step 2: the limit.} The sampled masses converge in $L^1$ to the
masses of the datum, because these are nondecreasing functions. By
\cref{lem:flow-step-limit}(b), the step flows converge uniformly to the flow of
the seven-stage datum, and in particular the values converge. By \cref{lem:flow-step-limit}(c),
applied twice, this flow is obtained by composing the flows of the three parts.
The final segment, with this terminal function, gives
$\sum_i\Phi(T,z_i)$: its matrix is $\mathrm{Id}$ and the terminal function is a sum of
functions of single coordinates; by \cref{lem:inp-steps}(c), its step flows are sums of
scalar recursions \eqref{eq:inp-scalar-recursion}, which converge to $\Phi$ by
the definition of $\Phi_\alpha$ (\cref{inp:ob-scalar}(i)). This function belongs
to $\mathcal C^2$, since $\abs u\le1$ and $0<\chi\le1$, and to $\mathcal C^\infty$
by \cref{lem:sc-derivatives}. The window then gives
$\mathcal B$, which belongs to $\mathcal C^2$ by \cref{lem:flow-step-limit}(d):
its terminal function has $\norm{D^2f}_\infty\le1$, and
$2\cdot3\,b\,\norm{D^2f}_\infty\sum_jh\kappa_j\abs{B_j}\le6bh_0\Lambda_*\le1$ by
\eqref{eq:cub-h0}. By \cref{rem:flow-diagonal} the initial
segment acts on $z\mapsto\mathcal B(z\mathbf e)$ as $\mathcal H$. We conclude
that the values converge to $\mathcal H(\mathcal B(\cdot\,\mathbf e))$. The penalties
converge as well. On the window, this is \cref{lem:flow-penalty}, with
$Q_\star=q_0\mathbf e\mathbf e^{\mathsf T}$. On the initial and final segments,
the penalties are Riemann--Stieltjes sums $\sum_im_i(\norm{Q_i}_F^2
-\norm{Q_{i-1}}_F^2)$, in which $m_i$ is the value of a nondecreasing function
at the left endpoint of a piece and $\norm Q_F^2$ is a continuous, piecewise
quadratic function of the scalar time; they converge to the corresponding
integrals. This shows that the limit \eqref{b:eq:os-trial} exists, and it
proves (b). The endpoints, values and penalties converge, and (a) follows from the
last sentence of \cref{inp:ob-matrix}.

\paragraph{Step 3: the baseline.} Let $\alpha_N$ be the step distribution
function equal to $\alpha$ at the left endpoints of the scalar partition. The
baseline finite path is the path of \cref{lem:inp-blocks}(b) for $\alpha_N$,
whose breakpoints contain $r$ and $q$: the common increments are those on
$[0,r]$, the $xy$-increments of stage $2$ precede the $v$-increments of stage
$3$ (their masses are ordered by \cref{lem:cub-path}), and the independent
increments are those on $[q,1]$. \Cref{lem:inp-blocks}(b) implies that its
trial value is $3\mathcal P_\beta(\alpha_N)$ and that its penalty is
$6\int_0^1t\alpha_N(t)\dd t$.
By \cref{inp:ob-scalar}(ii), these converge to $3\mathcal P_\beta(\alpha)$ and
$6\int_0^1t\alpha(t)\dd t$, and $\mathcal P_\beta(\alpha)=P_\beta$ by
\cref{inp:parisi,inp:unique}. The proof of \cref{lem:inp-blocks}(b) also shows
that the step flow of the baseline window, applied to $\sum_i\Phi_{\alpha_N}(T,z_i)$,
equals $3\Phi_{\alpha_N}(q_0,z)$ at $z\mathbf e$. By the last estimate of
\cref{lem:flow-step-limit}(a), by \cref{lem:flow-step-limit}(b), and by
\cref{inp:ob-scalar}(ii), the left side converges to $\mathcal B^\circ(q_0,h,k,z\mathbf e)$
and the right side to $3\Phi(q_0,z)$. This proves (c), and (d) follows from (b),
(c) and \eqref{eq:cub-prefix-scalar}.

\paragraph{Step 4: the penalty.} During stage $j$, at the fraction $\tau\in[0,1]$
of its scalar interval, the perturbed path is
$Q=q_0\mathbf e\mathbf e^{\mathsf T}+h(\widetilde Q_{j-1}+\kappa_j\tau B_j)$. Then
$\dd\norm Q_F^2=2Q\mathbin:\dd Q=2h\kappa_jQ\mathbin:B_j\dd\tau$, and the
mass is $\alpha(q_0+h(o_j+\kappa_j\tau))/\mathfrak n_j$. The term
$2h\kappa_jq_0\mathbf e\mathbf e^{\mathsf T}\mathbin:B_j=2h\kappa_jq_0\,
\mathbf e\cdot B_j\mathbf e$ is the same for the baseline path, by
\cref{lem:cub-identities}(a). The initial segments of the two paths coincide.
On $[T,1]$, both paths have $\dd\norm Q_F^2=2\tr Q\,\dd t$, and $\tr Q(T)$
is the same for both paths, because $\sum_j\kappa_jB_j$ and
$\sum_j\kappa_jB_j^\circ$ have the same diagonal
(\cref{lem:cub-identities}(b)). Subtracting the two penalties, we obtain
\eqref{eq:cub-penalty}.
\end{proof}

By \cref{lem:cub-trial}(a), every upper bound for $\mathcal T(q_0,h,k)$ is an
upper bound for the constrained pressure $p_{3,n}(D_\delta)$. The rest of the
section bounds $\mathcal T(q_0,h,k)-3P_\beta$ through \eqref{eq:cub-decomposition}.
In \eqref{eq:cub-decomposition}, the function $V$, the number $\Delta\Pi$, and
the part of $\Pi$ accumulated on $[q_0,1]$ depend on $\alpha$ only through its
restriction to $[q_0,1]$. Changing $\alpha$ on $[0,q_0)$
leaves them unchanged; we use this in \cref{sec:fractional}.

\section{Derivatives of the local block}\label{sec:cub-derivatives}

We now apply the estimates of \cref{sec:flow} to the local block. Let
$\mathcal U=[0,h_1]\times[-k_0,k_0]$. The stage datum of
\eqref{b:eq:os-vector-flow} satisfies the hypotheses \ref{H1}--\ref{H5} of
\cref{sec:flow}, with $\ell=3$ and $N=5$, for the following reasons.
\begin{itemize}
\item \ref{H1}: the matrices $B_j$ and $B_j^\circ$ do not depend on $h$, are
 polynomial in $k$, and are positive semidefinite (\cref{lem:cub-path}). Some
 of them are singular.
\item \ref{H2}: the durations are $h\kappa_j(k)$, where $\kappa_j\ge0$ is a
 polynomial in $k$. The coefficient $\kappa_5=4k^2L_\vartheta$ of the terminal
 correction vanishes at $k=0$.
\item \ref{H3}: the masses $m_j(\tau,h,k)=\mathfrak n_j^{-1}\alpha(q_0+h(o_j+\kappa_j(k)\tau))$
 take values in $[0,1]$. For $(h,k)\in\mathcal U$, the argument of $\alpha$ lies
 in $K'$, on which $\alpha$ is infinitely differentiable, and all derivatives
 of $m_j$ in $(h,k)$ exist and are continuous.
\item \ref{H4}: the terminal function is $f(h,k,\mathbf z)=\sum_i\Phi(q_0+h\tau_T(k),z_i)$.
 By \cref{lem:sc-derivatives} and \cref{b:lem:os-scalar-jets}(v), every
 derivative $\partial_h^{\gamma_h}\partial_k^{\gamma_k}D^\iota_{\mathbf z}f$ with
 $(\gamma_h,\gamma_k,\iota)\ne(0,0,0)$ exists, is continuous, and is bounded on
 $\mathcal U\times\R^3$.
\item \ref{H5}: since $0<\chi\le1$ and $f$ is a sum of functions of single
 coordinates, we have $\norm{D^2f}_\infty\le1$, and $2\ell bh_0\Lambda_*\norm{D^2f}_\infty
 \le6bh_0\Lambda_*\le1$ by \eqref{eq:cub-h0}.
\end{itemize}
The suprema that enter the constants of \cref{prop:flow-uniform}(ii) and
\cref{lem:flow-penalty} are bounded uniformly in $q_0\in K$, because they involve
only the derivatives of $\alpha$ and $\Phi$ on $K'$. By
\cref{rem:flow-uniformity}, the conclusions are uniform in $q_0\in K$.

\begin{lemma}[Derivatives of the local block]\label{b:lem:os-flow}
Let $h_0$ be as in \eqref{eq:cub-h0}. The following hold for $q_0\in K$,
$0\le h\le h_0$ and $\abs k\le k_0$, where derivatives in $h$ at $h=0$ are
right derivatives. All constants depend only on $\beta$, $\vartheta$, $k_0$, $K$
and the orders of the derivatives.
\begin{enumerate}[label=\textup{(\roman*)}]
\item For all integers $\gamma_h,\gamma_k,j\ge0$ with $\gamma_h+\gamma_k+j\ge1$, the derivatives
 $\partial_h^{\gamma_h}\partial_k^{\gamma_k}D_{\mathbf z}^j\mathcal B$ and
 $\partial_h^{\gamma_h}\partial_k^{\gamma_k}D_{\mathbf z}^j\mathcal B^\circ$ exist, are continuous,
 and are bounded uniformly in $\mathbf z\in\R^3$.
\item For all integers $\gamma_h,\gamma_k,j\ge0$, the derivative
 $\partial_h^{\gamma_h}\partial_k^{\gamma_k}\partial_z^jV(q_0,h,k,z)$ is bounded uniformly in $z$.
 Moreover,
 \[
  V(q_0,h,0,\cdot)=0,\qquad V(q_0,h,-k,\cdot)=V(q_0,h,k,\cdot),
 \]
 and $V(q_0,0,k,\cdot)=\partial_hV(q_0,0,k,\cdot)=0$.
\item There exists $C$ such that, for $0\le j\le3$ and all $z$,
 \begin{equation}\label{b:eq:os-V-bound}
  \abs{\partial_z^jV(q_0,h,k,z)}\le Ch^2k^2.
 \end{equation}
\item The function $\Delta\Pi$ is smooth in $(h,k)$, with bounded derivatives of
 every order, it is even in $k$, it vanishes at $k=0$, and
 $\abs{\Delta\Pi}\le Ch^2k^2$.
\end{enumerate}
\end{lemma}

We have $\norm{V}_{C^3_b}\le Ch^2k^2=C\delta^2$, which is at most
$C\delta^2/h$ for $h\le1$. Of the derivatives in (i) and (ii), we use those with
$\gamma_h\le4$, $\gamma_k\le2$, and $j\le3$; they have weight $j+2(\gamma_h+\gamma_k)\le15$ in the sense of
\cref{sec:flow}.

\begin{proof}
Part (i) is \cref{prop:flow-uniform}(ii), with constants uniform in $q_0\in K$
by \cref{rem:flow-uniformity}. The bounds include the right derivatives
at $h=0$ and the derivatives at $k=0$, where the terminal correction has zero
duration (\cref{rem:flow-regularization}). The value $\mathcal B$ itself has
linear growth, and (i) does not bound it.

For (ii), we first prove the identities. If $k=0$, then $B_j=B_j^\circ$ for $j\le4$ and
$\kappa_5=0$. The perturbed and baseline windows coincide, and
$V=\mathcal B^\circ(\cdot\,\mathbf e)-3\Phi(q_0,\cdot)=0$ by
\cref{lem:cub-trial}(c). For the evenness, let $\mathsf S$ be as in
\cref{lem:cub-identities}(d). The durations and masses are even in $k$, and the
terminal function is invariant under $\mathsf S$. If $F$ solves
\eqref{b:eq:os-vector-flow} at $k$, then $(\tau,\mathbf z)\mapsto F(\tau,\mathsf
S\mathbf z)$ solves it at $-k$, by \cref{lem:cub-identities}(d), and it has bounded
first and second spatial derivatives. By the uniqueness in
\cref{prop:flow-uniform}(i), $\mathcal B(q_0,h,-k,\mathbf z)
=\mathcal B(q_0,h,k,\mathsf S\mathbf z)$, and $\mathsf S(z\mathbf e)=z\mathbf e$.

At $h=0$ all durations vanish. Let $\varphi(\mathbf z)=\sum_i\Phi(q_0,z_i)$. Then
$\mathcal B(q_0,0,k,\mathbf z)=\varphi(\mathbf z)$ by \cref{rem:flow-first-order},
which implies that $V(q_0,0,k,\cdot)=0$. By \eqref{eq:flow-first-order}, in which the average mass of
stage $j$ at $h=0$ is $a/\mathfrak n_j$ with $a=\alpha(q_0)$,
\[
 \partial_h\mathcal B(q_0,0,k,\mathbf z)
 =\tau_T\sum_i\partial_t\Phi(q_0,z_i)
 +\frac b2\sum_{j=1}^5\kappa_j\Bigl\{B_j\mathbin:D^2\varphi(\mathbf z)
 +\frac a{\mathfrak n_j}\nabla\varphi(\mathbf z)\cdot B_j\nabla\varphi(\mathbf z)\Bigr\}.
\]
The first term comes from the dependence of $T$ on $h$. At $\mathbf
z=z\mathbf e$, we have $D^2\varphi=\chi(q_0,z)\mathrm{Id}$ and $\nabla\varphi=u(q_0,z)\mathbf
e$, and the sum equals
$\chi(q_0,z)\sum_j\kappa_j\tr B_j+u(q_0,z)^2a\sum_j\mathfrak n_j^{-1}\kappa_j\,
\mathbf e\cdot B_j\mathbf e$. By \cref{lem:cub-identities}(a) and (b), it does
not change when $B_j$ is replaced by $B_j^\circ$. Since $\mathcal B^\circ(q_0,h,k,z\mathbf
e)=3\Phi(q_0,z)$ does not depend on $h$, we conclude that
$\partial_hV(q_0,0,k,\cdot)=0$. The bounds in (ii) with $\gamma_h+\gamma_k+j\ge1$ follow
from (i), the formula for derivatives along the diagonal in
\cref{rem:flow-uniformity}, and \cref{lem:sc-derivatives}, since $V$ is the difference of
$\mathcal B(\cdot\,\mathbf e)$ and the function $3\Phi(q_0,\cdot)$, which does
not depend on $(h,k)$. Finally, $V$ itself is bounded, since
$V(q_0,0,k,\cdot)=0$ and $\partial_hV$ is bounded.

For (iii), let $0\le j\le3$. Taylor's formula and the identities
$V(q_0,0,k,\cdot)=\partial_hV(q_0,0,k,\cdot)=0$ yield
$\partial_z^jV(h)=\int_0^h(h-h')\,\partial_h^2\partial_z^jV(h')\dd h'$. The function
$\partial_h^2\partial_z^jV(h',k)$ vanishes at $k=0$ and is even in $k$, which
implies that $\abs{\partial_h^2\partial_z^jV(h',k)}\le\frac{k^2}2\sup\abs{\partial_k^2
\partial_h^2\partial_z^jV}$. Combining these bounds with (ii), we obtain
\eqref{b:eq:os-V-bound}.

For (iv), \cref{lem:flow-penalty} with $Q_\star=q_0\mathbf e\mathbf e^{\mathsf T}$
shows that the penalties of the two windows are smooth with bounded
derivatives of every order, and their contributions on $[0,q_0]$ and $[T,1]$
cancel in $\Delta\Pi$ (Step~4 of the proof of \cref{lem:cub-trial}). The
penalty of each path is invariant under $Q\mapsto\mathsf SQ\mathsf S$, since
$\norm{\mathsf SQ\mathsf S}_F=\norm Q_F$; by \cref{lem:cub-identities}(d), this
proves that $\Delta\Pi$ is even in $k$. At $k=0$ the two windows coincide, and
$\Delta\Pi=0$. Finally, $\Delta\Pi=h^2\Theta(h,k)$, where $\Theta$ is the sum in
\eqref{eq:cub-penalty} without the factor $h^2$; since $\Theta$ is smooth, even in
$k$ and zero at $k=0$, we have $\abs\Theta\le Ck^2$.
\end{proof}

\begin{corollary}[Reduction to the boundary perturbation]\label{cor:cub-prefix}
For $q_0\in K$, $0\le h\le h_0$, and $\abs k\le k_0$,
\[
 \mathcal T(q_0,h,k)-3P_\beta
 =\E V(q_0,h,k,Y_{q_0})-\frac b4\Delta\Pi(q_0,h,k)+\mathcal E_{\rm in},
 \qquad\abs{\mathcal E_{\rm in}}\le Ch^4k^4.
\]
\end{corollary}

\begin{proof}
By \cref{b:lem:os-flow}(ii), $V$, $V'$, and $V''$ are bounded, and
$\norm V_\infty\le Ch^2k^2$ by \eqref{b:eq:os-V-bound}. Using \cref{cor:flow-prefix}
with $\mathfrak n=3$, $t_\star=q_0$, and $\eps=Ch^2k^2$, we find that
\[
 \Bigl|\mathcal H\bigl(3\Phi(q_0,\cdot)+V\bigr)-3\Phi(0,0)
 -\E V(q_0,h,k,Y_{q_0})\Bigr|\le\frac{C^2h^4k^4}2.
\]
We have $3\Phi(0,0)=\mathcal H(3\Phi(q_0,\cdot))$ by \eqref{eq:cub-prefix-scalar},
and inserting this bound into \eqref{eq:cub-decomposition} completes the proof.
\end{proof}

The expectation $\E V(q_0,h,k,Y_{q_0})$ is taken under the law of the scalar
diffusion at the fixed time $q_0$, which does not depend on $h$ or $k$. This is
why we expand at a fixed left endpoint $q_0$ rather than at a fixed center $r$.

We expand $\Delta\Pi$ explicitly. We write $a=\alpha(q_0)$ and insert
$\alpha(q_0+h(o_j+\kappa_j\tau))=a+\alpha'(q_0)h(o_j+\kappa_j\tau)+O(h^2)$ in
\eqref{eq:cub-penalty}. Then the Taylor polynomial of degree three of
$\Delta\Pi$ in $h$ is
$ah^2N_\vartheta(k)+\alpha'(q_0)h^3N_\vartheta^{(1)}(k)$, where
\begin{equation}\label{eq:cub-N}
 N^{(i)}_\vartheta(k)=\sum_{j=1}^5\frac{2\kappa_j}{\mathfrak n_j}\int_0^1
 (o_j+\kappa_j\tau)^i\Bigl[(\widetilde Q_{j-1}+\kappa_j\tau B_j)\mathbin:B_j
 -(\widetilde Q^\circ_{j-1}+\kappa_j\tau B_j^\circ)\mathbin:B^\circ_j\Bigr]\dd\tau
\end{equation}
for $i\in\{0,1\}$, and $N_\vartheta=N_\vartheta^{(0)}$. The integrands are
polynomials in $\tau$, and we find
\begin{equation}\label{eq:cub-N-explicit}
 N_\vartheta(k)=\frac{8(1-\vartheta)}{(2-\vartheta)^2}k^2
 -\frac{2(3\vartheta^2+6\vartheta+4)}{3(2-\vartheta)^4}k^4,\qquad
 N_{1/2}(k)=\frac{16}9k^2-\frac{248}{243}k^4,
\end{equation}
and $N^{(1)}_{1/2}(k)=\frac{103}{108}k^2-\frac{7180}{729}k^4-\frac{21296}{729}k^6$.
The polynomial $N^{(1)}_\vartheta$ enters only through the cancellation in
\cref{b:prop:os-expansion}, and we do not use its value.

\section{Taylor coefficients at constant mass}\label{sec:cub-coefficients}

By \cref{b:lem:os-flow}, the function $h\mapsto V(q_0,h,k,z)$ has four
bounded derivatives, and it is determined up to an error $O(h^4)$ by its
Taylor polynomial of degree three. In this section and the next we compute
this polynomial. We write $[X]_n=\partial_h^nX|_{h=0}/n!$ for the Taylor
coefficients of a function $X$ of $h$ at $h=0$.

\subsection{The recursion for the coefficients}

By \cref{prop:flow-uniform}(iii), the derivatives $\partial_h^nF$ solve the
equations obtained by differentiating \eqref{b:eq:os-vector-flow} in $h$.
Evaluating them at $h=0$, we obtain a recursion for the Taylor coefficients
$[F]_n(\tau,\mathbf z)$ of the flow. We write
$[m_j]_l(\tau)=\alpha^{(l)}(q_0)\,(o_j+\kappa_j\tau)^l/(l!\,\mathfrak n_j)$ for the
Taylor coefficients of the masses. The right side of
\eqref{b:eq:os-vector-flow} contains the factor $h$, and taking the coefficient
of $h^n$ we find, for $n\ge1$ and $\tau\in[j-1,j]$,
\begin{equation}\label{eq:cub-coefficient-recursion}
\begin{aligned}
 [F]_n(\tau)=[F]_n(j)+\frac{b\kappa_j}2\int_\tau^j\Bigl\{&B_j\mathbin:D^2[F]_{n-1}(\tau')\\
 &+\sum_{l+p+p'=n-1}[m_j]_l(\tau'-j+1)\,\nabla[F]_p(\tau')\cdot B_j\nabla[F]_{p'}(\tau')\Bigr\}\dd\tau',
\end{aligned}
\end{equation}
with $[F]_0=f(0,k,\cdot)$ and
$[F]_n(5,\mathbf z)=\frac{\tau_T^n}{n!}\sum_i\partial_t^n\Phi(q_0,z_i)$.
We call each use of the right side of \eqref{b:eq:os-vector-flow} in this
recursion an \emph{insertion}. By \eqref{eq:sc-time-derivative} and its consequences in the proof of
\cref{b:lem:os-scalar-jets}, $\partial_t^n\Phi(q_0,\cdot)$ is a polynomial in
$\alpha(q_0),\dots,\alpha^{(n-1)}(q_0)$ and in the spatial derivatives of
$\Phi(q_0,\cdot)$ of orders one to $2n$. By induction on $n$, for every $n\ge1$ the coefficient
$[\mathcal B]_n(\mathbf z)$ is a sum of products of spatial
derivatives of positive order of $\Phi(q_0,\cdot)$ at $z_v,z_x,z_y$,
of total spatial order at most $2n$, with coefficients
that are polynomials in $k$ and in
$\alpha(q_0),\ldots,\alpha^{(n-1)}(q_0)$. Each power $h^l$ of the
expansion of a mass carries a factor $\alpha^{(l)}(q_0)$, and each insertion
of the right side of \eqref{b:eq:os-vector-flow} carries a factor $h$; in
particular, $\alpha'''(q_0)$ does not occur for $n\le3$. The same holds for
$\mathcal B^\circ$ and for $V$.

We call the \emph{constant-mass part} of $[V]_n$ the value of this polynomial
at $\alpha'(q_0)=\alpha''(q_0)=0$, with the other variables unchanged. It is
the value of the recursion when every mass $m_j$ is replaced by the constant
$a/\mathfrak n_j$, with $a=\alpha(q_0)$, and when the time derivatives of $\Phi$ at
$q_0$ are computed from the Parisi equation with the constant profile $a$. In
this section we compute the constant-mass part. The remaining part is computed
in \cref{sec:cub-profile}.

\subsection{Formal Gaussian steps}

With constant masses, each stage is a Gaussian integration step, and the
three stages of mass $a$ combine into one step by \cref{lem:inp-steps}(b). It
is convenient to subtract the scalar evolution of the terminal function from
this step. The result is a step whose covariance is not positive semidefinite,
and we treat it as a formal object.

Let $\ell\ge1$, let $\Sigma$ be a symmetric $\ell\times\ell$ matrix, let
$m\in[0,1]$, and let $f$ be a smooth function on $\R^\ell$. We define $f^{[0]}=f$
and, for $n\ge1$,
\begin{equation}\label{eq:cub-formal}
 f^{[n]}=\frac1{2n}\Bigl(\Sigma\mathbin:D^2f^{[n-1]}
 +m\sum_{p+p'=n-1}\nabla f^{[p]}\cdot\Sigma\nabla f^{[p']}\Bigr),
\end{equation}
and we write $\widehat T_{m,\varepsilon\Sigma}f=\sum_{n\ge0}\varepsilon^nf^{[n]}$
for the resulting formal power series in $\varepsilon$. It is the unique formal
solution of $\partial_\varepsilon F=\frac12\bigl(\Sigma\mathbin:D^2F
+m\nabla F\cdot\Sigma\nabla F\bigr)$ with $F=f$ at $\varepsilon=0$. By induction,
$f^{[n]}$ is a sum of products of derivatives of $f$ of positive order, of
total order $2n$, and it is a homogeneous polynomial of degree $n$ in the
entries of $\Sigma$. Formal series of functions are composed by substitution:
if the function in \eqref{eq:cub-formal} is itself a formal power series in
$\varepsilon$, we insert this series and collect the powers of $\varepsilon$. The coefficient of each power involves finitely many
terms. The relation with the integration steps of \cref{def:inp-step} is the
following.

\begin{lemma}[Formal steps]\label{lem:cub-formal}
Let $m\in[0,1]$, let $\Sigma,\Sigma'$ be symmetric $\ell\times\ell$ matrices, and
let $f$ be smooth with bounded derivatives of positive order.
\begin{enumerate}[label=\textup{(\alph*)}]
\item If $\Sigma\succeq0$, then $\varepsilon\mapsto T_{m,\varepsilon\Sigma}f(\mathbf z)$
 is infinitely differentiable on $[0,1]$, and $\widehat T_{m,\varepsilon\Sigma}f(\mathbf z)$ is its
 Taylor series at $\varepsilon=0$.
\item $\widehat T_{m,\varepsilon\Sigma}\widehat T_{m,\varepsilon\Sigma'}f
 =\widehat T_{m,\varepsilon(\Sigma+\Sigma')}f$.
\item For $c>0$ such that $m/c\in[0,1]$, we have $\widehat T_{m/c,\varepsilon\Sigma}(cf)=c\,\widehat
 T_{m,\varepsilon\Sigma}f$.
\item If $\Sigma$ is diagonal with entries $\sigma_1,\dots,\sigma_\ell$ and
 $f(\mathbf z)=\sum_i\psi_i(z_i)$, then $\widehat T_{m,\varepsilon\Sigma}f(\mathbf
 z)=\sum_i\widehat T_{m,\varepsilon\sigma_i}\psi_i(z_i)$.
\item For $c,u\in\R$ and $\mathbf e'\in\R^\ell$,
 \[
  \widehat T_{m,\varepsilon\Sigma}\bigl(c+u\,\mathbf e'\cdot\mathbf z+f\bigr)(\mathbf z)
  =c+u\,\mathbf e'\cdot\mathbf z+\frac{\varepsilon mu^2}2\,\mathbf e'\cdot\Sigma\mathbf e'
  +\bigl(\widehat T_{m,\varepsilon\Sigma}f\bigr)\bigl(\mathbf z+\varepsilon mu\Sigma\mathbf e'\bigr),
 \]
 where the last term is expanded in powers of $\varepsilon$ by Taylor's formula
 in the shift.
\end{enumerate}
\end{lemma}

\begin{proof}
For (a), let $\Sigma\succeq0$. By \cref{lem:flow-cole-hopf}(b) and (c), the
function $F(\varepsilon,\mathbf z)=T_{m,\varepsilon\Sigma}f(\mathbf z)$ is
infinitely differentiable on $[0,1]\times\R^\ell$ and solves
$\partial_\varepsilon F=\frac12(\Sigma\mathbin:D^2F+m\nabla F\cdot\Sigma\nabla F)$.
Differentiating this equation $n-1$ times in $\varepsilon$ and setting
$\varepsilon=0$, we find by induction that
$\partial_\varepsilon^nF(0,\cdot)=n!\,f^{[n]}$.

For (b), both sides are formal series whose coefficients are polynomials in
the entries of $\Sigma$ and $\Sigma'$ and in the derivatives of $f$. If
$\Sigma,\Sigma'\succeq0$, then \cref{lem:inp-steps}(b), part (a), and the chain
rule show that the two sides have the same coefficients. The set of pairs of
positive semidefinite matrices contains a nonempty open subset of the space of
pairs of symmetric matrices, and a polynomial that vanishes on a nonempty open
set vanishes identically. This proves (b). Parts (c) and (d) follow from
\eqref{eq:cub-formal} by induction on $n$: for (c), the function $(cf)^{[n]}$
computed with mass $m/c$ equals $c\,f^{[n]}$ computed with mass $m$, since the
quadratic term acquires the factor $(m/c)c^2=mc$; for (d), mixed derivatives of
$f$ vanish and the recursion splits into one recursion for each coordinate. For
(e), let $\Sigma\succeq0$ and $m>0$, and let $\mathbf X$ be a centered Gaussian
vector with covariance $\varepsilon\Sigma$. Multiplying the law of
$\mathbf X$ by $e^{mu\mathbf e'\cdot\mathbf X}/\E e^{mu\mathbf e'\cdot\mathbf X}$
shifts its mean to $\varepsilon mu\Sigma\mathbf e'$, and
$\E e^{mu\mathbf e'\cdot\mathbf X}=e^{\varepsilon m^2u^2\mathbf
e'\cdot\Sigma\mathbf e'/2}$. We obtain
\begin{equation}\label{b:eq:os-gauge}
 T_{m,\varepsilon\Sigma}\bigl(c+u\,\mathbf e'\cdot\mathbf z+f\bigr)(\mathbf z)
 =c+u\,\mathbf e'\cdot\mathbf z+\frac{\varepsilon mu^2}2\,\mathbf e'\cdot\Sigma\mathbf e'
 +\bigl(T_{m,\varepsilon\Sigma}f\bigr)\bigl(\mathbf z+\varepsilon mu\Sigma\mathbf e'\bigr).
\end{equation}
For $m=0$ this is immediate. Taking Taylor series in $\varepsilon$ and
arguing as in (b), we obtain (e) for every symmetric $\Sigma$.
\end{proof}

The identity \eqref{b:eq:os-gauge} is the Cameron--Martin formula for a
finite-dimensional Gaussian shift, obtained by completing the square as in
\cite[Section~A.3, (A.6)]{talagrand-2011}.

\subsection{The coefficient matrices}

We define
\begin{equation}\label{eq:cub-Lambda}
 \Lambda_0=\mathbf c_0\mathbf c_0^{\mathsf T},\qquad
 \Lambda_1=\vartheta\,\mathbf c_1\mathbf c_1^{\mathsf T},\qquad
 \Lambda_2=\mathbf c_v\mathbf c_v^{\mathsf T}
 +(1-\vartheta)\bigl(\mathbf c_x\mathbf c_x^{\mathsf T}+\mathbf c_y\mathbf c_y^{\mathsf T}\bigr)
 -2\mathrm{Id}-k^2L_\vartheta L_2.
\end{equation}
We have $\Lambda_0=\kappa_1B_1$, $\Lambda_1=\kappa_2B_2$, and
$\Lambda_2=\kappa_3B_3+\kappa_4B_4+\kappa_5B_5-\tau_T\mathrm{Id}$. The matrices
$\Lambda_0$ and $\Lambda_1$ are positive semidefinite, and $\Lambda_2$ is not:
its $(v,v)$ entry is $-1$. At $k=0$ we have $\Lambda_0=\mathbf e\mathbf
e^{\mathsf T}$, $\Lambda_1=\vartheta(\mathbf e_x+\mathbf e_y)(\mathbf e_x+\mathbf
e_y)^{\mathsf T}$ and $\Lambda_2=\vartheta\mathbf e_v\mathbf e_v^{\mathsf
T}-(1+\vartheta)\mathrm{Id}$, which are the same expressions for the baseline rates.
By \cref{lem:cub-identities}(b) and (c),
\begin{equation}\label{b:eq:os-matrix-cancellations}
 \operatorname{diag}(\Lambda_0+\Lambda_1+\Lambda_2)=0,\qquad
 \bigl(\tfrac13\Lambda_0+\tfrac12\Lambda_1+\Lambda_2\bigr)\mathbf e=0.
\end{equation}
Moreover, $\mathbf e\cdot\Lambda_0\mathbf e=9$,
$\mathbf e\cdot\Lambda_1\mathbf e=4\vartheta$,
$\mathbf e\cdot\Lambda_2\mathbf e=-3-2\vartheta$, and
\[
 \Lambda_0+\Lambda_1+\Lambda_2=\begin{pmatrix}0&1+k&1-k\\1+k&0&1+\vartheta\\
 1-k&1+\vartheta&0\end{pmatrix}.
\]
For $\vartheta=1/2$ we have $\omega_0=2/3$, $\omega_1=1/3$, $L_{1/2}=11/9$,
$\Lambda_0=\mathbf c_0\mathbf c_0^{\mathsf T}$ with
$\mathbf c_0=(1,1+\frac{2k}3,1-\frac{2k}3)^{\mathsf T}$, and
\[
 \Lambda_2=\begin{pmatrix}
 -1&\frac k3&-\frac k3\\[2pt]
 \frac k3&-\frac{2k^2}3-\frac{2k}3-\frac32&\frac{2k^2}3\\[2pt]
 -\frac k3&\frac{2k^2}3&-\frac{2k^2}3+\frac{2k}3-\frac32
 \end{pmatrix},
\]
\[
 \Lambda_0+\Lambda_1=\begin{pmatrix}
 1&1+\frac{2k}3&1-\frac{2k}3\\[2pt]
 1+\frac{2k}3&\frac{2k^2}3+\frac{2k}3+\frac32&\frac32-\frac{2k^2}3\\[2pt]
 1-\frac{2k}3&\frac32-\frac{2k^2}3&\frac{2k^2}3-\frac{2k}3+\frac32
 \end{pmatrix}.
\]

\begin{lemma}[Reduction to three formal steps]\label{lem:cub-three-steps}
Let $a=\alpha(q_0)$ and $\varphi(\mathbf z)=\sum_i\Phi(q_0,z_i)$. For $n\le3$,
the constant-mass part of $[\mathcal B]_n$ is the coefficient of $h^n$ in
\begin{equation}\label{eq:cub-three-steps}
 \widehat T_{a/3,\,bh\Lambda_0}\,\widehat T_{a/2,\,bh\Lambda_1}\,
 \widehat T_{a,\,bh\Lambda_2}\,\varphi,
\end{equation}
and the constant-mass part of $[\mathcal B^\circ]_n$ is the coefficient of $h^n$
in the same expression at $k=0$.
\end{lemma}

\begin{proof}
With the constant mass $a/\mathfrak n_j$, stage $j$ maps $F(j,\cdot)$ to
$F(j-1,\cdot)=T_{a/\mathfrak n_j,\,bh\kappa_jB_j}F(j,\cdot)$, by
\eqref{eq:flow-step-solution}. By \cref{lem:cub-formal}(a) and the chain rule,
the recursion \eqref{eq:cub-coefficient-recursion} for this stage is therefore
the recursion of the formal step $\widehat T_{a/\mathfrak n_j,\,bh\kappa_jB_j}$, with
$\varepsilon=h$. By \cref{lem:cub-formal}(b), the stages
$3$, $4$, and $5$, all of mass $a$, combine into
$\widehat T_{a,\,bh(\Lambda_2+\tau_T\mathrm{Id})}$. With the constant profile $a$, the
Parisi equation reads $\partial_t\Phi=-\frac b2(\partial_{zz}\Phi
+a(\partial_z\Phi)^2)$, and the Taylor series
$\sum_n(h\tau_T)^n\partial_t^n\Phi(q_0,\cdot)/n!$ is the formal solution of
this equation in the time $h\tau_T$, that is,
$\widehat T_{a,-bh\tau_T}\Phi(q_0,\cdot)$. By \cref{lem:cub-formal}(d), the
terminal function is therefore $\widehat T_{a,-bh\tau_T\mathrm{Id}}\varphi$. By
\cref{lem:cub-formal}(b) again,
$\widehat T_{a,bh(\Lambda_2+\tau_T\mathrm{Id})}\widehat T_{a,-bh\tau_T\mathrm{Id}}=\widehat
T_{a,bh\Lambda_2}$, which proves the first claim. The second is the same
computation with the baseline rates, whose coefficient matrices are the values of
$\Lambda_0,\Lambda_1,\Lambda_2$ at $k=0$.
\end{proof}

The scalar evolution of the terminal function between $q_0$ and $T$ appears in
\eqref{eq:cub-three-steps} through the term $-\tau_T\mathrm{Id}$ in $\Lambda_2$. It
accounts for the change of the curvature of the terminal function between the
times $T$ and $q_0$. This change is of order $h$, and it enters at order $h^3$
through the quadratic part of the terminal function; the term $-\tau_T\mathrm{Id}$ in $\Lambda_2$ retains this contribution in the coefficients computed below.

\subsection{Expansion at a centered point}

We now expand compositions of three formal steps applied to a separable function
whose linear part vanishes. For symmetric matrices $\Sigma_0,\Sigma_1,\Sigma_2$
we write $\Sigma_{\le j}=\Sigma_0+\dots+\Sigma_j$ and $\Sigma_{\le-1}=0$.

\begin{lemma}[Expansion at a centered point]\label{lem:cub-centered}
Let $\Sigma_0,\Sigma_1,\Sigma_2$ be symmetric $3\times3$ matrices with
$\operatorname{diag}(\Sigma_{\le2})=0$, and let $m_0,m_1,m_2\in(0,1]$. Let
$\psi\colon\R\to\R$ be smooth with $\psi(0)=\psi'(0)=0$, write
$\chi=\psi''(0)$ and $w=\psi'''(0)$, and set $\bar\psi(\mathbf y)=\sum_i\psi(y_i)$. Let
$\mathfrak E(\varepsilon)$ be the value at $\mathbf y=0$ of
$\widehat T_{m_0,\varepsilon\Sigma_0}\widehat T_{m_1,\varepsilon\Sigma_1}\widehat
T_{m_2,\varepsilon\Sigma_2}\bar\psi$. Then $[\mathfrak E]_0=[\mathfrak E]_1=0$,
\begin{equation}\label{b:eq:os-quadratic-norm}
 [\mathfrak E]_2=\frac{\chi^2}4\sum_{j=0}^2m_j\bigl(\norm{\Sigma_{\le j}}_F^2
 -\norm{\Sigma_{\le j-1}}_F^2\bigr),
\end{equation}
and
\begin{equation}\label{eq:cub-third}
 [\mathfrak E]_3=\chi^3\mathcal Q
 +\frac{w^2}{12}\sum_{j=0}^2m_j\bigl(\operatorname{cub}(\Sigma_{\le j})
 -\operatorname{cub}(\Sigma_{\le j-1})\bigr),
\end{equation}
where, with $\Upsilon_1=m_2\Sigma_2+m_1\Sigma_1$ and
$\Upsilon_2=m_2^2\Sigma_2^2+m_1m_2(\Sigma_1\Sigma_2+\Sigma_2\Sigma_1)+m_1^2\Sigma_1^2$,
\[
\begin{aligned}
 \mathcal Q={}&\frac{m_2^2}6\tr\Sigma_2^3+\frac{m_2^2}2\tr(\Sigma_1\Sigma_2^2)
 +\frac{m_1m_2}2\tr(\Sigma_1^2\Sigma_2)+\frac{m_1^2}6\tr\Sigma_1^3\\
 &+\frac12\tr(\Sigma_0\Upsilon_2)+\frac{m_0}2\tr(\Sigma_0^2\Upsilon_1)+\frac{m_0^2}6\tr\Sigma_0^3.
\end{aligned}
\]
\end{lemma}

The sum in \eqref{b:eq:os-quadratic-norm} is the squared-norm integral
\eqref{eq:inp-penalty} of the path with increments $\Sigma_0,\Sigma_1,\Sigma_2$
and masses $m_0,m_1,m_2$. In its cross terms
$2m_j\,\Sigma_{\le j-1}\mathbin:\Sigma_j$, the mass is that of the later
increment.

\begin{proof}
We first determine which products of derivatives of $\psi$ can occur. We then
compute the coefficients of the terms that are linear and quadratic in $\psi$,
and finally the term that is cubic in $\psi$. The computations in Steps~2
and~3 do not use the condition on the diagonal of $\Sigma_{\le2}$, which enters
only at the end of each step.

\paragraph{Step 1: the possible terms.} By the remark after
\eqref{eq:cub-formal}, applied to each of the three steps, $[\mathfrak E]_n$ is
a sum of products of derivatives of positive order of $\bar\psi$ at $0$, of total
order $2n$, with coefficients that are polynomials in the entries of the
$\Sigma_j$ and in the $m_j$. At $\mathbf y=0$, the derivatives of order one
vanish, because $\psi'(0)=0$, the mixed derivatives vanish, because $\bar\psi$ is
separable, and $\partial_i^p\bar\psi(0)=\psi^{(p)}(0)$. We conclude that
$[\mathfrak E]_0=\bar\psi(0)=0$ and
\[
 [\mathfrak E]_1=\mathsf c_1\chi,\qquad [\mathfrak E]_2=\mathsf c_2\psi^{(4)}(0)+\mathsf c_3\chi^2,\qquad
 [\mathfrak E]_3=\mathsf c_4\psi^{(6)}(0)+\mathsf c_5\chi\psi^{(4)}(0)+\mathsf c_6w^2+\mathsf c_7\chi^3,
\]
where the coefficients $\mathsf c_i$ are polynomials in the entries of
$\Sigma_0,\Sigma_1,\Sigma_2$ (with the masses fixed). These
coefficients depend on $\psi$ only through $\psi^{(p)}(0)$ for $2\le p\le6$, and
we may compute each of them with a convenient choice of $\psi$.

In Step~2 we take $\psi$ smooth with compact support, for instance the Taylor
polynomial of degree six of the given $\psi$ multiplied by a smooth function
with compact support that equals one near the origin. We also assume in Step~2
that $\Sigma_0,\Sigma_1,\Sigma_2$ are positive semidefinite. This is no loss of
generality: the triples of positive semidefinite matrices contain a nonempty
open subset of the triples of symmetric matrices, and an identity between
polynomials in the entries that holds on this subset holds for all symmetric
triples. For a parameter $\theta\in[-1,1]$, \cref{lem:flow-cole-hopf}(c),
applied to the three steps in turn, shows that the composition of the steps
$T_{m_j,\varepsilon\Sigma_j}$ applied to $\theta\bar\psi$ is infinitely
differentiable in $(\varepsilon,\theta)\in[0,1]\times[-1,1]$. By
\cref{lem:cub-formal}(a) and the chain rule, as in the proof of
\cref{lem:cub-formal}(b), its Taylor coefficients in $\varepsilon$ are those of
the formal composition. We use independent centered Gaussian vectors
$\mathbf X_0,\mathbf X_1,\mathbf X_2$ with covariances
$\varepsilon\Sigma_0,\varepsilon\Sigma_1,\varepsilon\Sigma_2$, and we set
$\mathbf S_j=\mathbf X_0+\dots+\mathbf X_j$, which has covariance
$\varepsilon\Sigma_{\le j}$.

\paragraph{Step 2: the terms linear and quadratic in $\psi$.} We replace $\psi$ by
$\theta\psi$, so that $\mathsf c_1,\mathsf c_2,\mathsf c_4$ are coefficients of $\theta$ and
$\mathsf c_3,\mathsf c_5,\mathsf c_6$ are coefficients of $\theta^2$. Let $U_3=\theta\bar\psi$ and
$U_j=T_{m_j,\varepsilon\Sigma_j}U_{j+1}$ for $j=2,1,0$, so that $U_0(0)$ is the
value of the composition. At $\theta=0$ all $U_j$ vanish. Differentiating
$U_j=m_j^{-1}\log\E_je^{m_jU_{j+1}}$ in $\theta$ at $\theta=0$, where $\E_j$ is
the expectation over $\mathbf X_j$, we obtain
$\partial_\theta U_j=\E_j\partial_\theta U_{j+1}$ and
$\partial_\theta^2U_j=\E_j\partial_\theta^2U_{j+1}+m_j\Var_j(\partial_\theta
U_{j+1})$. Let $M_j=\E[\bar\psi(\mathbf S_2)\mid\mathbf X_0,\dots,\mathbf X_{j-1}]$,
so that $M_0=\E\bar\psi(\mathbf S_2)$ and $M_3=\bar\psi(\mathbf S_2)$. These recursions imply
that
\[
 \partial_\theta U_0(0)\big|_{\theta=0}=\E\bar\psi(\mathbf S_2),\qquad
 \frac12\partial_\theta^2U_0(0)\big|_{\theta=0}
 =\sum_{j=0}^2\frac{m_j}2\bigl(\E M_{j+1}^2-\E M_j^2\bigr).
\]
The first expression is $\sum_i\E\psi(S_{2,i})$. Its Taylor coefficients in
$\varepsilon$ are polynomials in the diagonal entries $(\Sigma_{\le2})_{ii}$
without constant term. These polynomial identities hold for all symmetric
triples, and the coefficients vanish under the diagonal condition, which proves that
$\mathsf c_1=\mathsf c_2=\mathsf c_4=0$. For the second, we have
$M_{j+1}=(\mathsf H\bar\psi)(\mathbf S_j)$, where $\mathsf H$ is the heat average over
the Gaussian vector $\mathbf X_{j+1}+\dots+\mathbf X_2$, with covariance
$\varepsilon(\Sigma_{\le2}-\Sigma_{\le j})$. We write
$\psi=\psi_2+\dots+\psi_6+\psi_{>6}$, with $\psi_p(y)=\psi^{(p)}(0)y^p/p!$ and
$\abs{\psi_{>6}(y)}\le C\abs y^7$. All the Gaussian vectors involved have
covariances of order $\varepsilon$. The part of $M_{j+1}$ coming from $\psi_p$
with $p\le6$ is a polynomial in $\mathbf S_j$ of the parity of $p$, and it is
$O(\varepsilon^{p/2})$ in every $L^q$; the part coming from $\psi_{>6}$ is
$O(\varepsilon^{7/2})$. A product of the parts coming from $\psi_p$ and
$\psi_{p'}$ has expectation $O(\varepsilon^{(p+p')/2})$, with $p=7$
for $\psi_{>6}$, and this expectation vanishes if $p,p'\le6$ and $p+p'$ is
odd. The terms of order $\varepsilon^2$ and $\varepsilon^3$ of
$\E M_{j+1}^2$ come from the three products that we now compute. Let
$\Omega=\varepsilon\Sigma_{\le j}$, and let
$\Omega'=\varepsilon(\Sigma_{\le2}-\Sigma_{\le j})$, so that
$\Omega_{ii}+\Omega'_{ii}=\varepsilon(\Sigma_{\le2})_{ii}$.
The Gaussian moments in the following three computations are evaluated by
the Wick--Isserlis pairing formula \cite{isserlis-1918}.
\begin{itemize}
\item For $\psi_2$, we have $M_{j+1}=\frac\chi2\abs{\mathbf S_j}^2+{}$const and
 $\Var\abs{\mathbf S_j}^2=2\norm\Omega_F^2$. The means $\E M_j$ do not depend on
 $j$, and $\E M_{j+1}^2-\E M_j^2=\frac{\chi^2\varepsilon^2}2\bigl(\norm{\Sigma_{\le
 j}}_F^2-\norm{\Sigma_{\le j-1}}_F^2\bigr)$. This gives $\mathsf c_3$ and proves
 \eqref{b:eq:os-quadratic-norm}.
\item For $\psi_3$, we have $\mathsf H\psi_3(y_i)=\frac w6(y_i^3+3\Omega'_{ii}y_i)$.
 Using $\E S_iS_l=\Omega_{il}$, $\E S_i^3S_l=3\Omega_{ii}\Omega_{il}$ and
 $\E S_i^3S_l^3=6\Omega_{il}^3+9\Omega_{ii}\Omega_{ll}\Omega_{il}$ for $\mathbf S=\mathbf S_j$, we
 find
 \begin{equation}\label{eq:cub-wick}
  \E M_{j+1}^2=w^2\sum_{i,l}\Bigl\{\frac{\Omega_{il}^3}6
  +\frac{\Omega_{il}(\Omega_{ii}+\Omega'_{ii})(\Omega_{ll}+\Omega'_{ll})}4\Bigr\}.
 \end{equation}
 Under the diagonal condition the second term vanishes, and
 $\E M_{j+1}^2=\frac{w^2\varepsilon^3}6\operatorname{cub}(\Sigma_{\le j})$. Also
 $M_0=0$, because $\psi_3$ is odd. This gives $\mathsf c_6$.
\item For the product of $\psi_2$ and $\psi_4$, the means of the two parts of
 $M_j$ do not depend on $j$, and by the Wick formula,
 \[
  \Cov\bigl(M^{(2)}_{j+1},M^{(4)}_{j+1}\bigr)
  =\frac{\chi\psi^{(4)}(0)}4\sum_{i,l}\Omega_{il}^2(\Omega_{ll}+\Omega'_{ll}),
 \]
 where $M^{(p)}_{j+1}$ is the part of $M_{j+1}$ coming from $\psi_p$. Under
 the diagonal condition this vanishes, and $\mathsf c_5=0$.
\end{itemize}

\paragraph{Step 3: the term cubic in $\psi$.} It remains to compute $\mathsf c_7$, and
we may take $\psi(y)=\chi y^2/2$. For positive semidefinite covariances and
sufficiently small positive $\varepsilon$, the
logarithmic determinant below is the Gaussian quadratic-exponential integral,
obtained by diagonalizing and applying \cite[(A.11)]{talagrand-2011} in each
coordinate. We derive its formal extension by computing with the steps directly, for
symmetric $\Sigma_0,\Sigma_1,\Sigma_2$. Let $\Sigma$ and $H$ be symmetric,
$c\in\R$, and $m\in(0,1]$. For a quadratic function
$F=c(\varepsilon)+\frac12\mathbf y\cdot H(\varepsilon)\mathbf y$, we have
$\Sigma\mathbin:D^2F=\tr(\Sigma H(\varepsilon))$ and
$\nabla F\cdot\Sigma\nabla F=\mathbf y\cdot H(\varepsilon)\Sigma H(\varepsilon)\mathbf y$.
The equation $\partial_\varepsilon F=\frac12(\Sigma\mathbin:D^2F+m\nabla
F\cdot\Sigma\nabla F)$ with $F=c+\frac12\mathbf y\cdot H\mathbf y$ at
$\varepsilon=0$ reduces to $c'=\frac12\tr(\Sigma H(\varepsilon))$ and
$H'=mH(\varepsilon)\Sigma H(\varepsilon)$. Differentiating, we find that the
power series $H(\varepsilon)=H(\mathrm{Id}-\varepsilon m\Sigma H)^{-1}$ and
$c(\varepsilon)=c-(2m)^{-1}\log\det(\mathrm{Id}-\varepsilon m\Sigma H)$ solve these
equations. By the uniqueness of the formal solution,
\[
 \widehat T_{m,\varepsilon\Sigma}\Bigl(c+\tfrac12\mathbf y\cdot H\mathbf y\Bigr)
 =c-\frac1{2m}\log\det\bigl(\mathrm{Id}-\varepsilon m\Sigma H\bigr)
 +\tfrac12\mathbf y\cdot H(\mathrm{Id}-\varepsilon m\Sigma H)^{-1}\mathbf y,
\]
where the right side is expanded in powers of $\varepsilon$. By the
substitution rule for formal series, the same holds when $c$ and $H$ are
themselves formal power series in $\varepsilon$. Starting from $H_3=\chi \mathrm{Id}$, the three steps produce the matrices
$H_2=\chi(\mathrm{Id}-\varepsilon m_2\chi\Sigma_2)^{-1}$ and
$H_1=H_2(\mathrm{Id}-\varepsilon m_1\Sigma_1H_2)^{-1}=\chi \mathrm{Id}+\varepsilon\chi^2\Upsilon_1
+\varepsilon^2\chi^3\Upsilon_2+O(\varepsilon^3)$, and the value at $\mathbf y=0$ is
\[
 -\sum_{j=0}^2\frac1{2m_j}\log\det\bigl(\mathrm{Id}-\varepsilon m_j\Sigma_jH_{j+1}\bigr)
 =\sum_{j=0}^2\sum_{p\ge1}\frac{\varepsilon^pm_j^{p-1}}{2p}\tr\bigl((\Sigma_jH_{j+1})^p\bigr).
\]
Collecting the terms of order $\varepsilon^3$, we obtain $\chi^3\mathcal Q$. For
example, for $j=1$ the terms with $p=1$, $2$, and $3$ contribute
$\frac{m_2^2}2\tr(\Sigma_1\Sigma_2^2)$, $\frac{m_1m_2}2\tr(\Sigma_1^2\Sigma_2)$ and
$\frac{m_1^2}6\tr\Sigma_1^3$, respectively. The terms of order $\varepsilon$ and
$\varepsilon^2$ are $\frac\chi2\tr\Sigma_{\le2}$, which vanishes under the
diagonal condition, and \eqref{b:eq:os-quadratic-norm}. This completes the
proof.
\end{proof}

In \cref{lem:cub-centered} we take $\varepsilon=h$, $\Sigma_j=b\Lambda_j$ and
$(m_0,m_1,m_2)=(a/3,a/2,a)$. With these masses $\Upsilon_1=a(\Sigma_2+\Sigma_1/2)$ and
$\Upsilon_2=a^2(\Sigma_2+\Sigma_1/2)^2$. We define the trace polynomial
\begin{equation}\label{b:eq:os-R}
\begin{aligned}
 C_\chi(k)={}&\frac{\tr\Lambda_2^3}6+\frac{\tr(\Lambda_1\Lambda_2^2)}2
 +\frac{\tr(\Lambda_1^2\Lambda_2)}4+\frac{\tr\Lambda_1^3}{24}\\
 &+\frac{\tr\bigl(\Lambda_0(\Lambda_2+\tfrac12\Lambda_1)^2\bigr)}2
 +\frac{\tr\bigl(\Lambda_0^2(\Lambda_2+\tfrac12\Lambda_1)\bigr)}6
 +\frac{\tr\Lambda_0^3}{54},
\end{aligned}
\end{equation}
the Wick polynomial
\begin{equation}\label{b:eq:os-W}
\begin{aligned}
 C_w(k)=\frac1{12}\Bigl\{&\frac{\operatorname{cub}(\Lambda_0)}3
 +\frac{\operatorname{cub}(\Lambda_0+\Lambda_1)-\operatorname{cub}(\Lambda_0)}2\\
 &+\operatorname{cub}(\Lambda_0+\Lambda_1+\Lambda_2)-\operatorname{cub}(\Lambda_0+\Lambda_1)\Bigr\},
\end{aligned}
\end{equation}
and
\[
 N^{\rm f}_\vartheta(k)=\frac13\norm{\Lambda_0}_F^2
 +\frac12\bigl(\norm{\Lambda_0+\Lambda_1}_F^2-\norm{\Lambda_0}_F^2\bigr)
 +\norm{\Lambda_0+\Lambda_1+\Lambda_2}_F^2-\norm{\Lambda_0+\Lambda_1}_F^2.
\]
These are polynomials in $k$ whose coefficients are rational functions of
$\vartheta$, and they are even in $k$ by \cref{lem:cub-identities}(d).

\begin{proposition}[The constant-mass part]\label{prop:cub-frozen}
Let $a=\alpha(q_0)$, $\chi_0=\chi(q_0,z)$, and $w_0=w(q_0,z)$. The constant-mass
parts of $[V]_0,\dots,[V]_3$ are
\[
 0,\qquad 0,\qquad \frac{b^2\chi_0^2}4\,a\,N_\vartheta(k),\qquad
 a^2b^3\chi_0^3\,C_\chi(k)+ab^3w_0^2\,C_w(k),
\]
respectively. Moreover, $C_\chi(0)=C_w(0)=0$, and
$N^{\rm f}_\vartheta=N_\vartheta$.
\end{proposition}

\begin{proof}
Let $u_0=u(q_0,z)$ and, for $\mathbf y\in\R^3$, write
$\varphi(z\mathbf e+\mathbf y)=3\Phi(q_0,z)+u_0\,\mathbf e\cdot\mathbf y+\bar\psi(\mathbf y)$,
where $\bar\psi(\mathbf y)=\sum_i\psi(y_i)$ with
$\psi(y)=\Phi(q_0,z+y)-\Phi(q_0,z)-u_0y$. Then $\psi(0)=\psi'(0)=0$,
$\psi''(0)=\chi_0$ and $\psi'''(0)=w_0$. We use \cref{lem:cub-formal}(e) for
each of the three formal steps in \eqref{eq:cub-three-steps}, from the inside
out. Formal steps commute with translations, because their coefficients do not
depend on $\mathbf z$, and the translations produced by the inner steps pass
through the outer ones. The total translation is
$hbau_0(\frac13\Lambda_0+\frac12\Lambda_1+\Lambda_2)\mathbf e$, and the total
added constant is
$\frac12hbau_0^2\,\mathbf e\cdot(\frac13\Lambda_0+\frac12\Lambda_1+\Lambda_2)\mathbf
e$. Both vanish by \eqref{b:eq:os-matrix-cancellations}, which implies that the
value of \eqref{eq:cub-three-steps} at $z\mathbf e$ is $3\Phi(q_0,z)$ plus the value
$\mathfrak E(h)$ at $\mathbf y=0$ of the three formal steps applied to $\bar\psi$. By
\eqref{b:eq:os-matrix-cancellations}, \cref{lem:cub-centered} applies to
$\Sigma_j=b\Lambda_j$, and it gives $[\mathfrak E]_0=[\mathfrak E]_1=0$,
$[\mathfrak E]_2=\frac{b^2\chi_0^2}4aN^{\rm f}_\vartheta(k)$ and
$[\mathfrak E]_3=a^2b^3\chi_0^3C_\chi(k)+ab^3w_0^2C_w(k)$. The same holds for the
baseline, with the values at $k=0$. By \cref{lem:cub-three-steps}, the
constant-mass part of $[V]_n$ is the difference between these
expressions and their values at $k=0$.

At $k=0$ these values vanish. Indeed, at $k=0$ the composition
\eqref{eq:cub-three-steps} equals $\varphi$ on the diagonal, for every smooth
function $\Phi_0$ in place of $\Phi(q_0,\cdot)$ in the definition of
$\varphi$. To see this, note that $\Lambda_2=\operatorname{diag}(-1,-1-\vartheta,
-1-\vartheta)$ at $k=0$, and by \cref{lem:cub-formal}(d) the inner step maps
$\varphi$ to $\psi_v(z_v)+\psi_x(z_x)+\psi_x(z_y)$, with
$\psi_v=\widehat T_{a,-bh}\Phi_0$ and $\psi_x=\widehat T_{a,-bh(1+\vartheta)}\Phi_0$.
The covariance of the shared step is a multiple of $(\mathbf e_x+\mathbf
e_y)(\mathbf e_x+\mathbf e_y)^{\mathsf T}$, and its recursion \eqref{eq:cub-formal}
involves only derivatives in the direction $\mathbf e_x+\mathbf e_y$; on the
set $\{z_x=z_y\}$, it is the one-dimensional recursion with variance
$bh\vartheta$ and mass $a/2$ applied to $2\psi_x$, which gives
$2\widehat T_{a,bh\vartheta}\psi_x=2\widehat T_{a,-bh}\Phi_0$ by
\cref{lem:cub-formal}(b) and (c). Similarly, the common step acts on the
diagonal as the one-dimensional step with variance $bh$ and mass $a/3$ applied
to $\psi_v+2\widehat T_{a,-bh}\Phi_0=3\widehat T_{a,-bh}\Phi_0$, which gives
$3\Phi_0$. The identity holds for every smooth $\Phi_0$ with bounded
derivatives of positive order, so that the coefficients $\chi_0$ and $w_0$ in
the expansion above can be prescribed arbitrarily. We conclude that
$N_\vartheta^{\rm f}(0)=C_\chi(0)=C_w(0)=0$. These values can
also be checked directly from the matrices at $k=0$. For instance,
$\norm{\Lambda_0}_F^2=9$, $\norm{\Lambda_0+\Lambda_1}_F^2=5+4(1+\vartheta)^2$ and
$\norm{\Lambda_0+\Lambda_1+\Lambda_2}_F^2=4+2(1+\vartheta)^2$ at $k=0$, which
give $N^{\rm f}_\vartheta(0)=3+2(1+\vartheta)^2-2-1-2(1+\vartheta)^2=0$, and
the same computation with cubes in place of squares gives $C_w(0)=0$. This
proves the first claim.

It remains to show that $N^{\rm f}_\vartheta=N_\vartheta$. Integrating in
$\tau$ in \eqref{eq:cub-N} with $i=0$, we have
\[
 \sum_{j=1}^5\frac{2\kappa_j}{\mathfrak n_j}\int_0^1(\widetilde Q_{j-1}+\kappa_j\tau
 B_j)\mathbin:B_j\dd\tau
 =\sum_{j=1}^5\frac1{\mathfrak n_j}\bigl(\norm{\widetilde Q_j}_F^2-\norm{\widetilde Q_{j-1}}_F^2\bigr).
\]
Here $\mathfrak n_3=\mathfrak n_4=\mathfrak n_5=1$, $\widetilde Q_1=\Lambda_0$, $\widetilde
Q_2=\Lambda_0+\Lambda_1$ and $\widetilde Q_5=\Lambda_0+\Lambda_1+\Lambda_2+\tau_T\mathrm{Id}$.
The right side is $N^{\rm f}_\vartheta(k)$ with $\Lambda_2$ replaced by
$\Lambda_2+\tau_T\mathrm{Id}$, and the same holds for the baseline rates. We have
$\norm{U+\tau_T\mathrm{Id}}_F^2-\norm U_F^2=2\tau_T\tr U+3\tau_T^2$ and
$\tr(\Lambda_0+\Lambda_1+\Lambda_2)=0$ for both paths. The replacement
adds $3\tau_T^2$ to both expressions and does not change their difference. We conclude
that $N_\vartheta(k)=N_\vartheta^{\rm f}(k)-N_\vartheta^{\rm f}(0)=N_\vartheta^{\rm f}(k)$.
\end{proof}

\subsection{The coefficient table}

Substituting the matrices \eqref{eq:cub-Lambda} into \eqref{b:eq:os-R} and
\eqref{b:eq:os-W} gives
\begin{equation}\label{eq:cub-table}
\begin{array}{c|ccc}
 &k^2&k^4&k^6\\ \hline
 C_\chi(k)&\dfrac{2(\vartheta-1)(\vartheta^2+2)}{(\vartheta-2)^2}
 &\dfrac{12\vartheta^3+6\vartheta^2+\vartheta+8}{3(\vartheta-2)^4}
 &\dfrac{(3\vartheta+4)(9\vartheta^2+15\vartheta+10)}{27(\vartheta-2)^6}\\[8pt]
 C_w(k)&-\dfrac{(\vartheta-1)(\vartheta^2-\vartheta+2)}{(\vartheta-2)^2}
 &-\dfrac{3\vartheta^3+\vartheta^2+\vartheta+4}{2(\vartheta-2)^4}&0
\end{array}
\end{equation}
These are the coefficients of $k^2$, $k^4$ and $k^6$; the constant terms vanish
by \cref{prop:cub-frozen}, and the odd terms vanish because the polynomials are
even. The identities can be checked by hand from the explicit matrices, for
instance by multiplying by $(\vartheta-2)^6$ and comparing the coefficients
of $\vartheta^i k^j$. For $\vartheta=1/2$, the individual traces and cubic sums are
\[
\begin{aligned}
 \tr\Lambda_2^3&=-\tfrac{64}{27}k^6-\tfrac{32}3k^4-\tfrac{44}3k^2-\tfrac{31}4,\\
 \tr(\Lambda_1\Lambda_2^2)&=\tfrac{64}{81}k^6+\tfrac89k^4-\tfrac{11}9k^2+\tfrac94,\\
 \tr(\Lambda_1^2\Lambda_2)&=-\tfrac{64}{243}k^6-\tfrac{40}{81}k^4-\tfrac49k^2-\tfrac32,\\
 \tr\Lambda_1^3&=\tfrac{64}{729}k^6+\tfrac{16}{27}k^4+\tfrac43k^2+1,\\
 \tr\bigl(\Lambda_0(\Lambda_2+\tfrac12\Lambda_1)^2\bigr)
 &=\tfrac{800}{729}k^6+\tfrac{488}{81}k^4+\tfrac{22}3k^2+3,\\
 \tr\bigl(\Lambda_0^2(\Lambda_2+\tfrac12\Lambda_1)\bigr)
 &=-\tfrac{640}{729}k^6-\tfrac{464}{81}k^4-12k^2-9,\\
 \tr\Lambda_0^3&=\tfrac{512}{729}k^6+\tfrac{64}9k^4+24k^2+27,
\end{aligned}
\]
and
\[
\begin{gathered}
 \operatorname{cub}(\Lambda_0)=\tfrac{64}9k^4+16k^2+9,\qquad
 \operatorname{cub}(\Lambda_0+\Lambda_1)=\tfrac{88}9k^4+\tfrac{28}3k^2+\tfrac{37}2,\\
 \operatorname{cub}(\Lambda_0+\Lambda_1+\Lambda_2)=12k^2+\tfrac{43}4.
\end{gathered}
\]
Inserting these values into \eqref{b:eq:os-R} and \eqref{b:eq:os-W}, we obtain
\begin{equation}\label{eq:cub-half}
 C_\chi(k)=-k^2+\frac{184}{243}k^4+\frac{6952}{19683}k^6,\qquad
 C_w(k)=\frac7{18}k^2-\frac{41}{81}k^4\qquad(\vartheta=1/2).
\end{equation}
We also checked \eqref{eq:cub-table}, \eqref{eq:cub-half} and the values of
$N_\vartheta$ in \eqref{eq:cub-N-explicit} with computer algebra.

\section{The varying profile}\label{sec:cub-profile}

We now compute the part of the Taylor coefficients of $V$ that involves
$\alpha'(q_0)$ and $\alpha''(q_0)$. We show that it is exactly the
corresponding part of $\frac{b^2\chi(q_0,z)^2}4\Delta\Pi$. Together with
\cref{prop:cub-frozen}, this gives the pointwise expansion of
\cref{prop:cub-pointwise}.

The linear part of the terminal function now has a coefficient that changes
along the window. We remove it with the following form of
\eqref{b:eq:os-gauge}.

\begin{lemma}[Affine shift along a flow]\label{lem:cub-shift}
Let $(B_j,\lambda_j,m_j)_{j\le N}$ be a stage datum as in \cref{def:flow}, with
continuous masses, let $u\in\R$ and $\mathbf e'\in\R^\ell$, and let
$f\in\mathcal C^\infty$ satisfy $2\ell b\norm{D^2f}_\infty\sum_j\lambda_j\abs{B_j}\le1$.
Let $F_f$ and $F$ be the flows with terminal functions $f$ and
$u\,\mathbf e'\cdot\mathbf z+f$. For $\tau'\in(j-1,j)$, write
$\lambda(\tau')=\lambda_j$, $B(\tau')=B_j$ and $m(\tau')=m_j(\tau'-j+1)$, and set
\[
 \mathbf s(\tau)=ub\int_\tau^N\lambda m B\,\mathbf e'\dd\tau',\qquad
 \bar c(\tau)=\frac{u^2b}2\int_\tau^N\lambda m\,\mathbf e'\cdot B\mathbf e'\dd\tau'.
\]
Then $F(\tau,\mathbf z)=u\,\mathbf e'\cdot\mathbf z+\bar c(\tau)
+F_f\bigl(\tau,\mathbf z+\mathbf s(\tau)\bigr)$ for $0\le\tau\le N$.
\end{lemma}

\begin{proof}
Let $\widetilde F$ be the right side. By \cref{lem:flow-step-limit}(d),
$F_f$ is of class $C^{1,2}$ on each closed stage, with bounded first and
second spatial derivatives, and solves \eqref{eq:flow-equation}; the same
properties hold for $\widetilde F$, since $\mathbf s$ and $\bar c$ are continuous
and continuously differentiable on each closed stage. On a stage, with
$F_f$ and its derivatives evaluated at $(\tau,\mathbf z+\mathbf s(\tau))$,
\[
 -\partial_\tau\widetilde F=\frac{u^2b\lambda m}2\,\mathbf e'\cdot B\mathbf e'
 -\partial_\tau F_f+ub\lambda m\,B\mathbf e'\cdot\nabla F_f .
\]
Since $\nabla\widetilde F=u\mathbf e'+\nabla F_f$ and
$D^2\widetilde F=D^2F_f$, we have
\[
\begin{aligned}
 \frac{b\lambda}2\bigl\{B\mathbin:D^2\widetilde F+m\nabla\widetilde F\cdot B\nabla\widetilde F\bigr\}
 ={}&\frac{b\lambda}2\bigl\{B\mathbin:D^2F_f+m\nabla F_f\cdot B\nabla F_f\bigr\}\\
 &+\frac{b\lambda m}2\bigl(u^2\,\mathbf e'\cdot B\mathbf e'+2u\,\mathbf e'\cdot B\nabla F_f\bigr),
\end{aligned}
\]
which equals $-\partial_\tau\widetilde F$. We have $\widetilde F(N)=F(N)$, and the
terminal function $u\,\mathbf e'\cdot\mathbf z+f$ has the same Hessian as $f$.
The uniqueness statement of \cref{lem:flow-step-limit}(d) implies that
$\widetilde F=F$.
\end{proof}

At $\tau=0$ and for constant masses, this is \eqref{b:eq:os-gauge}, applied
to each step.

\begin{lemma}[Contribution of the variation of $\alpha$]\label{lem:cub-mass}
Fix $q_0\in K$, $k$ with $\abs k\le k_0$, and $z\in\R$, and write
$\chi_0=\chi(q_0,z)$. For $n\le3$, the difference between $[V]_n$ and its
constant-mass part equals the difference between
$\frac{b^2\chi_0^2}4[\Delta\Pi]_n$ and its constant-mass part. In other words,
it vanishes for $n\le2$, and for $n=3$ it equals
$\frac{b^2\chi_0^2}4\alpha'(q_0)N^{(1)}_\vartheta(k)$.
\end{lemma}

\begin{proof}
The proof has four steps. We remove the linear part of the terminal function
by \cref{lem:cub-shift}, we show that the resulting translations can be
replaced by a common one, we list the places where $\alpha'(q_0)$ and
$\alpha''(q_0)$ enter, and we compute the only term that survives. Throughout,
statements such as ``$X=O(h^p)$'' mean that $[X]_n=0$ for $n<p$, and $\alpha$
is regarded as a smooth function on $K'$ whose derivatives at $q_0$ are free
variables, as in the recursion \eqref{eq:cub-coefficient-recursion}.

\paragraph{Step 1: removal of the linear part.} Let $u_0=u(q_0,z)$, and write
the terminal function as $\sum_i\Phi(T,z_i)=u_0\,\mathbf e\cdot\mathbf
z+\bar\psi_h(\mathbf z)$. The function $\bar\psi_h$ has bounded derivatives of every positive
order and $\norm{D^2\bar\psi_h}_\infty\le1$. By \eqref{eq:cub-h0}, \cref{lem:cub-shift}
applies to the stage datum of \eqref{b:eq:os-vector-flow}, with
$\mathbf e'=\mathbf e$. At $\tau=0$, it gives
\[
\begin{gathered}
 \mathcal B(q_0,h,k,\mathbf z)=u_0\,\mathbf e\cdot\mathbf z+\bar c
 +\widetilde{\mathcal B}(\mathbf z+\mathbf s),\\
 \mathbf s=bu_0\sum_{j=1}^5\frac1{\mathfrak n_j}\Bigl(\int_{I_j}\alpha\Bigr)B_j\mathbf e,\qquad
 \bar c=\frac{bu_0^2}2\sum_{j=1}^5\frac1{\mathfrak n_j}\Bigl(\int_{I_j}\alpha\Bigr)\mathbf e\cdot B_j\mathbf e,
\end{gathered}
\]
where $\widetilde{\mathcal B}=F_{\bar\psi_h}(0,\cdot)$ is the output of the flow with terminal
function $\bar\psi_h$, and where we used that
$h\kappa_j\int_0^1m_j=\mathfrak n_j^{-1}\int_{I_j}\alpha$. The same holds for the
baseline with $\widetilde{\mathcal B}^\circ$, $\mathbf s^\circ$ and $\bar c^\circ$. By
\cref{lem:cub-identities}(a), $\bar c=\bar c^\circ$. Computing the vectors $B_j\mathbf e$
and $B^\circ_j\mathbf e$ as in the proof of \cref{lem:cub-identities}(c), we find
\[
\begin{gathered}
 \mathbf s^\circ=\mathfrak s\mathbf e,\qquad \mathfrak s=bu_0\int_{q_0}^T\alpha,\qquad
 \mathbf s-\mathbf s^\circ=bu_0k\zeta\,(\mathbf e_x-\mathbf e_y),\\
 \zeta=\omega_0J_0+(\omega_1-\omega_0)J_1+(\omega_1-2\omega_0)J_2,
\end{gathered}
\]
where $J_0=\int_{q_0}^r\alpha$, $J_1=\int_r^q\alpha$ and
$J_2=\int_q^{q_0+2h}\alpha$. Subtracting the two representations at $z\mathbf e$
and using $\mathcal B^\circ(z\mathbf e)=3\Phi(q_0,z)$, we obtain
\begin{equation}\label{eq:cub-V-shift}
 V(q_0,h,k,z)=\widetilde{\mathcal B}\bigl((z+\mathfrak s)\mathbf e+bu_0k\zeta(\mathbf e_x-\mathbf e_y)\bigr)
 -\widetilde{\mathcal B}^\circ\bigl((z+\mathfrak s)\mathbf e\bigr).
\end{equation}

\paragraph{Step 2: a common translation.} If $\alpha$ were constant on the
window, then $\zeta$ would vanish, since
$\omega_0+\vartheta(\omega_1-\omega_0)+(1-\vartheta)(\omega_1-2\omega_0)
=\omega_1-(1-\vartheta)\omega_0=0$. Since $\alpha$ is Lipschitz on
$K'$, we have $\abs\zeta\le Ch^2$, so that $\zeta=O(h^2)$. Next, the function
$\Delta_{xy}(h,z')=(\mathbf e_x-\mathbf e_y)\cdot\nabla\widetilde{\mathcal B}(z'\mathbf e)$
satisfies $\Delta_{xy}=O(h^2)$ for every $z'\in\R$. Indeed, at $h=0$ it equals
$\partial_z\Phi(q_0,z')-\partial_z\Phi(q_0,z')=0$. Its coefficient of order one
is the difference between the $x$ and $y$ components of the gradient of
\[
 \bigl[\bar\psi_h\bigr]_1+\frac b2\sum_j\kappa_j\Bigl\{B_j\mathbin:D^2\bar\psi_0
 +\frac a{\mathfrak n_j}\nabla\bar\psi_0\cdot B_j\nabla\bar\psi_0\Bigr\}
\]
at $z'\mathbf e$, where $\bar\psi_0=\bar\psi_h|_{h=0}=\varphi-u_0\,\mathbf e\cdot\mathbf z$ and
$[\bar\psi_h]_1$ is separable and symmetric in the coordinates. At $z'\mathbf e$, the
gradient of $B_j\mathbin:D^2\bar\psi_0$ has entries $(B_j)_{ii}w(q_0,z')$, and the
gradient of $\nabla\bar\psi_0\cdot B_j\nabla\bar\psi_0$ is
$2\chi(q_0,z')\gamma\,B_j\mathbf e$ with $\gamma=u(q_0,z')-u_0$. These differences
vanish after summation, by \cref{lem:cub-identities}(b) and (c). This proves
that $\Delta_{xy}=O(h^2)$. Expanding $\widetilde{\mathcal B}$ in the shift $bu_0k\zeta(\mathbf e_x-\mathbf
e_y)=O(h^2)$, we find that the first-order and second-order terms are
$O(h^4)$, so that
\[
 [V]_n=\bigl[\Delta\widetilde{\mathcal B}(z+\mathfrak s)\bigr]_n\quad(n\le3),\qquad
 \Delta\widetilde{\mathcal B}(z')=\widetilde{\mathcal B}(z'\mathbf e)-\widetilde{\mathcal B}^\circ(z'\mathbf e).
\]
For every $z'$, the function $\Delta\widetilde{\mathcal B}(z')$ is $O(h^2)$, with all its derivatives in
$z'$: its coefficients of order zero and one vanish by the computation in the
proof of \cref{b:lem:os-flow}(ii), with $\varphi$ replaced by $\bar\psi_0$. Since
$\mathfrak s=O(h)$, we obtain $[V]_n=[\Delta\widetilde{\mathcal B}(z)+\mathfrak s\,(\Delta\widetilde{\mathcal B})'(z)]_n$ for $n\le3$.

\paragraph{Step 3: where $\alpha'(q_0)$ and $\alpha''(q_0)$ enter.} In
$[\Delta\widetilde{\mathcal B}(z)+\mathfrak s\,(\Delta\widetilde{\mathcal B})'(z)]_n$, $n\le3$, these quantities enter through $\mathfrak s$,
through the terminal function $\bar\psi_h$, and through the masses.
\begin{itemize}
\item The part of $\mathfrak s$ that involves them is
 $bu_0\int_{q_0}^T(\alpha(t)-a)\dd t=O(h^2)$, and the part of $(\Delta\widetilde{\mathcal B})'(z)$ that
 involves them is $O(h^3)$ by the two items below. Since $(\Delta\widetilde{\mathcal B})'(z)=O(h^2)$ and
 $\mathfrak s=O(h)$, neither part contributes to $[\mathfrak s\,(\Delta\widetilde{\mathcal B})'(z)]_n$ for $n\le3$.
\item In $\bar\psi_h$, the time derivatives of $\Phi$ at $q_0$ involve
 $\alpha'(q_0)$ only from the second derivative on, and the part of $\bar\psi_h$ that
 involves $\alpha'(q_0)$ or $\alpha''(q_0)$ is $O(h^2)$. It is separable and
 symmetric in the coordinates. The first-order change of $\Delta\widetilde{\mathcal B}$ under a change
 $\delta\bar\psi$ of the terminal function of this kind is $O(h^2)\delta\bar\psi$: its
 coefficients of order zero and one vanish by the computation of Step~2,
 because $D^2\delta\bar\psi$ is a multiple of $\mathrm{Id}$ and $\nabla\delta\bar\psi$ is a multiple
 of $\mathbf e$ on the diagonal. This part contributes $O(h^4)$.
\item A mass enters \eqref{eq:cub-coefficient-recursion} only through the
 gradient term. Each insertion carries a factor $h$, and the coefficient $[m_j]_l$ of the mass carries the factor $h^l$ and
 the factor $\alpha^{(l)}(q_0)$. A term that contains $[m_j]_l$ with $l\ge1$
 and only one insertion contains the factor $\nabla\bar\psi_h\cdot B_j\nabla\bar\psi_h$ at
 $z\mathbf e$, because the stages without insertion act as the identity. Since
 $\nabla\bar\psi_0(z\mathbf e)=0$, we have $\nabla\bar\psi_h(z\mathbf e)=O(h)$, so such a
 term is $O(h^{3+l})$ and does not contribute for $n\le3$. A term with
 $[m_j]_2$ and two insertions is $O(h^4)$. The only remaining terms that
 involve $\alpha'(q_0)$ contain $[m_j]_1$ and exactly two insertions, and they
 are of order $h^3$; in them, everything else is evaluated at $h=0$: the
 terminal function is $\bar\psi_0$, the point is $z\mathbf e$, and the other insertion
 has the mass $a/\mathfrak n_i$.
\end{itemize}
We note that $\alpha''(q_0)$ does not enter $[V]_n$ for $n\le3$.

\paragraph{Step 4: the two-insertion term.} We use the variable
$\tau\in[0,5]$ of \eqref{b:eq:os-vector-flow}, and we write
$\Sigma'(\tau)=bh\kappa_jB_j$ and $m(\tau)=m_j(\tau-j+1)$ for $\tau\in(j-1,j)$.
Two insertions at $\tau_2<\tau_1$ correspond to an outer insertion at $\tau_2$
and an inner insertion at $\tau_1$. Since $\nabla\bar\psi_0(z\mathbf e)=0$ and
$D^2\bar\psi_0(z\mathbf e)=\chi_0\mathrm{Id}$, the only term of
\eqref{eq:cub-coefficient-recursion} with two insertions that depends on the
masses is the heat term of the outer insertion acting on the gradient term of the
inner one:
\[
\begin{aligned}
 &\frac12\int_0^5\!\!\int_{\tau_2}^5\Sigma'(\tau_2)\mathbin:D^2\Bigl(\frac{m(\tau_1)}2
 \nabla\bar\psi_0\cdot\Sigma'(\tau_1)\nabla\bar\psi_0\Bigr)(z\mathbf e)\dd\tau_1\dd\tau_2\\
 &\qquad=\frac{\chi_0^2}2\int_0^5\!\!\int_{\tau_2}^5m(\tau_1)\,\Sigma'(\tau_1)\mathbin:\Sigma'(\tau_2)
 \dd\tau_1\dd\tau_2 .
\end{aligned}
\]
Here we used that $D^2(\nabla\bar\psi_0\cdot\Sigma'\nabla\bar\psi_0)=2D^2\bar\psi_0\Sigma'D^2\bar\psi_0$ at
$z\mathbf e$. The terms in which the outer insertion is a gradient term
contain the factor $\nabla\bar\psi_0(z\mathbf e)=0$, and the terms in which both
insertions are heat terms do not depend on the masses. Let
$\Sigma^{\rm acc}(\tau)=\int_0^\tau\Sigma'$ be the covariance accumulated from
the outer end of the window. Then the right side equals
$\frac{\chi_0^2}2\int_0^5m(\tau)\,\Sigma'(\tau)\mathbin:\Sigma^{\rm acc}(\tau)\dd\tau
=\frac{\chi_0^2}4\int_0^5m(\tau)\,\partial_\tau\norm{\Sigma^{\rm acc}(\tau)}_F^2\dd\tau$,
which is $\frac{\chi_0^2}4$ times the squared-norm integral of the window path
(\eqref{eq:flow-penalty} with $Q_\star=0$); this is
\eqref{b:eq:os-quadratic-norm} for a continuous profile. By
\eqref{eq:cub-penalty}, its part linear in the coefficients $[m_j]_1$, for the
perturbed window minus the baseline window, is $\frac{b^2\chi_0^2}4$ times the
part of $[\Delta\Pi]_3$ that involves $\alpha'(q_0)$. By \eqref{eq:cub-N}, this
part is $\alpha'(q_0)N^{(1)}_\vartheta(k)$. Finally, the part of
$[\Delta\Pi]_n$ that involves $\alpha'(q_0)$ or $\alpha''(q_0)$ vanishes for
$n\le2$, which completes the proof.
\end{proof}

\begin{proposition}[Pointwise expansion of the boundary perturbation]\label{prop:cub-pointwise}
For $q_0\in K$, $0\le h\le h_0$ and $\abs k\le k_0$, write $a=\alpha(q_0)$,
$\chi_0(z)=\chi(q_0,z)$ and $w_0(z)=w(q_0,z)$. Then
\begin{equation}\label{eq:cub-A}
\begin{aligned}
 V(q_0,h,k,z)={}&\frac{b^2\chi_0(z)^2}4\,\Delta\Pi(q_0,h,k)\\
 &+h^3\Bigl\{a^2b^3\chi_0(z)^3C_\chi(k)+ab^3w_0(z)^2C_w(k)\Bigr\}+\mathcal R(q_0,h,k,z),
\end{aligned}
\end{equation}
where $\abs{\partial_z^j\mathcal R(q_0,h,k,z)}\le Ch^4k^2$ for $j\le2$ and all $z$.
\end{proposition}

The penalty $\Delta\Pi$ in \eqref{eq:cub-A} is the exact change of the
mass-weighted squared-norm integral of the path, with the Parisi distribution
function $\alpha$; it
is not replaced by its Taylor polynomial. By the expansion of $\Delta\Pi$ in
\cref{sec:cub-derivatives}, it contains the term
$\alpha'(q_0)h^3N^{(1)}_\vartheta(k)$. The function $V$ itself depends on
$\alpha'(q_0)$ at order $h^3$. This dependence disappears only after the
expectation is taken in \cref{b:prop:os-expansion}.

\begin{proof}
Let $\mathcal R$ be defined by \eqref{eq:cub-A}. Using \cref{prop:cub-frozen} and
\cref{lem:cub-mass}, together with the expansion $[\Delta\Pi]_2=aN_\vartheta(k)$,
$[\Delta\Pi]_3=\alpha'(q_0)N^{(1)}_\vartheta(k)$ of \cref{sec:cub-derivatives},
we find that $[\mathcal R]_n=0$ for $n\le3$, for every $k$ and $z$. By
\cref{b:lem:os-flow}(ii) and (iv), the functions $\partial_h^{\gamma_h}\partial_k^{\gamma_k}
\partial_z^j\mathcal R$ are bounded for $\gamma_h\le4$, $\gamma_k\le2$ and $j\le2$; for the
second and third terms of \eqref{eq:cub-A} this follows from the bounds on
$\chi$ and $w$ in \cref{lem:sc-derivatives}. Moreover, $\mathcal R$ vanishes at
$k=0$ and is even in $k$, because $V$, $\Delta\Pi$, $C_\chi$ and $C_w$ have these
properties. Fix $j\le2$ and $z$. The Taylor coefficients of
$\partial_k^{\gamma_k}\partial_z^j\mathcal R$ of order at most three in $h$ vanish, and
Taylor's formula implies that $\abs{\partial_k^{\gamma_k}\partial_z^j\mathcal R}\le Ch^4$
for $\gamma_k\le2$. Since $\partial_z^j\mathcal R$ and $\partial_k\partial_z^j\mathcal R$
vanish at $k=0$, we have
$\partial_z^j\mathcal R(k)=\int_0^k(k-k')\,\partial_k^2\partial_z^j\mathcal
R(k')\dd k'$, which implies that $\abs{\partial_z^j\mathcal R}\le Ch^4k^2$.
\end{proof}

The identity $[\mathcal R]_n=0$ for $n\le3$ can also be checked by expanding
the five stages of the perturbed path directly, with the derivatives of $\Phi$
at $(q_0,z)$, the values $\alpha(q_0)$, $\alpha'(q_0)$, $\alpha''(q_0)$, and $k$
as variables; we did so with computer algebra for several rational values of
$\vartheta$.

\section{The expansion of the trial value}\label{sec:cub-expansion}

We now take the expectation of \eqref{eq:cub-A} along the scalar diffusion.
Here we use the two stationarity identities. Let
\begin{equation}\label{eq:cub-M}
 M(t)=\alpha(t)^2\beta^6\E\chi(t,Y_t)^3,
\end{equation}
which is positive on $(0,q_\beta)$ and infinitely differentiable there by
\cref{b:lem:os-scalar-jets}. We define the even polynomial
\[
 P_\vartheta(k)=C_\chi(k)+2C_w(k)=c_2(\vartheta)k^2+c_4(\vartheta)k^4
 +c_6(\vartheta)k^6.
\]
By \eqref{eq:cub-table}, its coefficients are
\begin{equation}\label{eq:cub-c}
\begin{gathered}
 c_2(\vartheta)=\frac{2\vartheta(\vartheta-1)}{(\vartheta-2)^2},\qquad
 c_4(\vartheta)=\frac{(\vartheta-1)(3\vartheta^2+6\vartheta+4)}{3(\vartheta-2)^4},\\
 c_6(\vartheta)=\frac{(3\vartheta+4)(9\vartheta^2+15\vartheta+10)}{27(\vartheta-2)^6}.
\end{gathered}
\end{equation}
For $\vartheta\in(0,1)$ we have $c_2<0$, $c_4<0$ and $c_6>0$.
For $\vartheta=1/2$, by \eqref{eq:cub-half},
\begin{equation}\label{eq:cub-P-half}
 P_{1/2}(k)=-\frac29k^2-\frac{62}{243}k^4+\frac{6952}{19683}k^6.
\end{equation}
The polynomial $-\frac29k^2-\frac{62}{243}k^4+\frac{6952}{19683}k^6$ is the
sum of the trace polynomial \eqref{b:eq:os-R} and twice the Wick polynomial
\eqref{b:eq:os-W} at $\vartheta=1/2$; by \cref{b:prop:os-expansion}, it is the
coefficient of $M(q_0)h^3$ in the expansion of $\mathcal T(q_0,h,k)-3P_\beta$.

\begin{proposition}[Local three-replica expansion]\label{b:prop:os-expansion}
Let $\vartheta\in(0,1/2]$, $k_0>0$, and let $K\subset(0,q_\beta)$ be a compact
interval. There exist $h_0>0$ and $C<\infty$, depending only on $\beta$,
$\vartheta$, $k_0$ and $K$, such that for all $q_0\in K$, $0\le h\le h_0$ and
$\abs k\le k_0$, the trial value $\mathcal T(q_0,h,k)$ of \eqref{b:eq:os-trial}
is an upper bound for $p_{3,n}(D(r,q,\delta))$ whenever this matrix is
feasible, and
\begin{equation}\label{b:eq:finiteE-polynomial}
 \Bigl|\mathcal T(q_0,h,k)-3P_\beta-M(q_0)h^3P_\vartheta(k)\Bigr|\le Ch^4k^2,
\end{equation}
with $r,q,\delta$ as in \eqref{b:eq:os-parameters}. The same bound holds with
$M(q_0)$ replaced by $M(q_0+h)=M(r)$.
\end{proposition}

The constants are uniform in the left endpoint $q_0\in K$, in $h\le h_0$ and
in $\abs k\le k_0$; they are not uniform as $\vartheta\downarrow0$. In terms of
$d=\vartheta h$ and $\delta=hk$, the remainder is $Ch^4k^2=C\delta^2d^2/\vartheta^2$.

\begin{proof}
Let $h_0$ be as in \cref{b:lem:os-flow}. The first claim is
\cref{lem:cub-trial}(a). Using \cref{cor:cub-prefix} and
\cref{prop:cub-pointwise}, we obtain
\[
\begin{aligned}
 \mathcal T(q_0,h,k)-3P_\beta={}&\frac b4\bigl(b\,\E\chi(q_0,Y_{q_0})^2-1\bigr)\Delta\Pi\\
 &+h^3\Bigl\{a^2b^3\,\E\chi(q_0,Y_{q_0})^3C_\chi(k)+ab^3\,\E w(q_0,Y_{q_0})^2C_w(k)\Bigr\}\\
 &+\E\mathcal R+\mathcal E_{\rm in},
\end{aligned}
\]
with $a=\alpha(q_0)$. Since $q_0\in(0,q_\beta)$, the stationarity identities
\eqref{b:eq:os-contacts} hold at $t=q_0$: we have $b\,\E\chi(q_0,Y_{q_0})^2=1$ and
$\E w(q_0,Y_{q_0})^2=2a\,\E\chi(q_0,Y_{q_0})^3$. The first identity cancels the
penalty term exactly, and the second turns the braces into
$a^2b^3\E\chi(q_0,Y_{q_0})^3\bigl(C_\chi(k)+2C_w(k)\bigr)$, so that the second
term is $M(q_0)h^3P_\vartheta(k)$. Finally, $\abs{\E\mathcal R}\le Ch^4k^2$ by
\cref{prop:cub-pointwise}, and $\abs{\mathcal E_{\rm in}}\le Ch^4k^4\le
Ck_0^2h^4k^2$ by \cref{cor:cub-prefix}. This proves
\eqref{b:eq:finiteE-polynomial}.

For the last claim, recall that $M$ is continuously differentiable on $K'$. We
have $\abs{M(q_0+h)-M(q_0)}\le Ch$ and $\abs{P_\vartheta(k)}\le Ck^2$ for $\abs
k\le k_0$, and replacing $M(q_0)$ by $M(q_0+h)$ changes the left side of
\eqref{b:eq:finiteE-polynomial} by at most $Ch^4k^2$.
\end{proof}

\section{A uniform cubic deficit on a band}\label{sec:cub-band}

We now express the expansion in the variables $d=q-r$ and $\delta$, and we bound
the trial value on a band where both are comparable to a small scale $\rho$.
With $h=d/\vartheta$ and $k=\vartheta\delta/d$, we have
$-h^3P_\vartheta(k)=\mathcal G_\vartheta(d,\abs\delta)$, where
\begin{equation}\label{eq:cub-G-function}
 \mathcal G_\vartheta(d,x)=-\frac{c_2}\vartheta\,x^2d-\vartheta c_4\frac{x^4}d
 -\vartheta^3c_6\frac{x^6}{d^3}\qquad(d,x>0),
\end{equation}
and $c_j=c_j(\vartheta)$. Note that
$-c_2/\vartheta=2(1-\vartheta)/(2-\vartheta)^2>0$. For $E>0$ we set
\[
 G(\vartheta,E)=-\frac{c_2}\vartheta-\vartheta c_4E^2-\vartheta^3c_6E^4,
 \qquad
 \mu_1=-\frac{c_2}\vartheta+\vartheta c_4(2E)^2,\qquad
 \mu_2=-\frac{c_2}\vartheta-3\vartheta^3c_6(2E)^4,
\]
so that $\mathcal G_\vartheta(\rho,E\rho)=E^2G(\vartheta,E)\rho^3$. Since
$c_4<0<c_6$, we have $G(\vartheta,E)\ge\mu_2$.

\begin{lemma}[Monotonicity on the band]\label{lem:cub-band}
Let $\vartheta\in(0,1/2]$ and $E>0$ satisfy
\begin{equation}\label{b:eq:os-band-checks}
 -\frac{c_2(\vartheta)}\vartheta+\vartheta c_4(\vartheta)(2E)^2>0,\qquad
 -\frac{c_2(\vartheta)}\vartheta-3\vartheta^3c_6(\vartheta)(2E)^4>0,
\end{equation}
that is, $\mu_1>0$ and $\mu_2>0$. Let $0\le\sigma<\min\{\mu_1,\mu_2\}$ and $\rho>0$.
Then the function $(d,x)\mapsto\mathcal G_\vartheta(d,x)-\sigma x^2d$ is
increasing in each variable on the band $\{d\ge\rho,\ E\rho\le x\le2E\rho\}$,
and its minimum over the band is $E^2(G(\vartheta,E)-\sigma)\rho^3$, attained at
$(\rho,E\rho)$.
\end{lemma}

\begin{proof}
Differentiating \eqref{eq:cub-G-function}, we obtain
\[
 \frac{\partial_d\mathcal G_\vartheta}{x^2}
 =-\frac{c_2}\vartheta+\vartheta c_4\Bigl(\frac xd\Bigr)^2
 +3\vartheta^3c_6\Bigl(\frac xd\Bigr)^4,\qquad
 \frac{\partial_x\mathcal G_\vartheta}{2xd}
 =-\frac{c_2}\vartheta-2\vartheta c_4\Bigl(\frac xd\Bigr)^2
 -3\vartheta^3c_6\Bigl(\frac xd\Bigr)^4.
\]
On the band, $0<x/d\le2E\rho/\rho=2E$. Since $c_4<0<c_6$, the first
expression is at least $\mu_1$ and the second is at least $\mu_2$. Subtracting
$\sigma x^2d$ subtracts $\sigma$ from both normalized derivatives, and the partial
derivatives of $\mathcal G_\vartheta-\sigma x^2d$ in $d$ and $x$ are at least
$(\mu_1-\sigma)x^2>0$ and $2(\mu_2-\sigma)xd>0$. The band is the product
$[\rho,\infty)\times[E\rho,2E\rho]$, and so the minimum is attained at the corner
$(\rho,E\rho)$, where the value is
$\mathcal G_\vartheta(\rho,E\rho)-\sigma E^2\rho^3=E^2(G(\vartheta,E)-\sigma)\rho^3$.
\end{proof}

The lemma bounds $\mathcal G_\vartheta-\sigma x^2d$ below by its value at one
corner of the band. It does not say that $G(\vartheta,E)$ multiplies $x^2d$
throughout the band.

\begin{proposition}[A uniform cubic deficit]\label{prop:cub-band}
Let $I\subset(0,q_\beta)$ be a nonempty compact interval, and let $m_\beta>0$
satisfy $m_\beta\le\inf_{r\in I}M(r)$, for instance the constant of
\cref{cor:sc-mbeta} for an interval $I_\beta\supset I$. Let $\vartheta\in(0,1/2]$ and
$E>0$ satisfy \eqref{b:eq:os-band-checks}, and let $0<\sigma<\min\{\mu_1,\mu_2\}$.
There exists $d_0>0$, depending only on $\beta$, $I$, $\vartheta$, $E$ and $\sigma$,
such that the following holds. Let $0<\rho\le d_0$, and let $r\in I$, $d$ and
$\delta$ satisfy
\[
 \rho\le d\le d_0,\qquad E\rho\le\abs\delta\le2E\rho .
\]
Set $h=d/\vartheta$, $q_0=r-h$ and $k=\vartheta\delta/d$. Then $\mathcal
T=\mathcal T(q_0,h,k)$ is an upper bound for $p_{3,n}(D(r,r+d,\delta))$
whenever this matrix is feasible, and
\[
 \mathcal T\le3P_\beta-m_\beta\bigl\{\mathcal G_\vartheta(d,\abs\delta)-\sigma\delta^2d\bigr\}
 \le3P_\beta-m_\beta E^2\bigl(G(\vartheta,E)-\sigma\bigr)\rho^3.
\]
In particular, if $0<\bar\sigma<G(\vartheta,E)$ and $\sigma=\min\{\bar\sigma,\mu_1,\mu_2\}/2$,
then
\begin{equation}\label{b:eq:os-uniform-corner-gap}
 \mathcal T\le3P_\beta-m_\beta E^2\bigl\{G(\vartheta,E)-\bar\sigma\bigr\}\rho^3 .
\end{equation}
\end{proposition}

The parameters are chosen in the order of \cref{tab:cub-choices}, all before
$\rho$, and before $n$ in the later sections. The deficit
\eqref{b:eq:os-uniform-corner-gap} is uniform over the band, but it is a bound
by the value at the corner of the band; at other points of the band the
deficit $m_\beta\{\mathcal G_\vartheta(d,\abs\delta)-\sigma\delta^2d\}$ may be much
larger. A fixed $d=d_0$ does not make the remainder of
\eqref{b:eq:finiteE-polynomial} small compared with $\delta^2d$; it is the
choice of $d_0$ in the proof that makes it so.

\begin{table}[ht]
\centering
\begin{tabular}{@{}lll@{}}
\toprule
Parameter & Chosen in terms of & Purpose\\
\midrule
$I$, $m_\beta$ & fixed first & centers; scalar factor $M\ge m_\beta$ on $I$\\
$\vartheta$, $E$ & \eqref{b:eq:os-band-checks} & geometry of the band\\
$k_0=2\vartheta E$, $K$ & $\vartheta$, $E$, $I$ & range of $k$ and of $q_0$\\
$h_0$, $C$ & $\beta$, $\vartheta$, $k_0$, $K$ & \cref{b:prop:os-expansion}\\
$\sigma$ (or $\bar\sigma$) & $\mu_1$, $\mu_2$ (and $G(\vartheta,E)$) & slack in \cref{lem:cub-band}\\
$d_0$ & $h_0$, $C$, $\underline M_I$, $\sigma$, $I$ & $h\le h_0$, $q_0\in K$, absorption of the remainder\\
$\rho$ & $0<\rho\le d_0$ & scale of the band\\
\bottomrule
\end{tabular}
\caption{Order of the choices in \cref{prop:cub-band}.}\label{tab:cub-choices}
\end{table}

\begin{proof}
We first fix the constants, and then combine \cref{b:prop:os-expansion} with
\cref{lem:cub-band}.

\paragraph{Step 1: constants.} Let $k_0=2\vartheta E$ and
$K=[\frac12\min I,\max I]$, a compact subinterval of $(0,q_\beta)$. Let $h_0$
and $C$ be the constants of \cref{b:prop:os-expansion} for $\vartheta$, $k_0$
and $K$. By \cref{b:lem:os-scalar-jets}, the function $M$ is continuous and
positive on the compact interval $I$, and we set
\[
 \underline M_I=\min_{r\in I}M(r)>0,\qquad
 d_0=\min\Bigl\{\vartheta h_0,\ \frac\vartheta2\min I,\
 \frac{\underline M_I\sigma\vartheta^2}C\Bigr\}.
\]
These constants depend only on $\beta$, $I$, $\vartheta$, $E$ and $\sigma$.

\paragraph{Step 2: the expansion.} Let $\rho,r,d,\delta$ be as in the
statement. Then $h=d/\vartheta\le h_0$, $q_0=r-h\ge\min I-\frac12\min I$, so
that $q_0\in K$, and $\abs k=\vartheta\abs\delta/d\le2\vartheta E\rho/\rho=k_0$.
With these parameters, $q_0+h=r$, $q_0+(1+\vartheta)h=r+d$ and $hk=\delta$. The
endpoint of the path is $D(r,r+d,\delta)$, and the first claim is
\cref{b:prop:os-expansion}. By \eqref{b:eq:finiteE-polynomial} with $M(r)$,
\[
 \mathcal T\le3P_\beta-M(r)\,\mathcal G_\vartheta(d,\abs\delta)+\frac C{\vartheta^2}\delta^2d^2.
\]
Since $d\le d_0$ and $r\in I$, we have
\[
 \frac C{\vartheta^2}\delta^2d^2\le\underline M_I\sigma\delta^2d\le M(r)\sigma\delta^2d.
\]
By \cref{lem:cub-band}, the braces in the next display are at least
$E^2(G(\vartheta,E)-\sigma)\rho^3$, which is positive because
$G(\vartheta,E)\ge\mu_2>\sigma$. Since $M(r)\ge m_\beta$, we obtain
\[
 \mathcal T\le3P_\beta-M(r)\bigl\{\mathcal G_\vartheta(d,\abs\delta)-\sigma\delta^2d\bigr\}
 \le3P_\beta-m_\beta\bigl\{\mathcal G_\vartheta(d,\abs\delta)-\sigma\delta^2d\bigr\},
\]
and the lower bound on the braces gives the second inequality of the
proposition. For the last claim, $\sigma\le\bar\sigma/2<\bar\sigma$.
\end{proof}

\paragraph{An explicit admissible choice.} Let $\vartheta=1/2$ and $E=1/100$.
By \eqref{eq:cub-P-half}, $-c_2/\vartheta=4/9$,
$c_4=-62/243$ and $c_6=6952/19683$. These values satisfy $-1<c_4<0<c_6<1$,
which implies that
\[
 \mu_1=\frac49+2c_4E^2>\frac49-2E^2>0,\qquad
 \mu_2=\frac49-6c_6E^4>\frac49-6E^4>0,
\]
so \eqref{b:eq:os-band-checks} holds, and
\[
 G\Bigl(\frac12,E\Bigr)=\frac49+\frac{31}{243}E^2-\frac{869}{19683}E^4
 >\frac49-\frac{E^4}8>\frac13.
\]
The exact values are $\mu_1=269969/607500$, $\mu_2=18224999131/41006250000$ and
$G(1/2,1/100)=874825109131/1968300000000$. This choice is used in
\cref{sec:references}. It is compatible with every fixed $\beta>1$: the
constants of \cref{b:prop:os-expansion} depend on $\beta$, but
\eqref{b:eq:os-band-checks} does not.

\paragraph{Scope of the local construction.} \Cref{prop:cub-band} concerns only
matrices with $q-r\le d_0$. For $q-r>d_0$, \cref{sec:static} uses the quenched
comparison with a fixed width of \cref{lem:st-remote}, which does not require
the regularity of $\alpha$ near $q$. The bound of \cref{prop:cub-band} is a
bound on the quenched pressure. To bound the positive fractional moments for the
probability estimates, we use the separate interpolation of
\cref{sec:fractional}, which uses the trial path of this section with the
profile changed on $[0,q_0)$; by the remark after \cref{lem:cub-trial}, this
change does not affect $V$ or $\Delta\Pi$.
\chapter{Fractional moments of constrained partition functions}\label{sec:fractional}\label{b:sec:fractional-static}

In this section we bound positive fractional moments of the partition function of three configurations with prescribed overlaps. The main result is \cref{lem:fr-local-gap}. It states that, for every overlap matrix $D$ in the local band of \cref{sec:cubic}, the normalized fractional moment $(ns)^{-1}\log\E Z_{3,n}(D)^s$ of order $s=S\rho$ is at most $3P_\beta-g\rho^3$, uniformly in $n$. In \cref{sec:static} we use this bound and Markov's inequality to obtain a single disorder event on which all these constrained partition functions are small. That event is the static input of the dynamical argument in \cref{sec:references,sec:induction,sec:barrier-proof}.

We first explain why a fractional moment is needed. The cubic deficit of \cref{sec:cubic} bounds the quenched constrained pressure $p_{3,n}(D)=n^{-1}\E\log Z_{3,n}(D)$ by $3P_\beta$ minus a multiple of $\rho^3$. Gaussian concentration turns a mean deficit of order $\rho^3$ into a probability bound $\exp(-cn\rho^6)$, and for $\rho\asymp n^{-7/30}$ the exponent $n\rho^6\asymp n^{-2/5}$ tends to zero. With a bound of the same order on the moment of order $s\asymp\rho$, we obtain instead the probability bound $\exp(-cns\rho^3)=\exp(-cn\rho^4)$, and $n\rho^4\asymp n^{1/15}$. The proof has three ingredients.
\begin{enumerate}
\item \Cref{b:prop:fractional-matrix} is a matrix interpolation bound for the moment of order $s$ of a constrained partition function. Its right side is the trial value of the quenched comparison \cref{inp:ob-matrix}, which is \cite[Proposition~2.3]{oai-barriers}. However, the comparison path must have all its masses at least $s$, and the left side carries the correction $\beta^2\ell^2s/(4n)$ produced by the Gaussian completion of \cref{lem:model-completion}.
\item The three-replica path of \cref{sec:cubic} has masses $\alpha/3$ near the origin, and $\alpha(t)\to0$ as $t\to0$. We replace $\alpha$ by the profile $\max\{\alpha,3s\}$. Its cost is $O(s^5)$ by the estimate of Aronow and Lopatto recalled in \cref{inp:qp-floor}, and \cref{lem:fr-prefix-response} controls the effect of this change on the local covariance perturbation, which starts at a time bounded away from zero.
\item \Cref{lem:fr-local-gap} combines these estimates with the uniform cubic deficit \eqref{b:eq:os-uniform-corner-gap}.
\end{enumerate}
The fractional comparison and the local fractional bound are new relative to \cite{oai-barriers}. The comparison adapts the quenched bound \cite[Proposition~2.3]{oai-barriers}, and the positive-moment mechanism and the profile $\max\{\alpha,3s\}$ come from \cite{quantitative-parisi}; see \cref{rem:fr-history}.

\Cref{sec:fr-hierarchy} defines Gaussian hierarchies whose outermost integration has positive mass and computes their variance derivatives. \Cref{sec:fr-comparison} proves \cref{b:prop:fractional-matrix}. \Cref{sec:fr-floor,sec:fr-prefix} treat the profile $\max\{\alpha,3s\}$, and \cref{sec:fr-local} proves \cref{lem:fr-local-gap}.

\paragraph{Standing notation.} Throughout this section and the next, $\beta>1$ is fixed, and $\alpha=\alpha_\beta$, $\mu_\beta$, $q_\beta$, $\Phi$, $u$, $\chi$, $Y$ and $\Gamma$ are as in \cref{sec:inputs}. Recall that $\Gamma(t)=t$ for $0\le t\le q_\beta$ and that $\alpha$ is smooth on $[0,q_\beta)$, by \cref{cor:inp-identity}. We use the constrained partition functions $Z_{\ell,n}(D)$ and pressures $p_{\ell,n}(D)$ of \cref{def:inp-constrained}, the matrices $D_2(q)$ and $D(r,q,\delta)$ of \eqref{eq:inp-triple}, the integration steps $T_{m,\Sigma}$ of \cref{def:inp-step}, and the admissible paths and trial values $\mathcal T_D(Q,m,\lambda)$ of \cref{def:inp-hierarchy,def:inp-trial}. For real $\ell\times\ell$ matrices $U,U'$, not necessarily symmetric, $U\mathbin{:}U'=\sum_{a,a'}U_{aa'}U'_{aa'}$ and $\norm U_F^2=U\mathbin{:}U$. For a feasible $\ell\times\ell$ matrix $D$ and $0<s\le1$, we define the normalized fractional moment
\begin{equation}\label{eq:fr-moment}
 p^{(s)}_{\ell,n}(D)=\frac1{ns}\log\E Z_{\ell,n}(D)^s.
\end{equation}
By Jensen's inequality, we have $p_{\ell,n}(D)\le p^{(s)}_{\ell,n}(D)$, so an upper bound on $p^{(s)}_{\ell,n}(D)$ is stronger than the same upper bound on $p_{\ell,n}(D)$. Throughout, $s\in(0,1]$ denotes the order of a fractional moment; in \cref{sec:fr-floor} it is also the parameter of the profile $\max\{\alpha,3s\}$ of \cref{inp:qp-floor}.

\section{Hierarchies with a positive outer mass}\label{sec:fr-hierarchy}

In the hierarchies of \cref{inp:ob-hierarchy-derivative}, the quenched randomness is integrated by an ordinary expectation, which corresponds to mass zero. To control a fractional moment of order $s$, we instead integrate it at mass $s$.

\begin{definition}[Hierarchy with positive outer mass]\label{def:fr-positive-hierarchy}
Let $\mathcal S$ be a nonempty finite set, let $N\ge1$, let $s\in(0,1]$, and let $s\le m_1\le\dots\le m_N\le1$. Let $\omega,\mathbf g_1,\dots,\mathbf g_N$ be independent standard Gaussian vectors in finite-dimensional spaces, and for $x\in\mathcal S$ let $U(x)=U(x;\omega,\mathbf g_1,\dots,\mathbf g_N)$ be affine in $(\omega,\mathbf g_1,\dots,\mathbf g_N)$. We set
\[
 \Psi_N=\log\sum_{x\in\mathcal S}e^{U(x)},\qquad
 \Psi_{j-1}=\frac1{m_j}\log\E_je^{m_j\Psi_j}\quad(j=N,\dots,1),\qquad
 \Psi_{-1}=\frac1s\log\E_\omega e^{s\Psi_0},
\]
where $\E_j$ integrates $\mathbf g_j$ and $\E_\omega$ integrates $\omega$. Then $\Psi_j$ is a function of $(\omega,\mathbf g_1,\dots,\mathbf g_j)$, and the number $\Psi_{-1}$ is the value of the hierarchy. The path law is the law of $(\omega,\mathbf g_1,\dots,\mathbf g_N,\mathsf x)$ under which $\omega$ has density $e^{s(\Psi_0-\Psi_{-1})}$ with respect to its Gaussian law, given $(\omega,\mathbf g_1,\dots,\mathbf g_{j-1})$ the vector $\mathbf g_j$ has density $e^{m_j(\Psi_j-\Psi_{j-1})}$, and given all Gaussian variables the configuration $\mathsf x\in\mathcal S$ has the Gibbs law proportional to $e^{U(\mathsf x)}$. For $0\le l\le N$, $\E_l^{(2)}$ is the expectation under the pair law of $(\mathsf x^1,\mathsf x^2)$ sharing $(\omega,\mathbf g_1,\dots,\mathbf g_l)$, with conditionally independent continuations. We set $m_0=s$, $m_{N+1}=1$ and $\Delta m_l=m_{l+1}-m_l$ for $0\le l\le N$.
\end{definition}

Note that the recursion normalizes the densities, and that all integrals are finite since $\Psi_N$ is Lipschitz in the Gaussian variables. As $s\to0$, the outermost step becomes $\Psi_{-1}=\E_\omega\Psi_0$, the quenched root of \cref{inp:ob-hierarchy-derivative}, where $m_0=0$. The next lemma is the analogue of \cref{inp:ob-hierarchy-derivative}, which is \cite[Lemma~2.2]{oai-barriers}; the only change is that the weights now sum to $1-s$.

\begin{lemma}[Variance derivative]\label{lem:fr-derivative}
In the setting of \cref{def:fr-positive-hierarchy}, suppose that a standard Gaussian vector $\mathbf g$, which is either $\mathbf g_j$ for some $1\le j\le N$ or a component of $\omega$ (in which case $j=0$), enters $U$ only through an additive term $\sqrt v\,\mathbf g\cdot e(x)$, where $e$ is a fixed vector-valued function on $\mathcal S$. Then, with all other parameters fixed, for $v>0$,
\begin{equation}\label{b:eq:positive-tree-derivative}
 \frac{\partial\Psi_{-1}}{\partial v}
 =\frac12\Bigl(\E\norm{e(\mathsf x)}^2-\sum_{l=j}^N\Delta m_l\,\E^{(2)}_l\,e(\mathsf x^1)\cdot e(\mathsf x^2)\Bigr),
\end{equation}
where $\E$ is the expectation under the path law. The weights $\Delta m_0,\dots,\Delta m_N$ are nonnegative and sum to $1-s$.
\end{lemma}

\begin{proof}
We follow the proof of \cite[Lemma~2.2]{oai-barriers} and write it out, because the outermost level is now integrated at positive mass. The proof has three steps: a recursion for Hessians in an auxiliary deterministic field, the variance derivative at level $j$, and a telescoping sum.

\paragraph{Step 1: the Hessian recursion.} For a deterministic vector $\mathbf y$, let $\Psi_l(\mathbf y)$ be the values of the hierarchy when $\mathbf y\cdot e(x)$ is added to $U(x)$. For $0\le l\le N$, we write $\bar e_l=\E[e(\mathsf x)\mid\omega,\mathbf g_1,\dots,\mathbf g_l]$ under the path law, and let $H_l$ be the Hessian of $\Psi_l$ in $\mathbf y$ at $\mathbf y=0$. Note that $\nabla_{\mathbf y}\Psi_N=\bar e_N$ and that $H_N=\Cov(e(\mathsf x)\mid\omega,\mathbf g_1,\dots,\mathbf g_N)$, the covariance under the Gibbs law. For $1\le l\le N$, differentiating $\Psi_{l-1}=m_l^{-1}\log\E_le^{m_l\Psi_l}$ once and twice, we obtain
\[
 \nabla_{\mathbf y}\Psi_{l-1}=\bar e_{l-1},\qquad
 H_{l-1}=\E[H_l\mid\omega,\mathbf g_1,\dots,\mathbf g_{l-1}]+m_l\Cov(\bar e_l\mid\omega,\mathbf g_1,\dots,\mathbf g_{l-1}),
\]
where for the first identity we used the tower property of conditional expectation. Taking traces and expectations, we find
\begin{equation}\label{eq:fr-hessian-trace}
 \E\tr H_N=\E\norm{e(\mathsf x)}^2-\E\norm{\bar e_N}^2,\qquad
 \E\tr H_{l-1}=\E\tr H_l+m_l\bigl(\E\norm{\bar e_l}^2-\E\norm{\bar e_{l-1}}^2\bigr).
\end{equation}

\paragraph{Step 2: the derivative at level $j$.} The vector $\mathbf g$ enters $\Psi_j$ only through the shift $\mathbf y=\sqrt v\,\mathbf g$. We use the heat equation $\partial_v\E f(\sqrt v\,\mathbf g)=\frac12\E\Delta f(\sqrt v\,\mathbf g)$, valid for smooth $f$ of moderate growth. Applying it to $f=e^{m_j\Psi_j}$ in the step that integrates level $j$, at mass $m_j$ (with $m_0=s$ for $\omega$), we obtain
\[
 \partial_v\Psi_{j-1}=\frac12\E\bigl[\tr H_j+m_j\norm{\bar e_j}^2\bigm|\omega,\mathbf g_1,\dots,\mathbf g_{j-1}\bigr],
\]
which for $j=0$ is to be read without conditioning. At the earlier levels, differentiating $m^{-1}\log\E e^{m\Psi}$ in $v$ gives the tilted expectation of $\partial_v\Psi$; this holds at every level, including the outermost one at mass $s$. Iterating, we conclude that
\begin{equation}\label{eq:fr-derivative-step2}
 2\,\partial_v\Psi_{-1}=\E\tr H_j+m_j\E\norm{\bar e_j}^2.
\end{equation}

\paragraph{Step 3: telescoping.} Inserting \eqref{eq:fr-hessian-trace} for $l=N,\dots,j+1$ into \eqref{eq:fr-derivative-step2}, we have
\[
 2\,\partial_v\Psi_{-1}
 =\E\norm{e(\mathsf x)}^2-\E\norm{\bar e_N}^2+\sum_{l=j+1}^Nm_l\bigl(\E\norm{\bar e_l}^2-\E\norm{\bar e_{l-1}}^2\bigr)+m_j\E\norm{\bar e_j}^2.
\]
For $j\le l\le N$, the coefficient of $\E\norm{\bar e_l}^2$ is $m_l-m_{l+1}=-\Delta m_l$, where we used $m_{N+1}=1$. Two paths that share $(\omega,\mathbf g_1,\dots,\mathbf g_l)$ are conditionally independent, which implies that $\E\norm{\bar e_l}^2=\E^{(2)}_le(\mathsf x^1)\cdot e(\mathsf x^2)$. This proves \eqref{b:eq:positive-tree-derivative}. The weights are nonnegative because the masses are ordered, and they sum to $m_{N+1}-m_0=1-s$.

All differentiations under the integral sign are justified by dominated convergence. Indeed, $\mathcal S$ and the number of levels are finite, the gradients of $U$ in the Gaussian variables are bounded, and the values grow at most linearly in these variables.
\end{proof}

In the quenched case $m_0=0$, the term $m_0\E\norm{\bar e_0}^2$ is absent and the weights sum to one; this is the only difference from \cite[Lemma~2.2]{oai-barriers}. The same term appears in the differentiation of a positive fractional moment of the one-replica partition function by Aronow and Lopatto \cite[Section~2.3, (2.10)--(2.11)]{quantitative-parisi}, where it removes the prescribed mass at zero from the integral against the variational measure.

\section{The fractional matrix comparison}\label{sec:fr-comparison}

\begin{proposition}[Fractional matrix comparison]\label{b:prop:fractional-matrix}
Let $n\ge2$, $\ell\in\{2,3\}$ and $s\in(0,1]$, let $D$ be a feasible $\ell\times\ell$ matrix, let $(Q,m)$ be an admissible path of $\ell\times\ell$ matrices whose masses satisfy $m_1\ge s$, and let $\lambda$ be a symmetric $\ell\times\ell$ matrix. Then
\begin{equation}\label{b:eq:fractional-matrix-exact}
 p^{(s)}_{\ell,n}(D)+\frac{\beta^2\ell^2s}{4n}
 \le\mathcal T_D(Q,m,\lambda)
 =\Psi(Q,m,\lambda)-\lambda\mathbin{:}D+\frac{\beta^2}4\Bigl(\norm{D-Q_N}_F^2-\int m\dd\norm{Q}_F^2\Bigr).
\end{equation}
The same inequality holds for every trial value for $D$, in the sense of \cref{def:inp-trial}, that is a limit along admissible paths whose masses are all at least $s$.
\end{proposition}

The right side of \eqref{b:eq:fractional-matrix-exact} is exactly the right side of the quenched comparison \cref{inp:ob-matrix}. Every trial bound for $p_{\ell,n}(D)$ obtained from a path whose masses are all at least $s$ also bounds $p^{(s)}_{\ell,n}(D)+\beta^2\ell^2s/(4n)$.

\begin{proof}
We interpolate between the completed Hamiltonian and independent fields at the $n$ sites, as in the proof of \cite[Proposition~2.3]{oai-barriers}. The difference is that the Hamiltonian disorder is integrated at the outer mass $s$ instead of by an ordinary expectation. The proof has four steps: the covariances, the derivative of the interpolation, its endpoints, and the effect of the completion.

\paragraph{Step 1: covariances.} Let $J_0$ be a standard Gaussian variable independent of $J$, set $G=J_0/\sqrt2$, and let $H^c_n=H_n+G$ be the completed Hamiltonian of \cref{lem:model-completion}, so that $\E H^c_n(x)H^c_n(y)=nR(x,y)^2/2$. For two tuples $\bm x=(x^1,\dots,x^\ell)$ and $\bm y$ in $\Sigma_n^\ell$, let $R(\bm x,\bm y)$ be the cross-overlap matrix with entries $R(x^a,y^{a'})$; it need not be symmetric. Then
\begin{equation}\label{eq:fr-hamiltonian-covariance}
 \E\Bigl[\beta\sum_{a=1}^\ell H^c_n(x^a)\cdot\beta\sum_{a'=1}^\ell H^c_n(y^{a'})\Bigr]
 =\frac{n\beta^2}2\sum_{a,a'=1}^\ell R(x^a,y^{a'})^2=\frac{n\beta^2}2\norm{R(\bm x,\bm y)}_F^2.
\end{equation}
The process $\beta\sum_aH^c_n(x^a)$ is a linear form in the standard Gaussian vector $\omega=((J_{ij})_{i<j},J_0)$, as required in \cref{lem:fr-derivative}. For the site fields, let $\mathbf g_{j,i}$, $1\le j\le N$, $1\le i\le n$, be independent standard Gaussian vectors in $\R^\ell$, let $\mathbf g_j=(\mathbf g_{j,i})_{i\le n}$, and write $\bm x_i=(x^1_i,\dots,x^\ell_i)\in\{-1,1\}^\ell$ for the spins of $\bm x$ at site $i$. With $\Delta Q_j=Q_j-Q_{j-1}$, we have
\begin{equation}\label{eq:fr-field-covariance}
 \E\Bigl[\sum_{i=1}^n\beta\Delta Q_j^{1/2}\mathbf g_{j,i}\cdot\bm x_i\sum_{i=1}^n\beta\Delta Q_j^{1/2}\mathbf g_{j,i}\cdot\bm y_i\Bigr]
 =\beta^2\sum_{i=1}^n\bm x_i^{\mathsf T}\Delta Q_j\bm y_i
 =n\beta^2\,\Delta Q_j\mathbin{:}R(\bm x,\bm y).
\end{equation}

\paragraph{Step 2: the interpolation.} For $0\le t\le1$, consider the hierarchy of \cref{def:fr-positive-hierarchy} with $\mathcal S=\Sigma_D=\{\bm x\in\Sigma_n^\ell:R(\bm x,\bm x)=D\}$, the masses $s\le m_1\le\dots\le m_N$, the Gaussian vectors $\omega,\mathbf g_1,\dots,\mathbf g_N$, and
\[
 U_t(\bm x)=\sqrt t\,\beta\sum_{a=1}^\ell H^c_n(x^a)
 +\sqrt{1-t}\sum_{i=1}^n\sum_{j=1}^N\beta\Delta Q_j^{1/2}\mathbf g_{j,i}\cdot\bm x_i
 +\sum_{i=1}^n\bm x_i^{\mathsf T}\lambda\bm x_i.
\]
Let $\Theta(t)$ be its value divided by $n$. The Hamiltonian enters at level $0$ with variance parameter $t$, and the site fields of level $j$ enter with variance parameter $1-t$. For $\bm x\in\Sigma_D$ we have $R(\bm x,\bm x)=D$. Applying \cref{lem:fr-derivative} to each of these Gaussian vectors and using \eqref{eq:fr-hamiltonian-covariance} and \eqref{eq:fr-field-covariance}, we obtain, for $0<t<1$,
\begin{align*}
 \Theta'(t)
 &=\frac1{2n}\Bigl[\frac{n\beta^2}2\Bigl(\norm D_F^2-\sum_{l=0}^N\Delta m_l\E^{(2)}_l\norm R_F^2\Bigr)\\
 &\qquad\qquad-\sum_{j=1}^Nn\beta^2\Bigl(\Delta Q_j\mathbin{:}D-\sum_{l=j}^N\Delta m_l\E^{(2)}_l\,\Delta Q_j\mathbin{:}R\Bigr)\Bigr]\\
 &=\frac{\beta^2}4\Bigl[\norm D_F^2-2Q_N\mathbin{:}D-\sum_{l=0}^N\Delta m_l\E^{(2)}_l\bigl(\norm R_F^2-2Q_l\mathbin{:}R\bigr)\Bigr],
\end{align*}
where $R=R(\bm x^1,\bm x^2)$ is the cross-overlap matrix of the terminal tuples of the two paths, and in the last line we summed $\Delta Q_j$ over $j\le l$, using $Q_0=0$. For every matrix $R$ we have $\norm R_F^2-2Q_l\mathbin{:}R=\norm{R-Q_l}_F^2-\norm{Q_l}_F^2$. Since $\Delta m_l\ge0$, this implies that
\begin{equation}\label{eq:fr-phi-prime}
 \Theta'(t)\le\frac{\beta^2}4\Bigl(\norm D_F^2-2Q_N\mathbin{:}D+\sum_{l=0}^N\Delta m_l\norm{Q_l}_F^2\Bigr)
 =\frac{\beta^2}4\Bigl(\norm{D-Q_N}_F^2-\int m\dd\norm{Q}_F^2\Bigr).
\end{equation}
Here we used that $\Delta m_0=m_1-s\ge0$; this is where the condition $m_1\ge s$ enters. The equality in \eqref{eq:fr-phi-prime} is the summation by parts
\[
 \norm{Q_N}_F^2-\sum_{l=0}^N\Delta m_l\norm{Q_l}_F^2
 =\sum_{l=1}^Nm_l\norm{Q_l}_F^2-\sum_{l=1}^{N}m_l\norm{Q_{l-1}}_F^2
 =\sum_{j=1}^Nm_j\bigl(\norm{Q_j}_F^2-\norm{Q_{j-1}}_F^2\bigr),
\]
where we used $m_{N+1}=1$ and $Q_0=0$; the right side is the penalty \eqref{eq:inp-penalty}. The symmetry of $R$ was not used.

\paragraph{Step 3: the endpoints.} At $t=1$ the site fields do not appear, and $\sum_i\bm x_i^{\mathsf T}\lambda\bm x_i=n\lambda\mathbin{:}D$ on $\Sigma_D$. The integrations of $\mathbf g_N,\dots,\mathbf g_1$ act on a function of $\omega$ only and leave it unchanged, and the outermost integration gives
\[
 \Theta(1)=\frac1{ns}\log\E\bigl(Z^c_{\ell,n}(D)\bigr)^s+\lambda\mathbin{:}D,\qquad
 Z^c_{\ell,n}(D)=\sum_{\bm x\in\Sigma_D}\exp\Bigl(\beta\sum_{a=1}^\ell H^c_n(x^a)\Bigr).
\]
At $t=0$ the Hamiltonian does not appear. Replacing $\Sigma_D$ by $\Sigma_n^\ell$ increases $\Psi_N$, and the value increases with it, because every step of the recursion is monotone. For the unconstrained sum, $\Psi_N$ is a sum over $i\le n$ of functions of the independent families $(\mathbf g_{j,i})_{j\le N}$, and each step $m^{-1}\log\E e^{m(\cdot)}$ factorizes over such sums. The contribution of each site is the one-site value $\Psi(Q,m,\lambda)$ of \cref{def:inp-hierarchy}, and the outermost integration at mass $s$ acts on a constant. This shows that $\Theta(0)\le\Psi(Q,m,\lambda)$.

\paragraph{Step 4: the completion.} By \cref{lem:model-completion}, part (d), $Z^c_{\ell,n}(D)=e^{\beta\ell G}Z_{\ell,n}(D)$ and
\[
 \frac1{ns}\log\E\bigl(Z^c_{\ell,n}(D)\bigr)^s=p^{(s)}_{\ell,n}(D)+\frac{\beta^2\ell^2s}{4n};
\]
indeed $G\sim N(0,1/2)$ is independent of $J$, and $\E e^{s\beta\ell G}=e^{s^2\beta^2\ell^2/4}$. Combining Steps~2--4 with $\Theta(1)\le\Theta(0)+\int_0^1\Theta'(t)\dd t$, we obtain \eqref{b:eq:fractional-matrix-exact}. The function $\Theta$ is continuous on $[0,1]$ and differentiable on $(0,1)$, since all Gaussian integrals are finite-dimensional and $U_t$ is Lipschitz in the Gaussian variables. Singular increments $\Delta Q_j$ cause no difficulty, since only $\Delta Q_j^{1/2}$ is used. Finally, the left side of \eqref{b:eq:fractional-matrix-exact} does not depend on the path, and the inequality passes to limits along which the right side converges.
\end{proof}

\begin{remark}\label{rem:fr-history}
Gaussian interpolation for the free energy is due to Guerra \cite{guerra-2003}. Constrained comparisons for several coupled replicas go back to Talagrand \cite{talagrand-2007}, and the vector-spin formulation is due to Panchenko \cite{panchenko-2018}; see the discussion at the start of \cite[Section~2]{oai-barriers}. The quenched arbitrary-endpoint bound used in this paper is \cref{inp:ob-matrix}. Positive fractional moments of the SK partition function were studied by Talagrand \cite{talagrand-2007-large-deviations}. For fixed $0<s<1$, the limiting normalized fractional moment is given by the one-level case of the multiscale formula of Contucci and Mingione \cite[Theorem~2.1]{contucci-mingione-2019}, and by the constrained Parisi formula of Chen, Guionnet, Ko, Lacroix-A-Chez-Toine and Mourrat \cite[Theorem~2.1]{chen-guionnet-ko-lacroix-mourrat-2026}, in which the order of the moment is a lower bound on the mass at zero. The condition $m_1\ge s$ in \cref{b:prop:fractional-matrix} is the finite-volume form of this constraint. Quantitative finite-size versions for the one-replica partition function, with $s$ depending on $n$, are due to Aronow and Lopatto \cite[Section~2.3]{quantitative-parisi}. The replica-matrix version with a free endpoint in \cref{b:prop:fractional-matrix} is a direct adaptation of these arguments and is not stated in \cite{oai-barriers}. The completion correction $\beta^2\ell^2s/(4n)$ is nonnegative, and below we discard it.
\end{remark}

\section{The profile \texorpdfstring{$\max\{\alpha,3s\}$}{max(alpha, 3s)}}\label{sec:fr-floor}

For $s>0$ we define $\alpha_s(t)=\max\{\alpha(t),3s\}$ for $0\le t\le1$. The factor three ensures that a block of three coordinates sharing a field, which is integrated at mass $\alpha_s/3$ (\cref{lem:inp-blocks}(b)), has mass at least $s$.

\begin{lemma}[Cost of the profile]\label{lem:fr-floor}
There exist $s_0\in(0,1/3]$ and $C_{\rm fl}<\infty$, depending only on $\beta$, such that for every $0<s\le s_0$ the function $\alpha_s$ is a distribution function and
\begin{equation}\label{b:eq:specific-floor-cost}
 0\le\mathcal P_\beta(\alpha_s)-P_\beta\le C_{\rm fl}s^5,\qquad
 \alpha_s=\alpha\ \text{on }[C_{\rm fl}s,1],\qquad
 \int_0^1\abs{\alpha_s(t)-\alpha(t)}\dd t\le C_{\rm fl}s^2.
\end{equation}
\end{lemma}

\begin{proof}
Let $T_*,c_1,s_0',C$ be the constants of \cref{inp:qp-floor}, and let $s_0=\min\{s_0',c_1T_*/3\}$. For $0<s\le s_0$, part (ii) of \cref{inp:qp-floor} states that $\alpha_s$ is a distribution function whose cost is at most $Cs^5$; this is \cite[Lemma~7.2, (7.3)]{quantitative-parisi} for this particular profile, in the fixed-temperature normalization of \cite[Proposition~7.12]{quantitative-parisi}. Since $3s\le c_1T_*$, \cref{rem:inp-floor} implies that $\alpha_s=\alpha$ on $[3s/c_1,1]$ and that the $L^1$ distance is at most $9s^2/c_1$. The lemma holds with $C_{\rm fl}=\max\{C,3/c_1,9/c_1\}$.
\end{proof}

The cost bound concerns the explicit profile $\alpha_s$, and not only the infimum of $\mathcal P_\beta$ over profiles with mass at least $3s$ at the origin. We need this distinction below, because we use the same profile in the comparison path for three replicas.

\section{A change of profile before the local window}\label{sec:fr-prefix}

The local covariance perturbation of \cref{sec:cubic} starts at a time $q_0>0$, and on $[0,q_0]$ the three coordinates share their field, integrated at mass $\alpha/3$. For $s$ sufficiently small that $C_{\rm fl}s<q_0$, replacing $\alpha$ by $\alpha_s$ changes only this initial segment, by \cref{lem:fr-floor}. We now control the effect of that change on the perturbation.

Fix $q_0\in(0,1]$. For a nondecreasing $\gamma\colon[0,q_0]\to[0,1]$ and a function $F\colon\R\to\R$ with bounded continuous derivatives of orders one to three, let $\Phi^\gamma_F$ be the solution on $[0,q_0]\times\R$ of
\begin{equation}\label{eq:fr-prefix-pde}
 -\partial_t\Phi^\gamma_F=\frac{\beta^2}2\Bigl(\partial_{zz}\Phi^\gamma_F+\frac{\gamma(t)}3(\partial_z\Phi^\gamma_F)^2\Bigr),\qquad
 \Phi^\gamma_F(q_0,\cdot)=F,
\end{equation}
and write $\mathcal H_\gamma(F)=\Phi^\gamma_F(0,0)$. For a step function $\gamma$, we interpret the equation as the recursion $\Phi^\gamma_F(t,\cdot)=T_{m,\beta^2(t_2-t)}\Phi^\gamma_F(t_2,\cdot)$ for $t_1\le t\le t_2$, on every interval $[t_1,t_2)$ where $\gamma=3m$. This is the recursion of three coordinates that share every field increment on $[0,q_0]$: on the line spanned by $(1,1,1)$, such an increment acts on a function of the common value of the three fields (\cref{lem:inp-steps}(e)). For general $\gamma$, $\Phi^\gamma_F$ is the limit along step functions converging to $\gamma$ in $L^1([0,q_0])$; the opening paragraph of the proof of \cref{lem:fr-prefix-response}, using the bound of Step~2 there, shows that this limit exists and does not depend on the approximating sequence. If $\gamma$ is the restriction to $[0,q_0]$ of a distribution function, also denoted $\gamma$, then
\begin{equation}\label{eq:fr-block}
 \mathcal H_\gamma\bigl(3\Phi_\gamma(q_0,\cdot)\bigr)=3\Phi_\gamma(0,0).
\end{equation}
For step functions this is \cref{lem:inp-steps}(d) with $k=3$, applied at each step as in the proof of \cref{lem:inp-blocks}(b). The general case follows by approximation. Here we use \cref{inp:ob-scalar}(ii) for $\Phi_\gamma$, part (a) of the next lemma, and the bound $\abs{\mathcal H_\gamma(F_1)-\mathcal H_\gamma(F_2)}\le\norm{F_1-F_2}_\infty$, which holds by \cref{lem:inp-steps}(a).

\begin{lemma}[Response to a change of the initial profile]\label{lem:fr-prefix-response}
There exists $C_{\rm r}<\infty$, depending only on $\beta$, with the following property. Let $\gamma,\widetilde\gamma\colon[0,q_0]\to[0,1]$ be nondecreasing, and let $\varphi,V\colon\R\to\R$ have bounded continuous derivatives of orders one to three, with
\[
 \norm{\varphi'}_\infty\le3,\qquad\norm{\varphi''}_\infty\le3,\qquad\norm{V'}_\infty+\norm{V''}_\infty\le1.
\]
Then:
\begin{enumerate}[label=\textup{(\alph*)}]
\item for $F\in\{\varphi,\varphi+V\}$, $\abs{\mathcal H_\gamma(F)-\mathcal H_{\widetilde\gamma}(F)}\le C_{\rm r}\norm{\gamma-\widetilde\gamma}_{L^1([0,q_0])}$;
\item the response of $\mathcal H$ to the terminal perturbation $V$ changes by at most
\begin{equation}\label{b:eq:prefix-floor-response}
 \Bigl|\bigl[\mathcal H_{\widetilde\gamma}(\varphi+V)-\mathcal H_{\widetilde\gamma}(\varphi)\bigr]-\bigl[\mathcal H_\gamma(\varphi+V)-\mathcal H_\gamma(\varphi)\bigr]\Bigr|
 \le C_{\rm r}\norm{V'}_\infty\norm{\gamma-\widetilde\gamma}_{L^1([0,q_0])}.
\end{equation}
\end{enumerate}
\end{lemma}

The quantity bounded in \eqref{b:eq:prefix-floor-response} is a mixed difference: it vanishes when $V=0$ and when $\gamma=\widetilde\gamma$. A bound on the full difference $\mathcal H_{\widetilde\gamma}(\varphi+V)-\mathcal H_\gamma(\varphi+V)$ would be of order $\norm{\gamma-\widetilde\gamma}_{L^1}$ without the factor $\norm{V'}_\infty$. In the application, that full difference is accounted for by the scalar cost \eqref{b:eq:specific-floor-cost}.
The proof adapts the scalar profile-stability argument of
\cite[Section~2.4, Lemma~14]{jagannath-tobasco-2016}: subtract the two Parisi
equations and represent the resulting linear equation by a diffusion. Here we
also compare terminal functions, retaining the factor $\norm{V'}_\infty$ in the
mixed difference; this refinement is proved below.

\begin{proof}
It suffices to prove (a) and (b) for step functions $\gamma,\widetilde\gamma$, with constants independent of the number of steps. Indeed, let $\gamma$ be general and let $\gamma_k\to\gamma$ in $L^1$ be step functions. The bound of Step~2 below, applied to the whole functions, shows that $\Phi^{\gamma_k}_F(t,\cdot)$ is a Cauchy sequence in the supremum norm for each $t$. The limit exists, it does not depend on the sequence, and the bounds pass to it. We take a common refinement, so that $\gamma$ and $\widetilde\gamma$ are constant on the same intervals. On each such interval the solutions are smooth in $z$, with bounded derivatives of orders one to three, and solve \eqref{eq:fr-prefix-pde} classically; at the finitely many endpoints they are continuous in $t$.

We use the following maximum principle, which follows from the Feynman--Kac representation \cite[Theorem~5.7.6]{karatzas-shreve-1991} for a diffusion with generator $\frac{\beta^2}2\partial_{zz}+a\,\partial_z$, after an exponential rescaling to make the killing rate nonnegative. Let $a,c,\mathfrak s$ be bounded continuous functions on $[t_1,t_2]\times\R$, with $a$ Lipschitz in $z$, and let $f$ be bounded and solve $-\partial_tf=\frac{\beta^2}2f_{zz}+af_z+cf+\mathfrak s$. Then
\begin{equation}\label{eq:fr-max-principle}
 \norm{f(t,\cdot)}_\infty\le e^{\norm c_\infty(t_2-t)}\Bigl(\norm{f(t_2,\cdot)}_\infty+\int_t^{t_2}\norm{\mathfrak s(t',\cdot)}_\infty\dd t'\Bigr)\qquad(t_1\le t\le t_2).
\end{equation}
Applying \eqref{eq:fr-max-principle} on consecutive intervals, we see that it holds on $[0,q_0]$ for coefficients that are continuous on each interval where $\gamma$ and $\widetilde\gamma$ are constant.

The functions to which we apply \eqref{eq:fr-max-principle} are bounded. Indeed, fix a step profile, and let $F$ be $L$-Lipschitz. Each integration step $T_{m,v}$ maps $L$-Lipschitz functions to $L$-Lipschitz functions, by \cref{lem:inp-steps}(a), and for $0\le m\le1$ it satisfies $\norm{T_{m,v}f-f}_\infty\le\log\E e^{L\sqrt v\abs\zeta}$ for $L$-Lipschitz $f$, where $\zeta$ is a standard Gaussian variable; the upper bound follows from $f(z+\sqrt v\,\zeta)-f(z)\le L\sqrt v\abs\zeta$ and the monotonicity of $m\mapsto m^{-1}\log\E e^{mX}$, and the lower bound from Jensen's inequality. Summing over the finitely many steps, we find that $\Phi^\gamma_F(t,\cdot)-F$ is bounded. The differences $\Delta$ and $\widetilde{\mathcal D}-\mathcal D$ of Steps~2 and~3 below are bounded, and so are the derivatives to which we apply \eqref{eq:fr-max-principle}, by Step~1. The resulting estimates do not depend on the number of steps.

\paragraph{Step 1: derivative bounds.} We use the tilted-chain martingale argument of \cite[proof of Lemma~3.1]{oai-barriers}, with terminal function $F$ in place of $\log(2\cosh z)$. Let $F\in\{\varphi,\varphi+V\}$, so that $\norm{F'}_\infty\le4$ and $\norm{F''}_\infty\le4$. We claim that
\begin{equation}\label{eq:fr-prefix-derivatives}
 \abs{\partial_z\Phi^\gamma_F}\le4,\qquad\abs{\partial_{zz}\Phi^\gamma_F}\le20\qquad\text{on }[0,q_0]\times\R.
\end{equation}
For one step $\psi=T_{m,v}f$ with $m\in[0,1]$, we have $\psi'(z)=\E^Wf'(z+\sqrt v\,\zeta)$ and $\psi''(z)=\E^Wf''(z+\sqrt v\,\zeta)+m\Var^W(f'(z+\sqrt v\,\zeta))$, where $\E^W$ and $\Var^W$ refer to the law of the standard Gaussian $\zeta$ tilted by $e^{mf(z+\sqrt v\,\zeta)}$; this follows by differentiating under the integral. Iterating over the steps, as in Step~1 of the proof of \cref{lem:fr-derivative}, we find that $\partial_z\Phi^\gamma_F(t,z)$ is the expectation of $F'$ at the final time under the tilted chain started from $z$ at time $t$, and that $\partial_{zz}\Phi^\gamma_F(t,z)$ is the expectation of $F''$ at the final time plus $\sum_im_i\E\Var(M_{i+1}\mid\text{steps }1,\dots,i)$, where $M_i$ is the conditional expectation of the final value of $F'$ given the first $i$ steps. The masses $m_i=\gamma/3$ lie in $[0,1]$, and the sum of the expected conditional variances is at most the variance of the final value of $F'$, by orthogonality of martingale increments. We conclude that $-\norm{F''}_\infty\le\partial_{zz}\Phi^\gamma_F\le\norm{F''}_\infty+\norm{F'}_\infty^2$, which implies \eqref{eq:fr-prefix-derivatives}. These bounds do not depend on the number of steps.

\paragraph{Step 2: part (a).} Let $\Delta=\Phi^\gamma_F-\Phi^{\widetilde\gamma}_F$. Subtracting the two equations \eqref{eq:fr-prefix-pde}, we have $\Delta(q_0,\cdot)=0$ and
\[
 -\partial_t\Delta=\frac{\beta^2}2\Delta_{zz}+\frac{\beta^2\gamma}6\bigl(\partial_z\Phi^\gamma_F+\partial_z\Phi^{\widetilde\gamma}_F\bigr)\Delta_z+\frac{\beta^2}6(\gamma-\widetilde\gamma)\bigl(\partial_z\Phi^{\widetilde\gamma}_F\bigr)^2.
\]
Then \eqref{eq:fr-max-principle} with $c=0$ and \eqref{eq:fr-prefix-derivatives} imply that $\norm{\Delta(t,\cdot)}_\infty\le\frac{8\beta^2}3\int_t^{q_0}\abs{\gamma-\widetilde\gamma}$, which proves (a).

We also need the analogous bound for the first derivatives. Let $\Delta_z=\partial_z\Delta=\partial_z\Phi^\gamma_F-\partial_z\Phi^{\widetilde\gamma}_F$. Differentiating \eqref{eq:fr-prefix-pde} in $z$ and subtracting, we find $\Delta_z(q_0,\cdot)=0$ and
\[
 -\partial_t\Delta_z=\frac{\beta^2}2(\Delta_z)_{zz}+\frac{\beta^2\gamma}3\,\partial_z\Phi^\gamma_F\,(\Delta_z)_z+\frac{\beta^2\gamma}3\,\partial_{zz}\Phi^{\widetilde\gamma}_F\,\Delta_z+\frac{\beta^2}3(\gamma-\widetilde\gamma)\,\partial_z\Phi^{\widetilde\gamma}_F\,\partial_{zz}\Phi^{\widetilde\gamma}_F.
\]
By \eqref{eq:fr-prefix-derivatives}, the coefficient of $\Delta_z$ is bounded by $20\beta^2/3$ and the last term by $\frac{80\beta^2}3\abs{\gamma-\widetilde\gamma}$. By \eqref{eq:fr-max-principle} and $q_0\le1$, we have
\begin{equation}\label{eq:fr-gradient-stability}
 \norm{\Delta_z(t,\cdot)}_\infty\le\frac{80\beta^2}3e^{7\beta^2}\norm{\gamma-\widetilde\gamma}_{L^1([0,q_0])}\qquad(0\le t\le q_0).
\end{equation}

\paragraph{Step 3: part (b).} We write $\Phi^0=\Phi^\gamma_\varphi$, $\Phi^1=\Phi^\gamma_{\varphi+V}$, $\widetilde\Phi^0=\Phi^{\widetilde\gamma}_\varphi$ and $\widetilde\Phi^1=\Phi^{\widetilde\gamma}_{\varphi+V}$, and let $\mathcal D=\Phi^1-\Phi^0$ and $\widetilde{\mathcal D}=\widetilde\Phi^1-\widetilde\Phi^0$. The left side of \eqref{b:eq:prefix-floor-response} is $\abs{\widetilde{\mathcal D}(0,0)-\mathcal D(0,0)}$. Using $(\Phi^1_z)^2-(\Phi^0_z)^2=(\Phi^1_z+\Phi^0_z)\mathcal D_z$, we find that $\mathcal D$ solves the linear equation
\[
 -\partial_t\mathcal D=\frac{\beta^2}2\mathcal D_{zz}+a\,\mathcal D_z,\qquad
 a=\frac{\beta^2\gamma}6\bigl(\Phi^1_z+\Phi^0_z\bigr),\qquad\mathcal D(q_0,\cdot)=V,
\]
and $\widetilde{\mathcal D}$ solves the same equation with $\widetilde a=\frac{\beta^2\widetilde\gamma}6(\widetilde\Phi^1_z+\widetilde\Phi^0_z)$. Differentiating in $z$, the function $\mathcal D_z$ solves $-\partial_t\mathcal D_z=\frac{\beta^2}2(\mathcal D_z)_{zz}+a(\mathcal D_z)_z+a_z\mathcal D_z$ with $\mathcal D_z(q_0,\cdot)=V'$. By \eqref{eq:fr-prefix-derivatives}, we have $\abs{a_z}\le\frac{\beta^2}6\cdot40\le7\beta^2$, and \eqref{eq:fr-max-principle} implies that $\norm{\mathcal D_z(t,\cdot)}_\infty\le e^{7\beta^2}\norm{V'}_\infty$. Next, the difference $\widetilde{\mathcal D}-\mathcal D$ vanishes at $q_0$ and solves
\[
 -\partial_t(\widetilde{\mathcal D}-\mathcal D)=\frac{\beta^2}2(\widetilde{\mathcal D}-\mathcal D)_{zz}+\widetilde a\,(\widetilde{\mathcal D}-\mathcal D)_z+(\widetilde a-a)\mathcal D_z.
\]
By \eqref{eq:fr-max-principle} with $c=0$, we have
\[
 \abs{\widetilde{\mathcal D}(0,0)-\mathcal D(0,0)}\le e^{7\beta^2}\norm{V'}_\infty\int_0^{q_0}\norm{\widetilde a(t,\cdot)-a(t,\cdot)}_\infty\dd t.
\]
Finally, $\widetilde a-a=\frac{\beta^2}6(\widetilde\gamma-\gamma)(\widetilde\Phi^1_z+\widetilde\Phi^0_z)+\frac{\beta^2\gamma}6\bigl((\widetilde\Phi^1_z-\Phi^1_z)+(\widetilde\Phi^0_z-\Phi^0_z)\bigr)$. Using \eqref{eq:fr-prefix-derivatives} for the first term and \eqref{eq:fr-gradient-stability} for the second, we find $\int_0^{q_0}\norm{\widetilde a-a}_\infty\dd t\le C\norm{\gamma-\widetilde\gamma}_{L^1([0,q_0])}$ with $C$ depending only on $\beta$. This proves \eqref{b:eq:prefix-floor-response}.
\end{proof}

\section{The local fractional bound}\label{sec:fr-local}

We now combine \cref{b:prop:fractional-matrix}, \cref{lem:fr-floor} and \cref{lem:fr-prefix-response} with the cubic deficit of \cref{sec:cubic}. We first recall what we use from that section. Fix the ratio $\vartheta\in(0,1/2]$ and the amplitude $E>0$ of \cref{sec:cub-band}, satisfying the two strict inequalities \eqref{b:eq:os-band-checks}; in the dynamical argument, $\vartheta=1/2$ and $E=1/100$. Let $G(\vartheta,E)>0$ be the corner coefficient of \cref{sec:cub-band}. Fix compact intervals $\mathcal J\subset\operatorname{int}I$ with $I\subset(0,q_\beta)$, and let $m_\beta>0$ be a lower bound on $I$ for the scalar factor $M(r)=\alpha(r)^2\beta^6\E\chi(r,Y_r)^3$ of \cref{b:prop:os-expansion}, as in \eqref{b:eq:main-analytic-constants}. For $D=D(r,q,\delta)$ with $d=q-r>0$, we use the parameters of \eqref{b:eq:os-parameters}, namely $h=d/\vartheta$, $q_0=r-h$ and $k=\delta/h$. Let $\mathcal T(q_0,h,k)$ be the trial value \eqref{b:eq:os-trial}, let $V=V(q_0,h,k,\cdot)$ be the boundary perturbation \eqref{eq:cub-V}, and let $\Pi=\Pi(q_0,h,k)$ be the penalty of \cref{lem:cub-trial}. We use the following three facts.
\begin{enumerate}
\item[(T1)] By \eqref{b:eq:os-trial} and \cref{lem:cub-trial}, $\mathcal T(q_0,h,k)$ is the limit of the trial values of admissible paths $(Q^{(N)},m^{(N)})$ with endpoint $D$ and multiplier zero. On $[0,q_0]$ these paths have the increments $\mathbf e\mathbf e^{\mathsf T}\dd t$ at the masses $\alpha/3$, evaluated at the left endpoints of a partition; on $[q_0,1]$ their masses are at least $\alpha(q_0)/3$; and all masses are nondecreasing. Moreover
\[
 \mathcal T(q_0,h,k)=\mathcal H_\alpha\bigl(3\Phi(q_0,\cdot)+V\bigr)-\frac{\beta^2}4\Pi,
\]
where $\mathcal H_\alpha$ is the operator of \cref{sec:fr-prefix} on $[0,q_0]$ with $\gamma=\alpha$. The function $V$ and the penalty $\Pi_+=\Pi-6\int_0^{q_0}t\alpha(t)\dd t$ of the path on $[q_0,1]$ depend on $\alpha$ only through its restriction to $[q_0,1]$.
\item[(T2)] By the spatial regularity in \cref{b:lem:os-flow}, $V$ has bounded continuous derivatives of orders one to three. By \eqref{b:eq:os-V-bound}, $\norm V_{C^2_b}\le C_Vh^2k^2=C_V\delta^2$ for $h\le h_0$. We use only the consequence $\norm V_{C^2_b}\le C_V\delta^2/h$ for $h\le\min\{h_0,1\}$; no smallness of the third derivative is needed.
\item[(T3)] By \cref{prop:cub-band}, applied to the interval $\mathcal J$ of centers (note that $m_\beta\le\inf_{\mathcal J}M$), for every $\bar\sigma\in(0,G(\vartheta,E))$ there exists $d_0>0$ such that $\mathcal T(q_0,h,k)\le3P_\beta-m_\beta E^2\{G(\vartheta,E)-\bar\sigma\}\rho^3$, which is \eqref{b:eq:os-uniform-corner-gap}, for every $0<\rho\le d_0$ and every feasible $D(r,q,\delta)$ with $r\in\mathcal J$, $\rho\le q-r\le d_0$ and $E\rho\le\abs\delta\le2E\rho$; for these matrices also $h\le h_0$. The window is $[q_0,T]$ with $T=q_0+h\tau_T$ and $\tau_T=2+4k^2L_\vartheta$, by \eqref{eq:cub-T}. If $\abs\delta\le2Ed$, then $\abs k\le2\vartheta E$, and the window has length $T-q_0\le C_{\rm win}d$ with $C_{\rm win}=(2+16\vartheta^2E^2L_\vartheta)/\vartheta$.
\end{enumerate}
By \eqref{eq:fr-block}, $\mathcal H_\alpha(3\Phi(q_0,\cdot))=3\Phi(0,0)$.

Now let $\gamma\colon[0,1]\to[0,1]$ be nondecreasing, with $\gamma=\alpha$ on $[q_0,1]$ and $\gamma\le\alpha(q_0)$ on $[0,q_0]$. We write $\mathcal T_\gamma(q_0,h,k)$ for the limit of the trial values of the paths of (T1) with the masses $\alpha/3$ on $[0,q_0]$ replaced by the left-endpoint values of $\gamma/3$. These paths are admissible, with the same endpoint, and on $[q_0,1]$ nothing changes. The penalty of the segment $[0,q_0]$ converges to $\int_0^{q_0}\frac\gamma3\,\dd(9t^2)=6\int_0^{q_0}t\gamma(t)\dd t$. By \cref{lem:fr-prefix-response}(a) and the bound $\abs{\mathcal H_\gamma(F_1)-\mathcal H_\gamma(F_2)}\le\norm{F_1-F_2}_\infty$, the values converge as well, and with $\varphi=3\Phi(q_0,\cdot)$ we find
\begin{equation}\label{eq:fr-trial-split}
 \mathcal T_\gamma(q_0,h,k)=\mathcal H_\gamma(\varphi+V)-\frac{3\beta^2}2\int_0^{q_0}t\gamma(t)\dd t-\frac{\beta^2}4\Pi_+.
\end{equation}
For $\gamma=\alpha$, this is the identity of (T1). If $\gamma$ is a distribution function, then by \eqref{eq:fr-block} we have
\begin{equation}\label{eq:fr-scalar-split}
 3\mathcal P_\beta(\gamma)=\mathcal H_\gamma(\varphi)-\frac{3\beta^2}2\int_0^{q_0}t\gamma(t)\dd t-\frac{3\beta^2}2\int_{q_0}^1t\alpha(t)\dd t.
\end{equation}
Subtracting \eqref{eq:fr-trial-split} and \eqref{eq:fr-scalar-split} for $\alpha$ from the same identities for $\gamma$, and using $\mathcal P_\beta(\alpha)=P_\beta$, we obtain
\begin{multline}\label{eq:fr-trial-difference}
 \mathcal T_\gamma(q_0,h,k)-\mathcal T(q_0,h,k)
 =3\bigl(\mathcal P_\beta(\gamma)-P_\beta\bigr)\\
 +\bigl[\mathcal H_\gamma(\varphi+V)-\mathcal H_\gamma(\varphi)\bigr]-\bigl[\mathcal H_\alpha(\varphi+V)-\mathcal H_\alpha(\varphi)\bigr].
\end{multline}

\begin{lemma}[Local fractional bound]\label{lem:fr-local-gap}
Let $\mathcal J\subset\operatorname{int}I$ and $I\subset(0,q_\beta)$ be compact intervals, let $\vartheta,E,m_\beta$ be as above, and let $0<g<m_\beta E^2G(\vartheta,E)$. There exist $d_0,S,\rho_0>0$, depending only on $\beta$, $\mathcal J$, $I$, $\vartheta$, $E$ and $g$, such that the following holds. For every $0<\rho\le\rho_0$, every $n\ge2$ and every feasible $D=D(r,q,\delta)$ with
\[
 r\in\mathcal J,\qquad\rho\le q-r\le d_0,\qquad E\rho\le\abs\delta\le2E\rho,
\]
the window $[q_0,T]$ of the local perturbation lies in $I$, and with $s=S\rho$ we have
\begin{equation}\label{b:eq:local-fractional-gap}
 \frac1{ns}\log\E Z_{3,n}(D)^s\le3P_\beta-g\rho^3.
\end{equation}
\end{lemma}

The constants $d_0$, $S$ and $\rho_0$ do not depend on $n$, on $\rho$, or on $(r,q,\delta)$ in the stated domain. The order in which they are chosen is recorded in \cref{tab:fr-choices}.

\begin{table}[ht]
\centering
\begin{tabular}{@{}lll@{}}
\toprule
Parameter & Chosen in terms of & Purpose\\
\midrule
$\mathcal J$, $I$, $\vartheta$, $E$, $m_\beta$ & fixed first & centers; geometry of \cref{sec:cubic}\\
$g<g'$ & $m_\beta E^2G(\vartheta,E)$ & two strict margins \eqref{b:eq:static-gap-choice}\\
$d_0$ & $g'$, $\mathcal J$, $I$, $\vartheta$, $E$ & cubic deficit (T3); window in $I$; $h\le1$\\
$S$ & $g$, $g'$, $\vartheta$, $E$ & profile response at most $\frac12(g'-g)\rho^3$\\
$\rho_0$ & all of the above & $\alpha_s$ admissible; $V$ small; cost $s^5$\\
$n\ge2$ & arbitrary & the bound is uniform in $n$\\
\bottomrule
\end{tabular}
\caption{Order of the choices in the proof of \cref{lem:fr-local-gap}.}\label{tab:fr-choices}
\end{table}

\begin{proof}
The proof has three steps. We first fix the constants. We then apply \cref{b:prop:fractional-matrix} to the comparison path with the profile $\alpha_s$ on the common segment, and finally we compare its trial value with $\mathcal T(q_0,h,k)$ by \eqref{eq:fr-trial-difference}.

\paragraph{Step 1: choice of constants.} Fix $g'$ with
\begin{equation}\label{b:eq:static-gap-choice}
 0<g<g'<m_\beta E^2G(\vartheta,E).
\end{equation}
Let $d_0$ be the constant of (T3) for $\bar\sigma=G(\vartheta,E)-g'/(m_\beta E^2)$, decreased so that also $d_0\le\vartheta$ and $(1/\vartheta+C_{\rm win})d_0\le\operatorname{dist}(\mathcal J,\R\setminus I)$. Then, for every $0<\rho\le d_0$ and every $D$ in the domain of the lemma,
\begin{equation}\label{b:eq:local-static-input}
 \mathcal T(q_0,h,k)\le3P_\beta-g'\rho^3,
\end{equation}
moreover $h=d/\vartheta\le1$, and the window $[q_0,T]\subset[r-d/\vartheta,r-d/\vartheta+C_{\rm win}d]$ lies in $I$, so that $q_0\ge a_*$ with $a_*=\min I>0$. Let $C_{\rm fl}$, $s_0$ and $C_{\rm r}$ be the constants of \cref{lem:fr-floor,lem:fr-prefix-response}, let $C'=3C_{\rm fl}$ and $C''=C_{\rm r}C_VC_{\rm fl}$, and choose $S\in(0,1/3]$ so small that
\begin{equation}\label{eq:fr-S}
 4\vartheta E^2C''S^2\le\tfrac12(g'-g).
\end{equation}
Finally choose $\rho_0\in(0,d_0]$ so small that
\[
 S\rho_0\le s_0,\qquad C_{\rm fl}S\rho_0<a_*,\qquad 8\vartheta E^2C_V\rho_0\le1,\qquad C'S^5\rho_0^2\le\tfrac12(g'-g).
\]
Now fix $0<\rho\le\rho_0$, $n\ge2$ and $D$ in the domain of the lemma, and set $s=S\rho$.

\paragraph{Step 2: the fractional comparison.} Since $s\le s_0$ and $C_{\rm fl}s<a_*\le q_0$, \cref{lem:fr-floor} shows that $\alpha_s$ is a distribution function with $\alpha_s=\alpha$ on $[q_0,1]$. Moreover $\alpha_s\le\alpha_s(q_0)=\alpha(q_0)$ on $[0,q_0]$, and $\mathcal T_{\alpha_s}(q_0,h,k)$ is defined. By its definition, it is the limit of the trial values of admissible paths with endpoint $D$ and multiplier zero, whose masses are left-endpoint values of $\alpha_s/3$ on $[0,q_0]$ and are at least $\alpha(q_0)/3$ on $[q_0,1]$. All these masses are at least $s$, because $\alpha_s\ge3s$ and $\alpha(q_0)=\alpha_s(q_0)\ge3s$. By \cref{b:prop:fractional-matrix} with $\ell=3$ for these paths, we obtain
\begin{equation}\label{eq:fr-step2}
 p^{(s)}_{3,n}(D)+\frac{9\beta^2s}{4n}\le\mathcal T_{\alpha_s}(q_0,h,k).
\end{equation}

\paragraph{Step 3: comparison with $\mathcal T(q_0,h,k)$.} We apply \eqref{eq:fr-trial-difference} with $\gamma=\alpha_s$. The first term on its right side is at most $3C_{\rm fl}s^5=C's^5$ by \eqref{b:eq:specific-floor-cost}. For the second, note that $\varphi=3\Phi(q_0,\cdot)$ satisfies $\norm{\varphi'}_\infty\le3$ and $\norm{\varphi''}_\infty\le3$, since $\abs u\le1$ and $0<\chi\le1$ by \cref{inp:ob-scalar}(i), and that $\norm{V'}_\infty+\norm{V''}_\infty\le2C_V\delta^2/h\le8\vartheta E^2C_V\rho\le1$ by (T2), since $\abs\delta\le2E\rho$ and $h=d/\vartheta\ge\rho/\vartheta$. \Cref{lem:fr-prefix-response}(b) on $[0,q_0]$, with $\widetilde\gamma=\alpha_s$ and $\gamma=\alpha$, and \eqref{b:eq:specific-floor-cost} then bound the mixed difference by $C_{\rm r}\norm{V'}_\infty\norm{\alpha_s-\alpha}_{L^1}\le C_{\rm r}C_V(\delta^2/h)C_{\rm fl}s^2$. We have shown that the change of profile costs at most
\begin{equation}\label{b:eq:coarse-floor-error}
 \mathcal T_{\alpha_s}(q_0,h,k)-\mathcal T(q_0,h,k)\le C's^5+C''\frac{\delta^2s^2}h.
\end{equation}
Since $\abs\delta\le2E\rho$, $h=d/\vartheta$ and $d\ge\rho$, we have
\[
 \frac{\delta^2s^2}h=\frac{\vartheta\,\delta^2S^2\rho^2}d\le4\vartheta E^2S^2\rho^3.
\]
Inserting \eqref{b:eq:local-static-input} and \eqref{b:eq:coarse-floor-error} into \eqref{eq:fr-step2}, and discarding the nonnegative term $9\beta^2s/(4n)$, we obtain
\[
 p^{(s)}_{3,n}(D)\le3P_\beta-g'\rho^3+C'S^5\rho^5+4\vartheta E^2C''S^2\rho^3\le3P_\beta-g\rho^3,
\]
where in the last step we used \eqref{eq:fr-S} and $C'S^5\rho^2\le\frac12(g'-g)$. This is \eqref{b:eq:local-fractional-gap}.
\end{proof}

\begin{remark}\label{rem:fr-local-scope}
The bound \eqref{b:eq:local-fractional-gap} is a deterministic statement about the Gaussian moments $\E Z_{3,n}(D)^s$. It holds for every $n\ge2$ once $\rho\le\rho_0$, and it requires no relation between $\rho$ and $n$. The separation $q_0\ge a_*>0$ between the local window and the origin is what makes the change of profile harmless: since $\alpha_s=\alpha$ on $[q_0,1]$, the local window, its terminal correction and its penalty are identical for the two profiles. The scalar cost of order $s^5$ is much smaller than needed; a cost of order $s^3$ would also suffice, after decreasing $S$. Since $\norm V_{C^2_b}\le C_V\delta^2$ by \eqref{b:eq:os-V-bound}, the response term is in fact of order $\delta^2s^2\le4E^2S^2\rho^4$; we used only the weaker bound $C_V\delta^2/h$, for which the small choice of $S$ is needed. The local bound concerns only $q-r\le d_0$. For $q-r>d_0$ the construction of \cref{sec:cubic} is not used, and \cref{sec:static} uses instead the quenched comparison of \cref{lem:st-remote}, whose deficit is a multiple of $\delta^2$, much larger than $\rho^3$.
\end{remark}
\chapter{A simultaneous estimate for overlap constraints}\label{sec:static}

In this section we prove \cref{b:lem:static-event}. For deterministic scales $\rho_n\to0$ and $\eta_n$ satisfying the conditions \eqref{eq:st-S1}--\eqref{eq:st-S5} below, it provides a disorder event $\mathcal A_n$ of probability tending to one on which three families of overlap configurations have small Gibbs probability simultaneously: triples whose overlaps lie in a narrow band around a center $r$, triples violating a small-overlap constraint across a large overlap, and pairs whose absolute overlap $q=\abs{R(x,y)}$ has $\abs{q-\Gamma(q)}$ bounded away from zero. These estimates are the static input of the dynamical argument: \cref{sec:references} uses them to choose families of reference configurations (\cref{lem:ref-single-event}), and \cref{sec:induction,sec:barrier-proof} use them along trajectories of the heat-bath dynamics. The conditions \eqref{eq:st-S1}--\eqref{eq:st-S5} hold at the scales of \cref{thm:barrier-730}, and more generally under \eqref{eq:ref-exponents} and \eqref{eq:ref-mean-input}; see \cref{rem:st-scales}.

The narrow triples split into two regimes. When the larger overlap $q$ is close to the center $r$, we use the local fractional bound \cref{lem:fr-local-gap}, which rests on the cubic deficit of \cref{sec:cubic}. When $q-r$ is bounded below, we use a quadratic pressure deficit, which we derive in \cref{lem:st-remote} from the three-replica comparison of \cite[Section~7]{oai-barriers}, with a fixed width and a fixed tolerance. The pair bound and the fixed-scale bound are the pressure estimates \cref{inp:ob-pair-gap,inp:ob-fixed-gap}, which are \cite[Lemma~4.3]{oai-barriers} and \cite[(4.20)]{oai-barriers}. In all these regimes the mechanisms are those of \cite{oai-barriers}; what changes are the scales at which the pressure deficits are converted into probability bounds. The simultaneous event and its probability estimate are new relative to \cite{oai-barriers}: they replace the conversion \cite[Lemma~4.2]{oai-barriers}, whose finite-size error $n^{-1/24}$ is far too large at the scale $\rho_n^3$.

\paragraph{Two levels of probability.} We keep two kinds of error separate, and the later sections depend on this distinction. The first is the disorder probability $\PP(\mathcal A_n^c)$, for which we obtain the upper bound $O(n^{-2/15})$ at the scales of \cref{thm:barrier-730}; it is paid once. The second is the Gibbs probability of the forbidden configurations at a fixed disorder in $\mathcal A_n$, which for narrow triples is bounded above by $C(n+1)^3e^{-cn^{3/10}}$ at the same scales. Only errors of the second kind are multiplied by the number of reference configurations and paths in \cref{sec:references,sec:induction,sec:barrier-proof}; see \cref{tab:st-errors} and \cref{rem:st-two-levels}.

\paragraph{Upper bound on the mean bias.} The proof of \cref{b:lem:static-event} uses the mean bias $\Delta_{n,\beta}=P_\beta-p_n(\beta)$ only through the hypothesis \eqref{eq:st-S1}. The bound $\Delta_{n,\beta}\le C_bn^{-7/10}$ of \cref{inp:hb-rate} gives the scale $\rho_n\asymp n^{-7/30}$ of \cref{thm:barrier-730}. The variance bound of \cref{inp:qp-rate} is used only for the lower bound (A1) on $F_n$ below, through Chebyshev's inequality and the hypothesis \eqref{eq:st-S2}.

\paragraph{Standing notation.} We keep the notation of \cref{sec:fractional}; in particular we do not abbreviate $\beta^2$. Triples are taken in the order $(v,x,y)$, and for a triple we write
\[
 r=\frac{R(v,x)+R(v,y)}2,\qquad\delta=\frac{R(v,x)-R(v,y)}2,\qquad q=R(x,y),
\]
so that its overlap matrix is the matrix $D(r,q,\delta)$ of \eqref{eq:inp-triple}. We also write $\chi_t=\chi(t,Y_t)$. Constants $C,c$ depend only on $\beta$ and on the fixed parameters named in each statement; they never depend on $n$, on the overlap parameters $(r,q,\delta)$, or on the scales $\rho_n,\eta_n$.

\section{Orientations, pairs and the fixed scale}\label{sec:st-pair-fixed}

Every overlap takes one of the $n+1$ values $-1,-1+2/n,\dots,1$, so there are at most $n+1$ feasible matrices $D_2(\bar q)$ and at most $(n+1)^3$ feasible matrices $D(r,q,\delta)$. Let $\mathfrak O_\ell$ be the group of maps $\mathsf O\colon(x^a)_{a\le\ell}\mapsto(\epsilon_ax^{\varpi(a)})_{a\le\ell}$ of \cref{rem:inp-orientation}, where $\varpi$ is a permutation of $\{1,\dots,\ell\}$ and $\epsilon\in\{-1,1\}^\ell$; the group $\mathfrak O_3$ has $48$ elements. By \cref{rem:inp-orientation}, $Z_{\ell,n}(\mathsf OD)=Z_{\ell,n}(D)$ for every disorder and every orientation $\mathsf O$. In addition, every $\mathsf O\in\mathfrak O_\ell$ is a bijection of $\Sigma_n^\ell$ that preserves the product measure $(\pi^J)^{\otimes\ell}$, because $\pi^J(-x)=\pi^J(x)$. We have
\begin{equation}\label{eq:st-orientation}
 (\pi^J)^{\otimes\ell}(\mathsf OB)=(\pi^J)^{\otimes\ell}(B)\qquad(B\subset\Sigma_n^\ell,\ \mathsf O\in\mathfrak O_\ell).
\end{equation}
All statements below are proved for canonical orientations. A disorder bound for $Z_{\ell,n}(D)$ holds verbatim for $Z_{\ell,n}(\mathsf OD)$, a Gibbs bound for a set $B$ holds for $\mathsf OB$, and a union over all orientations of a triple costs at most a factor $48$.

\begin{lemma}[Pair constraints]\label{lem:st-pair}
For every $n\ge2$ and every feasible $D_2(\bar q)$,
\begin{equation}\label{b:eq:pair-static-gap}
 p_{2,n}\bigl(D_2(\bar q)\bigr)\le2P_\beta-\frac12\bigl(\abs{\bar q}-\Gamma(\abs{\bar q})\bigr)^2.
\end{equation}
Moreover, there exist $q_{\rm tail}<1$ and $c_{\rm tail}>0$, depending only on $\beta$, such that $q-\Gamma(q)\ge c_{\rm tail}$ for $q_{\rm tail}\le q\le1$.
\end{lemma}

\begin{proof}
For $\bar q\ge0$, \eqref{b:eq:pair-static-gap} is \cref{inp:ob-pair-gap}, valid for every $n$ and every feasible pair constraint; for $\bar q<0$ it follows from $Z_{2,n}(D_2(\bar q))=Z_{2,n}(D_2(\abs{\bar q}))$. By \cref{inp:ob-scalar}(iv), $\Gamma$ is continuous, and $\Gamma(1)=\E\tanh^2(Y_1)<1$ because $u(1,z)=\tanh z$. Since $q-\Gamma(q)$ is continuous and positive at $q=1$, there exist $q_{\rm tail}$ and $c_{\rm tail}$ as claimed.
\end{proof}

\begin{lemma}[The fixed scale]\label{lem:st-fixed}
For every $b_*\in(0,1)$ there exist $\underline b(b_*),\nu_{\rm fix}(b_*),c_{\rm fix}(b_*)>0$, depending only on $\beta$ and $b_*$, such that
\begin{equation}\label{b:eq:fixed-pressure-gap}
 p_{3,n}\bigl(D(r,q,\delta)\bigr)\le3P_\beta-c_{\rm fix}(b_*)\,\delta^2
\end{equation}
for every $n\ge2$ and every feasible $D(r,q,\delta)$ with $b_*\le q\le1$, $\abs{r+\delta}\le\underline b(b_*)$, $\abs{r-\delta}\le\underline b(b_*)$ and $\abs{q-\Gamma(q)}\le\nu_{\rm fix}(b_*)$. The same holds after decreasing $\underline b(b_*)$ or $\nu_{\rm fix}(b_*)$.
\end{lemma}

\begin{proof}
Let $r_0,\delta_0,\nu,c_0$ be the constants of \cref{inp:ob-fixed-gap} for the cutoff $b_*$, and set $\underline b(b_*)=\min\{r_0,\delta_0\}$, $\nu_{\rm fix}(b_*)=\nu$ and $c_{\rm fix}(b_*)=c_0$. If $\abs{r\pm\delta}\le\underline b(b_*)$, then $\abs r\le\underline b(b_*)$ and $\abs\delta\le\underline b(b_*)$. For $r\ge0$ the bound is \cref{inp:ob-fixed-gap}. For $r<0$, reversing $v$ maps $D(r,q,\delta)$ to $D(-r,q,-\delta)$, and the two constrained partition functions are equal by \cref{rem:inp-orientation}. Decreasing $\underline b(b_*)$ or $\nu_{\rm fix}(b_*)$ shrinks the set of matrices to which the bound is applied.
\end{proof}

Here we use only the pressure bound \cite[(4.20)]{oai-barriers}. The fixed-scale probability estimate \cite[Lemma~4.1]{oai-barriers}, obtained using the conversion of \cite[Lemma~4.2]{oai-barriers}, concerns overlap imbalances of size $n^{-1/100}$; below we derive the probability bound at overlap imbalances of size $\eta_n$. No restriction $r\ge\abs\delta$ is needed, so the two small overlaps $r\pm\delta$ may have opposite signs.

\section{Separated overlaps}\label{sec:st-remote}

We now treat triples $D(r,q,\delta)$ with $r$ in a compact subinterval of $(0,q_\beta)$ and $q-r$ bounded below. The argument is that of \cite[Section~7, Steps~1--8]{oai-barriers}, recalled in \cref{inp:ob-remote}. There it is carried out with the shrinking parameters $h=\rho_{\rm OB}^8$ and $\theta=\rho_{\rm OB}^2$ and a restriction on the Parisi mass of $[r-h,r+h]$. We need it with fixed parameters, so we write the argument out in this form; \cref{rem:inp-remote-scales} lists the places where the scales enter. The only change in the computation is that the oscillation of $\alpha$ near $r$ is kept as an explicit term instead of being bounded by $\rho_{\rm OB}^4$. We begin with the scalar estimate that produces the deficit.

\begin{lemma}[Decay of the mixed susceptibility]\label{lem:st-susceptibility}
Let $\mathcal J\subset(0,q_\beta)$ be a compact interval. There exists $c_\chi>0$, depending only on $\beta$ and $\mathcal J$, such that for every $r\in\mathcal J$ and $r\le q\le1$,
\begin{equation}\label{eq:st-susceptibility}
 \beta^2\E[\chi_q\chi_r]=1-\beta^4\int_r^q\alpha(t)\,\E[\chi_t^2\chi_r]\,\dd t\le1-c_\chi(q-r).
\end{equation}
\end{lemma}

\begin{proof}
By \cref{cor:inp-identity}, the hypotheses of \cref{inp:ob-remote} hold for every $t_0\in(0,q_\beta/3]$; note that then $3t_0\le q_\beta<1$. The intervals $(t_0/2,3t_0)$ with $t_0\in(0,q_\beta/3]$ cover $(0,q_\beta)$, and finitely many of them cover the compact set $\mathcal J$. By part (ii) of \cref{inp:ob-remote}, which is \cite[(7.31)]{oai-barriers}, the bound \eqref{eq:st-susceptibility} holds for $r$ in each of these intervals, with a constant depending on $t_0$, and we take $c_\chi$ to be the smallest of these constants.
\end{proof}

The identity in \eqref{eq:st-susceptibility} follows from the conditional evolution of $\chi$ in \cref{inp:ob-scalar}(iv), multiplied by $\chi_r$, and from $\beta^2\E\chi_r^2=\Gamma'(r)=1$. The inequality comes from restricting the scalar diffusion to an event of positive probability on which $\abs{Y_t}$ stays bounded, where $\chi$ has a positive minimum; no lower bound on $\chi$ on the whole real line is used. The next lemma is the fixed-parameter form of \cite[(7.29)--(7.30)]{oai-barriers}.

\begin{lemma}[Comparison with a triangular covariance perturbation of fixed width]\label{lem:st-remote-expansion}
Let $r,q\in(0,1]$, $\bar h\in(0,1]$ and $\bar\theta\in(0,1/2]$ satisfy $\bar h<r$ and $r+\bar h<q$, and set
\[
 \bar W=\frac{2(1-\bar\theta)^2+\bar\theta^2}{\bar h},\qquad\operatorname{osc}_r=\alpha(r+\bar h)-\alpha(r-\bar h).
\]
There exist $\widetilde C,\widehat C<\infty$, depending only on $\beta$, such that for every $n\ge2$ and every feasible $D(r,q,\delta)$ with $8\beta^2\bar W\delta^2\le1$,
\begin{equation}\label{eq:st-remote-expansion}
\begin{split}
 p_{3,n}\bigl(D(r,q,\delta)\bigr)\le3P_\beta+\delta^2\Bigl\{&\beta^2\bar W\alpha(q)\bigl(q-\Gamma(q)\bigr)
 +2(1-\bar\theta)\bar\theta\beta^2\alpha(r)\bigl(\beta^2\E[\chi_q\chi_r]-1\bigr)\\
 &+\widetilde C(\operatorname{osc}_r+\bar\theta^2)\Bigr\}+\widehat C\bigl(\abs\delta^3\bar h^{-3/2}+\delta^4\bar h^{-2}\bigr).
\end{split}
\end{equation}
\end{lemma}

\begin{proof}
We perturb the base path of \cref{lem:inp-blocks}(b) for three replicas by a contrast of fixed width $\bar h$ around $r$, keep the resulting endpoint mismatch of order $\delta^2$, and evaluate the change of the one-site value with Gaussian integration by parts. The proof follows \cite[Section~7, Steps~1--7]{oai-barriers} and has seven steps. Steps~1--6 are carried out for step approximations of $\alpha$, and Step~7 passes to the limit and applies \eqref{eq:inp-stationarity}.

\paragraph{Step 1: step data and the base path.} Fix partitions of $[0,1]$ whose mesh tends to zero and which contain $r-\bar h$, $r$, $r+\bar h$ and $q$. On each interval $[t_{i-1},t_i)$ of such a partition let the step function take the value $\alpha(t_{i-1})$, and let it equal $1$ at $t=1$. These step functions are distribution functions, they converge to $\alpha$ in $L^1([0,1])$, they agree with $\alpha$ at $r$ and $q$, and on $[r-\bar h,r+\bar h]$ they take values in $[\alpha(r-\bar h),\alpha(r+\bar h)]$. In Steps~1--6 we fix one of them and denote it, and the scalar objects built from it, again by $\alpha$, $\Phi$, $u$, $\chi$; its oscillation on $[r-\bar h,r+\bar h]$ is at most $\operatorname{osc}_r$. The bounds $\abs u\le1$, $0<\chi\le1$ and $\abs{\chi(t,z)-\chi(t,z')}\le C\abs{z-z'}$ of \cref{inp:ob-scalar}(i) hold uniformly over distribution functions, and all constants below are independent of the partition.

Let $(Q,m)$ be the base path of \cref{lem:inp-blocks}(b) for this step function. On $[0,r]$ the three coordinates share every increment, at mass $\alpha/3$. On $[r,q]$ the coordinate $v$ has its own increments at mass $\alpha$, and the coordinates $x,y$ share increments at mass $\alpha/2$. On $[q,1]$ the three coordinates have independent increments at mass $\alpha$. By \cref{lem:inp-blocks}(b), its endpoint is $D(r,q,0)$ and its trial value for $D(r,q,0)$ is $3\mathcal P_\beta(\alpha)$ \cite[(7.7)]{oai-barriers}. Let $z^x$ and $z^v$ be the raw coordinates of the cumulative fields of $x$ and of $v$, sums of the Gaussian increments, which agree on $[0,r]$. We write $\E_0$ for the path law of the base path. Under the Gaussian law of the increments, $z^x$ and $z^v$ are Brownian motions with variance $\beta^2$ per unit time, observed at the partition points; under $\E_0$, they are the paths $Y^x$ and $Y^v$ of \cref{lem:inp-blocks}(d). Under $\E_0$, each is a scalar diffusion for $\alpha$, they coincide on $[0,r]$, and conditionally on the common path through $r$ their continuations are independent \cite[(4.6)]{oai-barriers}.

\paragraph{Step 2: the perturbation and its endpoint.} We define the Gaussian linear combinations
\[
 \mathsf Y_0=\frac{(z^x_r-z^x_{r-\bar h})-(z^x_{r+\bar h}-z^x_r)}{\bar h},\qquad
 \mathsf Y_1=\frac{z^v_{r+\bar h}-z^v_r}{\bar h},\qquad
 \mathsf Y=(1-\bar\theta)\mathsf Y_0+\bar\theta\mathsf Y_1,
\]
which are \cite[(7.8)]{oai-barriers} with $h=\bar h$ and $\theta=\bar\theta$. At every increment of the first two stages, we add $\delta$ times its contribution to $\mathsf Y$ to the field of $x$ and subtract it from the field of $y$; the field of $v$ is unchanged. Every modified increment is a linear image of the original Gaussian increment, so it is positive semidefinite, increments remain independent, and the masses are unchanged. At the end of the second stage the added fields are $(0,\delta\mathsf Y,-\delta\mathsf Y)$. In units of $\beta^2$, the variance of $\mathsf Y$ is $\bar W$, its covariance with the final field of $v$ is $(1-\bar\theta)+\bar\theta=1$, and its covariance with the final fields of $x$ and of $y$ is $(1-\bar\theta)-(1-\bar\theta)=0$. Let $W_c$, $W_v$ and $W_x$ be the corresponding accumulated quantities along the path, and let
\[
 L_1=\begin{pmatrix}0&W_v&-W_v\\ W_v&2W_x&0\\ -W_v&0&-2W_x\end{pmatrix},\qquad
 L_2=(\mathbf e_x-\mathbf e_y)(\mathbf e_x-\mathbf e_y)^{\mathsf T}.
\]
Then the perturbed covariance is $\widetilde Q=Q+\delta L_1+\delta^2W_cL_2$ along the path, and its endpoint is $\widetilde Q_N=D(r,q,\delta)+\delta^2\bar WL_2$ \cite[(7.10), (7.12)]{oai-barriers}. Its diagonal is not equal to one, which the arbitrary-endpoint comparison \cref{inp:ob-matrix} allows; the mismatch cost is
\begin{equation}\label{eq:st-mismatch}
 \frac{\beta^2}4\norm{D(r,q,\delta)-\widetilde Q_N}_F^2=\beta^2\delta^4\bar W^2.
\end{equation}

\paragraph{Step 3: the penalty.} Note that $Q_{vx}=Q_{vy}$ and $Q_{xx}=Q_{yy}$ along the base path, which implies that $Q\mathbin{:}L_1=0$; also $L_1\mathbin{:}L_2=0$, $\norm{L_1}_F^2=4W_v^2+8W_x^2$ and $\norm{L_2}_F^2=4$. Expanding, we have
\[
 \norm{\widetilde Q}_F^2-\norm Q_F^2=\delta^2\bigl(4W_v^2+8W_x^2+2W_c\,Q\mathbin{:}L_2\bigr)+4\delta^4W_c^2.
\]
In the first two stages the base fields of $x$ and $y$ coincide and $Q\mathbin{:}L_2=0$; in the last stage, $W_c=\bar W$ and $Q\mathbin{:}L_2=2(t-q)$. The last term has nonnegative increments, because $W_c$ is nondecreasing. We conclude that the increase of the penalty is at least
\begin{equation}\label{eq:st-penalty}
 \frac{\beta^2}4\Bigl(\int m\dd\norm{\widetilde Q}_F^2-\int m\dd\norm Q_F^2\Bigr)\ge\beta^2\delta^2\Bigl(\bar W\int_q^1\alpha(t)\dd t+\mathcal I\Bigr),\qquad
 \mathcal I=\int m\,\dd\bigl(W_v^2+2W_x^2\bigr).
\end{equation}
The integral $\mathcal I$ has three pieces. On $[r-\bar h,r]$, the common increments carry mass $\alpha/3$ and $W_v=W_x$ grows from $0$ to $1-\bar\theta$; this piece is $\int\frac\alpha3\,\dd(3W_v^2)$. On the $v$-branch over $[r,r+\bar h]$, the mass is $\alpha$ and $W_v$ grows from $1-\bar\theta$ to $1$. On the shared branch of $x,y$ over $[r,r+\bar h]$, the mass is $\alpha/2$ and $W_x$ decreases from $1-\bar\theta$ to $0$, and this piece is $\int\frac\alpha2\,\dd(2W_x^2)$. After cancelling the block-size factors, the three pieces are integrals of $\alpha$ against $\dd(W_v^2)$, $\dd(W_v^2)$ and $\dd(W_x^2)$, respectively. Their integrators are monotone, with total variations $(1-\bar\theta)^2$, $2\bar\theta-\bar\theta^2$ and $(1-\bar\theta)^2$, each at most one, and on these pieces $\alpha$ differs from $\alpha(r)$ by at most $\operatorname{osc}_r$. Replacing $\alpha$ by $\alpha(r)$ therefore changes their sum by at most $3\operatorname{osc}_r$:
\[
 \Bigl|\mathcal I-\alpha(r)\bigl\{(1-\bar\theta)^2+(2\bar\theta-\bar\theta^2)-(1-\bar\theta)^2\bigr\}\Bigr|\le3\operatorname{osc}_r,
\]
so that $\mathcal I\ge\alpha(r)(2\bar\theta-\bar\theta^2)-3\operatorname{osc}_r$.
This is \cite[(7.14)--(7.15)]{oai-barriers}, with $3\operatorname{osc}_r$ in place of the error $O(\rho_{\rm OB}^4)$.

\paragraph{Step 4: the one-site value.} Let $\Psi_0$ and $\widetilde\Psi$ be the one-site values of the base and perturbed paths. The last stage is the same for both, and integrating it leaves the sum of $\Phi(q,\cdot)$ over the three fields. With $z=z^x_q$ the common field of $x,y$ at the end of the second stage, the two boundary functions differ by
\[
 \Upsilon=\Phi(q,z+\delta\mathsf Y)+\Phi(q,z-\delta\mathsf Y)-2\Phi(q,z),
\]
a function of the Gaussian increments of the first two stages. We claim that
\begin{equation}\label{eq:st-kernel}
 \widetilde\Psi-\Psi_0\le\log\E_0e^{\Upsilon}.
\end{equation}
Indeed, both hierarchies integrate the same Gaussian increments at the same masses. Let $\Delta_i$ be the difference of their partial values after the increments beyond $i$ have been integrated, as a function of the increments through $i$, and let $\mathcal K_i$ be the tilted kernel of the base path at increment $i$. Taking the ratio of the two defining integrals, we find $\Delta_{i-1}=m^{-1}\log\mathcal K_ie^{m\Delta_i}$ if increment $i$ has mass $m>0$, and $\Delta_{i-1}=\mathcal K_i\Delta_i$ if $m=0$. Jensen's inequality, together with the concavity of $x\mapsto x^m$ for $0<m\le1$, implies that $\Delta_{i-1}\le\log\mathcal K_ie^{\Delta_i}$ in both cases. If $\Delta_i\le\log\E_0[e^\Upsilon\mid\text{increments through }i]$, then the tower property gives the same inequality at $i-1$. At the end of the second stage we have $\Delta=\Upsilon$, and induction proves \eqref{eq:st-kernel}. This is the kernel comparison \cite[(4.11), (7.17)]{oai-barriers}; it uses no inverse of the field transformation, and singular increments cause no difficulty.

Using convexity of $\Phi(q,\cdot)$, the bound $\chi\le1$ and the Lipschitz bound on $\chi$, we find by the second-order Taylor formula with integral remainder that
\begin{equation}\label{eq:st-U-bounds}
 0\le\Upsilon\le\delta^2\mathsf Y^2,\qquad\bigl|\Upsilon-\delta^2\chi_q\mathsf Y^2\bigr|\le C\abs{\delta\mathsf Y}^3,\qquad\chi_q=\chi(q,z^x_q).
\end{equation}
Under $\E_0$, the diffusion representation gives $\mathsf Y=\mathsf Y^{\rm G}+\mathsf Y^{\rm B}$, where $\mathsf Y^{\rm G}$ is a centered Gaussian variable of variance $\varsigma^2=\beta^2\bar W$ and $\abs{\mathsf Y^{\rm B}}\le2\beta^2$, since every drift is bounded by $\beta^2$, is integrated over an interval of length $\bar h$, and is then divided by $\bar h$. Since $\bar\theta\le1/2$, we have $\varsigma^2\le2\beta^2/\bar h$, and the hypothesis $8\beta^2\bar W\delta^2\le1$ means $4\delta^2\varsigma^2\le\frac12$. Using $\mathsf Y^2\le2(\mathsf Y^{\rm G})^2+8\beta^4$, $\abs\delta\le1$ and the Gaussian identities $\E e^{a(\mathsf Y^{\rm G})^2}=(1-2a\varsigma^2)^{-1/2}$ and $\E(\mathsf Y^{\rm G})^4e^{a(\mathsf Y^{\rm G})^2}=3\varsigma^4(1-2a\varsigma^2)^{-5/2}$ with $a=2\delta^2$, we find
\[
 \E_0\abs\mathsf Y^3\le C\bar h^{-3/2},\qquad\E_0\bigl[\mathsf Y^4e^{\delta^2\mathsf Y^2}\bigr]\le C\bar h^{-2}.
\]
Using $e^\Upsilon\le1+\Upsilon+\frac12\Upsilon^2e^\Upsilon$ for $\Upsilon\ge0$, $\log(1+z)\le z$, \eqref{eq:st-kernel} and \eqref{eq:st-U-bounds}, we obtain
\begin{equation}\label{eq:st-exponential-error}
 \widetilde\Psi-\Psi_0\le\delta^2\E_0\bigl[\chi_q\mathsf Y^2\bigr]+C\bigl(\abs\delta^3\bar h^{-3/2}+\delta^4\bar h^{-2}\bigr),
\end{equation}
which is \cite[(7.19)]{oai-barriers}.

\paragraph{Step 5: the triangular function on the first branch.} It remains to estimate $\E_0[\chi_q\mathsf Y^2]$. Let $0=s_0<s_1<\dots<s_M=q$ be the partition points in $[0,q]$. Under the Gaussian law of the increments, $z^x$ is a Brownian motion of variance $\beta^2$ per unit time at these points, and under $\E_0$ its law through $q$ has density $e^{\mathfrak d_x}$ with respect to it, where
\[
 \mathfrak d_x=\sum_{j=0}^{M-1}\alpha(s_j)\bigl(\Phi(s_{j+1},z^x_{s_{j+1}})-\Phi(s_j,z^x_{s_j})\bigr)
\]
\cite[(7.21)]{oai-barriers}; the masses here are the scalar masses $\alpha(s_j)$, because the block factorization of \cref{lem:inp-blocks} cancels the division of the masses by the block sizes. Let $\psi_0(t)=(1-\abs{t-r}/\bar h)_+$ be the triangular function supported on $[r-\bar h,r+\bar h]$, and let $\partial_0$ denote the derivative in the direction that shifts every coordinate $z^x_t$ by $\psi_0(t)$. Under the Gaussian law, $\Cov(\mathsf Y_0,z^x_t)=\beta^2\psi_0(t)$, $\Var(\mathsf Y_0)=2\beta^2/\bar h$ and $\partial_0\mathsf Y_0=2/\bar h$. Writing $\E_{\rm G}$ for the Gaussian expectation, finite-dimensional Gaussian integration by parts gives
\begin{equation}\label{eq:st-tent-ibp}
 \E_{\rm G}[\mathsf Y_0f]=\beta^2\E_{\rm G}[\partial_0f],\qquad
 \E_{\rm G}[\mathsf Y_0^2f]=\frac{2\beta^2}{\bar h}\E_{\rm G}f+\beta^4\E_{\rm G}[\partial_0^2f]
\end{equation}
for every smooth function $f$ of the coordinates with moderate growth. Summing by parts in $\mathfrak d_x$ and using $\psi_0(0)=\psi_0(q)=0$, we have
\begin{align*}
 \partial_0\mathfrak d_x&=\sum_{j=1}^{M-1}\bigl(\alpha(s_{j-1})-\alpha(s_j)\bigr)\psi_0(s_j)\,u(s_j,z^x_{s_j}),\\
 \partial_0^2\mathfrak d_x&=\sum_{j=1}^{M-1}\bigl(\alpha(s_{j-1})-\alpha(s_j)\bigr)\psi_0(s_j)^2\,\chi(s_j,z^x_{s_j}).
\end{align*}
Only the points $s_j\in(r-\bar h,r+\bar h)$ contribute. As the partition contains $r-\bar h$ and the step function is nondecreasing, the sum of $\abs{\alpha(s_{j-1})-\alpha(s_j)}$ over these points is at most $\operatorname{osc}_r$. Together with $\abs u\le1$ and $0<\chi\le1$, this implies that $\abs{\partial_0\mathfrak d_x}\le\operatorname{osc}_r$ and $\abs{\partial_0^2\mathfrak d_x}\le\operatorname{osc}_r$ \cite[(7.23)]{oai-barriers}.

Since $\psi_0(q)=0$, we have $\partial_0\chi_q=0$. Applying \eqref{eq:st-tent-ibp} to $f=e^{\mathfrak d_x}\chi_q$, we obtain
\begin{equation}\label{eq:st-Y0-square}
 \E_0[\chi_q\mathsf Y_0^2]=\frac{2\beta^2}{\bar h}\E_0\chi_q+\beta^4\E_0\bigl[\chi_q\bigl((\partial_0\mathfrak d_x)^2+\partial_0^2\mathfrak d_x\bigr)\bigr]
 =\frac{2\beta^2}{\bar h}\E_0\chi_q+\mathcal E_1,\qquad\abs{\mathcal E_1}\le2\beta^4\operatorname{osc}_r.
\end{equation}
With $u_r=u(r,z^x_r)$ and $\chi_r=\chi(r,z^x_r)$, we have $\partial_0u_r=\psi_0(r)\chi_r=\chi_r$. Applying the first identity in \eqref{eq:st-tent-ibp} to $f=e^{\mathfrak d_x}\chi_qu_r$, we find
\begin{equation}\label{eq:st-Y0-linear}
 \E_0[\chi_q\mathsf Y_0u_r]=\beta^2\E_0[\chi_q\chi_r]+\beta^2\E_0[\chi_qu_r\,\partial_0\mathfrak d_x]
 =\beta^2\E_0[\chi_q\chi_r]+\mathcal E_2,\qquad\abs{\mathcal E_2}\le\beta^2\operatorname{osc}_r.
\end{equation}
These are \cite[(7.24)--(7.25)]{oai-barriers}. The integrations by parts are legitimate for every fixed partition, since the scalar values grow at most linearly, their derivatives are bounded, and the density grows at most exponentially in the Gaussian variables.

\paragraph{Step 6: the ramp on the second branch.} We condition on the whole path $z^x$ through $q$. By the conditional independence in Step~1, the continuation of $v$ after $r$ is the scalar diffusion started from $z^x_r$ at time $r$, as in \cref{inp:ob-scalar}(v). Its law on $[r,r+\bar h]$ has a density $e^{\mathfrak d_v}$ of the same form as $\mathfrak d_x$, over the partition points of $[r,r+\bar h]$. Let $\psi_1(t)=\min\{(t-r)_+/\bar h,1\}$ be the ramp, and $\partial_1$ the corresponding derivative in the coordinates $z^v_t$, $t\ge r$. Under the conditional Gaussian law, $\Cov(\mathsf Y_1,z^v_t)=\beta^2\psi_1(t)$, $\Var(\mathsf Y_1)=\beta^2/\bar h$ and $\partial_1\mathsf Y_1=1/\bar h$. Summing by parts, $\partial_1\mathfrak d_v$ consists of the terminal term $\alpha(s_{\rm last})u(r+\bar h,z^v_{r+\bar h})$ and internal terms whose coefficients have total absolute value at most one, so $\abs{\partial_1\mathfrak d_v}\le2$ and $\abs{\partial_1^2\mathfrak d_v}\le2$. By the second identity of \eqref{eq:st-tent-ibp}, in conditional form, we have $\E_0[\mathsf Y_1^2\mid z^x]=\beta^2/\bar h+\beta^4\E_0[(\partial_1\mathfrak d_v)^2+\partial_1^2\mathfrak d_v\mid z^x]$, and since $\chi_q$ is a function of $z^x$ with $0<\chi_q\le1$,
\begin{equation}\label{eq:st-Y1-square}
 \E_0[\chi_q\mathsf Y_1^2]=\frac{\beta^2}{\bar h}\E_0\chi_q+\mathcal E_3,\qquad\abs{\mathcal E_3}\le6\beta^4.
\end{equation}
For the mixed moment we need the conditional mean of $\mathsf Y_1$. The Brownian part of $z^v_{r+\bar h}-z^v_r$ has conditional mean zero, and $u(t,Y^v_t)$ is a martingale for the diffusion started at time $r$, by \cref{inp:ob-scalar}(iv)--(v). With $\bar\alpha=\bar h^{-1}\int_r^{r+\bar h}\alpha(t)\dd t$,
\[
 \E_0[\mathsf Y_1\mid z^x]=\frac{\beta^2}{\bar h}\int_r^{r+\bar h}\alpha(t)\,\E_0\bigl[u(t,Y^v_t)\mid z^x\bigr]\dd t=\beta^2\bar\alpha\,u_r
\]
\cite[(7.27)]{oai-barriers}. The variables $\mathsf Y_0$, $\chi_q$ and $u_r$ are functions of $z^x$. Combining the last display with \eqref{eq:st-Y0-linear} and $\abs{\bar\alpha-\alpha(r)}\le\operatorname{osc}_r$, we find
\begin{equation}\label{eq:st-mixed}
 \E_0[\chi_q\mathsf Y_0\mathsf Y_1]=\beta^2\bar\alpha\,\E_0[\chi_q\mathsf Y_0u_r]=\beta^4\alpha(r)\E_0[\chi_q\chi_r]+\mathcal E_4,\qquad\abs{\mathcal E_4}\le2\beta^4\operatorname{osc}_r.
\end{equation}
Expanding $\mathsf Y^2=(1-\bar\theta)^2\mathsf Y_0^2+2(1-\bar\theta)\bar\theta\mathsf Y_0\mathsf Y_1+\bar\theta^2\mathsf Y_1^2$ and using \eqref{eq:st-Y0-square}, \eqref{eq:st-Y1-square} and \eqref{eq:st-mixed}, we have
\begin{equation}\label{eq:st-quadratic}
 \E_0[\chi_q\mathsf Y^2]\le\beta^2\bar W\E_0\chi_q+2(1-\bar\theta)\bar\theta\beta^4\alpha(r)\E_0[\chi_q\chi_r]+6\beta^4(\operatorname{osc}_r+\bar\theta^2),
\end{equation}
where we used $(1-\bar\theta)^2\cdot2\beta^2/\bar h+\bar\theta^2\beta^2/\bar h=\beta^2\bar W$. This is \cite[(7.20)]{oai-barriers}. The triangular function contributes errors of order $\operatorname{osc}_r$, while the error of order one from the ramp is multiplied by $\bar\theta^2$.

\paragraph{Step 7: assembly and the limit.} By \cref{inp:ob-matrix}, applied to the perturbed path with zero multiplier and endpoint $\widetilde Q_N$, the constrained pressure $p_{3,n}(D)$ is at most the trial value of the perturbed path. The base path has trial value $3\mathcal P_\beta(\alpha)$ for $D(r,q,0)$. Combining this with \eqref{eq:st-mismatch}, \eqref{eq:st-penalty}, \eqref{eq:st-exponential-error} and the lower bound on $\mathcal I$, we obtain
\begin{multline*}
 p_{3,n}(D)\le3\mathcal P_\beta(\alpha)+\delta^2\E_0[\chi_q\mathsf Y^2]+C\bigl(\abs\delta^3\bar h^{-3/2}+\delta^4\bar h^{-2}\bigr)\\
 -\beta^2\delta^2\Bigl(\bar W\int_q^1\alpha(t)\dd t+\alpha(r)(2\bar\theta-\bar\theta^2)-3\operatorname{osc}_r\Bigr)+\beta^2\delta^4\bar W^2.
\end{multline*}
We insert \eqref{eq:st-quadratic} and note that
\begin{multline*}
 2(1-\bar\theta)\bar\theta\beta^4\alpha(r)\E_0[\chi_q\chi_r]-\beta^2\alpha(r)(2\bar\theta-\bar\theta^2)\\
 =2(1-\bar\theta)\bar\theta\beta^2\alpha(r)\bigl(\beta^2\E_0[\chi_q\chi_r]-1\bigr)-\beta^2\alpha(r)\bar\theta^2.
\end{multline*}
We discard the last term, which is nonpositive, and use $\beta^2\delta^4\bar W^2\le4\beta^2\delta^4\bar h^{-2}$. With $\widetilde C=6\beta^4+3\beta^2$, this proves \eqref{eq:st-remote-expansion} for the step approximation, with $3\mathcal P_\beta(\alpha)$ in place of $3P_\beta$ and with $\E_0\chi_q-\int_q^1\alpha(t)\dd t$ in place of $\alpha(q)(q-\Gamma(q))$; this is \cite[(7.29)]{oai-barriers}. The constants do not depend on the partition.

We now keep $n,r,q,\delta,\bar h,\bar\theta$ fixed and let the mesh tend to zero. The left side does not depend on the approximation. By the stability estimate \cref{inp:ob-scalar}(ii), the scalar values and their first two spatial derivatives converge uniformly. Driving the diffusions by the same Brownian motion, Gronwall's inequality and the uniform Lipschitz bound on the drifts show that the paths converge. The expectations $\E_0\chi_q$ and $\E_0[\chi_q\chi_r]$ converge to $\E\chi_q$ and $\E[\chi_q\chi_r]$ for the scalar diffusion $Y$ of the minimizer, $\int_q^1\alpha$ and $\mathcal P_\beta(\alpha)$ converge to their values for the minimizer, the latter being $P_\beta$, and the values at $r$ and $q$ are unchanged. Finally, by \eqref{eq:inp-stationarity}, we have $\E\chi_q-\int_q^1\alpha(t)\dd t=\alpha(q)(q-\Gamma(q))$. This proves \eqref{eq:st-remote-expansion}.
\end{proof}

We take the limit in Step~7 at fixed $\bar h$ and $\bar\theta$; no approximation rate uniform in these parameters is needed. Only the values of $\alpha$ at $r\pm \bar h$ enter; no regularity of $\alpha$ at the distant overlap $q$ is used, and $q$ may be close to one. We now choose the parameters.

\begin{lemma}[Separated overlaps]\label{lem:st-remote}
Let $\mathcal J\subset(0,q_\beta)$ be a compact interval and $d_0>0$. There exist $\nu_{\rm sep},\delta_{\rm sep},c_{\rm sep}>0$, depending only on $\beta$, $\mathcal J$ and $d_0$, such that
\begin{equation}\label{b:eq:remote-pressure-gap}
 p_{3,n}\bigl(D(r,q,\delta)\bigr)\le3P_\beta-c_{\rm sep}\delta^2
\end{equation}
for every $n\ge2$ and every feasible $D(r,q,\delta)$ with $r\in\mathcal J$, $r+d_0\le q\le1$, $\abs{q-\Gamma(q)}\le\nu_{\rm sep}$ and $\abs\delta\le\delta_{\rm sep}$.
\end{lemma}

\begin{proof}
We may assume $d_0\le1$. Let $r_-=\min\mathcal J$ and $r_+=\max\mathcal J$. By \cref{cor:inp-identity}, $\alpha$ is smooth on $[0,q_\beta)$ and in particular Lipschitz, with some constant $L_\alpha$, on the compact interval $\mathcal J'=[r_-/2,(r_++q_\beta)/2]\subset(0,q_\beta)$. Let $c_\chi$ be the constant of \cref{lem:st-susceptibility}, set $c_{\rm mix}=\beta^2\alpha(r_-)c_\chi$, which is positive by \cref{cor:inp-identity}, and let $\widetilde C,\widehat C$ be the constants of \cref{lem:st-remote-expansion}. We choose, in this order,
\begin{align*}
 &\bar\theta\in(0,1/2]\ \text{with}\ \widetilde C\bar\theta\le\tfrac18c_{\rm mix}d_0;\\
 &\bar h\in(0,1]\ \text{with}\ \bar h\le\min\{\tfrac12r_-,\tfrac12(q_\beta-r_+),\tfrac12d_0\}\ \text{and}\ 2\widetilde CL_\alpha\bar h\le\tfrac18c_{\rm mix}\bar\theta d_0;\\
 &\nu_{\rm sep}>0\ \text{with}\ 2\beta^2\nu_{\rm sep}\le\tfrac18c_{\rm mix}\bar\theta d_0\bar h;\\
 &\delta_{\rm sep}\in(0,1]\ \text{with}\ 16\beta^2\delta_{\rm sep}^2\le \bar h\ \text{and}\ \widehat C\bigl(\delta_{\rm sep}\bar h^{-3/2}+\delta_{\rm sep}^2\bar h^{-2}\bigr)\le\tfrac18c_{\rm mix}\bar\theta d_0.
\end{align*}
Let $D=D(r,q,\delta)$ be as in the lemma. Then $\bar h<r$ and $q-r\ge d_0>\bar h$, and $\bar W\le2/\bar h$ because $\bar\theta\le1/2$. We have $8\beta^2\bar W\delta^2\le16\beta^2\delta^2/\bar h\le1$, and \cref{lem:st-remote-expansion} applies. We bound the terms in the braces of \eqref{eq:st-remote-expansion}. Since $\alpha(q)\le1$ and $\abs{q-\Gamma(q)}\le\nu_{\rm sep}$, the first term is at most $2\beta^2\nu_{\rm sep}/\bar h$. By \cref{lem:st-susceptibility}, $2(1-\bar\theta)\ge1$, $\alpha(r)\ge\alpha(r_-)$ and $q-r\ge d_0$, the second term is at most $-\bar\theta\beta^2\alpha(r_-)c_\chi d_0=-c_{\rm mix}\bar\theta d_0$. Moreover $\operatorname{osc}_r\le2L_\alpha \bar h$, because $[r-\bar h,r+\bar h]\subset\mathcal J'$. Altogether,
\begin{multline}\label{b:eq:remote-tent-estimate}
 p_{3,n}(D)\le3P_\beta+\delta^2\Bigl\{\frac{2\beta^2\nu_{\rm sep}}{\bar h}-c_{\rm mix}\bar\theta d_0+\widetilde C\bigl[\alpha(r+\bar h)-\alpha(r-\bar h)+\bar\theta^2\bigr]\Bigr\}\\
 +\widehat C\bigl(\abs\delta^3\bar h^{-3/2}+\delta^4\bar h^{-2}\bigr).
\end{multline}
By the choices above, the first term in the braces, the terms $2\widetilde CL_\alpha\bar h$ and $\widetilde C\bar\theta^2$, and the last term divided by $\delta^2$ are each at most $\frac18c_{\rm mix}\bar\theta d_0$. This proves \eqref{b:eq:remote-pressure-gap} with $c_{\rm sep}=\frac12c_{\rm mix}\bar\theta d_0$.
\end{proof}

We chose the constants in the order $d_0$, then $\bar\theta$, then $\bar h$, then $\nu_{\rm sep}$, then $\delta_{\rm sep}$. The condition $\abs\delta\le\delta_{\rm sep}$ cannot be dropped: \eqref{b:eq:remote-pressure-gap} is a statement about small overlap imbalances. Decreasing $\nu_{\rm sep}$ later only shrinks the domain of \eqref{b:eq:remote-pressure-gap}.

\section{Families of constraints}\label{sec:st-families}

We now fix the data of the simultaneous estimate. All of the following are chosen before $n\to\infty$, in the order listed.
\begin{enumerate}
\item[(F1)] Compact intervals $\mathcal J\subset\operatorname{int}I$ with $I\subset(0,q_\beta)$, and a nonempty finite set $\mathfrak B\subset(0,1)$ of large-overlap cutoffs. In the dynamical argument, $\mathcal J=J_\beta$ and $I=I_\beta$ are the intervals of \eqref{eq:ref-intervals}, and $\mathfrak B=\{q_*,b_1\}$; the fixed-scale exceptional sets of \cref{sec:references} are then subsets of the set $\mathcal W_n$ defined below.
\item[(F2)] The band ratio $\vartheta$ and amplitude $E$ of \cref{sec:fr-local}, the constant $m_\beta$ for $I$, a coefficient $0<g<m_\beta E^2G(\vartheta,E)$, and the constants $d_0,S,\rho_0$ of \cref{lem:fr-local-gap} for $\mathcal J$, $I$, $\vartheta$, $E$ and $g$.
\item[(F3)] The constants $\nu_{\rm sep},\delta_{\rm sep},c_{\rm sep}$ of \cref{lem:st-remote} for $\mathcal J$ and $d_0$, and, for each $b_*\in\mathfrak B$, the constants $\underline b(b_*),\nu_{\rm fix}(b_*),c_{\rm fix}(b_*)$ of \cref{lem:st-fixed}.
\item[(F4)] The tolerance and the two deficits
\[
 \nu=\min\Bigl\{\nu_{\rm sep},\tfrac12c_{\rm tail},\min_{b_*\in\mathfrak B}\nu_{\rm fix}(b_*)\Bigr\},\qquad
 c_{\rm fix}=\min_{b_*\in\mathfrak B}c_{\rm fix}(b_*),\qquad c_{\rm pair}=\tfrac12\nu^2,
\]
with $c_{\rm tail}$ from \cref{lem:st-pair}.
\item[(F5)] The fraction $\varkappa\in(0,1)$ of the local deficit that is retained in the disorder event.
\end{enumerate}
Let $(\rho_n)$ and $(\eta_n)$ be deterministic positive sequences with $\rho_n\to0$, and write $\varepsilon_n=E\rho_n$. We define four deterministic families of feasible overlap matrices, the local family, the separated-overlap family, the fixed-scale family, and the excluded pairs:
\begin{align*}
 \mathcal D^{\rm loc}_n&=\bigl\{D(r,q,\delta):r\in\mathcal J,\ \rho_n\le q-r\le d_0,\ \varepsilon_n\le\abs\delta\le2\varepsilon_n\bigr\},\\
 \mathcal D^{\rm sep}_n&=\bigl\{D(r,q,\delta):r\in\mathcal J,\ q-r>d_0,\ \abs{q-\Gamma(q)}\le\nu,\ \varepsilon_n\le\abs\delta\le2\varepsilon_n\bigr\},\\
 \mathcal D^{\rm fix}_n&=\bigcup_{b_*\in\mathfrak B}\bigl\{D(r,q,\delta):q\ge b_*,\ \abs{r\pm\delta}\le\underline b(b_*),\ \abs{q-\Gamma(q)}\le\nu,\ \abs\delta\ge\eta_n\bigr\},\\
 \mathcal D^{\rm pair}_n&=\bigl\{D_2(\bar q):\bigl|\abs{\bar q}-\Gamma(\abs{\bar q})\bigr|>\nu\bigr\},
\end{align*}
where only feasible matrices are included. There are at most $(n+1)^3$ matrices in $\mathcal D^{\rm loc}_n\cup\mathcal D^{\rm sep}_n$, at most $\abs{\mathfrak B}(n+1)^3$ in $\mathcal D^{\rm fix}_n$, and at most $n+1$ in $\mathcal D^{\rm pair}_n$. Since $\nu<c_{\rm tail}$, the family $\mathcal D^{\rm pair}_n$ contains every feasible $D_2(\bar q)$ with $\abs{\bar q}\ge q_{\rm tail}$. For a family $\mathcal D$ of $\ell\times\ell$ matrices, we write $\{R\in\mathcal D\}$ for the set of $\ell$-tuples whose overlap matrix lies in $\mathcal D$. The corresponding sets of configurations are the narrow triples, the fixed-scale triples and the excluded pairs,
\begin{align*}
 \mathcal N_n&=\bigl\{(v,x,y)\in\Sigma_n^3:r\in\mathcal J,\ q-r\ge\rho_n,\ \varepsilon_n\le\abs\delta\le2\varepsilon_n\bigr\},\\
 \mathcal W_n&=\bigcup_{b_*\in\mathfrak B}\bigl\{(v,x,y)\in\Sigma_n^3:R(x,y)\ge b_*,\ \abs{R(v,x)},\abs{R(v,y)}\le\underline b(b_*),\\
 &\hspace{14em}\abs{R(v,x)-R(v,y)}\ge2\eta_n\bigr\},\\
 \mathcal P_n&=\{R\in\mathcal D^{\rm pair}_n\}=\bigl\{(x,y)\in\Sigma_n^2:\bigl|\abs{R(x,y)}-\Gamma(\abs{R(x,y)})\bigr|>\nu\bigr\},
\end{align*}
with $r,\delta,q$ the coordinates of the triple.

\begin{lemma}[Covering]\label{lem:st-cover}
If $(v,x,y)\in\mathcal N_n$, then $(x,y)\in\mathcal P_n$ or the overlap matrix of $(v,x,y)$ lies in $\mathcal D^{\rm loc}_n\cup\mathcal D^{\rm sep}_n$. If $(v,x,y)\in\mathcal W_n$, then $(x,y)\in\mathcal P_n$ or the overlap matrix of $(v,x,y)$ lies in $\mathcal D^{\rm fix}_n$.
\end{lemma}

\begin{proof}
Let $(v,x,y)\in\mathcal N_n$. Then $q=R(x,y)\ge r+\rho_n>0$, since $r\in\mathcal J\subset(0,1)$. If $\abs{q-\Gamma(q)}>\nu$, then $(x,y)\in\mathcal P_n$. Otherwise $D(r,q,\delta)$ lies in $\mathcal D^{\rm loc}_n$ if $q-r\le d_0$ and in $\mathcal D^{\rm sep}_n$ if $q-r>d_0$. Next let $(v,x,y)\in\mathcal W_n$, for the cutoff $b_*\in\mathfrak B$. Then $q=R(x,y)\ge b_*>0$, $\abs{r\pm\delta}\le\underline b(b_*)$ and $\abs\delta\ge\eta_n$, so either $(x,y)\in\mathcal P_n$ or $D(r,q,\delta)\in\mathcal D^{\rm fix}_n$.
\end{proof}

By \cref{lem:st-cover}, every narrow triple is covered by the local regime, the regime of separated overlaps, or the pair regime. The local regime uses the cubic deficit of \cref{sec:cubic} through \cref{lem:fr-local-gap}; the other two use the quadratic deficits of \cref{lem:st-remote,lem:st-pair}. No fractional moment bound at a distant overlap $q$ is needed, and the construction of \cref{sec:cubic} is used only for $q-r\le d_0$.

\section{The simultaneous event}\label{sec:st-event}

The conversion of constrained-pressure bounds to Gibbs probabilities is
treated in \cite[Lemma~4.2]{oai-barriers}, where the logarithm of the Gibbs
probability is concentrated about its mean. Here we bound the constrained
numerators and the full partition-function denominator separately.
For the local numerators we use the positive fractional-moment mechanism of
\cite[Section~2.3]{quantitative-parisi}, through \cref{lem:fr-local-gap}. This
controls their deviations at the required shrinking scale. The simultaneous event
below combines this with a variance bound for the denominator and retains
separate disorder and conditional Gibbs errors at the shrinking scales.

We impose the following conditions on the scales. Recall that $\Delta_{n,\beta}=P_\beta-p_n(\beta)$.
\begin{align}
 &\limsup_{n\to\infty}\frac{\Delta_{n,\beta}}{\rho_n^3}<\frac{\varkappa g}3,\tag{S1}\label{eq:st-S1}\\
 &\lim_{n\to\infty}\frac{n^{7/15}}{n^2\rho_n^6}=0,\tag{S2}\label{eq:st-S2}\\
 &\lim_{n\to\infty}\frac{n\rho_n^4}{\log n}=\infty,\tag{S3}\label{eq:st-S3}\\
 &\lim_{n\to\infty}\frac{n\eta_n^4}{\log n}=\infty,\tag{S4}\label{eq:st-S4}\\
 &\lim_{n\to\infty}\frac{\rho_n^3}{\eta_n^2}=0.\tag{S5}\label{eq:st-S5}
\end{align}
Condition \eqref{eq:st-S1} says that the mean bias is absorbed by the retained part of the cubic deficit, with a strict constant margin. Given the variance bound $\Var F_n\le C_vn^{7/15}$ of \cref{inp:qp-rate}, \eqref{eq:st-S2} makes the Chebyshev bound for the denominator tend to zero. Conditions \eqref{eq:st-S3} and \eqref{eq:st-S4} make the numerator bounds summable over polynomially many constraints, and \eqref{eq:st-S5} makes the loss in the denominator negligible against the fixed-scale deficit. By \eqref{eq:st-S1}, we may fix $a_0$ and $c_0$ with
\begin{equation}\label{b:eq:denominator-margin}
 \limsup_{n\to\infty}\frac{\Delta_{n,\beta}}{\rho_n^3}<a_0<c_0<\frac{\varkappa g}3,
\end{equation}
for instance $c_0=\frac12(a_0+\varkappa g/3)$.

We define $\mathcal A_n$ as the set of disorders $J$ for which all of the following hold:
\begin{enumerate}
\item[(A1)] $F_n\ge n(P_\beta-c_0\rho_n^3)$;
\item[(A2)] $n^{-1}\log Z_{3,n}(D)\le3P_\beta-\varkappa g\rho_n^3$ for every $D\in\mathcal D^{\rm loc}_n\cup\mathcal D^{\rm sep}_n$;
\item[(A3)] $n^{-1}\log Z_{3,n}(D)\le3P_\beta-\frac12c_{\rm fix}\eta_n^2$ for every $D\in\mathcal D^{\rm fix}_n$;
\item[(A4)] $n^{-1}\log Z_{2,n}(D)\le2P_\beta-\frac12c_{\rm pair}$ for every $D\in\mathcal D^{\rm pair}_n$.
\end{enumerate}
The event $\mathcal A_n$ is measurable, since all constrained partition functions are finite sums of continuous functions of the disorder.

\begin{lemma}[Simultaneous static event]\label{b:lem:static-event}
Fix the data \textup{(F1)--(F5)}, and let $(\rho_n)$, $(\eta_n)$ satisfy \eqref{eq:st-S1}--\eqref{eq:st-S5}. There exist $C,c>0$ and $n_0$, depending only on $\beta$, on the data \textup{(F1)--(F5)} and on the sequences, such that for all $n\ge n_0$,
\begin{equation}\label{b:eq:static-event-failure}
 \PP(\mathcal A_n^c)\le C\frac{n^{7/15}}{n^2\rho_n^6}+C(n+1)^3\bigl(e^{-cn\rho_n^4}+e^{-cn\eta_n^4}\bigr)+C(n+1)e^{-cn},
\end{equation}
and the right side tends to zero. Moreover, for all $n\ge n_0$ and every disorder $J\in\mathcal A_n$,
\begin{equation}\label{b:eq:conditional-static-errors}
\begin{gathered}
 (\pi^J)^{\otimes3}\bigl(\{R\in\mathcal D^{\rm loc}_n\cup\mathcal D^{\rm sep}_n\}\bigr)\le C(n+1)^3e^{-cn\rho_n^3},\qquad
 (\pi^J)^{\otimes3}(\mathcal N_n)\le C(n+1)^3e^{-cn\rho_n^3},\\
 (\pi^J)^{\otimes3}\bigl(\{R\in\mathcal D^{\rm fix}_n\}\bigr)\le C(n+1)^3e^{-cn\eta_n^2},\qquad
 (\pi^J)^{\otimes3}(\mathcal W_n)\le C(n+1)^3e^{-cn\eta_n^2},\\
 (\pi^J)^{\otimes2}(\mathcal P_n)\le C(n+1)e^{-cn}.
\end{gathered}
\end{equation}
The same bounds hold for the images of these sets under every element of $\mathfrak O_3$, respectively $\mathfrak O_2$, and the set of triples $(v,x,y)$ with $(x,y)\in\mathcal P_n$ has $(\pi^J)^{\otimes3}$-probability $(\pi^J)^{\otimes2}(\mathcal P_n)$.
\end{lemma}

\begin{proof}
We bound the probability that each of (A1)--(A4) fails, take a union bound over the deterministic families, and then divide the bounds on the numerators by the bound on the denominator, on $\mathcal A_n$. Let $n_0$ be so large that, for $n\ge n_0$, we have $\Delta_{n,\beta}\le a_0\rho_n^3$, $\Var F_n\le C_vn^{7/15}$, $\rho_n\le\min\{\rho_0,1\}$, $2\varepsilon_n\le\delta_{\rm sep}$, $\varkappa g\rho_n\le\frac12c_{\rm sep}E^2$, $12c_0\rho_n^3\le c_{\rm fix}\eta_n^2$ and $8c_0\rho_n^3\le c_{\rm pair}$. This is possible by \eqref{b:eq:denominator-margin}, \cref{inp:qp-rate}, $\rho_n\to0$ and \eqref{eq:st-S5}.

\paragraph{Step 1: the denominator.} Since $\E F_n=n(P_\beta-\Delta_{n,\beta})\ge n(P_\beta-a_0\rho_n^3)$, the failure of (A1) implies $F_n-\E F_n<-n(c_0-a_0)\rho_n^3$. By Chebyshev's inequality,
\[
 \PP\bigl(F_n<n(P_\beta-c_0\rho_n^3)\bigr)\le\frac{C_vn^{7/15}}{(c_0-a_0)^2n^2\rho_n^6}.
\]

\paragraph{Step 2: local numerators.} Let $D\in\mathcal D^{\rm loc}_n$ and $s=S\rho_n$. By Markov's inequality applied to $Z_{3,n}(D)^s$ and by \eqref{b:eq:local-fractional-gap},
\[
 \PP\bigl(n^{-1}\log Z_{3,n}(D)>3P_\beta-\varkappa g\rho_n^3\bigr)
 \le\E Z_{3,n}(D)^s\,e^{-ns(3P_\beta-\varkappa g\rho_n^3)}
 \le e^{-(1-\varkappa)gS\,n\rho_n^4}.
\]
This is the only place where a fractional moment enters. A Gaussian concentration bound at a mean deficit of order $\rho_n^3$ would give the exponent $n\rho_n^6$, which is $R^6n^{-2/5}$ at the scale $\rho_n=Rn^{-7/30}$ and tends to zero.

\paragraph{Step 3: the other numerators.} By \cref{rem:inp-lipschitz}, $\log Z_{\ell,n}(D)$ is Lipschitz in $(J_{ij})_{i<j}$ with constant $\beta\ell\sqrt{(n-1)/2}$. By the Gaussian concentration inequality \eqref{eq:inp-concentration} of \cref{inp:ob-concentration}, for every $t>0$,
\begin{equation}\label{eq:st-concentration}
 \PP\bigl(\log Z_{\ell,n}(D)-\E\log Z_{\ell,n}(D)\ge nt\bigr)\le2\exp\Bigl(-\frac{n^2t^2}{\beta^2\ell^2(n-1)}\Bigr)\le2\exp\Bigl(-\frac{nt^2}{\beta^2\ell^2}\Bigr).
\end{equation}
For $D\in\mathcal D^{\rm sep}_n$, we have $\varepsilon_n\le\abs\delta\le2\varepsilon_n\le\delta_{\rm sep}$, and \cref{lem:st-remote} implies that $p_{3,n}(D)\le3P_\beta-c_{\rm sep}E^2\rho_n^2$. Since $\varkappa g\rho_n^3\le\frac12c_{\rm sep}E^2\rho_n^2$, the threshold in (A2) exceeds the mean $p_{3,n}(D)$ by at least $\frac12c_{\rm sep}E^2\rho_n^2$. For $D\in\mathcal D^{\rm fix}_n$, we see from \cref{lem:st-fixed} and $\abs\delta\ge\eta_n$ that $p_{3,n}(D)\le3P_\beta-c_{\rm fix}\eta_n^2$, and the threshold in (A3) exceeds the mean by at least $\frac12c_{\rm fix}\eta_n^2$. For $D\in\mathcal D^{\rm pair}_n$, we have $p_{2,n}(D)<2P_\beta-\frac12\nu^2=2P_\beta-c_{\rm pair}$ by \cref{lem:st-pair}, and the threshold in (A4) exceeds the mean by at least $\frac12c_{\rm pair}$. Applying \eqref{eq:st-concentration} with $\ell=3,3,2$ and these values of $t$, we obtain, for a single matrix of the respective family, the failure probabilities
\[
 2e^{-c_{\rm sep}^2E^4n\rho_n^4/(36\beta^2)},\qquad
 2e^{-c_{\rm fix}^2n\eta_n^4/(36\beta^2)},\qquad
 2e^{-c_{\rm pair}^2n/(16\beta^2)}.
\]

\paragraph{Step 4: the union bound.} Recall that the families are deterministic and that their sizes were bounded in \cref{sec:st-families}. Summing the bounds of Steps~1--3 over them, we have
\begin{multline*}
 \PP(\mathcal A_n^c)\le\frac{C_v}{(c_0-a_0)^2}\frac{n^{7/15}}{n^2\rho_n^6}
 +(n+1)^3\Bigl(e^{-(1-\varkappa)gS n\rho_n^4}+2e^{-c_{\rm sep}^2E^4n\rho_n^4/(36\beta^2)}\Bigr)\\
 +2\abs{\mathfrak B}(n+1)^3e^{-c_{\rm fix}^2n\eta_n^4/(36\beta^2)}+2(n+1)e^{-c_{\rm pair}^2n/(16\beta^2)},
\end{multline*}
which is \eqref{b:eq:static-event-failure}. The first term tends to zero by \eqref{eq:st-S2}, and the others by \eqref{eq:st-S3} and \eqref{eq:st-S4}, since $(n+1)^3e^{-cn\rho_n^4}=\exp\{3\log(n+1)-cn\rho_n^4\}$, and similarly for $\eta_n$.

\paragraph{Step 5: Gibbs probabilities on $\mathcal A_n$.} Fix $J\in\mathcal A_n$. By (A1), $Z_n^\ell\ge\exp\{\ell n(P_\beta-c_0\rho_n^3)\}$. The ratio $Z_{\ell,n}(D)/Z_n^\ell$ is the Gibbs probability of $\{R=D\}$. For $D\in\mathcal D^{\rm loc}_n\cup\mathcal D^{\rm sep}_n$, (A2) gives
\[
 \frac{Z_{3,n}(D)}{Z_n^3}\le e^{-n(\varkappa g-3c_0)\rho_n^3},
\]
and the coefficient $\varkappa g-3c_0$ is positive by \eqref{b:eq:denominator-margin}. For $D\in\mathcal D^{\rm fix}_n$ and $D\in\mathcal D^{\rm pair}_n$, (A3), (A4) and the choice of $n_0$ give, respectively,
\[
 \frac{Z_{3,n}(D)}{Z_n^3}\le e^{-n(c_{\rm fix}\eta_n^2/2-3c_0\rho_n^3)}\le e^{-c_{\rm fix}n\eta_n^2/4},\qquad
 \frac{Z_{2,n}(D)}{Z_n^2}\le e^{-n(c_{\rm pair}/2-2c_0\rho_n^3)}\le e^{-c_{\rm pair}n/4}.
\]
Summing over the families gives the first three bounds of \eqref{b:eq:conditional-static-errors}, with $c=\min\{\varkappa g-3c_0,\frac14c_{\rm fix},\frac14c_{\rm pair}\}$ and $C=\abs{\mathfrak B}$. An event that depends only on $(x,y)$ has the same probability under $(\pi^J)^{\otimes3}$ as under $(\pi^J)^{\otimes2}$, which is the last assertion of the lemma. By \cref{lem:st-cover}, $\mathcal N_n$ is contained in the union of $\{R\in\mathcal D^{\rm loc}_n\cup\mathcal D^{\rm sep}_n\}$ and the set of triples with $(x,y)\in\mathcal P_n$, so
\[
 (\pi^J)^{\otimes3}(\mathcal N_n)\le(n+1)^3e^{-n(\varkappa g-3c_0)\rho_n^3}+(n+1)e^{-c_{\rm pair}n/4},
\]
and similarly $(\pi^J)^{\otimes3}(\mathcal W_n)\le\abs{\mathfrak B}(n+1)^3e^{-c_{\rm fix}n\eta_n^2/4}+(n+1)e^{-c_{\rm pair}n/4}$. We may assume $\eta_n\le1$, since otherwise $\mathcal D^{\rm fix}_n$ and $\mathcal W_n$ are empty. As also $\rho_n\le1$, the term $e^{-c_{\rm pair}n/4}$ is at most $e^{-c_{\rm pair}n\rho_n^3/4}$ and at most $e^{-c_{\rm pair}n\eta_n^2/4}$. We obtain the last two bounds of \eqref{b:eq:conditional-static-errors} with $C=\abs{\mathfrak B}+1$. The statements for the images under $\mathfrak O_3$ and $\mathfrak O_2$ follow from \eqref{eq:st-orientation}.
\end{proof}

\begin{table}[ht]
\centering
\small
\begin{tabular}{@{}llcc@{}}
\toprule
Estimate & Bound & Kind & \cref{thm:barrier-730}\\
\midrule
(A1), denominator & $n^{7/15}/(n^2\rho_n^6)$ & disorder & $n^{-2/15}$\\
(A2), local numerators & $(n+1)^3e^{-cn\rho_n^4}$ & disorder & $e^{-cn^{1/15}}$\\
(A2), separated overlaps & $(n+1)^3e^{-cn\rho_n^4}$ & disorder & $e^{-cn^{1/15}}$\\
(A3), fixed-scale numerators & $(n+1)^3e^{-cn\eta_n^4}$ & disorder & $e^{-cn^{1/21}}$\\
(A4), pair numerators & $(n+1)e^{-cn}$ & disorder & $e^{-cn}$\\
\midrule
narrow triples & $(n+1)^3e^{-cn\rho_n^3}$ & Gibbs, on $\mathcal A_n$ & $e^{-cn^{3/10}}$\\
fixed-scale triples & $(n+1)^3e^{-cn\eta_n^2}$ & Gibbs, on $\mathcal A_n$ & $e^{-cn^{11/21}}$\\
excluded pairs & $(n+1)e^{-cn}$ & Gibbs, on $\mathcal A_n$ & $e^{-cn}$\\
\bottomrule
\end{tabular}
\caption{The error terms of \cref{b:lem:static-event}, up to constant factors, at the scales $\rho_n=Rn^{-7/30}$ and $\eta_n=n^{-5/21}$ of \cref{thm:barrier-730}. The first five rows bound the disorder probability $\PP(\mathcal A_n^c)$; they are paid once. The last three rows are Gibbs probabilities at a fixed disorder in $\mathcal A_n$; only these are multiplied by counts in \cref{sec:references,sec:induction,sec:barrier-proof}. The last column omits polynomial factors.}\label{tab:st-errors}
\end{table}

\begin{remark}[The scales of the barrier theorem]\label{rem:st-scales}
Fix the data (F1)--(F5), and write $C_b$ for the constant of the upper bound on the mean bias that is used.
\begin{enumerate}
\item[(a)] By \cref{inp:hb-rate}, $0\le\Delta_{n,\beta}\le C_bn^{-7/10}$ for $n\ge2$. We choose
\begin{equation}\label{b:eq:critical-static-scales}
 \rho_n=Rn^{-7/30},\qquad\varepsilon_n=E\rho_n,\qquad\eta_n=n^{-5/21},\qquad s_n=S\rho_n,\qquad R^3>\frac{3C_b}{\varkappa g}.
\end{equation}
Then $\Delta_{n,\beta}/\rho_n^3\le C_b/R^3<\varkappa g/3$, which is \eqref{eq:st-S1}. Moreover $n^{7/15}/(n^2\rho_n^6)=R^{-6}n^{-2/15}$, $n\rho_n^4=R^4n^{1/15}$, $n\eta_n^4=n^{1/21}$ and $\rho_n^3/\eta_n^2=R^3n^{-47/210}$, which proves \eqref{eq:st-S2}--\eqref{eq:st-S5}. The Gibbs probabilities decay on the scales $n\rho_n^3=R^3n^{3/10}$ and $n\eta_n^2=n^{11/21}$.
\item[(b)] More generally, let $\rho_n=Rn^{-\lambda}$ and $\eta_n=n^{-\xi}$, where the scale exponents $\lambda$ and $\xi$ satisfy \eqref{eq:ref-exponents}. Suppose the mean-bias bound \eqref{eq:ref-mean-input} and $R^3>3C_b/(\varkappa g)$. Then $\Delta_{n,\beta}/\rho_n^3\le C_b/R^3<\varkappa g/3$ for large $n$, which is \eqref{eq:st-S1}. We have \eqref{eq:st-S2} because $n^{7/15}/(n^2\rho_n^6)=R^{-6}n^{7/15-2+6\lambda}$ and $6\lambda<3/2<23/15$. We have \eqref{eq:st-S3} and \eqref{eq:st-S4} because $1-4\lambda>0$ and $1-4\xi>0$, and \eqref{eq:st-S5} holds because $\rho_n^3/\eta_n^2=R^3n^{2\xi-3\lambda}$. Part (a) is the case $(\lambda,\xi)=(7/30,5/21)$, with \cref{inp:hb-rate} as the bound \eqref{eq:ref-mean-input}.
\end{enumerate}
The dynamical argument needs in addition the conditions \eqref{eq:ind-scales}: $\eta_n/\rho_n\to E'$ for a fixed $E'\in[0,E/10]$, and $n^{-1}=o(\varepsilon_n)$. In part (b) we have $E'=0$, since $\eta_n/\varepsilon_n=(ER)^{-1}n^{\lambda-\xi}$ and $\lambda<\xi$; in part (a), this ratio is $(ER)^{-1}n^{-1/210}$. Also $1/(n\varepsilon_n)=(ER)^{-1}n^{\lambda-1}\to0$. The conditions \eqref{eq:st-S1}--\eqref{eq:st-S3} do not involve $\eta_n$. For $\eta_n=\rho_n/1000$, as in \cref{sec:bp-strong}, where $E'=1/1000=E/10$ because $E=1/100$, conditions \eqref{eq:st-S4} and \eqref{eq:st-S5} follow from \eqref{eq:st-S3} and $\rho_n\to0$, since $n\eta_n^4=10^{-12}n\rho_n^4$ and $\rho_n^3/\eta_n^2=10^6\rho_n$. The variance input $n^{7/15}$ of \cref{inp:qp-rate} and the profile cost $s^5$ of \cref{lem:fr-floor} do not depend on the choice of the scales, and $s_n^5/\rho_n^3=S^5R^2n^{-2\lambda}$. Within this argument the exponent $\lambda$ is limited both by one third of the exponent in the upper bound on the mean bias and by $1/4$, the latter through \eqref{eq:st-S3}.
\end{remark}

\begin{remark}[How the estimate is used]\label{rem:st-two-levels}
The later sections use \cref{b:lem:static-event} in the following form. If a bad event at a fixed disorder is a union of at most $K_n$ events, each contained in one of the sets of \eqref{b:eq:conditional-static-errors}, in one of their images under $\mathfrak O_3$, or in a lift of $\mathcal P_n$ or of one of its images to more configurations, then its probability under the joint law of the disorder and the Gibbs samples is at most
\[
 \PP(\mathcal A_n^c)+K_n\sup_{J\in\mathcal A_n}\bigl\{(\pi^J)^{\otimes3}(\mathcal N_n)+(\pi^J)^{\otimes3}(\mathcal W_n)+(\pi^J)^{\otimes2}(\mathcal P_n)\bigr\}
\]
up to a fixed factor for the orientations. The count $K_n$ multiplies only the Gibbs probabilities. For example, with $K_n=e^{CL_n}$ and $L_n=An^{7/30}$, we have $K_n(n+1)^3e^{-cR^3n^{3/10}}\to0$ for every fixed $A$ and $C$. No such count may be applied to the disorder probability $\PP(\mathcal A_n^c)$, for which this section gives only the bound $O(n^{-2/15})$ at the scales of \cref{rem:st-scales}(a). In \cref{sec:references}, $\mathcal A_n$ is intersected with the disorder event of \cref{b:lem:coverage-extension}; the intersection has probability tending to one by a union bound, and no independence between the two events is needed.
\end{remark}
\part{Dynamics: proof of the lower bound}\label{part:dynamics}
\chapter{Overlap neighborhoods and reference configurations}\label{sec:references}

This section collects the probabilistic input of the dynamical argument in
\cref{sec:induction,sec:barrier-proof}. It has three parts. In
\cref{sec:ref-neighborhoods}, we prove \cref{b:lem:coverage-extension}, the
overlap-neighborhood mass bound. It states that outside an event of
vanishing disorder probability, for every set $S\subseteq\Sigma_n$, every
$x\in S$ outside a set of Gibbs mass less than $e^{-D_1L_n}$ has the property
that the configurations $z\in S$ with $|R(x,z)|\ge q_*$ carry Gibbs mass at
least $e^{-D_3L_n}$;
here $D_1>0$ is arbitrary, and $D_3$ depends only on $\beta$ and $D_1$.
This is the argument of \cite[Proposition~6.1]{oai-barriers}, carried out at
every diverging sublinear scale $L_n$. In \cref{sec:ref-parameters}, we fix the
parameters of the dynamical argument and the order in which they are chosen
(\cref{tab:ref-parameters}). We then define the set $\mathcal B_n$ of
exceptional triples as an explicit union and deduce from
\cref{b:lem:static-event} that $\pi^{\otimes3}(\mathcal B_n)$ is small on
one disorder event (\cref{lem:ref-single-event}). In
\cref{sec:ref-families}, we combine these two inputs. At each disorder in an
event of probability tending to one, we fix one family of Gibbs reference
configurations and one set $V$ of small Gibbs mass such that the four
properties used in \cref{sec:induction} hold along every path that avoids
$V$ (\cref{lem:ref-families}).

The section uses \cref{b:lem:static-event} and the pressure estimate
\eqref{b:eq:fixed-pressure-gap} from \cref{sec:static}, the scalar
constant of \eqref{b:eq:main-analytic-constants} from \cref{sec:scalar},
the band of \eqref{b:eq:os-band-checks} from \cref{sec:cubic}, and the
imported results \cref{inp:ob-energy-tail,inp:ob-rotation,inp:ob-concentration}.
\Cref{lem:ref-families} is the only point at which the static
estimates of \cref{part:static} enter the induction of \cref{sec:induction};
its bound on $\pi(V)$ enters the stationary sign test of
\cref{sec:barrier-proof}. Although \cref{thm:barrier-730} uses only the scale
exponent $7/30$, we state the construction for general exponents, because this
shows which conditions on the scales each step uses
(see \cref{tab:bp-exponents}).

Throughout this section, $\beta>1$ is fixed, and we write $\pi=\pi^J$. For
a vector $\bar J=(\bar J_{ij})_{1\le i<j\le n}$ of real numbers, we write
$H_n^{\bar J}$ and $Z_n^{\bar J}$ for the Hamiltonian and the partition
function in which $J$ is replaced by $\bar J$. For $k\ge1$ and a
set $\mathcal U\subseteq\Sigma_n^k$, $\pi^{\otimes k}(\mathcal U)$ is the
probability, at fixed disorder, that $k$ independent samples from $\pi$ form
a $k$-tuple in $\mathcal U$; we also write
$\pi^{\otimes2}(|R|\ge q)$ for
$\pi^{\otimes2}(\{(x,y):|R(x,y)|\ge q\})$. The Gibbs measure is invariant
under the spin flip $x\mapsto-x$, because $H_n^J(-x)=H_n^J(x)$, and $\pi^{\otimes k}$
is invariant under independently reversing whole replicas and under permutations of
the replicas. All constants depend on $\beta$ and on the fixed parameters
listed in \cref{tab:ref-parameters}, and never on $n$ or on the disorder.

\section{Overlap neighborhoods at sublinear scales}\label{sec:ref-neighborhoods}

The bound below was proved in \cite[Proposition~6.1]{oai-barriers} at the
scale $L_n=n^{1/10000}$ used there; this is \cref{inp:ob-coverage}. Its proof
uses the scale only through the conditions $L_n\to\infty$ and $L_n=o(n)$. Our
time scale is different, and the constants of the bound enter the
construction of \cref{sec:ref-families}. For these reasons we state the
extension and write out the proof. The ingredients are those of
\cite{oai-barriers}: the upper
tail of the energy of a Gibbs sample (\cref{inp:ob-energy-tail}, whose proof
in \cite[Lemma~5.1]{oai-barriers} uses the Brascamp--Lieb inequality
\cite{brascamp-lieb-1976} in the form of \cite[Lemma~2.1]{oai-barriers}), the
disorder rotation estimate (\cref{inp:ob-rotation};
\cite[Lemma~5.2]{oai-barriers}), Gaussian concentration
for Lipschitz functions (\cref{inp:ob-concentration};
\cite[Lemma~2.1]{oai-barriers}), and the
Paley--Zygmund inequality.

\begin{lemma}[Overlap-neighborhood mass bound at sublinear scales]
\label{b:lem:coverage-extension}
There exists $q_*=q_*(\beta)\in(0,1)$ with the following property. Let
$(L_n)$ be a deterministic sequence of positive numbers with $L_n\to\infty$
and $L_n/n\to0$, and let $D_1>0$. Then there exist $D_3>0$ and $D_4>0$, depending only
on $\beta$ and $D_1$, and disorder events $\mathcal C_n(D_1)$ such that
$\PP(\mathcal C_n(D_1)^c)\le3e^{-D_4L_n}$ for all sufficiently large $n$,
and such that on $\mathcal C_n(D_1)$,
\begin{equation}\label{b:eq:general-coverage}
 \pi\bigl(\bigl\{x\in S:
   \pi(\{z\in S:|R(x,z)|\ge q_*\})<e^{-D_3L_n}\bigr\}\bigr)
 <e^{-D_1L_n}
 \qquad\text{for every }S\subseteq\Sigma_n .
\end{equation}
\end{lemma}

Both masses in \eqref{b:eq:general-coverage} are masses under $\pi$ itself,
not under $\pi$ conditioned on $S$. The inner mass is the probability that a
single Gibbs sample lands in the overlap neighborhood
$\{z\in S:|R(x,z)|\ge q_*\}$ of $x$ in $S$; in \cref{sec:ref-families} it is
the success probability of one reference sample. The overlap $q_*$ is chosen
before $D_1$ and before the sequence $(L_n)$, and $D_3$ and $D_4$ do not
depend on the sequence either. Since $L_n\to\infty$, the lemma implies that
$\PP(\mathcal C_n(D_1))\to1$. The event $\mathcal C_n(D_1)$ and the threshold beyond
which the estimates below hold depend on $(L_n)$; the lemma does not provide
one event valid for all sequences at once. For $L_n=n^{1/10000}$,
\cref{b:lem:coverage-extension} contains \cref{inp:ob-coverage}.

\begin{proof}
The proof has three steps. In the first, we show that outside an event of
vanishing probability, every set of Gibbs mass at least $e^{-D_1L_n}$
contains a fraction at least $e^{-D_2L_n}$ of pairs with overlap at least
$q_*$ in absolute value. The idea, from \cite[Section~6]{oai-barriers}, is
that if a set has few such pairs, then a second moment computation shows
that a small rotation of the disorder increases the partition function
restricted to this set by nearly its full Gaussian factor. In the second
step, we show that this gain outweighs the loss caused by shrinking the
original disorder, which the upper energy tail controls, and that the
rotation estimate makes the resulting increase of $\log Z_n^J$ unlikely. In the
third step, we apply the pair bound to the set of exceptional targets
itself.

\paragraph{Constants.}
By \cref{inp:ob-energy-tail}, there exist $e_1\in(0,\beta/2)$ and $c_E>0$
such that the event
\[
 \mathcal E_n=\bigl\{\pi(\{x:H_n^J(x)>ne_1\})\le e^{-c_En}\bigr\}
\]
satisfies $\PP(\mathcal E_n^c)\le e^{-c_En}$ for all sufficiently large $n$.
We define
\[
 g_0=\frac{\beta^2}4-\frac{\beta e_1}2,
\]
which is positive because $e_1<\beta/2$. We fix $q_*\in(0,1)$ with
$\beta^2q_*<g_0/8$. These choices depend only on $\beta$. Given $D_1>0$, we
fix constants $\Lambda$, $D_2$, and $D_3$ such that
\begin{equation}\label{eq:ref-cov-constants}
 \Lambda>\frac{8(D_1+1)}{g_0},\qquad
 D_2>\frac{\beta^2\Lambda}2+1,\qquad
 D_3>D_1+D_2 .
\end{equation}
They depend only on $\beta$ and $D_1$. Now let $(L_n)$ be as in the
statement, write $L=L_n$, and let
\[
 \omega=\omega_n=\frac{\Lambda L_n}n .
\]
Then $\omega_n\to0$ and $n\omega_n=\Lambda L_n\to\infty$. We have
$0<\omega<1/2$ for all sufficiently large $n$; we assume this from now
on.

\paragraph{Step 1: pairs with large overlap in every large set.}
Let $\mathcal V_n$ be the event that some $S\subseteq\Sigma_n$ satisfies
\begin{equation}\label{eq:ref-pair-violation}
 \pi(S)\ge e^{-D_1L}\qquad\text{and}\qquad
 \pi(\cdot\mid S)^{\otimes2}(|R|\ge q_*)<e^{-D_2L}.
\end{equation}
This event is measurable, since it is a finite union of events, each
defined by finitely many continuous functions of the disorder. Off
$\mathcal V_n$, we have
\begin{equation}\label{b:eq:general-coverage-pairs}
 \pi(\cdot\mid S)^{\otimes2}(|R|\ge q_*)\ge e^{-D_2L}
 \qquad\text{for every }S\subseteq\Sigma_n\text{ with }
 \pi(S)\ge e^{-D_1L}.
\end{equation}
We will show that $\PP(\mathcal V_n)\to0$.

Fix $J\in\mathcal E_n\cap\mathcal V_n$ and a set $S$ satisfying
\eqref{eq:ref-pair-violation}. We first remove the configurations of high
energy from $S$. Let
\[
 S'=S\cap\{x:H_n^J(x)\le ne_1\}.
\]
For large $n$, we have $e^{-c_En}\le e^{-D_1L}/2\le\pi(S)/2$, because
$D_1L=o(n)$. This implies that
\[
 \pi(S')\ge\pi(S)-\pi(\{x:H_n^J(x)>ne_1\})\ge\pi(S)/2 .
\]
Restricting the pairs to $S'\times S'$ and dividing by $\pi(S')^2$ instead of
$\pi(S)^2$, we find
\begin{equation}\label{eq:ref-trimmed}
 \pi(\cdot\mid S')^{\otimes2}(|R|\ge q_*)
 \le\Bigl(\frac{\pi(S)}{\pi(S')}\Bigr)^2
   \pi(\cdot\mid S)^{\otimes2}(|R|\ge q_*)
 <4e^{-D_2L}.
\end{equation}

Let $\widetilde J=(\widetilde J_{ij})_{1\le i<j\le n}$ be a standard Gaussian vector independent of
$J$, and define
\[
 W=\sum_{x\in S'}\pi(x\mid S')\,e^{\beta\sqrt\omega H_n^{\widetilde J}(x)},\qquad
 J'=\sqrt{1-\omega}\,J+\sqrt\omega\,\widetilde J .
\]
We write $\PP_{\widetilde J}$ and $\E_{\widetilde J}$ for probability and expectation over $\widetilde J$ with
$J$ fixed. Recall from \eqref{eq:common-covariance} that
$\E H_n^{\widetilde J}(x)H_n^{\widetilde J}(y)=(nR(x,y)^2-1)/2$. Using $\E e^X=e^{\E X^2/2}$ for a
centered Gaussian variable $X$, we find
\[
 \E_{\widetilde J}W=e^{\beta^2\omega(n-1)/4},\qquad
 \frac{\E_{\widetilde J}W^2}{(\E_{\widetilde J}W)^2}
 =\sum_{x,y\in S'}\pi(x\mid S')\,\pi(y\mid S')\exp\Bigl\{\frac{\beta^2\omega}2
     \bigl(nR(x,y)^2-1\bigr)\Bigr\}.
\]
We split the last sum according to whether $|R(x,y)|<q_*$. On the
first part, the exponent is at most
\[
 \Theta=\frac{\beta^2n\omega q_*^2}2 .
\]
On the second part, the exponent is at most $\beta^2n\omega/2=\beta^2\Lambda L/2$,
and we use \eqref{eq:ref-trimmed}. Since $D_2>\beta^2\Lambda/2+1$, this
gives
\[
 \frac{\E_{\widetilde J}W^2}{(\E_{\widetilde J}W)^2}
 \le e^{\Theta}+4e^{(\beta^2\Lambda/2-D_2)L}
 \le e^{\Theta}+4e^{-L}
 \le 2e^{\Theta}
\]
once $L\ge\log4$. The Paley--Zygmund inequality
$\PP(W\ge\E W/2)\ge(\E W)^2/(4\E W^2)$, valid for every nonnegative random
variable $W$ with finite second moment, then implies that
\begin{equation}\label{eq:ref-PZ}
 \PP_{\widetilde J}\bigl(W\ge\E_{\widetilde J}W/2\bigr)\ge\frac{e^{-\Theta}}8 .
\end{equation}

Next, we raise this probability to $1/2$ at the cost of a lower threshold.
Let $f(\widetilde J)=\log W$. Differentiating the finite sum defining $W$, we have
\[
 \frac{\partial f}{\partial\widetilde J_{ij}}=\frac{\beta}{W}\sqrt{\frac\omega n}
   \sum_{x\in S'}\pi(x\mid S')\,e^{\beta\sqrt\omega H_n^{\widetilde J}(x)}x_ix_j .
\]
The right side is $\beta\sqrt{\omega/n}$ times an average of the numbers
$x_ix_j\in\{-1,1\}$, so each coordinate of $\nabla f$ is at most
$\beta\sqrt{\omega/n}$ in absolute value. Summing
over the $n(n-1)/2$ coordinates, we find
$|\nabla f|^2\le\beta^2\omega(n-1)/2=\sigma_W^2$; in particular, $f$ is
$\sigma_W$-Lipschitz. Let $\bar f=\E_{\widetilde J}f$. By the second bound in
\eqref{eq:inp-concentration} (\cref{inp:ob-concentration}), we have $\E_{\widetilde J}\exp\{\theta(f-\bar f)\}\le e^{\theta^2\sigma_W^2/2}$ for every
$\theta\in\R$. The exponential Markov inequality then implies the two
one-sided bounds
\begin{equation}\label{eq:ref-one-sided}
 \PP_{\widetilde J}(f-\bar f\ge a)\le e^{-a^2/(2\sigma_W^2)},\qquad
 \PP_{\widetilde J}(f-\bar f\le-a)\le e^{-a^2/(2\sigma_W^2)}\qquad(a\ge0).
\end{equation}
Note that $W\ge\E_{\widetilde J}W/2$ if and only if $f\ge\log(\E_{\widetilde J}W/2)$. If
$\log(\E_{\widetilde J}W/2)>\bar f$, then \eqref{eq:ref-PZ} and the first bound in
\eqref{eq:ref-one-sided} with $a=\log(\E_{\widetilde J}W/2)-\bar f$ give
\[
 \frac{e^{-\Theta}}8\le\PP_{\widetilde J}\bigl(f\ge\log(\E_{\widetilde J}W/2)\bigr)
 \le\exp\Bigl\{-\frac{(\log(\E_{\widetilde J}W/2)-\bar f)^2}{2\sigma_W^2}\Bigr\},
\]
so that $\bar f\ge\log(\E_{\widetilde J}W/2)-\sigma_W\sqrt{2(\Theta+\log8)}$. This lower
bound for $\bar f$ also holds if $\log(\E_{\widetilde J}W/2)\le\bar f$. The second
bound in \eqref{eq:ref-one-sided} with $a=\sigma_W\sqrt{2\log2}$ then implies
that
\[
 \PP_{\widetilde J}\Bigl(f>\log(\E_{\widetilde J}W/2)-\sigma_W\sqrt{2(\Theta+\log8)}
     -\sigma_W\sqrt{2\log2}\Bigr)\ge\frac12 .
\]
We estimate the loss in the threshold using $\sigma_W^2\le\beta^2n\omega/2$,
$\sqrt{u+v}\le\sqrt u+\sqrt v$, and
$\sqrt{\log8}+\sqrt{\log2}=(1+\sqrt3)\sqrt{\log2}<3$:
\[
 \sigma_W\sqrt{2\Theta}
   +\sigma_W\bigl(\sqrt{2\log8}+\sqrt{2\log2}\bigr)
 \le\frac{\beta^2n\omega q_*}{\sqrt2}+3\beta\sqrt{n\omega} .
\]
Recalling that $\log(\E_{\widetilde J}W/2)=\beta^2\omega(n-1)/4-\log2$, we conclude
that, with conditional probability at least $1/2$,
\begin{equation}\label{b:eq:general-coverage-W}
 \log W\ge\frac{\beta^2\omega(n-1)}4-\beta^2n\omega q_*
   -3\beta\sqrt{n\omega}-\log2 .
\end{equation}
The constants in \eqref{b:eq:general-coverage-W}, and the threshold for $n$
beyond which it holds, do not depend on $J$ or $S$.

\paragraph{Step 2: comparison with the rotation estimate.}
By the linearity of $H_n^J$ in $J$, we have
$H_n^{J'}=\sqrt{1-\omega}\,H_n^J+\sqrt\omega\,H_n^{\widetilde J}$. Because
$1-\sqrt{1-\omega}>0$ and $H_n^J\le ne_1$ on $S'$, we have
\[
 \frac{Z_n^{J'}}{Z_n^J}
 =\pi\Bigl(\exp\bigl\{-\beta\bigl(1-\sqrt{1-\omega}\bigr)H_n^J
    +\beta\sqrt\omega H_n^{\widetilde J}\bigr\}\Bigr)
 \ge\pi(S')\,e^{-\beta(1-\sqrt{1-\omega})ne_1}\,W .
\]
For $0\le\omega\le1$, we have
$1-\sqrt{1-\omega}=\omega/(1+\sqrt{1-\omega})\le\omega/(2-\omega)
\le\omega/2+\omega^2/2$; the last inequality is equivalent to
$\omega(1-\omega)\ge0$. We take logarithms and insert
$\pi(S')\ge e^{-D_1L}/2$, \eqref{b:eq:general-coverage-W}, and
$\beta^2\omega(n-1)/4\ge\beta^2n\omega/4-\beta^2/8$. Recalling that
$n\omega=\Lambda L$, we obtain, with conditional probability at least $1/2$,
\[
 \log Z_n^{J'}-\log Z_n^J
 \ge n\omega\Bigl(\frac{\beta^2}4-\frac{\beta e_1}2-\beta^2q_*\Bigr)
   -D_1L-\mathrm{Err}_n
 =\bigl[(g_0-\beta^2q_*)\Lambda-D_1\bigr]L-\mathrm{Err}_n,
\]
where
\[
 \mathrm{Err}_n=\frac{\beta e_1\Lambda^2L^2}{2n}+3\beta\sqrt{\Lambda L}
   +\frac{\beta^2}8+2\log2 .
\]
By the choices of $q_*$ and $\Lambda$, we have $\beta^2q_*<g_0/8$ and
$D_1<g_0\Lambda/8-1$, so that
\[
 (g_0-\beta^2q_*)\Lambda-D_1>\frac78g_0\Lambda-\frac18g_0\Lambda
 =\frac34g_0\Lambda .
\]
Since $L\to\infty$ and $L/n\to0$, we have $\mathrm{Err}_n=o(L)$; this is
where the two conditions on the scale are used together. We have
$\mathrm{Err}_n\le g_0\Lambda L/4$ for large $n$, and then
\begin{equation}\label{eq:ref-gain}
 \PP_{\widetilde J}\Bigl(\log Z_n^{J'}-\log Z_n^J\ge\frac{g_0\Lambda L}2\Bigr)\ge\frac12
 \qquad\text{for every }J\in\mathcal E_n\cap\mathcal V_n .
\end{equation}
The threshold for $n$ depends only on $\beta$, $D_1$, and the sequence
$(L_n)$.

Since the event in \eqref{eq:ref-gain} does not involve $S$, we need no
measurable choice of $S$ as a function of $J$ (in
\cite[Section~6]{oai-barriers}, the first violating set in a fixed order of
the subsets is chosen; either way works). By Fubini's theorem, the left side
of \eqref{eq:ref-gain} is a measurable function of $J$. Integrating it over
$J$, we obtain
\[
 \frac12\PP(\mathcal E_n\cap\mathcal V_n)
 \le\PP_{J,\widetilde J}\Bigl(\log Z_n^{J'}-\log Z_n^J\ge\frac{g_0\Lambda L}2\Bigr).
\]
The right side is a probability under the joint law of $(J,\widetilde J)$, which is
the setting of \cref{inp:ob-rotation}. Since $0<\omega<1/2$, the last bound
of \cref{inp:ob-rotation}, with $t=\omega$ and $u=g_0\Lambda L/2=g_0n\omega/2$,
bounds it by $\exp\{-u^2/(2\beta^2n\omega)\}=\exp\{-g_0^2\Lambda L/(8\beta^2)\}$.
We conclude that
\[
 \PP(\mathcal V_n)\le\PP(\mathcal E_n^c)+\PP(\mathcal E_n\cap\mathcal V_n)
 \le e^{-c_En}+2\exp\Bigl\{-\frac{g_0^2\Lambda L}{8\beta^2}\Bigr\}
 \longrightarrow0 .
\]
This bound holds simultaneously for all subsets, without a union bound over
them.

\paragraph{Step 3: from pairs to targets.}
Let $\mathcal C_n(D_1)=\mathcal V_n^c$, fix $J\in\mathcal C_n(D_1)$ and
$S\subseteq\Sigma_n$, and let $U_S$ be the set of $x\in S$ with
$\pi(\{z\in S:|R(x,z)|\ge q_*\})<e^{-D_3L}$. Suppose that
$\pi(U_S)\ge e^{-D_1L}$. Since $U_S\subseteq S$, every $x\in U_S$
satisfies $\pi(\{y\in U_S:|R(x,y)|\ge q_*\})<e^{-D_3L}$. This implies that
\begin{align*}
 \pi(\cdot\mid U_S)^{\otimes2}(|R|\ge q_*)
 &=\frac1{\pi(U_S)^2}\sum_{x\in U_S}\pi(x)\,
   \pi(\{y\in U_S:|R(x,y)|\ge q_*\})\\
 &<\frac{e^{-D_3L}}{\pi(U_S)}
 \le e^{-(D_3-D_1)L}<e^{-D_2L},
\end{align*}
where in the last step we used $D_3>D_1+D_2$. Then $U_S$ satisfies
\eqref{eq:ref-pair-violation}, which contradicts $J\notin\mathcal V_n$.
We conclude that $\pi(U_S)<e^{-D_1L}$ for every $S$, which is
\eqref{b:eq:general-coverage}. Finally, for all sufficiently large $n$, the
last display of Step~2 and $L\le n$ give
\[
 \PP\bigl(\mathcal C_n(D_1)^c\bigr)=\PP(\mathcal V_n)
 \le e^{-c_En}+2\exp\Bigl\{-\frac{g_0^2\Lambda L}{8\beta^2}\Bigr\}
 \le3e^{-D_4L},\qquad
 D_4=\min\Bigl\{c_E,\frac{g_0^2\Lambda}{8\beta^2}\Bigr\}.
\]
The constant $D_4$ depends only on $\beta$ and $D_1$. This completes the
proof.
\end{proof}

For $D>0$, we write $c(D)$ for the constant $D_3$ given by
\cref{b:lem:coverage-extension} with $D_1=D$. It depends only on $\beta$ and
$D$.

\section{Parameters and exceptional triples}\label{sec:ref-parameters}

\subsection{Exponents}

We fix exponents $\lambda$ and $\xi$ of the scales such that
\begin{equation}\label{eq:ref-exponents}
 0<\lambda<\xi<\frac14,\qquad 2\xi<3\lambda,
\end{equation}
and we assume the following upper bound on the mean bias
$\Delta_{n,\beta}=P_\beta-p_n(\beta)$: there exists $C_b<\infty$ such that
\begin{equation}\label{eq:ref-mean-input}
 0\le P_\beta-p_n(\beta)\le C_bn^{-3\lambda}
 \qquad\text{for all sufficiently large }n.
\end{equation}
For \cref{thm:barrier-730}, we take $\lambda=7/30$ and $\xi=5/21$, and
\eqref{eq:ref-mean-input} is \cref{inp:hb-rate}, with exponent $7/10=3\lambda$.
Then \eqref{eq:ref-exponents} holds because $7/30<5/21<1/4$ and
$2\xi=10/21<7/10=3\lambda$. The barrier exponent is one
third of the exponent of the mean-bias input. \Cref{tab:ref-exponents}
lists the powers of $n$ that appear in this section and in
\cref{sec:induction,sec:barrier-proof}; the error terms of the static
estimates are tabulated in \cref{sec:static}.

\subsection{The fixed parameters}

Let $q_*=q_*(\beta)$ be the overlap of \cref{b:lem:coverage-extension}.
Recall from \cref{lem:st-fixed} that for each fixed large-overlap cutoff
$b_*\in(0,1)$, there exists a small-overlap cutoff $\underline b(b_*)>0$, depending only
on $\beta$ and $b_*$, such that the three-replica pressure deficit
\eqref{b:eq:fixed-pressure-gap} holds for all feasible overlap matrices
$D(r,q,\delta)$ with $q\ge b_*$, $|r+\delta|\le\underline b(b_*)$,
$|r-\delta|\le\underline b(b_*)$, and $|q-\Gamma(q)|\le\nu_{\rm fix}(b_*)$. We fix signed-overlap cutoffs
$b_1$ and $b_2$ with
\[
 0<b_1<1,\quad b_1\le\underline b(q_*),\qquad
 0<b_2<b_1,\quad b_2\le\underline b(b_1),
\]
and then $t_0>0$ with
\begin{equation}\label{b:dyn:fixed-cutoffs}
 4t_0<\min\{q_*,b_2,b_1^2/2\},\qquad 3t_0<q_\beta .
\end{equation}
In \cref{sec:induction}, the thresholds lie in $[t_0,2t_0]$, and the cells
that we define below use the cutoff $3t_0$. We use the first condition in
\eqref{b:dyn:fixed-cutoffs} to keep these values below $q_*$ and $b_2$, with
room for the error terms of \cref{sec:induction}. It also gives
$3t_0<b_1^2/2$, which we use there to bound the number of deleted
references. The second condition ensures that the intervals $J_\beta$ and
$I_\beta$ chosen next lie inside $(0,q_\beta)$, where $\Gamma(t)=t$ by
\eqref{b:eq:identity-interval}. We fix compact intervals $J_\beta$ and
$I_\beta$ such that
\begin{equation}\label{eq:ref-intervals}
 [t_0,2t_0]\subseteq\operatorname{int}J_\beta,\qquad
 J_\beta\subseteq\operatorname{int}I_\beta,\qquad
 I_\beta\subseteq(t_0/2,3t_0).
\end{equation}
The centers of all three-replica constraints met in \cref{sec:induction}
lie in $J_\beta$. The local comparison of
\cref{sec:fractional,sec:static} is applied only at centers in $J_\beta$,
with its local cutoff chosen so small that all its windows lie in
$I_\beta$. Let $m_\beta>0$ be the lower
bound of \eqref{b:eq:main-analytic-constants} for
$\alpha(r)^2\beta^6\E\chi(r,Y_r)^3$ on $I_\beta$; it is positive because
$I_\beta$ is a compact subset of $(0,q_\beta)$.

We fix the band of the local comparison of \cref{sec:cubic} at
$\vartheta=1/2$ and $E=1/100$. Here $\vartheta$ is the ratio $(q-r)/h$ of
that comparison and $E\rho_n\le|\delta|\le2E\rho_n$ is its range of
imbalances. At these values \eqref{b:eq:os-band-checks} holds and
$G(\vartheta,E)>1/3$. We fix the pressure coefficient $g$, a fraction
$\varkappa\in(0,1)$, and the scale constant $R>0$ with
\begin{align}
 0<g&<m_\beta E^2G(\vartheta,E),\label{b:dyn:finite-band-g}\\
 R^3&>\frac{3C_b}{\varkappa g},\label{b:dyn:R-condition}
\end{align}
where $C_b$ is the constant of \eqref{eq:ref-mean-input}. Here $g$ is the
coefficient of $\rho_n^3$ in the local fractional gap
\eqref{b:eq:local-fractional-gap}, and $\varkappa$ is the fraction of this gap that
is kept in the probability bounds of \cref{sec:static}. The scales are
\begin{equation}\label{b:dyn:scales}
 \rho_n=Rn^{-\lambda},\qquad \varepsilon_n=E\rho_n,\qquad
 \eta_n=n^{-\xi},
\end{equation}
which are the scales of \cref{rem:st-scales}. With these choices, the data
of the simultaneous estimate in \cref{sec:st-families} are the intervals
$J_\beta\subseteq\operatorname{int}I_\beta$, the large-overlap cutoffs
$q_*$ and $b_1$, the band $(\vartheta,E)$, the coefficient $g$, and the
fraction $\varkappa$. The remaining constants of the static estimates (the local
cutoff, the tolerance $\nu$ of the pair exclusions, the pressure deficits of
the separated-overlap, fixed-scale, and pair regimes, the coefficient of the
fractional tilt, and the denominator margin \eqref{b:eq:denominator-margin})
are fixed in \cref{sec:fractional,sec:static} with positive slack. They
depend only on $\beta$ and on the parameters above, and not on $n$.

Finally, we fix integers
\begin{equation}\label{b:dyn:BH}
 H=\lceil2/b_1^2\rceil,\qquad B\ge H+2 .
\end{equation}
Here $H$ bounds the number of references deleted at the start of a block in
\cref{sec:induction}, and $B$ is the number of levels in a block; $B$ is
chosen in \cref{sec:barrier-proof}. After $B$, the constants
$c_s,C_s$ of the schedule \eqref{eq:ref-schedule} below are fixed, and then
the coefficient $A>0$ of the logarithmic time scale
\[
 L_n=An^\lambda
\]
is chosen in \cref{sec:barrier-proof}, with $A=A_\beta$ for \cref{thm:barrier-730}; the time horizon is $e^{L_n}$. The sequence $L_n$ tends to infinity
and is $o(n)$, because $0<\lambda<1$; in particular,
\cref{b:lem:coverage-extension} applies to it.

\begin{table}[t]
\centering\small
\caption{The fixed parameters of the dynamical argument, in the order in
which they are chosen. Every entry is fixed before $n\to\infty$; each
depends only on $\beta$ and on the entries above it.}
\label{tab:ref-parameters}
\begin{tabular}{@{}>{$}l<{$}p{7.4cm}p{4.0cm}@{}}
\toprule
\text{Parameter} & Choice & Reference\\
\midrule
q_\beta & right end of $\operatorname{supp}\mu_\beta=[0,q_\beta]$ & \cref{inp:full-rsb}\\
q_* & overlap of the overlap-neighborhood mass bound & \cref{b:lem:coverage-extension}\\
b_1,\ b_2 & $b_1\le\underline b(q_*)$, $b_2\le\underline b(b_1)$, $0<b_2<b_1<1$ & \eqref{b:eq:fixed-pressure-gap}\\
t_0 & $4t_0<\min\{q_*,b_2,b_1^2/2\}$, $3t_0<q_\beta$ & \eqref{b:dyn:fixed-cutoffs}\\
J_\beta,\ I_\beta & compact intervals around $[t_0,2t_0]$ & \eqref{eq:ref-intervals}\\
m_\beta & lower bound for the scalar coefficient on $I_\beta$ & \eqref{b:eq:main-analytic-constants}\\
\vartheta,\ E & $\vartheta=1/2$, $E=1/100$ & \eqref{b:eq:os-band-checks}\\
g & $0<g<m_\beta E^2G(\vartheta,E)$ & \eqref{b:dyn:finite-band-g}\\
\varkappa & $0<\varkappa<1$ & \cref{sec:static}\\
C_b & constant of the mean-bias input; for \cref{thm:barrier-730}, that of \cref{inp:hb-rate} & \eqref{eq:ref-mean-input}\\
R & $R^3>3C_b/(\varkappa g)$ & \eqref{b:dyn:R-condition}\\
\text{static constants} & local cutoff, $\nu$, deficits, fractional tilt, denominator margin & \cref{sec:fractional,sec:static}\\
H & $H=\lceil2/b_1^2\rceil$ & \eqref{b:dyn:BH}\\
\text{threshold constants} & widths and spacing of the threshold intervals & \cref{sec:induction}\\
B & integer, $B\ge H+2$, block length & \cref{sec:barrier-proof}\\
c_s,\ C_s & schedule of list sizes, $0\le s<B$ & \eqref{eq:ref-schedule}\\
A & coefficient of $L_n=An^\lambda$ & \cref{sec:barrier-proof}\\
\bottomrule
\end{tabular}
\end{table}

We note that the quantities that depend on $n$ are the scales $\rho_n$,
$\varepsilon_n$, and $\eta_n$ of \eqref{b:dyn:scales}, the logarithmic time scale $L_n$, the
number $N_{\rm lev}$ of levels (fixed in \cref{sec:induction}), the list
sizes $K_s$ of \eqref{eq:ref-schedule}, and the total number $N$ of list
entries. We never let a fixed parameter vary with $n$, and the threshold
$\tau$ for the total variation distance does not enter until the last step
of \cref{sec:barrier-proof}.

\begin{table}[t]
\centering\small
\caption{Powers of $n$ at the scales \eqref{b:dyn:scales} and the logarithmic
time scale $L_n=An^\lambda$, for general exponents and for \cref{thm:barrier-730}.}
\label{tab:ref-exponents}
\begin{tabular}{@{}>{$}l<{$}>{$}l<{$}>{$}l<{$}p{3.9cm}@{}}
\toprule
\text{Quantity} & \text{General} & \text{\cref{thm:barrier-730}} & Role\\
\midrule
P_\beta-p_n(\beta) & O(n^{-3\lambda}) & O(n^{-7/10}) & mean-bias input\\
\rho_n & Rn^{-\lambda} & Rn^{-7/30} & overlap gap\\
\eta_n & n^{-\xi} & n^{-5/21} & fixed-scale tolerance\\
L_n & An^{\lambda} & A_\beta n^{7/30} & logarithmic time scale\\
n\rho_n^3 & R^3n^{1-3\lambda} & R^3n^{3/10} & narrow triples\\
n\eta_n^2 & n^{1-2\xi} & n^{11/21} & fixed-scale triples\\
n\rho_n^3/L_n & (R^3/A)n^{1-4\lambda} & (R^3/A_\beta)n^{1/15} & \eqref{eq:ref-absorb}\\
n\eta_n^2/L_n & n^{1-2\xi-\lambda}/A & n^{61/210}/A_\beta & \eqref{eq:ref-absorb}\\
\eta_n/\rho_n & n^{\lambda-\xi}/R & n^{-1/210}/R & \cref{sec:induction}\\
\bottomrule
\end{tabular}
\end{table}

\subsection{Exceptional triples}

The overlap constraints that we use in \cref{sec:induction} are of two
kinds. The first kind states that when $x$ and ${\widetilde v}$ have a large overlap,
their small overlaps with a third configuration $v$ are nearly equal. For a
large-overlap cutoff $b_*\in(0,1)$ and a small-overlap cutoff $b'\in(0,1)$, we
define
\begin{equation}\label{eq:ref-fixed-violation}
 \begin{aligned}
 \mathcal B_n(b_*,b')=\bigl\{(v,x,{\widetilde v})\in\Sigma_n^3:{}
   &R(x,{\widetilde v})\ge b_*,\ |R(v,x)|\le b',\ |R(v,{\widetilde v})|\le b',\\
   &|R(v,x)-R(v,{\widetilde v})|>2\eta_n\bigr\}.
 \end{aligned}
\end{equation}
This is the set excluded in \cite[Lemma~4.1]{oai-barriers}, with the
tolerance $2n^{-1/100}$ there replaced by $2\eta_n$. We bound its Gibbs
probability at this tolerance in \cref{sec:static} through the pressure
estimate \eqref{b:eq:fixed-pressure-gap}, and not through
\cite[Lemma~4.1]{oai-barriers}. A triple
$(v,x,{\widetilde v})\notin\mathcal B_n(b_*,b')$ such that $R(x,{\widetilde v})\ge b_*$,
$|R(v,x)|\le b'$, and $|R(v,{\widetilde v})|\le b'$ satisfies
$|R(v,x)-R(v,{\widetilde v})|\le2\eta_n$.

The second kind concerns overlaps near a level $r$ in the compact interval
$J_\beta$. For $(v,x,{\widetilde v})\in\Sigma_n^3$, we write
\begin{equation}\label{b:dyn:narrow-coordinates}
 r=\frac{R(v,x)+R(v,{\widetilde v})}2,\qquad
 \delta=\frac{R(v,x)-R(v,{\widetilde v})}2,\qquad
 q=R(x,{\widetilde v}),
\end{equation}
so that the Gram matrix of $(v,x,{\widetilde v})$ is the matrix $D(r,q,\delta)$ of
\cref{sec:static}, with off-diagonal entries $r+\delta$, $r-\delta$, and $q$.
Let $\mathcal N_n$ be the set of triples with
\[
 r\in J_\beta,\qquad \varepsilon_n\le|\delta|\le2\varepsilon_n,\qquad
 q\ge r+\rho_n .
\]
A triple outside $\mathcal N_n$ with $r\in J_\beta$ and
$\varepsilon_n\le|\delta|\le2\varepsilon_n$ satisfies $R(x,{\widetilde v})<r+\rho_n$: if
the overlaps of $v$ with $x$ and with ${\widetilde v}$ lie near a common level $r$ and
differ by an amount of order $\varepsilon_n$, then the overlap of $x$ and ${\widetilde v}$
cannot exceed $r$ by $\rho_n$. The set $\mathcal N_n$ corresponds to the set
excluded in \cite[Lemma~7.1]{oai-barriers}. There, the gap is
$\rho=n^{-1/10000}$, the imbalance is of order $\rho^{20}$, and the centers
are restricted by the condition $\mu_\beta([r-\rho^8,r+\rho^8])\le\rho^4$ on
the Parisi measure. Here the gap $\rho_n$ and the imbalance
$\varepsilon_n=E\rho_n$ are of the same order, and there is no condition on
the Parisi measure, because the local comparison of \cref{sec:fractional} is
uniform over centers in $J_\beta$. The exceptional set is
\begin{equation}\label{eq:ref-exceptional}
 \mathcal B_n=\mathcal N_n\cup\mathcal B_n(q_*,b_1)\cup\mathcal B_n(b_1,b_2).
\end{equation}
It is a deterministic subset of $\Sigma_n^3$, and it is not closed under
permutations or spin flips of the coordinates. In the dynamical argument, we
evaluate $\mathcal B_n$ on triples in several orders and orientations; since
$\pi^{\otimes3}$ is invariant under these operations, each of them costs only
a fixed factor in \cref{lem:ref-families}.

The Gibbs probability of $\mathcal B_n$ is controlled in \cref{sec:static},
where different parts of it are handled by different estimates. The set
$\mathcal N_n$ is the set of narrow triples of \cref{sec:st-families} for the
interval of centers $J_\beta$. By \cref{lem:st-cover}, a narrow triple either
has its pair $(x,{\widetilde v})$ excluded by the pair deficit
\eqref{b:eq:pair-static-gap}, or its Gram matrix lies in the local family,
for which we use the positive fractional moment
\eqref{b:eq:local-fractional-gap}, or in the separated-overlap family, for
which we use the stronger pressure deficit \eqref{b:eq:remote-pressure-gap}
and ordinary Gaussian concentration. The sets $\mathcal B_n(q_*,b_1)$ and
$\mathcal B_n(b_1,b_2)$ are contained in the set $\mathcal W_n$ of
fixed-scale triples of \cref{sec:st-families}, which is controlled by
\eqref{b:eq:fixed-pressure-gap} and the pair deficit. We need
no fractional moment bound at a distant overlap $q$.

\begin{lemma}[One disorder event for the exceptional triples]
\label{lem:ref-single-event}
Let $(\rho_n)$ and $(\eta_n)$ be deterministic positive sequences with
$\rho_n\to0$ that satisfy \eqref{eq:st-S1}--\eqref{eq:st-S5}, and let
$\varepsilon_n=E\rho_n$. Let
$\mathcal B_n$ be the set \eqref{eq:ref-exceptional} defined with these
sequences, and let $\mathcal A_n$ be the disorder event of
\cref{b:lem:static-event} for these sequences and for the data of
\cref{sec:st-families} fixed in \cref{tab:ref-parameters} (in particular the
intervals $J_\beta$ and $I_\beta$ and the large-overlap cutoffs $q_*$ and
$b_1$). Then $\PP(\mathcal A_n)\to1$, and there exist constants $C,c>0$ such
that, for all sufficiently large $n$ and every $J\in\mathcal A_n$,
\begin{equation}\label{b:dyn:uniform-gibbs}
 \pi^{\otimes3}(\mathcal B_n)\le q_n,\qquad
 q_n=C(n+1)^3\bigl(e^{-cn\rho_n^3}+e^{-cn\eta_n^2}+e^{-cn}\bigr).
\end{equation}
\end{lemma}

\begin{proof}
By \cref{b:lem:static-event}, we have $\PP(\mathcal A_n^c)\to0$. Moreover,
for large $n$ and every $J\in\mathcal A_n$,
\eqref{b:eq:conditional-static-errors} bounds the Gibbs probability of the set
$\mathcal N_n$ of narrow triples of \cref{sec:st-families} by
$C(n+1)^3e^{-cn\rho_n^3}$, and that of the set $\mathcal W_n$ of fixed-scale
triples of \cref{sec:st-families} by $C(n+1)^3e^{-cn\eta_n^2}$.

It remains to compare $\mathcal B_n$ with these two sets, where our triples
$(v,x,{\widetilde v})$ are the triples $(v,x,y)$ of \cref{sec:static}. The set
$\mathcal N_n$ in \eqref{eq:ref-exceptional} is the set of narrow triples for
the interval of centers $J_\beta$. Let $(v,x,{\widetilde v})\in\mathcal B_n(b_*,b')$ with
$(b_*,b')=(q_*,b_1)$ or $(b_*,b')=(b_1,b_2)$. Then $R(x,{\widetilde v})\ge b_*$,
$|R(v,x)|\le b'\le\underline b(b_*)$, $|R(v,{\widetilde v})|\le b'\le\underline b(b_*)$,
and $|R(v,x)-R(v,{\widetilde v})|>2\eta_n$. Since $b_*$ is one of the cutoffs $q_*$ and
$b_1$, the triple satisfies the conditions that define $\mathcal W_n$. This
shows that $\mathcal B_n(q_*,b_1)\cup\mathcal B_n(b_1,b_2)\subseteq\mathcal W_n$.
Summing the two bounds of the previous paragraph, we obtain
$\pi^{\otimes3}(\mathcal B_n)\le C(n+1)^3(e^{-cn\rho_n^3}+e^{-cn\eta_n^2})$,
which implies \eqref{b:dyn:uniform-gibbs}.
\end{proof}

We apply \cref{lem:ref-single-event} with the scales \eqref{b:dyn:scales}.
Since $\lambda>0$, we have $\rho_n=Rn^{-\lambda}\to0$. These scales also satisfy
\eqref{eq:st-S1}--\eqref{eq:st-S5}, as in
\cref{rem:st-scales}(b). Indeed, by \eqref{eq:ref-mean-input} and
\eqref{b:dyn:R-condition}, we have
$\Delta_{n,\beta}/\rho_n^3\le C_b/R^3<\varkappa g/3$ for large $n$, which is
\eqref{eq:st-S1}. Further, $n^{7/15}/(n^2\rho_n^6)=R^{-6}n^{6\lambda-23/15}$
tends to zero because $6\lambda<3/2<23/15$; the quantities
$n\rho_n^4=R^4n^{1-4\lambda}$ and $n\eta_n^4=n^{1-4\xi}$ grow faster than
$\log n$ because $\lambda,\xi<1/4$; and $\rho_n^3/\eta_n^2=R^3n^{2\xi-3\lambda}$
tends to zero because $2\xi<3\lambda$. From now on, $\mathcal B_n$,
$\mathcal A_n$, and $q_n$ denote the objects of \cref{lem:ref-single-event} at
the scales \eqref{b:dyn:scales}. In \cref{sec:bp-strong}, the lemma is
applied with $\eta_n=\rho_n/1000$ instead.

\begin{remark}[Two kinds of error]\label{rem:ref-two-errors}
In the proof of \cref{thm:barrier-730}, we use two kinds of
small quantities, and we never mix them. The bound
\eqref{b:eq:static-event-failure} on the disorder probability
$\PP(\mathcal A_n^c)$ includes a polynomial term from Chebyshev's inequality,
and therefore it does not remain small after multiplication by a factor
$e^{CL_n}$. For $\lambda=7/30$ and $\xi=5/21$, the terms
$e^{-cn\rho_n^4}$ and $e^{-cn\eta_n^4}$ in it, which are
$e^{-cR^4n^{1/15}}$ and $e^{-cn^{1/21}}$, also decay more slowly than
$e^{-CL_n}=e^{-CAn^{7/30}}$ for every fixed $C>0$. We remove this disorder probability,
together with the probabilities of the complements of the events of
\cref{b:lem:coverage-extension}, once, before counting reference
configurations or path positions. The conditional Gibbs bound $q_n$ in
\eqref{b:dyn:uniform-gibbs}, in contrast, holds for every
$J\in\mathcal A_n$, and it absorbs every factor
$e^{CL_n}$. \Cref{tab:ref-exponents} shows that the exponents
$n\rho_n^3$, $n\eta_n^2$, and $n$ exceed $L_n$ by a power of $n$. We verify
in the proof of \cref{lem:ref-families} that
\begin{equation}\label{eq:ref-absorb}
 \lim_{n\to\infty}e^{CL_n}q_n=0\qquad\text{for every fixed }C>0 .
\end{equation}
In \cref{lem:ref-families}, only $q_n$ is multiplied by the number of
reference configurations. The deterministic set $\mathcal B_n$ contains every
exceptional triple that can arise for the allowed thresholds and paths, and \eqref{b:dyn:uniform-gibbs} bounds its mass under
$\pi^{\otimes3}$ by $q_n$. We impose no condition
on the disorder that depends on the thresholds of \cref{sec:induction}.
\end{remark}

\section{Fixed families of reference configurations}\label{sec:ref-families}

The induction of \cref{sec:induction} selects references from lists of
independent Gibbs samples, organized in blocks of $B$ consecutive levels.
This is the construction of \cite[Section~8.3]{oai-barriers}. The selection
cell at a level is determined by the references chosen at the preceding
levels of the same block. The list sizes are chosen recursively over the $B$
positions of a block, so that the depth of the recursion is the fixed number
$B$ even though the number of levels tends to infinity. In
\cite[Section~8.3]{oai-barriers}, the required properties of the lists are
proved with probability tending to one jointly over the disorder, the lists,
and the path. The proof there uses union bounds over path positions and over
triples, and the bounds for the triples are averaged over the disorder. At
our scales, the
averaged bound available from \cref{b:lem:static-event} contains the disorder
failure probability of \cref{rem:ref-two-errors}, and this term does not
remain small after multiplication by the number $e^{O(L_n)}$ of reference
configurations. We therefore fix the
disorder first, in the event of \cref{lem:ref-single-event}, and use only
the conditional bound \eqref{b:dyn:uniform-gibbs}. At such a disorder, we fix
one realization of the lists, together with a set $V$ of exceptional
targets, before any path is considered.

\subsection{Lists, cells, and list sizes}

Let $N_{\rm lev}=N_{\rm lev}(n)\ge1$ be the number of levels; it is fixed in
\cref{sec:induction}. Level $j\in\{0,\ldots,N_{\rm lev}-1\}$ belongs to block
$\lfloor j/B\rfloor$ and has position $s(j)=j-B\lfloor j/B\rfloor$ in its
block. The list at level $j$ is an ordered tuple
\[
 \mathcal F_j=(y_{j,1},\ldots,y_{j,K_{s(j)}})\in\Sigma_n^{K_{s(j)}},
\]
whose coordinates we call the entries of $\mathcal F_j$; the sizes $K_s$ are
defined in \eqref{eq:ref-schedule} below. We write
\[
 \mathcal F_j^\pm=\bigl\{\sigma y:\sigma\in\{-1,1\},\ y\text{ an entry of }
   \mathcal F_j\bigr\}
\]
for the signed list. The family of reference configurations is
$\mathcal F=(\mathcal F_0,\ldots,\mathcal F_{N_{\rm lev}-1})$, and
$N=\sum_{j<N_{\rm lev}}K_{s(j)}$ is its total number of entries.

For a level $j$, let $\mathcal T_j$ be the set of tuples $T=(u_i)_{i\in\mathcal I}$,
where $\mathcal I$ is a set of levels of the block of $j$ that precede $j$, and $u_i$
is an entry of $\mathcal F_i$ for each $i\in\mathcal I$. The cell of $T$ is
\begin{equation}\label{eq:ref-cell}
 S_T=\bigl\{x\in\Sigma_n:|R(u,x)|\le3t_0\text{ for every entry }u
   \text{ of }T\bigr\}.
\end{equation}
The empty tuple has cell $S_\emptyset=\Sigma_n$, and it is the only element
of $\mathcal T_j$ when $s(j)=0$. Spin flips of the entries do not change the
cells. The references selected in \cref{sec:induction} at the levels of a
block that precede $j$, after removing their signs, form a tuple in $\mathcal T_j$, and the cells used
there are among the cells $S_T$.

The list sizes are chosen by the following schedule over positions
$s=0,\ldots,B-1$. Recall that $c(D)$ denotes the constant $D_3$ of
\cref{b:lem:coverage-extension} for $D_1=D$. We set $C_0=0$ and, recursively,
\begin{equation}\label{eq:ref-schedule}
 c_s=c(C_s+5),\qquad C_{s+1}=C_s+c_s+2,\qquad
 K_s=\bigl\lceil e^{(c_s+1)L_n}\bigr\rceil .
\end{equation}
We have $C_s=\sum_{s'<s}(c_{s'}+2)$. The constants $c_s$ and $C_s$ depend only
on $\beta$ and $s$, and only $K_s$ depends on $n$ and $A$. For $D>0$, let
$\mathcal C_n(D)$ be the event of \cref{b:lem:coverage-extension} for the
sequence $(L_n)$. On $\mathcal C_n(D)$, \eqref{b:eq:general-coverage} reads
\begin{equation}\label{b:dyn:coverage}
 \pi\bigl(\bigl\{x\in S:\pi(\{y\in S:|R(x,y)|\ge q_*\})<e^{-c(D)L_n}\bigr\}
   \bigr)<e^{-DL_n}\qquad\text{for every }S\subseteq\Sigma_n .
\end{equation}
Only $B$ of these events are needed. We define the disorder event
\begin{equation}\label{eq:ref-good-event}
 \mathcal G_n=\mathcal A_n\cap\bigcap_{s=0}^{B-1}\mathcal C_n(C_s+5).
\end{equation}
It is measurable, and $\PP(\mathcal G_n)\to1$ by
\cref{lem:ref-single-event,b:lem:coverage-extension}, since it is the
intersection of a fixed number of events whose probabilities tend to one.

\subsection{The family}

\begin{lemma}[Fixed families of reference configurations]
\label{lem:ref-families}
Let $B\ge1$ be a fixed integer, let $(L_n)$ be a deterministic sequence with
$L_n\to\infty$ and $L_n/n\to0$, and let $K_s$ and $\mathcal G_n$ be given by
\eqref{eq:ref-schedule} and \eqref{eq:ref-good-event}. Let
$N_{\rm lev}=N_{\rm lev}(n)\ge1$ be integers with
$N_{\rm lev}\le e^{L_n}$, and assume that $q_n$ satisfies
\eqref{eq:ref-absorb}. Then, for all sufficiently large $n$ and every
$J\in\mathcal G_n$, there exist lists
$\mathcal F_j\in\Sigma_n^{K_{s(j)}}$, $0\le j<N_{\rm lev}$, and a set
$V\subseteq\Sigma_n$ with
\begin{equation}\label{b:dyn:fixed-bank}
 \pi(V)\le3e^{-4L_n},
\end{equation}
such that the following four properties hold.
\begin{enumerate}[label=\textup{(\roman*)}]
\item \emph{Overlap neighborhoods in every cell of the current block.} For
every level $j$,
every $T\in\mathcal T_j$, and every $x\in S_T\setminus V$, there exists an
entry $y$ of $\mathcal F_j$ with $y\in S_T$ and $|R(x,y)|\ge q_*$.
\item \emph{Comparison across the overlap $q_*$.} For every
$x\in\Sigma_n\setminus V$, all levels $i\ne j$, and all
$v\in\mathcal F_i^\pm$ and ${\widetilde v}\in\mathcal F_j^\pm$ with
\[
 R(x,{\widetilde v})\ge q_*,\qquad |R(v,x)|\le b_1,\qquad |R(v,{\widetilde v})|\le b_1,
\]
we have $|R(v,x)-R(v,{\widetilde v})|\le2\eta_n$.
\item \emph{Comparison across the overlap $b_1$ for triples of entries.}
For all pairwise distinct levels $i$, $a$, and $j$, and all
$v\in\mathcal F_a^\pm$, $y\in\mathcal F_i^\pm$, and ${\widetilde v}\in\mathcal F_j^\pm$
with
\[
 |R(y,{\widetilde v})|\ge b_1,\qquad |R(v,y)|\le b_2,\qquad |R(v,{\widetilde v})|\le b_2,
\]
we have $|R(v,y)-\sigma R(v,{\widetilde v})|\le2\eta_n$, where $\sigma\in\{-1,1\}$ is the
sign of $R(y,{\widetilde v})$. In particular,
$\bigl||R(v,y)|-|R(v,{\widetilde v})|\bigr|\le2\eta_n$.
\item \emph{No exceptional narrow triple.} For every
$x\in\Sigma_n\setminus V$, all levels $i\ne j$, and all
$v\in\mathcal F_i^\pm$ and ${\widetilde v}\in\mathcal F_j^\pm$, we have
$(v,x,{\widetilde v})\notin\mathcal N_n$. That is, if the coordinates
\eqref{b:dyn:narrow-coordinates} of $(v,x,{\widetilde v})$ satisfy $r\in J_\beta$ and
$\varepsilon_n\le|\delta|\le2\varepsilon_n$, then $R(x,{\widetilde v})<r+\rho_n$.
\end{enumerate}
If $L_n=An^\lambda$ for a fixed $A>0$, with $\lambda$ and $\xi$ as in
\eqref{eq:ref-exponents}, then $L_n\to\infty$, $L_n/n\to0$, and
\eqref{eq:ref-absorb} holds.
\end{lemma}

The threshold for $n$ depends on $\beta$, on the parameters of
\cref{tab:ref-parameters}, and on the sequences $(L_n)$ and
$(N_{\rm lev})$, but not on $J$. In \cref{sec:induction,sec:barrier-proof},
the lemma is applied with $L_n=An^\lambda$ and the number of levels fixed
there, which is at most a constant times $n^\lambda$.

\begin{proof}
We adapt the independent-list construction of
\cite[Section~8.3]{oai-barriers} by estimating exceptional Gibbs mass at
each fixed disorder in $\mathcal G_n$, before considering any path.
The proof is an expectation argument over independent samples of the lists.
We bound the expected Gibbs mass of the targets missed by the lists, that is,
of the configurations $x$ such that, at some level $j$, some cell $S_T$
containing $x$ contains no entry $y$ of $\mathcal F_j$ with
$|R(x,y)|\ge q_*$. We also bound the probability that three list entries
form an exceptional triple, and the expected Gibbs mass of the configurations
that form an exceptional triple with two list entries. We then choose a realization at which the sum of the Gibbs mass of the union of these two sets and the indicator of that event is at most its mean.

\paragraph{Step 1: random lists.}
Fix $n$ and $J\in\mathcal G_n$, and write $L=L_n$. We let the entries
$y_{j,k}$, $0\le j<N_{\rm lev}$, $1\le k\le K_{s(j)}$, be independent random
configurations with law $\pi$, defined on an auxiliary probability space with
probability $\PP_{\mathcal F}$ and expectation $\E_{\mathcal F}$. The
disorder is fixed throughout, and the space $\Sigma_n^N$ of realizations is
finite.

\paragraph{Step 2: targets missed by the lists.}
For a level $j$ and $T\in\mathcal T_j$, let $U_{j,T}$ be the set of
$x\in S_T$ such that no entry $y$ of $\mathcal F_j$ satisfies $y\in S_T$ and
$|R(x,y)|\ge q_*$, and let $V_1=\bigcup_j\bigcup_{T\in\mathcal T_j}
U_{j,T}$. Fix $j$, let $s=s(j)$, and condition on the lists at the levels of
the block of $j$ that precede $j$. This fixes $\mathcal T_j$ and the cells
$S_T$, and leaves the $K_s$ entries of $\mathcal F_j$ independent with law
$\pi$. For $T\in\mathcal T_j$ and $x\in S_T$, let
$a_T(x)=\pi(\{y\in S_T:|R(x,y)|\ge q_*\})$. Then
\[
 \E_{\mathcal F}\bigl[\pi(U_{j,T})\bigm|\text{earlier lists}\bigr]
 =\sum_{x\in S_T}\pi(x)\bigl(1-a_T(x)\bigr)^{K_s}.
\]
We split the sum according to whether $a_T(x)<e^{-c_sL}$. Recall that
$J\in\mathcal C_n(C_s+5)$ and $c_s=c(C_s+5)$. By \eqref{b:dyn:coverage} with
$S=S_T$, the first group has Gibbs mass less than $e^{-(C_s+5)L}$. For the
second group, $(1-a_T(x))^{K_s}\le\exp\{-a_T(x)K_s\}\le
\exp\{-e^{-c_sL}e^{(c_s+1)L}\}=e^{-e^L}$. The number of tuples is
\[
 |\mathcal T_j|\le\prod_{s'<s}\bigl(K_{s'}+1\bigr)
 \le\prod_{s'<s}e^{(c_{s'}+2)L}=e^{C_sL},
\]
where we used $K_{s'}+1\le e^{(c_{s'}+1)L}+2\le e^{(c_{s'}+2)L}$, valid
once $e^L\ge3$. Summing over $T\in\mathcal T_j$ and taking expectations, we
obtain
\[
 \E_{\mathcal F}\,\pi\Bigl(\bigcup_{T\in\mathcal T_j}U_{j,T}\Bigr)
 \le e^{C_sL}\bigl(e^{-(C_s+5)L}+e^{-e^L}\bigr)\le2e^{-5L},
\]
where the last inequality holds once $e^L\ge(C_s+5)L$ for every $s<B$.
Summing over the $N_{\rm lev}\le e^L$ levels, we find
\begin{equation}\label{eq:ref-miss-mass}
 \E_{\mathcal F}\,\pi(V_1)\le2e^{-4L}.
\end{equation}

\paragraph{Step 3: exceptional triples.}
Let $c_{\max}=\max_{s<B}c_s$. Using $N_{\rm lev}\le e^L$ and
$K_s\le e^{(c_s+2)L}$, we find $N\le e^{C_NL}$ with $C_N=c_{\max}+3$.

Let $\mathcal O$ be the event that there exist entries $y_1$, $y_2$, and
$y_3$ of lists at three distinct levels, and signs
$\sigma_1,\sigma_2,\sigma_3\in\{-1,1\}$, such that
$(\sigma_1y_1,\sigma_2y_2,\sigma_3y_3)\in\mathcal B_n$. Fix a choice of three entries, specified by their levels and
indices, and of signs. The three entries are independent with law $\pi$,
because they belong to distinct levels, and $\pi$ is invariant under spin
flips. The triple therefore has law $\pi^{\otimes3}$, and it lies in
$\mathcal B_n$ with probability at most $q_n$ by
\eqref{b:dyn:uniform-gibbs}. There are at most $N^3$ ordered choices of
three entries and $8$ choices of signs, which gives
\[
 \PP_{\mathcal F}(\mathcal O)\le8N^3q_n .
\]
The configurations themselves need not be distinct; only their sampling
indices are.

Let $V_2$ be the set of $x\in\Sigma_n$ for which there exist entries
$y$ and $y'$ of lists at two distinct levels, signs $\sigma,\sigma',
\sigma''\in\{-1,1\}$, and an ordering of $\sigma y$, $\sigma'y'$, and
$\sigma''x$ as a triple, such that this triple lies in $\mathcal B_n$. Let
$x$ be a configuration with law $\pi$, independent of the lists. For a fixed
ordered pair of entries at distinct levels, fixed signs, and a fixed
ordering, the resulting triple has law $\pi^{\otimes3}$, and it lies in
$\mathcal B_n$ with probability at most $q_n$. There are at most $N^2$
ordered pairs of entries, $8$ choices of signs, and $6$ orderings, and a
union bound yields
\[
 \E_{\mathcal F}\,\pi(V_2)
 =\PP_{\mathcal F\otimes\pi}(x\in V_2)\le48N^2q_n .
\]

\paragraph{Step 4: choice of one realization.}
Let $V=V_1\cup V_2$. Combining \eqref{eq:ref-miss-mass} with
the two bounds of Step~3, we conclude that
\[
 \E_{\mathcal F}\bigl[\mathbf 1_{\mathcal O}+\pi(V)\bigr]
 \le2e^{-4L}+\bigl(8N^3+48N^2\bigr)q_n
 \le2e^{-4L}+56e^{3C_NL}q_n\le3e^{-4L},
\]
where the last inequality holds for all sufficiently large $n$ by
\eqref{eq:ref-absorb} with $C=3C_N+4$. On the finite probability space of
Step~1, some realization of the lists satisfies
$\mathbf 1_{\mathcal O}+\pi(V)\le\E_{\mathcal F}[\mathbf 1_{\mathcal O}+\pi(V)]\le3e^{-4L}$.
We fix it. Then $\pi(V)\le3e^{-4L}$, which is \eqref{b:dyn:fixed-bank}, and
the realization does not belong to $\mathcal O$, since
$\mathbf 1_{\mathcal O}\in\{0,1\}$ and $3e^{-4L}<1$.

\paragraph{Step 5: the four properties.}
Property~(i) holds because $V\supseteq V_1$. For~(ii), let $x$, $v$,
and ${\widetilde v}$ be as there. The triple $(v,x,{\widetilde v})$ is one of the orderings in the
definition of $V_2$, with $\sigma''=1$. Since $x\notin V_2$,
it does not lie in $\mathcal B_n\supseteq\mathcal B_n(q_*,b_1)$. By the
definition \eqref{eq:ref-fixed-violation} and the hypotheses of~(ii), this
means that $|R(v,x)-R(v,{\widetilde v})|\le2\eta_n$. The same argument with
$\mathcal N_n\subseteq\mathcal B_n$ proves~(iv). For~(iii), let $v$, $y$,
${\widetilde v}$, and $\sigma$ be as there. Then $R(y,\sigma {\widetilde v})=|R(y,{\widetilde v})|\ge b_1$,
$|R(v,y)|\le b_2$, and $|R(v,\sigma {\widetilde v})|\le b_2$. Since the realization does not belong to $\mathcal O$, the triple
$(v,y,\sigma {\widetilde v})$ of signed entries from three distinct levels does not lie in
$\mathcal B_n(b_1,b_2)$. This implies that
$|R(v,y)-\sigma R(v,{\widetilde v})|\le2\eta_n$, and the last claim of~(iii) follows from
$\bigl||R(v,y)|-|R(v,{\widetilde v})|\bigr|\le|R(v,y)-\sigma R(v,{\widetilde v})|$.

\paragraph{Step 6: the time scale $L_n=An^\lambda$.}
If $L_n=An^\lambda$ with $0<\lambda<1$, then $L_n\to\infty$ and
$L_n/n\to0$. For a fixed $C>0$, we have
\[
 e^{CL_n}q_n=C'(n+1)^3\Bigl(e^{CAn^\lambda-cR^3n^{1-3\lambda}}
   +e^{CAn^\lambda-cn^{1-2\xi}}+e^{CAn^\lambda-cn}\Bigr),
\]
with $C'$ the constant of \eqref{b:dyn:uniform-gibbs}. By
\eqref{eq:ref-exponents}, we have $1-3\lambda>\lambda$ and
$1-2\xi>1/2>\lambda$. Hence there exists $\lambda'>0$ such that each of the
three exponents is at most $-n^{\lambda'}$ for all large $n$, and
$(n+1)^3e^{-n^{\lambda'}}\to0$. This proves \eqref{eq:ref-absorb} and
completes the proof.
\end{proof}

The proof gives slightly more than (ii)--(iv). For every $x\notin V$, no
ordering of $x$ or $-x$ with two signed entries from distinct levels lies in
$\mathcal B_n$, and no ordering of three signed entries from distinct levels
lies in $\mathcal B_n$. Properties (ii)--(iv) are the instances of these
statements used in \cref{sec:induction}.

\begin{remark}[The same family for every path avoiding $V$]
\label{rem:ref-every-path}
The family $\mathcal F$ and the set $V$ of \cref{lem:ref-families} depend
only on $J$, and properties (i)--(iv) are deterministic statements about
$\mathcal F$, $V$, and configurations outside $V$. No path enters their
construction. We may therefore apply them along any finite sequence
$x_0,\ldots,x_m$ in $\Sigma_n\setminus V$ and for any choice of the
thresholds of \cref{sec:induction}: properties (i), (ii), and (iv) hold at
every $x_k$, simultaneously for all $k$, all levels, and all entries, and
(iii) holds for all entries. This is the form in which we use the lemma in
the induction of \cref{sec:induction}.

The order of choices matters for the stationary sign test of
\cref{sec:barrier-proof}. There, we bound the probability that the
stationary discrete heat-bath chain visits $V$ at one of the indices
$0,\ldots,m$ by $(m+1)\pi(V)$, using only that its law is $\pi$ at each fixed
integer index. This union bound requires that $V$ is fixed before the chain
is run. Note also that a
reference selected along a path depends on the path and need not have law
$\pi$; for this reason we state (ii)--(iv) for all entries of all lists, and
not only for selected references. Finally, we choose the family separately
at each $J\in\mathcal G_n$ and assert no measurable dependence on $J$.
None is needed, because the quantities that we integrate over the disorder
in \cref{sec:barrier-proof} do not depend on the family: they are the Gibbs
masses of initial states whose transition law is close to $\pi$ in total
variation.
\end{remark}
\chapter{Stable overlap constraints along a trajectory}\label{sec:induction}

This section contains the deterministic part of the dynamical argument. We fix the disorder, the families of reference configurations constructed in \cref{sec:references}, an array of thresholds, and a sequence of configurations with small overlap increments that avoids the exceptional set $V$ of \cref{sec:references}. From these data, an explicit algorithm (\cref{def:ind-algorithm}) selects one reference at each level and finds one crossing time at each level, searching backward from the previous crossing. The main result is \cref{prop:ind-induction}: if the overlap of the sequence with its first reference becomes nonpositive, then every crossing exists, and the errors in all overlap constraints stay of order $\varepsilon_n$. \Cref{cor:ind-witness} extracts the consequence used in \cref{sec:barrier-proof}: at one common time, the overlap of the sequence with the first reference of every block lies within a fixed multiple of $\varepsilon_n$ of the corresponding threshold. In \cref{sec:barrier-proof}, the thresholds are made random, and we show by a counting bound that this cannot happen before the time in \cref{thm:barrier-730}.

We use the four properties of the families of reference configurations proved in \cref{lem:ref-families}, which we recall as (P1)--(P4) in \cref{sec:ind-setting}, and elementary geometry of overlaps. No probability is used in this section.

The construction is that of \cite[Sections~8.3 and~8.4]{oai-barriers}. There, references are selected from fixed lists of Gibbs samples inside cells determined by the earlier references of the current block. The cells are reset to the whole space at the start of each block, references with a large overlap with the new reference are deleted, and each crossing time is found by a backward search from the previous one. It is also proved in \cite[Section~8.4]{oai-barriers} that the deleted references form a final segment, with at most $H$ elements, of the retained references ordered by level, and that older blocks are protected by the complete block that precedes each reset; we give these arguments in \cref{lem:ind-deletion}. What is new is the form of the induction hypothesis. In \cite[(8.19)--(8.20)]{oai-barriers}, after $j$ levels, the overlaps of every retained reference with the current target and with the later references are controlled within $6j\varepsilon$, where $\varepsilon=\rho^{20}$ is the scale of the narrow overlap constraints there. These errors accumulate over the levels, and the final count of \cite[(8.29)]{oai-barriers} requires $N_{\rm lev}\varepsilon$ to be much smaller than $\rho$ for $N_{\rm lev}$ levels. Here we compare each retained reference with the next retained reference. This comparison changes only when a deletion removes the next retained reference, which happens at most once for each reference. The error of every constraint stays bounded by a fixed multiple of $\varepsilon$ at every level (\cref{def:ind-invariant}). The argument uses that the first two levels of every block are never deleted, which follows from the same protection argument. As a result, $\varepsilon_n=E\rho_n$ with a fixed $E>0$ is allowed.

\section{Standing assumptions and the threshold geometry}\label{sec:ind-setting}

Throughout this section and \cref{sec:barrier-proof}, $\beta>1$ is fixed, $n$ is large, and the disorder $J$ is fixed; we write $\pi=\pi^J$. We use the parameters of \cref{sec:references}, which are listed in \cref{tab:ref-parameters}. The overlap $q_*\in(0,1)$ is that of the overlap-neighborhood mass bound, \cref{b:lem:coverage-extension}. The signed-overlap cutoffs $0<b_2<b_1<1$ and the level $t_0>0$ satisfy \eqref{b:dyn:fixed-cutoffs}, and in particular
\[
 4t_0<\min\{q_*,b_2,b_1^2/2\}.
\]
This implies that $3t_0<b_1^2/2<b_1$, since $b_1<1$. The compact interval $J_\beta$ of \eqref{eq:ref-intervals} contains $[t_0,2t_0]$ in its interior. The integer $H=\lceil2/b_1^2\rceil$ is that of \eqref{b:dyn:BH}, and $B\ge H+2$ is a fixed integer, the number of levels in a block; it is chosen in \cref{sec:barrier-proof}. The scales of \eqref{b:dyn:scales} are
\[
 \rho=\rho_n=Rn^{-\lambda},\qquad \varepsilon=\varepsilon_n=E\rho_n,\qquad \eta_n=n^{-\xi},
\]
where $R>0$ and $E=1/100$ are fixed, and the exponents satisfy $0<\lambda<\xi<1/4$ by \eqref{eq:ref-exponents}. \Cref{thm:barrier-730} uses $\lambda=7/30$ and $\xi=5/21$. In \cref{sec:bp-strong} we use $\eta_n=\rho_n/1000$ in place of $\eta_n=n^{-\xi}$ with $\xi>\lambda$; this is why the hypotheses below allow a positive limit of $\eta_n/\rho_n$. We suppress $n$ from $\rho_n$ and $\varepsilon_n$, and we abbreviate
\[
 \zeta=2\eta_n,
\]
which is the tolerance of the overlap comparisons in \cref{lem:ref-families}. For the deterministic induction in this section, the asymptotic scale assumptions needed are $\varepsilon=E\rho$ and
\begin{equation}\label{eq:ind-scales}
 \rho\to0,\qquad \frac{\eta_n}{\rho}\to E',\qquad \frac1{n\rho}\to0,
\end{equation}
where $E'\in[0,E/10]$ is fixed. For \cref{thm:barrier-730}, these hold with $E'=0$ because $0<\lambda<\xi$ and $\lambda<1$; at $\lambda=7/30$ and $\xi=5/21$, we have $\eta_n/\rho_n=n^{-1/210}/R$. In \cref{sec:bp-strong}, they hold with $E'=E/10=1/1000$. The counting estimate in \cref{sec:barrier-proof} also retains $\rho_n=Rn^{-\lambda}$; the additional static scale conditions for \cref{sec:bp-strong} are checked there.

\paragraph{Reference families.} The levels are the integers $0\le i<N_{\rm lev}$, where $N_{\rm lev}=MB$ and $M$ is defined in \eqref{b:dyn:threshold-intervals} below. As in \cref{sec:ref-families}, level $i$ belongs to the block ${\ell}(i)=\lfloor i/B\rfloor$; we say that it is the first level of its block if $i=B\ell(i)$. A family of reference configurations consists of an ordered list $\mathcal F_i$ of configurations for each level $i$, whose coordinates are called the entries of $\mathcal F_i$, together with a set $V\subseteq\Sigma_n$. We call a configuration $\sigma y$, where $y$ is an entry of $\mathcal F_i$ and $\sigma\in\{-1,1\}$, a signed entry of level $i$; the set of signed entries of level $i$ is $\mathcal F_i^\pm$. For a finite tuple $T$ of configurations, the cell $S_T$ is the set of $x\in\Sigma_n$ with $|R(y,x)|\le3t_0$ for every $y$ in $T$, as in \eqref{eq:ref-cell}, and $S_T=\Sigma_n$ for the empty tuple. Reversing the sign of an element of $T$ does not change $S_T$.

By \cref{lem:ref-families}, applied with the logarithmic time scale $L_n=An^\lambda$ of \cref{sec:barrier-proof} and with $N_{\rm lev}\le e^{L_n}$, for all sufficiently large $n$ and every disorder in the event $\mathcal G_n$ of \eqref{eq:ref-good-event}, there exists a family with $\pi(V)\le3e^{-4L_n}$ and with the following four properties. They are consequences of properties (i)--(iv) of \cref{lem:ref-families}, as we explain after the list.
\begin{enumerate}[label=(P\arabic*)]
\item \emph{Overlap neighborhoods in every cell.} Let $j$ be a level, let $T$ be a tuple consisting of one signed entry of each earlier level of the block of $j$, and let $x\in S_T\setminus V$. Then there exists an entry $y$ of $\mathcal F_j$ with $y\in S_T$ and $|R(x,y)|\ge q_*$.
\item \emph{Comparison across the overlap $q_*$.} Let $x\notin V$, and let $v$ and ${\widetilde v}$ be signed entries of two distinct levels. If $R(x,{\widetilde v})\ge q_*$, $|R(v,x)|\le b_1$, and $|R(v,{\widetilde v})|\le b_1$, then $|R(v,x)-R(v,{\widetilde v})|\le\zeta$.
\item \emph{Comparison across the overlap $b_1$.} Let $v$, $y$, and ${\widetilde v}$ be signed entries of three distinct levels. If $|R(y,{\widetilde v})|\ge b_1$, $|R(v,y)|\le b_2$, and $|R(v,{\widetilde v})|\le b_2$, then $\bigl||R(v,y)|-|R(v,{\widetilde v})|\bigr|\le\zeta$.
\item \emph{No narrow triples.} Let $x\notin V$, and let $v$ and ${\widetilde v}$ be signed entries of two distinct levels. Let $r=(R(v,x)+R(v,{\widetilde v}))/2$ and $\delta=(R(v,x)-R(v,{\widetilde v}))/2$. Then it is not the case that
\[
 r\in J_\beta,\qquad \varepsilon\le|\delta|\le2\varepsilon,\qquad R({\widetilde v},x)\ge r+\rho;
\]
that is, the triple $(v,x,{\widetilde v})$ does not belong to the set $\mathcal N_n$ of narrow triples defined after \eqref{b:dyn:narrow-coordinates}.
\end{enumerate}
Property (P1) is the case of property (i) in which the tuple has one entry at each earlier level of the block of $j$; here we used that the cell of a tuple of signed entries equals the cell of the corresponding tuple of entries. Properties (P2) and (P4) are properties (ii) and (iv), and (P3) is the last assertion of property (iii). Properties (P2) and (P3) say that, across a large overlap between $x$ and ${\widetilde v}$ (or between $y$ and ${\widetilde v}$), a small overlap with $v$ is transferred from one configuration to the other with error at most $\zeta$. The family and $V$ do not depend on any path or threshold, and the four properties are deterministic statements about them (see \cref{rem:ref-every-path}).

\paragraph{Paths.} We call a finite sequence $x_0,\dots,x_m$ in $\Sigma_n$ a path if
\begin{equation}\label{eq:ind-jump}
 |R(y,x_{k+1})-R(y,x_k)|\le2/n\qquad\text{for all }y\in\Sigma_n\text{ and }0\le k<m.
\end{equation}
We say that a path avoids $V$ if $x_k\notin V$ for $0\le k\le m$. If $x_{k+1}$ and $x_k$ differ in at most one coordinate, then \eqref{eq:ind-jump} holds; in particular, every trajectory of the discrete heat-bath chain is a path.

\paragraph{Thresholds.} We set
\begin{equation}\label{b:dyn:threshold-intervals}
 M=\Bigl\lfloor\frac{t_0}{3B\rho}\Bigr\rfloor,\qquad N_{\rm lev}=MB,\qquad
 I_i=\bigl[t_0+3i\rho,\ t_0+(3i+1)\rho\bigr]\quad(0\le i<N_{\rm lev}).
\end{equation}
For all sufficiently large $n$, we have $M\ge1$. Each interval $I_i$ has length $\rho$ and lies in $[t_0,2t_0]$, since $t_0+(3(N_{\rm lev}-1)+1)\rho\le t_0+3MB\rho\le2t_0$. We call a vector $\mathbf r=(r_0,\dots,r_{N_{\rm lev}-1})$ such that $r_i\in I_i$ for every $i$ a threshold array. For every threshold array and all levels $i<a$, we have
\begin{equation}\label{b:dyn:geometry}
 r_a-r_i\ge\bigl(3(a-i)-1\bigr)\rho\ge2\rho.
\end{equation}

\paragraph{Radii and margins.} We define
\begin{equation}\label{b:dyn:radii}
 \begin{aligned}
 \psi_0&=\zeta+\frac2n, & \psi_1&=2\varepsilon+2\zeta+\frac2n,\\
 \Psi_0&=\psi_0+2\varepsilon=2\varepsilon+\zeta+\frac2n,\qquad &
 \Psi_1&=\psi_1+2\varepsilon=4\varepsilon+2\zeta+\frac2n,\qquad
 \Psi_2=\Psi_1+\zeta.
 \end{aligned}
\end{equation}
In the induction, $\psi_0$ and $\psi_1$ bound the error, relative to the threshold, of the overlap of a retained reference with its next retained reference; $\Psi_0$ and $\Psi_1$ bound the corresponding error for the overlap with the current target; and $\Psi_2$ bounds it for the overlaps between retained references. We use the following inequalities:
\begin{equation}\label{b:dyn:finite-margins}
 \begin{gathered}
 \zeta<2\varepsilon,\qquad \frac1n\le\varepsilon,\qquad
 2t_0+\Psi_2<\min\{3t_0,b_2\},\qquad 2t_0+\Psi_1<q_*,\\
 2\Psi_2+\zeta<2\rho,\qquad 5\varepsilon+3\zeta+\frac7n\le\rho,
 \end{gathered}
\end{equation}
and the closed neighborhood of $[t_0,2t_0]$ of radius $3\varepsilon+2\zeta+3/n$ is contained in $J_\beta$.

\begin{lemma}[The margins hold for large $n$]\label{lem:ind-margins}
Assume \eqref{eq:ind-scales}. There exists $n_0$, depending only on the fixed parameters $q_*$, $b_2$, $t_0$, $J_\beta$, $R$, $E$, $E'$, and the sequences $(\rho_n)$ and $(\eta_n)$, such that \eqref{b:dyn:finite-margins} and the inclusion of the neighborhood in $J_\beta$ hold for all $n\ge n_0$.
\end{lemma}

\begin{proof}
By \eqref{b:dyn:radii}, we have
\[
 \frac{\zeta}{\rho}\to2E',\qquad
 \frac{2\Psi_2+\zeta}{\rho}=8E+\frac{7\zeta}{\rho}+\frac4{n\rho}\to8E+14E',\qquad
 \frac{5\varepsilon+3\zeta+7/n}{\rho}\to5E+6E',
\]
where we used \eqref{eq:ind-scales} and $\varepsilon=E\rho$. Since $E'\le E/10$ and $E=1/100$, the three limits are at most $E/5<2E$, $9.4E<2$, and $5.6E<1$, respectively. The inequalities in the second line of \eqref{b:dyn:finite-margins} hold for large $n$, and so do $\zeta<2\varepsilon$ and $1/n\le\varepsilon$. The radii $\Psi_2$ and $\Psi_1$ tend to zero, while $2t_0<3t_0$, $2t_0<b_2$, and $2t_0<q_*$ by \eqref{b:dyn:fixed-cutoffs}. This gives the remaining two inequalities. Finally, the compact set $[t_0,2t_0]$ lies in the interior of $J_\beta$ and has positive distance from its complement, while the radius $3\varepsilon+2\zeta+3/n$ tends to zero.
\end{proof}

We record three consequences of \eqref{b:dyn:finite-margins} that are used repeatedly. First, we have $\Psi_1<\Psi_2<t_0$, and every overlap within $\Psi_2$ of a threshold $r_i\in[t_0,2t_0]$ is positive and has absolute value less than $2t_0+\Psi_2<3t_0$. Second, such an overlap has absolute value less than $b_2<b_1$, because $2t_0+\Psi_2<b_2$. Third, every threshold is smaller than $q_*$, because $r_i\le2t_0<q_*$.

\section{The selection and crossing algorithm}\label{sec:ind-algorithm}

We now define the references and the crossing times. The algorithm is that of \cite[Section~8.4]{oai-barriers}, including its conventions on failure, with zero-based levels.

\begin{definition}[Selection and crossing algorithm]\label{def:ind-algorithm}
Fix a family of reference configurations, a path $x_0,\dots,x_m$, and a threshold array $\mathbf r$.
\begin{enumerate}[label=(\roman*)]
\item \emph{Level $0$.} If some entry $y$ of $\mathcal F_0$ satisfies $|R(y,x_0)|\ge q_*$, let $y$ be the first such entry and set $v_0=\sigma y$, where $\sigma$ is the sign of $R(y,x_0)$, so that $R(v_0,x_0)\ge q_*$. If the set $\{0\le k\le m:R(v_0,x_k)\le r_0\}$ is also nonempty, let $k_0$ be its smallest element and set $\mathcal R_0=\{0\}$; level $0$ then succeeds. If no entry qualifies, let $v_0$ be the first entry of $\mathcal F_0$; level $0$ then fails, as it does when $v_0$ is selected but the set defining $k_0$ is empty.
\item \emph{Level $j\ge1$, when levels $0,\dots,j-1$ have succeeded.} Write ${\bar k}=k_{j-1}$, let $T_j$ be the tuple $(v_l)$ over the levels $l<j$ of the block of $j$ (empty if $j$ is the first level of its block), and write $S_j=S_{T_j}$.
\begin{enumerate}[label=(\alph*)]
\item \emph{Selection.} If some entry $y$ of $\mathcal F_j$ satisfies $y\in S_j$ and $|R(y,x_{\bar k})|\ge q_*$, let $y$ be the first such entry and set $v_j=\sigma y$, where $\sigma$ is the sign of $R(y,x_{\bar k})$, so that $R(v_j,x_{\bar k})\ge q_*$. Otherwise let $v_j$ be the first entry of $\mathcal F_j$; level $j$ then fails.
\item \emph{Deletion.} The conflicts at level $j$ are the elements of $\mathcal D_j=\{i\in\mathcal R_{j-1}:|R(v_i,v_j)|>b_1\}$, and we set $\mathcal R_j=(\mathcal R_{j-1}\setminus \mathcal D_j)\cup\{j\}$. The elements of $\mathcal R_{j-1}\setminus \mathcal D_j$ are called compatible.
\item \emph{Crossing.} If the set $\{0\le k\le {\bar k}:R(v_j,x_k)\le r_j\}$ is nonempty, let $k_j$ be its largest element; level $j$ then succeeds. Otherwise level $j$ fails, and $v_j$ is kept as the reference of level $j$.
\end{enumerate}
\item \emph{After a failure.} If some level fails, the algorithm stops, and for every later level $i$ we let $v_i$ be the first entry of $\mathcal F_i$, without a change of sign.
\end{enumerate}
We say that the algorithm succeeds through level $j$ if levels $0,\dots,j$ all succeed. We write $v(x_0)$ for the reference $v_0$ of level $0$.
\end{definition}

The reference $v(x_0)$ depends only on $x_0$ and $\mathcal F_0$. The set $\mathcal R_j$ is the set of retained levels after level $j$, and the references $v_i$ with $i\in\mathcal R_j$ are the retained references. We order $\mathcal R_j$ increasingly; its largest element is $j$, the level of the newest reference. The crossing time at level $0$ is the first time $k$ with $R(v_0,x_k)\le r_0$, and the crossing time at level $j\ge1$ is the last time $k\le k_{j-1}$ with $R(v_j,x_k)\le r_j$; in particular $k_0\ge k_1\ge\cdots$. These crossing times are not stopping times of the path, and nothing below requires them to be.

The convention in (ii)(c), that a selected reference whose crossing fails is kept, matters for the following lemma. If $v_j$ were replaced by a default after its crossing failed, it would depend on its own threshold $r_j$.

\begin{lemma}[References depend only on earlier thresholds]\label{lem:ind-predictable}
Fix the family and the path. For every level $i$, the reference $v_i$ is a function of $(r_0,\dots,r_{i-1})$ alone, with finitely many values, each taken on a Borel set; the reference $v_0=v(x_0)$ does not depend on $\mathbf r$.
\end{lemma}

\begin{proof}
Let $\mathrm{ok}_j$ be the indicator that the algorithm succeeds through level $j$. We show by induction on $j$ that $v_j$ is a finite-valued Borel function of $(r_0,\dots,r_{j-1})$, that $\mathrm{ok}_j$ is a Borel function of $(r_0,\dots,r_j)$, and that $k_j$ and $\mathcal R_j$, restricted to $\{\mathrm{ok}_j=1\}$, are finite-valued Borel functions of $(r_0,\dots,r_j)$. For $j=0$, the selection uses only $x_0$ and $\mathcal F_0$, and $v_0$ is constant. If no entry qualifies, then $\mathrm{ok}_0=0$ identically. Otherwise $v_0$ is fixed, and the event $\{\mathrm{ok}_0=1,k_0=k\}$ is described by the finitely many inequalities $R(v_0,x_l)>r_0$ for $l<k$ and $R(v_0,x_k)\le r_0$; on it, $\mathcal R_0=\{0\}$. For $j\ge1$, on $\{\mathrm{ok}_{j-1}=0\}$ the reference $v_j$ is the fixed default and $\mathrm{ok}_j=0$. On $\{\mathrm{ok}_{j-1}=1\}$, the cell $S_j$, the target $x_{k_{j-1}}$, and therefore the selection, depend only on $v_0,\dots,v_{j-1}$ and $k_{j-1}$, which are Borel functions of $(r_0,\dots,r_{j-1})$ there by the induction hypothesis. The threshold $r_j$ enters only through the crossing, which does not change $v_j$, and for fixed earlier data the event $\{\mathrm{ok}_j=1,k_j=k\}$ is described by finitely many inequalities between $r_j$ and the overlaps $R(v_j,x_l)$. On $\{\mathrm{ok}_j=1\}$, the set $\mathcal R_j$ is determined by $\mathcal R_{j-1}$ and $v_0,\dots,v_j$.
\end{proof}

\paragraph{Next retained references.} Suppose that the algorithm succeeds through level $j-1$ and that the selection at level $j$ succeeds, so that $\mathcal R_j$ is defined (for $j=0$, suppose that level $0$ succeeds). For $i\in\mathcal R_j\setminus\{j\}$, the index of the next retained reference of $i$ after level $j$ is
\[
 a_j(i)=\min\{a\in\mathcal R_j:a>i\},
\]
the next retained reference of $i$ is $v_{a_j(i)}$, and we write $c_j(i)=R(v_i,v_{a_j(i)})$. We say that the next retained reference of $i$ is unchanged if $a_j(i)=i+1$, and replaced otherwise. In \cref{def:ind-invariant} below, the overlap $c_j(i)$ is the center around which the overlaps of $v_i$ with the configurations of the path are controlled.

\begin{definition}[The invariant after level $j$]\label{def:ind-invariant}
Suppose that the algorithm succeeds through level $j$. The invariant $\mathbf I(j)$ consists of the following five statements.
\begin{enumerate}[label=(\alph*)]
\item \emph{Blocks.} Every level of the block of $j$ that is at most $j$ belongs to $\mathcal R_j$. For every block ${\ell}<{\ell}(j)$, the set of levels of block ${\ell}$ that belong to $\mathcal R_j$ is $\{{\ell}B,{\ell}B+1,\dots,{\ell}B+B-d-1\}$ for some $0\le d\le H$.
\item \emph{The newest reference.} We have
\begin{equation}\label{b:dyn:newest-band}
 R(v_j,x_{k_j})\in[r_j-2/n,\,r_j].
\end{equation}
\item \emph{Centers.} For every $i\in\mathcal R_j\setminus\{j\}$, we have $|c_j(i)-r_i|\le\psi_0$ if the next retained reference of $i$ is unchanged. Otherwise $|c_j(i)-r_i|\le\psi_1$, and $a_j(i)=B({\ell}(i)+1)$ is the first level of the block that follows the block of $i$.
\item \emph{Windows.} For every $i\in\mathcal R_j\setminus\{j\}$, we have
\begin{equation}\label{b:dyn:anchor-invariant}
 |R(v_i,x_{k_j})-c_j(i)|<2\varepsilon.
\end{equation}
\item \emph{Pairs.} For all $i<a$ in $\mathcal R_j$, we have
\begin{equation}\label{b:dyn:pair-invariant}
 |R(v_i,v_a)-r_i|\le\Psi_2.
\end{equation}
\end{enumerate}
\end{definition}

Statement (a) implies that $0\in\mathcal R_j$: level $0$ is the first level of block $0$, and since $B-H\ge2$, the first two levels of every earlier block ${\ell}<{\ell}(j)$ are retained. Combining (c) and (d), we find that every $i\in\mathcal R_j\setminus\{j\}$ satisfies
\begin{equation}\label{eq:ind-target-error}
 |R(v_i,x_{k_j})-r_i|<
 \begin{cases}
 \Psi_0&\text{if the next retained reference of $i$ is unchanged},\\
 \Psi_1&\text{if it is replaced}.
 \end{cases}
\end{equation}

\section{One level of the algorithm}\label{b:sec:dyn-anchors}

The following five lemmas carry out one step of the induction. They share these hypotheses, which we call $(\mathrm H_j)$: the inequalities \eqref{b:dyn:finite-margins} and the inclusion of the neighborhood in $J_\beta$ hold; the family satisfies (P1)--(P4); the path $x_0,\dots,x_m$ avoids $V$; $\mathbf r$ is a threshold array; $1\le j<N_{\rm lev}$; the algorithm succeeds through level $j-1$; and $\mathbf I(j-1)$ holds. We write ${\bar k}=k_{j-1}$ for the previous crossing time and call $x_{\bar k}$ the target of level $j$. Statements (a), (c), and (e) of $\mathbf I(j)$ concern only the set $\mathcal R_j$ and the references; in the lemmas below we use them as soon as $\mathcal R_j$ is defined, before the crossing at level $j$ is known to exist.

\begin{lemma}[Selection and transfer]\label{lem:ind-transfer}
Assume $(\mathrm H_j)$.
\begin{enumerate}[label=(\alph*)]
\item For every $i\in\mathcal R_{j-1}$, we have $|R(v_i,x_{\bar k})-r_i|\le\Psi_1$. More precisely, the error is at most $2/n$ if $i=j-1$, and less than $\Psi_0$ if $i<j-1$ and the next retained reference of $i$ after level $j-1$ is unchanged.
\item The selection at level $j$ succeeds, and $R(v_j,x_{\bar k})\ge q_*$.
\item For every compatible $i\in\mathcal R_{j-1}$, we have
\begin{equation}\label{b:dyn:target-transfer}
 |R(v_i,v_j)-R(v_i,x_{\bar k})|\le\zeta,\qquad |R(v_i,v_j)-r_i|\le\Psi_2.
\end{equation}
\end{enumerate}
\end{lemma}

\begin{proof}
Part (a) is \eqref{b:dyn:newest-band} at level $j-1$ for $i=j-1$, and \eqref{eq:ind-target-error} at level $j-1$ otherwise; note that $2/n\le\Psi_0<\Psi_1$.

For part (b), if the tuple $T_j$ is nonempty, then $j$ is not the first level of its block, the block of $j$ is the block of $j-1$, and every level $l$ in $T_j$ belongs to $\mathcal R_{j-1}$ by $\mathbf I(j-1)$(a). Using part (a) and \eqref{b:dyn:finite-margins}, we find $|R(v_l,x_{\bar k})|\le r_l+\Psi_1\le2t_0+\Psi_1<3t_0$ for every such $l$. Therefore $x_{\bar k}\in S_j$. Since also $x_{\bar k}\notin V$, property (P1), applied with the tuple $T_j$, gives an entry $y$ of $\mathcal F_j$ with $y\in S_j$ and $|R(y,x_{\bar k})|\ge q_*$, and the selection succeeds.

For part (c), let $i\in\mathcal R_{j-1}$ be compatible, which means that $|R(v_i,v_j)|\le b_1$. By part (a) and \eqref{b:dyn:finite-margins}, we have $|R(v_i,x_{\bar k})|\le2t_0+\Psi_1<b_1$. Further, $R(x_{\bar k},v_j)\ge q_*$ by part (b), and $v_i$ and $v_j$ are signed entries of the distinct levels $i<j$. Property (P2), applied with $x=x_{\bar k}$, $v=v_i$, and ${\widetilde v}=v_j$, implies the first inequality in \eqref{b:dyn:target-transfer}. The second follows from the first and part (a), since $\zeta+\Psi_1=\Psi_2$.
\end{proof}

\begin{lemma}[Deletion and protection of initial segments]\label{lem:ind-deletion}
Assume $(\mathrm H_j)$.
\begin{enumerate}[label=(\alph*)]
\item The set $\mathcal D_j$ of conflicts is a final segment of $\mathcal R_{j-1}$: if $i<a$ belong to $\mathcal R_{j-1}$ and $i\in \mathcal D_j$, then $a\in \mathcal D_j$.
\item We have $|\mathcal D_j|<2/b_1^2\le H$.
\item If $j$ is not the first level of its block, then $\mathcal D_j=\emptyset$.
\item If $j$ is the first level of its block, then $\mathcal D_j$ consists of the last $d$ levels of the block ${\ell}(j)-1$, for some $0\le d\le H$. In particular, the first $B-H\ge2$ levels of the block ${\ell}(j)-1$ belong to $\mathcal R_j$, and no level of a block ${\ell}<{\ell}(j)-1$ is deleted.
\end{enumerate}
As a consequence, $\mathbf I(j)$(a) holds.
\end{lemma}

\begin{proof}
The final-segment, conflict-count, and block-protection arguments are those
of \cite[Section~8.4, ``Conflicts occupy a bounded final segment'']{oai-barriers},
applied here with the bounds of $\mathbf I(j-1)$.
For part (a), suppose to the contrary that $i<a$ belong to $\mathcal R_{j-1}$, that $i\in \mathcal D_j$ (that is, $|R(v_i,v_j)|>b_1$), and that $a$ is compatible. By \eqref{b:dyn:pair-invariant} at level $j-1$, we have $|R(v_i,v_a)-r_i|\le\Psi_2$, and by \eqref{b:dyn:target-transfer}, we have $|R(v_a,v_j)-r_a|\le\Psi_2$. By the consequences of \eqref{b:dyn:finite-margins} recorded after \cref{lem:ind-margins}, both overlaps are positive and have absolute value less than $b_2$. Since $v_a$, $v_i$, and $v_j$ are signed entries of the three distinct levels $a$, $i$, and $j$, property (P3), applied with $v=v_a$, $y=v_i$, and ${\widetilde v}=v_j$, gives $|R(v_a,v_j)|-|R(v_a,v_i)|\le\zeta$. On the other hand, using \eqref{b:dyn:geometry} and \eqref{b:dyn:finite-margins}, we have
\[
 |R(v_a,v_j)|-|R(v_a,v_i)|=R(v_a,v_j)-R(v_i,v_a)\ge(r_a-\Psi_2)-(r_i+\Psi_2)\ge2\rho-2\Psi_2>\zeta,
\]
which is a contradiction.

For part (b), we may assume that $d=|\mathcal D_j|\ge1$. For $i\in \mathcal D_j$, let $\sigma_i$ be the sign of $R(v_i,v_j)$, and consider the unit vectors $\mathbf e=v_j/\sqrt n$ and $\mathbf e_i=\sigma_iv_i/\sqrt n$ in $\R^n$. Then $\langle\mathbf e_i,\mathbf e\rangle=|R(v_i,v_j)|>b_1$, and for distinct $i,i'\in \mathcal D_j$, \eqref{b:dyn:pair-invariant} at level $j-1$ and the consequences of \eqref{b:dyn:finite-margins} recorded after \cref{lem:ind-margins} give $|\langle\mathbf e_i,\mathbf e_{i'}\rangle|=|R(v_i,v_{i'})|<3t_0$. By the Cauchy--Schwarz inequality,
\[
 d^2b_1^2<\Bigl\langle\sum_{i\in \mathcal D_j}\mathbf e_i,\mathbf e\Bigr\rangle^2\le\Bigl\|\sum_{i\in \mathcal D_j}\mathbf e_i\Bigr\|^2
 =d+\sum_{i\neq i'}\langle\mathbf e_i,\mathbf e_{i'}\rangle\le d+3t_0d(d-1).
\]
Dividing by $d$, we obtain $d(b_1^2-3t_0)<1-3t_0$. Since $3t_0<b_1^2/2$, this implies
\[
 d<\frac{1-3t_0}{b_1^2-3t_0}<\frac2{b_1^2}\le H,
\]
where the middle inequality is equivalent to $3t_0(2-b_1^2)<b_1^2$.

For part (c), the level $j-1$ belongs to the block of $j$. Then $v_{j-1}$ is an element of $T_j$, and since $v_j\in S_j$, we have $|R(v_{j-1},v_j)|\le3t_0<b_1$. In other words, the largest element $j-1$ of $\mathcal R_{j-1}$ is compatible, and the final segment $\mathcal D_j$ is empty by part (a).

For part (d), let $j={\ell}B$ with ${\ell}\ge1$. The level $j-1$ is the last level of block ${\ell}-1$, which is the block of $j-1$. By $\mathbf I(j-1)$(a), all $B$ levels of block ${\ell}-1$ belong to $\mathcal R_{j-1}$, and they are its $B$ largest elements. By parts (a) and (b), $\mathcal D_j$ is a final segment of $\mathcal R_{j-1}$ with fewer than $H<B$ elements. It therefore consists of the last $d=|\mathcal D_j|\le H$ levels of block ${\ell}-1$.

Finally, we verify $\mathbf I(j)$(a). If $j$ is not the first level of its block, nothing is deleted by part (c), and $\mathcal R_j=\mathcal R_{j-1}\cup\{j\}$; statement (a) at level $j-1$ gives statement (a) at level $j$. If $j$ is the first level of its block, the block of $j$ meets $\mathcal R_j$ in $\{j\}$, the retained levels of block ${\ell}(j)-1$ form an initial segment of length $B-d$ with $d\le H$ by part (d), and the earlier blocks are unchanged.
\end{proof}

Part (d) is the protection of initial segments: once the reset at the first level of block ${\ell}+1$ has occurred, the retained levels of block ${\ell}$ form an initial segment of length at least $B-H\ge2$ that is never changed afterward. Indeed, a later deletion removes only levels of the block immediately preceding the current reset, and that block is complete when the reset occurs. This is proved by induction along the run of the algorithm, using only $\mathbf I(j-1)$.

\begin{lemma}[Next retained references after a deletion]\label{lem:ind-successor}
Assume $(\mathrm H_j)$, so that $\mathcal R_j$ is defined by \cref{lem:ind-transfer}(b).
\begin{enumerate}[label=(\alph*)]
\item Let $i\in\mathcal R_j\setminus\{j\}$ with $i\neq j-1$. If $a_j(i)\neq a_{j-1}(i)$, then $j$ is the first level of its block, $\mathcal D_j\neq\emptyset$, $i$ is the largest element of $\mathcal R_{j-1}\setminus \mathcal D_j$ and lies in block ${\ell}(j)-1$, $a_{j-1}(i)=i+1$, and $a_j(i)=j$.
\item For every $i\in\mathcal R_j\setminus\{j\}$, the bounds of $\mathbf I(j)$(c) hold, and $|R(v_i,x_{\bar k})-c_j(i)|<2\varepsilon$.
\item The statement $\mathbf I(j)$(e) holds.
\end{enumerate}
\end{lemma}

\begin{proof}
For part (a), note that $i$ is not the largest element $j-1$ of $\mathcal R_{j-1}$, and that $a_{j-1}(i)$ is defined. Since $\mathcal R_j$ is obtained from $\mathcal R_{j-1}$ by removing $\mathcal D_j$ and adding $j$, which exceeds every element of $\mathcal R_{j-1}$, the next retained reference of $i$ changes only if $a_{j-1}(i)\in \mathcal D_j$. Then $\mathcal D_j\neq\emptyset$, and by \cref{lem:ind-deletion}(c), $j$ is the first level of its block. Since $\mathcal D_j$ is a final segment of $\mathcal R_{j-1}$ by \cref{lem:ind-deletion}(a), every element of $\mathcal R_{j-1}$ larger than $i$ lies in $\mathcal D_j$. This means that $i$ is the largest element of $\mathcal R_{j-1}\setminus \mathcal D_j$ and that $a_j(i)=j$. By \cref{lem:ind-deletion}(d), the first $B-H$ levels of block ${\ell}(j)-1$ survive, and $i$ lies in this block. Since this block was complete in $\mathcal R_{j-1}$, we have $a_{j-1}(i)=i+1$.

For part (b), there are three cases.
\emph{Case 1: $i\neq j-1$ and $a_j(i)=a_{j-1}(i)$.} Then $c_j(i)=c_{j-1}(i)$, the next retained reference of $i$ is unchanged after level $j$ if and only if it is unchanged after level $j-1$, and the claims are $\mathbf I(j-1)$(c) and (d).
\emph{Case 2: $i=j-1$.} Then $a_j(i)=j=i+1$, and the next retained reference of $i$ is unchanged. Using \eqref{b:dyn:target-transfer} and \eqref{b:dyn:newest-band} at level $j-1$, we find
\[
 |c_j(i)-R(v_i,x_{\bar k})|\le\zeta,\qquad |c_j(i)-r_i|\le\zeta+\frac2n=\psi_0,
\]
and $|R(v_i,x_{\bar k})-c_j(i)|\le\zeta<2\varepsilon$ by \eqref{b:dyn:finite-margins}.
\emph{Case 3: $i\neq j-1$ and $a_j(i)\neq a_{j-1}(i)$.} By part (a), $a_{j-1}(i)=i+1$ and $a_j(i)=j=B({\ell}(i)+1)$, which is the first level of the block that follows the block of $i$. Since $i+1\in \mathcal D_j$, we have $j\neq i+1$, and the next retained reference of $i$ is replaced. Before the deletion, we have $|c_{j-1}(i)-r_i|\le\psi_0$ and $|R(v_i,x_{\bar k})-c_{j-1}(i)|<2\varepsilon$ by $\mathbf I(j-1)$(c) and (d). By \eqref{b:dyn:target-transfer}, $|c_j(i)-R(v_i,x_{\bar k})|\le\zeta$. Combining these three bounds, we obtain
\[
 |c_j(i)-r_i|<\zeta+2\varepsilon+\psi_0=\psi_1,
\]
and $|R(v_i,x_{\bar k})-c_j(i)|\le\zeta<2\varepsilon$.

For part (c), a pair $i<a$ of elements of $\mathcal R_j$ with $a<j$ is a pair of elements of $\mathcal R_{j-1}$, and \eqref{b:dyn:pair-invariant} at level $j-1$ applies. If $a=j$, then $i$ is compatible, and \eqref{b:dyn:target-transfer} gives $|R(v_i,v_j)-r_i|\le\Psi_2$.
\end{proof}

The next lemma treats the backward search for the crossing at level $j$. Its main point is that the windows of all old references are preserved simultaneously. To show this, we take the largest index at which some window fails, for all references together. At that index every window is still almost preserved, and in particular so is the window of the next retained reference of the failing reference. This gives the lower bound on the overlap $q$ in part (b) of the lemma.

\begin{lemma}[The backward search]\label{lem:ind-search}
Assume $(\mathrm H_j)$. Let $\underline k=k_j$ if the set defining $k_j$ in \cref{def:ind-algorithm}(ii)(c) is nonempty, and $\underline k=0$ otherwise.
\begin{enumerate}[label=(\alph*)]
\item For every $k$ with $\underline k\le k\le {\bar k}$, we have $R(v_j,x_k)\ge r_j-2/n$. If the crossing exists, then $k_j<{\bar k}$.
\item Suppose that $|R(v_i,x_k)-c_j(i)|\ge2\varepsilon$ for some $k\in[\underline k,{\bar k}]$ and some $i\in\mathcal R_j\setminus\{j\}$. Let $k$ be the largest such index, let $i$ be such that the inequality holds at $k$, write $a=a_j(i)$, and set
\[
 r=\frac{R(v_i,x_k)+c_j(i)}2,\qquad \delta=\frac{R(v_i,x_k)-c_j(i)}2,\qquad q=R(v_a,x_k).
\]
Then
\begin{equation}\label{b:dyn:first-exit-gap}
 \varepsilon\le|\delta|\le2\varepsilon,\qquad
 |r-r_i|\le3\varepsilon+2\zeta+\frac3n,\qquad
 q-r\ge2\rho-5\varepsilon-3\zeta-\frac7n\ge\rho,
\end{equation}
so that $(v_i,x_k,v_a)$ is a narrow triple in the sense of (P4).
\item For every $k$ with $\underline k\le k\le {\bar k}$ and every $i\in\mathcal R_j\setminus\{j\}$, we have $|R(v_i,x_k)-c_j(i)|<2\varepsilon$.
\end{enumerate}
\end{lemma}

\begin{proof}
For part (a), suppose first that the crossing exists. By the definition of $k_j$ as a largest element, we have $R(v_j,x_k)>r_j$ for $k_j<k\le {\bar k}$. By \cref{lem:ind-transfer}(b), we have $R(v_j,x_{\bar k})\ge q_*>r_j$, which implies that $k_j<{\bar k}$. Then \eqref{eq:ind-jump} gives $R(v_j,x_{k_j})\ge R(v_j,x_{k_j+1})-2/n>r_j-2/n$. If the crossing does not exist, then $R(v_j,x_k)>r_j$ for every $0\le k\le {\bar k}$.

For part (b), the windows hold at $k={\bar k}$ by \cref{lem:ind-successor}(b). This implies that $k<{\bar k}$ and $k+1\in[\underline k,{\bar k}]$. By the maximality of $k$, every window holds strictly at $k+1$, and \eqref{eq:ind-jump} gives
\begin{equation}\label{eq:ind-almost}
 |R(v_{i'},x_k)-c_j(i')|<2\varepsilon+\frac2n\qquad\text{for every }i'\in\mathcal R_j\setminus\{j\}.
\end{equation}
For the chosen $i$, this means that $2\varepsilon\le2|\delta|<2\varepsilon+2/n$, and \eqref{b:dyn:finite-margins} implies that $\varepsilon\le|\delta|<\varepsilon+1/n\le2\varepsilon$. We have $|r-r_i|\le|c_j(i)-r_i|+|\delta|$, and the bounds on the centers in $\mathbf I(j)$(c), which hold by \cref{lem:ind-successor}(b), give
\[
 |r-r_i|<
 \begin{cases}
 \psi_0+\varepsilon+1/n=\varepsilon+\zeta+3/n&\text{if $a=i+1$},\\
 \psi_1+\varepsilon+1/n=3\varepsilon+2\zeta+3/n&\text{otherwise}.
 \end{cases}
\]
We now bound $q$ from below, distinguishing three cases.
\emph{Case 1: $a=j$.} By part (a), $q=R(v_j,x_k)\ge r_j-2/n$.
\emph{Case 2: $a<j$ and $a=i+1$.} Then $a\in\mathcal R_j\setminus\{j\}$. Using \eqref{eq:ind-almost} for $i'=a$ and $|c_j(a)-r_a|\le\psi_1$, we find
\[
 q>c_j(a)-2\varepsilon-\frac2n\ge r_a-\psi_1-2\varepsilon-\frac2n=r_a-4\varepsilon-2\zeta-\frac4n.
\]
\emph{Case 3: $a<j$ and $a\neq i+1$.} By $\mathbf I(j)$(c), $a$ is the first level of a block. The level $a+1$ belongs to the same block and satisfies $a+1\le j$. If this block is the block of $j$, then $a+1\in\mathcal R_j$ by $\mathbf I(j)$(a); otherwise $a+1\in\mathcal R_j$ because the retained initial segment of the block has length at least $B-H\ge2$. The next retained reference of $a$ is then unchanged, and $|c_j(a)-r_a|\le\psi_0$. By \eqref{eq:ind-almost} for $i'=a$, we have
\[
 q>r_a-\psi_0-2\varepsilon-\frac2n=r_a-2\varepsilon-\zeta-\frac4n.
\]
In Case 2 we use the first bound on $|r-r_i|$, and in Cases 1 and 3 the second. In all three cases, we obtain
\[
 q-r\ge r_a-r_i-5\varepsilon-3\zeta-\frac7n\ge2\rho-5\varepsilon-3\zeta-\frac7n\ge\rho,
\]
where we used \eqref{b:dyn:geometry} and the last inequality in \eqref{b:dyn:finite-margins}. This proves \eqref{b:dyn:first-exit-gap}. Further, $r_i\in[t_0,2t_0]$, and the bound on $|r-r_i|$ together with the inclusion of the neighborhood of $[t_0,2t_0]$ in $J_\beta$ implies that $r\in J_\beta$. Finally, $R(v_i,x_k)=r+\delta$, $R(v_i,v_a)=c_j(i)=r-\delta$, and $R(v_a,x_k)=q\ge r+\rho$, which are the conditions in (P4) for the triple $(v_i,x_k,v_a)$.

For part (c), suppose that the conclusion fails. Then part (b) produces a narrow triple $(v_i,x_k,v_a)$ in which $x_k\notin V$, because the path avoids $V$, and $v_i$ and $v_a$ are signed entries of the distinct levels $i<a$. This contradicts (P4).
\end{proof}

\begin{lemma}[The crossing exists]\label{lem:ind-crossing}
Assume $(\mathrm H_j)$. Then the crossing at level $j$ exists, the algorithm succeeds through level $j$, and \eqref{b:dyn:newest-band} holds at level $j$.
\end{lemma}

\begin{proof}
Suppose that the set defining $k_j$ is empty, so that $\underline k=0$ in \cref{lem:ind-search}. The level $0$ belongs to $\mathcal R_j$ by \cref{lem:ind-deletion}, and $0\neq j$. By \cref{lem:ind-search}(c) at $k=0$ and by $\mathbf I(j)$(c),
\[
 R(v_0,x_0)<c_j(0)+2\varepsilon\le r_0+\psi_1+2\varepsilon=r_0+\Psi_1\le2t_0+\Psi_1<q_*,
\]
where the last inequality is \eqref{b:dyn:finite-margins}. Recall that the selection of $v_0$ at level $0$ gives $R(v_0,x_0)\ge q_*$. This contradiction shows that the crossing exists. By \cref{lem:ind-search}(a), $R(v_j,x_{k_j})\ge r_j-2/n$, and $R(v_j,x_{k_j})\le r_j$ by the definition of $k_j$.
\end{proof}

\section{The induction}\label{sec:ind-induction}

\begin{proposition}[Overlap constraints along a path]\label{prop:ind-induction}
Assume that \eqref{b:dyn:finite-margins} and the inclusion of the neighborhood in $J_\beta$ hold, and that the family of reference configurations satisfies (P1)--(P4). Let $x_0,\dots,x_m$ be a path that avoids $V$, let $\mathbf r$ be a threshold array, and run the algorithm of \cref{def:ind-algorithm}. If
\[
 R\bigl(v(x_0),x_m\bigr)\le0,
\]
then the algorithm succeeds through level $N_{\rm lev}-1$, and $\mathbf I(j)$ holds for every $0\le j<N_{\rm lev}$. Moreover, the next retained reference of every reference is replaced at most once: if $i\in\mathcal R_j\setminus\{j\}$ and $a_j(i)\neq i+1$, then $i\in\mathcal R_{j'}$ and $a_{j'}(i)=a_j(i)$ for every $j'$ with $j\le j'<N_{\rm lev}$.
\end{proposition}

\begin{proof}
We first treat level $0$. Since $x_0\notin V$, property (P1), applied with $j=0$ and the empty tuple, gives an entry $y$ of $\mathcal F_0$ with $|R(y,x_0)|\ge q_*$. Then $v_0=v(x_0)$ is selected, and $R(v_0,x_0)\ge q_*>r_0$. Since $R(v_0,x_m)\le0<r_0$, the set defining $k_0$ is nonempty, and $k_0\ge1$. The minimality of $k_0$ gives $R(v_0,x_{k_0-1})>r_0$, and \eqref{eq:ind-jump} then implies that $R(v_0,x_{k_0})\in[r_0-2/n,r_0]$. Level $0$ succeeds, and $\mathbf I(0)$ holds: $\mathcal R_0=\{0\}$, statement (b) was just shown, and statements (c)--(e) are empty.

Suppose that $1\le j<N_{\rm lev}$, that the algorithm succeeds through level $j-1$, and that $\mathbf I(j-1)$ holds; these are the hypotheses $(\mathrm H_j)$. By \cref{lem:ind-transfer}(b), the selection at level $j$ succeeds, and by \cref{lem:ind-crossing}, so does the crossing, with \eqref{b:dyn:newest-band} at level $j$. Statement $\mathbf I(j)$(a) holds by \cref{lem:ind-deletion}, statements (c) and (e) hold by \cref{lem:ind-successor}, and statement (d) is \cref{lem:ind-search}(c) at $k=k_j$, which lies in $[\underline k,{\bar k}]$ with $\underline k=k_j$. This completes the induction.

For the last assertion, let $a=a_j(i)\neq i+1$. By $\mathbf I(j)$(c), $a=B({\ell}(i)+1)$ is the first level of the block that follows the block of $i$, and $a\le j$ since $a\in\mathcal R_j$. Let $j'>j$. If $j'$ is not the first level of its block, no deletion occurs at level $j'$ by \cref{lem:ind-deletion}(c). Otherwise $j'$ and $a$ are distinct first levels of blocks with $j'>a$, so that ${\ell}(j')-1\ge {\ell}(a)>{\ell}(i)$. By \cref{lem:ind-deletion}(d), a deletion at level $j'$ can remove only levels of the block ${\ell}(j')-1$ other than its first $B-H$ levels. It follows that $i$ is never deleted, and $a$, which is the first level of its block, is never deleted either. The levels strictly between $i$ and $a$ do not belong to $\mathcal R_j$, and they never belong to a later set $\mathcal R_{j'}$, since the levels added after level $j$ are larger than $j$. We conclude that $a_{j'}(i)=a$ for every $j'\ge j$.
\end{proof}

The windows of the replaced references are needed inside the induction, because the corresponding references carry constraints through later backward searches; for the count in \cref{sec:barrier-proof}, only the first level of each block is used.

\begin{corollary}[A common time with $M$ narrow windows]\label{cor:ind-witness}
Under the hypotheses of \cref{prop:ind-induction}, let $k_*=k_{N_{\rm lev}-1}$. Then $0\le k_*\le m$ and
\[
 |R(v_{{\ell}B},x_{k_*})-r_{{\ell}B}|<\Psi_0\qquad\text{for every }{\ell}\in\{0,1,\dots,M-1\}.
\]
\end{corollary}

\begin{proof}
Let $j=N_{\rm lev}-1=MB-1$, the last level of block $M-1$. Let ${\ell}\le M-1$. If ${\ell}<M-1$, then by $\mathbf I(j)$(a) the retained levels of block ${\ell}$ form an initial segment of length at least $B-H\ge2$, so that ${\ell}B$ and ${\ell}B+1$ belong to $\mathcal R_j$. If ${\ell}=M-1$, then all levels of block $M-1$ belong to $\mathcal R_j$, and ${\ell}B+1\le j$ since $B\ge2$. In both cases ${\ell}B\in\mathcal R_j\setminus\{j\}$ and $a_j({\ell}B)={\ell}B+1$; that is, the next retained reference of ${\ell}B$ is unchanged. By statements (c) and (d) of $\mathbf I(j)$, we conclude that $|R(v_{{\ell}B},x_{k_*})-r_{{\ell}B}|<\psi_0+2\varepsilon=\Psi_0$.
\end{proof}
\chapter{Proof of the barrier theorem}\label{sec:barrier-proof}

In this section, we prove \cref{thm:barrier-730} and \cref{b:cor:subcritical}, and, in \cref{sec:bp-strong}, the quantitative and almost-sure forms of \cref{thm:barrier-strong}. We first make the thresholds of \cref{sec:induction} random and show, using \cref{cor:ind-witness}, that a path avoiding $V$ keeps a positive overlap with its first reference during the first $T_n=\lceil n^2e^{L}\rceil$ update attempts (\cref{prop:bp-obstruction,lem:bp-coefficient}). This conclusion is deterministic: it holds for every path that avoids $V$, and the thresholds are only a device in its proof. Combining it with stationarity, spin-flip symmetry, and a Poisson clock, we obtain bounds on the Gibbs mass of the initial states from which the dynamics is close to equilibrium (\cref{prop:bp-quenched}). Integration over the disorder event of \cref{sec:references} then proves \cref{thm:barrier-730}.

The main inputs are \cref{lem:ind-predictable}, \cref{cor:ind-witness}, the families of reference configurations of \cref{sec:references} with the bound $\pi(V)\le3e^{-4L}$ (see \eqref{b:dyn:fixed-bank}), and the properties of the heat-bath dynamics in \cref{lem:model-dynamics}. The probability of the disorder event of \cref{sec:references} enters only once, in the last step of the proof of \cref{thm:barrier-730}. The quantities that are multiplied by the number of time indices are the Gibbs mass $\pi(V)$ and probabilities in an auxiliary space of thresholds, both at a fixed disorder.

The argument follows \cite[Sections~8.1 and~8.5]{oai-barriers}. The product bound for predictable thresholds is the elimination argument following \cite[(8.28)]{oai-barriers}. The reduction of total variation to a sign passage, with the half-space $\{y:R(v,y)\le0\}$ of Gibbs mass at least $1/2$ and a Poisson clock, is \cite[(8.1)--(8.3)]{oai-barriers}, together with the remark on discrete time at the end of \cite[Section~8.5]{oai-barriers}. There are three changes. First, the windows of \cref{cor:ind-witness} have radius of order $\varepsilon$ and sit at deterministic positions. We therefore count only the $M$ windows at the first levels of the blocks, in place of the union over subsets of size $N_{\rm lev}/B$ in \cite[(8.29)]{oai-barriers}. Second, the static estimates of \cref{sec:static} are uniform over the compact interval $J_\beta$, and we need no random event restricting the thresholds, such as \cite[(8.12)]{oai-barriers}. As a result, the sign obstruction holds for every path avoiding $V$, and not only with high probability. Third, the budget $T_n$ of update attempts has logarithm $L+O(\log n)$, in place of $2L$, and every fixed $\tau<1/2$ is allowed, in place of $\tau=1/4$.

We keep the notation and the standing assumptions of \cref{sec:ind-setting}. Recall that $\rho=Rn^{-\lambda}$, $\varepsilon=E\rho$ with $E=1/100$, $\zeta=2\eta_n$, the integers $M$ and $N_{\rm lev}=MB$ and the intervals $I_i$ are given by \eqref{b:dyn:threshold-intervals}, and the radius $\Psi_0=2\varepsilon+\zeta+2/n$ is given by \eqref{b:dyn:radii}.

\section{Predictable thresholds}\label{b:sec:dyn-count}

Fix the disorder, a family of reference configurations, and a path $x_0,\dots,x_m$. On the product $I_0\times\cdots\times I_{N_{\rm lev}-1}$, let $\PP^{\rm thr}$ be the product of the uniform probability measures on the intervals. Under $\PP^{\rm thr}$, the coordinates $r_0,\dots,r_{N_{\rm lev}-1}$ of a threshold array are independent and $r_i$ is uniform on $I_i$. This is an auxiliary probability space; it is not related to the randomness of the dynamics. By \cref{lem:ind-predictable}, for every fixed time $k\le m$ the overlap
\[
 z_i(k)=R\bigl(v_i,x_k\bigr)
\]
of the reference of level $i$ with $x_k$ is a Borel function of $(r_0,\dots,r_{i-1})$. The next lemma is the elimination argument of \cite[Section~8.5]{oai-barriers}.

\begin{lemma}[Windows with predictable centers]\label{lem:bp-product}
Let $I_0,\dots,I_{\bar N-1}$ be intervals of length $h>0$, and let $r_0,\dots,r_{\bar N-1}$ be independent random variables with $r_i$ uniform on $I_i$. Let $D\subseteq\{0,\dots,\bar N-1\}$, and for $i\in D$ let $a_i\ge0$ and let $z_i$ be a Borel function of $(r_0,\dots,r_{i-1})$ (a constant if $i=0$). Then
\[
 \PP\bigl\{|z_i-r_i|\le a_i\text{ for every }i\in D\bigr\}\le\prod_{i\in D}\frac{2a_i}{h}.
\]
\end{lemma}

\begin{proof}
We argue by induction on the number of elements of $D$; the case $D=\emptyset$ is trivial. Let $i_*$ be the largest element of $D$, let $W$ be the event $\{|z_i-r_i|\le a_i$ for every $i\in D\setminus\{i_*\}\}$, and let $\mathscr H$ be the $\sigma$-algebra generated by $r_0,\dots,r_{i_*-1}$. Then $W$ and $z_{i_*}$ are $\mathscr H$-measurable, and $r_{i_*}$ is independent of $\mathscr H$. It follows that
\[
 \PP\bigl\{|z_{i_*}-r_{i_*}|\le a_{i_*}\bigm|\mathscr H\bigr\}
 =\frac{\bigl|[z_{i_*}-a_{i_*},z_{i_*}+a_{i_*}]\cap I_{i_*}\bigr|}{h}\le\frac{2a_{i_*}}{h},
\]
where $|\cdot|$ denotes length. Multiplying by $\mathbf 1_W$ and taking expectations, we obtain
\[
 \PP\bigl(W\cap\{|z_{i_*}-r_{i_*}|\le a_{i_*}\}\bigr)\le\frac{2a_{i_*}}{h}\,\PP(W),
\]
and the induction hypothesis applied to $D\setminus\{i_*\}$ completes the proof.
\end{proof}

The order of integration matters: the largest index is removed first, because the later centers may depend on the earlier thresholds. In the lemma, we condition only on earlier thresholds. We do not condition on the event that the algorithm succeeds, nor on the sets $\mathcal R_j$ of retained levels, which depend on all thresholds. A bound $2a_i/h$ larger than one is allowed.

\section{A deterministic sign obstruction}\label{sec:bp-obstruction}

We define
\begin{equation}\label{b:dyn:window-weights}
 \gamma_n=\frac{2\Psi_0}{\rho}=4E+\frac{2\zeta}{\rho}+\frac4{n\rho},
\end{equation}
which is the ratio of the length of a window of radius $\Psi_0$ to the length $\rho$ of a threshold interval. By \eqref{eq:ind-scales}, we have $\gamma_n\to4(E+E')\le11/250$, and $\gamma_n\to4E=1/25$ when $E'=0$. For integers $m\ge0$ we set
\begin{equation}\label{b:dyn:prefix-count}
 C_n(m)=(m+1)\gamma_n^M.
\end{equation}

\begin{proposition}[Sign obstruction for paths avoiding $V$]\label{prop:bp-obstruction}
Assume that \eqref{b:dyn:finite-margins} and the inclusion of the neighborhood in $J_\beta$ hold, and that the family of reference configurations satisfies (P1)--(P4). Let $m\ge0$ be an integer with $C_n(m)<1$. Then
\begin{equation}\label{b:dyn:deterministic-obstruction}
 R\bigl(v(x_0),x_m\bigr)>0\qquad\text{for every path }x_0,\dots,x_m\text{ that avoids }V.
\end{equation}
\end{proposition}

\begin{proof}
Fix a path $x_0,\dots,x_m$ that avoids $V$, and suppose that $R(v(x_0),x_m)\le0$. Run the algorithm of \cref{def:ind-algorithm} with an arbitrary threshold array $\mathbf r$. Recall that the reference $v(x_0)$ of level $0$ does not depend on $\mathbf r$. The hypothesis of \cref{prop:ind-induction} therefore holds for every threshold array. By \cref{cor:ind-witness}, every threshold array belongs to the union over $0\le k\le m$ of the events
\[
 W_k=\bigl\{\mathbf r:|z_{\ell B}(k)-r_{\ell B}|\le\Psi_0\text{ for every }\ell\in\{0,\dots,M-1\}\bigr\},
\]
where $z_i(k)=R(v_i,x_k)$ and $v_i=v_i(r_0,\dots,r_{i-1})$ is the reference of level $i$ produced by the algorithm. By \cref{lem:ind-predictable}, each $z_{\ell B}(k)$ is a Borel function of $(r_0,\dots,r_{\ell B-1})$. Applying \cref{lem:bp-product} under $\PP^{\rm thr}$, with $h=\rho$, $D=\{0,B,\dots,(M-1)B\}$, and $a_i=\Psi_0$, we obtain $\PP^{\rm thr}(W_k)\le\gamma_n^M$. Summing over $k$, we find
\[
 1=\PP^{\rm thr}\Bigl(\bigcup_{k=0}^mW_k\Bigr)\le(m+1)\gamma_n^M=C_n(m)<1,
\]
which is a contradiction.
\end{proof}

The events $W_k$ use deterministic positions and deterministic radii. Their union contains the event produced by the algorithm, but we do not assert that the algorithm produces any particular configuration of retained references, and we do not condition on such an assertion. The proof also uses no property of the crossing times other than those established in \cref{sec:induction}; in particular, no distributional statement about the path at a crossing time is involved.

\section{The choice of the coefficient}\label{sec:bp-coefficient}

The count $C_n(m)$ is small when $\log(m+1)$ is smaller than $M\log(1/\gamma_n)$. Since $\gamma_n\to4(E+E')<1$ and $M$ is of order $n^\lambda$, this allows $\log m$ of order $n^\lambda$. Fix a coefficient $A$ with
\begin{equation}\label{b:dyn:coefficient}
 0<A<\frac{t_0\log\bigl(1/(4E+4E')\bigr)}{3BR};
\end{equation}
when $E'=0$, as in the proof of \cref{thm:barrier-730}, the right side is $t_0\log25/(3BR)$. We set
\begin{equation}\label{eq:bp-horizon}
 L=L_n=An^{\lambda},\qquad T_n=\bigl\lceil n^2e^{L}\bigr\rceil.
\end{equation}
Here $L_n$ is the logarithmic time scale and $e^{L_n}$ is the time horizon. The integer $T_n$ is a budget of update attempts: the continuous-time dynamics makes on average $ne^{L_n}$ attempts up to time $e^{L_n}$, and $T_n$ exceeds this number by a factor $n$.

\begin{lemma}[The count for paths of exponential length]\label{lem:bp-coefficient}
Assume \eqref{eq:ind-scales}, and let $A$ satisfy \eqref{b:dyn:coefficient}. There exists $n_0$, depending only on $A$, $t_0$, $B$, $R$, $E$, $E'$, and the sequences $(\rho_n)$ and $(\eta_n)$, such that $C_n(m)<1$ for every $n\ge n_0$ and every integer $0\le m\le n^3e^{L}$. In particular, this holds for every $0\le m\le T_n$.
\end{lemma}

\begin{proof}
By \eqref{b:dyn:coefficient}, there exists $\gamma'\in(4E+4E',1)$ with $A<t_0\log(1/\gamma')/(3BR)$; we fix such a $\gamma'$. Since $\gamma_n\to4E+4E'$, we have $\gamma_n\le\gamma'$ for all sufficiently large $n$. By \eqref{b:dyn:threshold-intervals}, we have $M\ge t_0/(3B\rho)-1=t_0n^\lambda/(3BR)-1$. Since $C_n(m)$ is increasing in $m$, it suffices to consider the largest integer $m$ with $m\le n^3e^L$, for which $m+1\le2n^3e^{L}$. We find
\[
 \begin{aligned}
 \log C_n(m)&\le\log2+3\log n+An^\lambda-\Bigl(\frac{t_0n^\lambda}{3BR}-1\Bigr)\log\frac1{\gamma'}\\
 &=-\Bigl(\frac{t_0\log(1/\gamma')}{3BR}-A\Bigr)n^\lambda+3\log n+\log2+\log\frac1{\gamma'}.
 \end{aligned}
\]
The coefficient of $n^\lambda$ is negative and $\lambda>0$, so the right side tends to $-\infty$. Finally, $T_n\le n^2e^L+1\le n^3e^L$ because $n\ge2$.
\end{proof}

We make no attempt to optimize the admissible coefficient in \eqref{b:dyn:coefficient}; only its positivity matters for \cref{thm:barrier-730}.

\section{Stationarity, the clocks, and total variation}\label{b:sec:dyn-TV}

The following lemma turns a sign obstruction for paths into a bound on the Gibbs mass of the initial states from which the dynamics is close to equilibrium. It abstracts the sign-passage reduction of \cite[Section~8.1, (8.1)--(8.3)]{oai-barriers}. It is stated for a general kernel and uses only stationarity at deterministic times, spin-flip symmetry, and the Poisson representation of the continuous-time semigroup; for the latter representation, see \cite[Section~20.1, (20.5)--(20.6)]{levin-peres-2017}.

\begin{lemma}[From a sign obstruction to total variation]\label{lem:bp-sign-test}
Let $Q$ be a Markov kernel on $\Sigma_n$ with stationary law $\pi$, and assume that $\pi(-x)=\pi(x)$ for every $x\in\Sigma_n$. Let $v\colon\Sigma_n\to\Sigma_n$ be a map, let $V\subseteq\Sigma_n$, and let $m\ge0$ be an integer. Assume that every sequence $x_0,\dots,x_k$ with $k\le m$, with $Q(x_{a-1},x_a)>0$ for $1\le a\le k$, and with $x_a\notin V$ for $0\le a\le k$, satisfies $R(v(x_0),x_k)>0$. Then for every $\tau\in[0,1/2)$ and every integer $0\le k\le m$,
\begin{equation}\label{eq:bp-sign-discrete}
 \pi\bigl\{x:\norm{Q^k(x,\cdot)-\pi}_\TV\le\tau\bigr\}\le\frac{(k+1)\pi(V)}{1/2-\tau}.
\end{equation}
Moreover, for every $t\ge0$, the kernel $P_t=e^{nt(Q-I)}$ satisfies
\begin{equation}\label{eq:bp-sign-continuous}
 \pi\bigl\{x:\norm{P_t(x,\cdot)-\pi}_\TV\le\tau\bigr\}\le\frac{(m+1)\pi(V)+\PP\{\mathsf N>m\}}{1/2-\tau},
\end{equation}
where $\mathsf N$ has the Poisson law with mean $nt$.
\end{lemma}

\begin{proof}
For $x\in\Sigma_n$, let $G_x=\{y\in\Sigma_n:R(v(x),y)\le0\}$. The map $y\mapsto-y$ preserves $\pi$ and exchanges the sets $\{y:R(v(x),y)<0\}$ and $\{y:R(v(x),y)>0\}$. These two sets therefore have the same Gibbs mass, and
\[
 \pi(G_x)=\frac{1+\pi\{y:R(v(x),y)=0\}}2\ge\frac12.
\]
Let $\nu_x$ be a probability measure on $\Sigma_n$ for each $x$. If $\norm{\nu_x-\pi}_\TV\le\tau$, then $\nu_x(G_x)\ge\pi(G_x)-\tau\ge1/2-\tau$. Multiplying by $\pi(x)$ and summing over $x$, we obtain
\begin{equation}\label{eq:bp-half-space}
 \Bigl(\frac12-\tau\Bigr)\,\pi\bigl\{x:\norm{\nu_x-\pi}_\TV\le\tau\bigr\}\le\sum_x\pi(x)\,\nu_x(G_x).
\end{equation}
The half-space $G_x$ is determined by the initial state $x$ alone. The bound \eqref{eq:bp-half-space} is a statement about the law of the dynamics started at a fixed state, averaged over that state afterward.

Let $(X_a)_{a\ge0}$ be the Markov chain with kernel $Q$ and $X_0\sim\pi$, and set $g_k=\sum_x\pi(x)Q^k(x,G_x)=\PP\{R(v(X_0),X_k)\le0\}$. Almost surely, $Q(X_{a-1},X_a)>0$ for every $a\ge1$. For $k\le m$, the hypothesis then implies that $\{R(v(X_0),X_k)\le0\}\subseteq\{X_a\in V\text{ for some }0\le a\le k\}$, up to a null set. Since $\pi Q=\pi$, the law of $X_a$ is $\pi$ for every deterministic $a$, and by a union bound we have
\begin{equation}\label{b:dyn:stationary-sign}
 g_k\le\sum_{a=0}^k\PP\{X_a\in V\}=(k+1)\pi(V)\qquad(0\le k\le m).
\end{equation}
Together with \eqref{eq:bp-half-space} for $\nu_x=Q^k(x,\cdot)$, this proves \eqref{eq:bp-sign-discrete}. The matrices $ntQ$ and $ntI$ commute, which gives $P_t=\sum_{k\ge0}\PP\{\mathsf N=k\}Q^k$ and
\[
 \sum_x\pi(x)P_t(x,G_x)=\sum_{k\ge0}\PP\{\mathsf N=k\}\,g_k\le(m+1)\pi(V)+\PP\{\mathsf N>m\},
\]
where we used \eqref{b:dyn:stationary-sign} for $k\le m$ and $g_k\le1$ for $k>m$. With \eqref{eq:bp-half-space} for $\nu_x=P_t(x,\cdot)$, this proves \eqref{eq:bp-sign-continuous}.
\end{proof}

The union bound \eqref{b:dyn:stationary-sign} uses stationarity only at the deterministic times $0,\dots,k$, and it does not use independence along the chain. The set $V$ and the map $v$ are fixed before the chain is run.

We now apply \cref{lem:bp-sign-test} to the heat-bath kernel at a fixed disorder. Recall the time scale $L$ and the budget $T_n$ of \eqref{eq:bp-horizon}.

\begin{proposition}[Gibbs mass of initial states close to equilibrium]\label{prop:bp-quenched}
Assume \eqref{eq:ind-scales}, let $A$ satisfy \eqref{b:dyn:coefficient}, and let $L=An^\lambda$ and $T_n=\lceil n^2e^L\rceil$. There exists $n_0$, depending only on the fixed parameters, such that the following holds for every $n\ge n_0$. Let $J$ be a disorder for which there exists a family of reference configurations with properties (P1)--(P4) and with $\pi^J(V)\le3e^{-4L}$. Then for every $\tau\in[0,1/2)$,
\begin{align}
 \pi^J\bigl\{x:d^J(x,e^{L})\le\tau\bigr\}
 &\le\frac{3(T_n+1)e^{-4L}+1/n}{1/2-\tau},\label{b:dyn:continuous-mixed}\\
 \pi^J\bigl\{x:d^J_{\rm disc}(x,\lfloor e^{L}\rfloor)\le\tau\bigr\}
 &\le\frac{3(T_n+1)e^{-4L}}{1/2-\tau}.\label{b:dyn:discrete-mixed}
\end{align}
\end{proposition}

\begin{proof}
Let $n_0$ be so large that the conclusions of \cref{lem:ind-margins} and \cref{lem:bp-coefficient} hold for $n\ge n_0$, and fix a family as in the statement. We apply \cref{lem:bp-sign-test} with $Q=K^J$, the Gibbs measure $\pi=\pi^J$, the map $v(x)$ of \cref{def:ind-algorithm}, the set $V$ of the family, and $m=T_n$. The kernel $K^J$ preserves $\pi^J$ by \cref{lem:model-dynamics}(a), and $\pi^J(-x)=\pi^J(x)$ because $H_n(-x)=H_n(x)$. If $K^J(x_{a-1},x_a)>0$, then $x_a$ and $x_{a-1}$ differ in at most one coordinate by \eqref{b:eq:discrete-kernel}. Every sequence in the hypothesis of \cref{lem:bp-sign-test} is a path in the sense of \eqref{eq:ind-jump}. For $k\le T_n$, we have $C_n(k)<1$ by \cref{lem:bp-coefficient}, and \cref{prop:bp-obstruction}, applied to the path $x_0,\dots,x_k$, implies that $R(v(x_0),x_k)>0$ when the path avoids $V$. We have verified the hypothesis of \cref{lem:bp-sign-test}.

In continuous time, we have $P^J_t=e^{nt(K^J-I)}$ by \eqref{eq:common-semigroup}, and we take $t=e^L$. Then $\mathsf N$ has mean $ne^L$, and Markov's inequality gives
\begin{equation}\label{b:dyn:clock-tail}
 \PP\{\mathsf N>T_n\}\le\frac{ne^L}{T_n}\le\frac{ne^L}{n^2e^L}=\frac1n.
\end{equation}
Inserting \eqref{b:dyn:clock-tail} and $\pi^J(V)\le3e^{-4L}$ into \eqref{eq:bp-sign-continuous}, we obtain \eqref{b:dyn:continuous-mixed}. In discrete time, $k=\lfloor e^L\rfloor\le T_n$, and by \eqref{eq:bp-sign-discrete} we have
\[
 \pi^J\bigl\{x:d^J_{\rm disc}(x,\lfloor e^L\rfloor)\le\tau\bigr\}\le\frac{(\lfloor e^L\rfloor+1)\,3e^{-4L}}{1/2-\tau},
\]
which implies \eqref{b:dyn:discrete-mixed}.
\end{proof}

For every fixed $A>0$, the right sides of \eqref{b:dyn:continuous-mixed} and \eqref{b:dyn:discrete-mixed} tend to zero. Indeed, $T_n+1\le n^2e^L+2$, and we have
\begin{equation}\label{eq:bp-mixed-limit}
 3(T_n+1)e^{-4L}\le3n^2e^{-3An^\lambda}+6e^{-4An^\lambda}\longrightarrow0.
\end{equation}

\section{The parameters and the exponents}\label{b:sec:completion}

We first list the parameters of the proof of \cref{thm:barrier-730} in the order in which they are chosen. All of them are fixed before $n\to\infty$ and before the threshold $\tau$.

\begin{table}[ht]
\centering
\small
\begin{tabular}{@{}c>{\raggedright\arraybackslash}p{0.47\textwidth}>{\raggedright\arraybackslash}p{0.33\textwidth}@{}}
\toprule
Step & Parameter & Condition and reference \\
\midrule
1 & the parameters of \cref{tab:ref-parameters} up to $H$: $q_\beta$, $q_*$, $b_1$, $b_2$, $t_0$, $J_\beta$, $I_\beta$, $m_\beta$, $\vartheta$, $E$, $g$, $\varkappa$, $C_b$, $R$, the constants of the static estimates, and $H$ & \cref{sec:ref-parameters}; $C_b$ from \cref{inp:hb-rate} \\
2 & the threshold geometry: intervals of length $\rho_n$ and spacing $3\rho_n$ & \eqref{b:dyn:threshold-intervals} \\
3 & the block length $B$ & any integer $B\ge H+2$, \eqref{b:dyn:BH} \\
4 & the constants $c_s$, $C_s$ of the list sizes & \eqref{eq:ref-schedule} \\
5 & the coefficient $A_\beta$ & $0<A_\beta<t_0\log25/(3BR)$, \eqref{b:dyn:coefficient} \\
6 & for each $n$: $\rho_n$, $\varepsilon_n$, $\eta_n$, $L_n=A_\beta n^{7/30}$, $M$, $N_{\rm lev}$, the list sizes $K_s$, and $T_n$ & \eqref{b:dyn:scales}, \eqref{b:dyn:threshold-intervals}, \eqref{eq:ref-schedule}, \eqref{eq:bp-horizon} \\
7 & the threshold $\tau\in(0,1/2)$ & enters only through $1/(1/2-\tau)$ \\
\bottomrule
\end{tabular}
\caption{The parameters in the proof of \cref{thm:barrier-730}, in the order in which they are chosen. Each parameter of Steps 1--5 depends only on $\beta$ and on the parameters chosen before it.}
\label{tab:bp-order}
\end{table}

We use the exponents $\lambda=7/30$ and $\xi=5/21$. \Cref{tab:bp-exponents} lists every condition on the exponents used in the proof, with its value at these exponents; the corresponding powers of $n$ are listed in \cref{tab:ref-exponents}.

\begin{table}[ht]
\centering
\small
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.64\textwidth}c@{}}
\toprule
Condition (where it is used) & Value at $\lambda=7/30$ \\
\midrule
$3\lambda\le a$, where $Cn^{-a}$ bounds the mean bias (denominator margin, \cref{sec:static}) & $a=7/10=3\lambda$ \\
$7/15-2+6\lambda<0$ (Chebyshev bound for the denominator) & $-2/15$ \\
$1-4\lambda>0$ (local and separated-overlap numerators, $e^{-cn\rho_n^4}$) & $1/15$ \\
$1-4\xi>0$ (fixed-cutoff numerators, $e^{-cn\eta_n^4}$) & $1/21$ \\
$3\lambda-2\xi>0$ (so that $\rho_n^3=o(\eta_n^2)$) & $47/210$ \\
$1-3\lambda>\lambda$ (conditional narrow mass $e^{-cn\rho_n^3}$ against $e^{CL_n}$) & $3/10>7/30$ \\
$1-2\xi>\lambda$ (conditional fixed-cutoff mass against $e^{CL_n}$) & $11/21>7/30$ \\
$\lambda<1$ (overlap-neighborhood bound needs $L_n=o(n)$; $1/(n\rho_n)\to0$) & yes \\
$\xi-\lambda>0$ (so that $\eta_n=o(\rho_n)$, \cref{sec:induction}) & $1/210$ \\
\bottomrule
\end{tabular}
\caption{The conditions on the exponents, with $\xi=5/21$. The entries in the last column are the values of the left sides at $\lambda=7/30$; for instance, the probability that the denominator bound fails is $O(n^{7/15-2+6\lambda})=O(n^{-2/15})$, and the probability that the bound for a single local numerator fails is at most $e^{-cn^{1-4\lambda}}=e^{-cn^{1/15}}$. In the first row, the mean-bias bound of \cref{inp:hb-rate} has $a=3\lambda$, and the strict margin in \eqref{eq:st-S1} comes from the choice of $R$ in \eqref{b:dyn:R-condition}.}
\label{tab:bp-exponents}
\end{table}

The conditions of \cref{tab:bp-exponents} also explain why the argument gives no exponent of $1/4$ or more. For an exponent $a$ of the mean-bias bound and an exponent $\lambda\le a/3$ with $\lambda<1/4$, all rows hold if $\xi$ is chosen strictly between $\lambda$ and $\min\{1/4,3\lambda/2\}$, which is the condition \eqref{eq:ref-exponents}; the choice $\xi=5/21$ works for $\lambda=7/30$. At $\lambda=1/4$, the third row fails, since then $n\rho_n^4=R^4$ does not tend to infinity.

\section{Proof of \texorpdfstring{\cref{thm:barrier-730}}{the barrier theorem}}\label{sec:bp-730}

\Cref{sec:induction} and \cref{sec:bp-obstruction,sec:bp-coefficient,b:sec:dyn-TV} were written for a general exponent $\lambda$. They use the deterministic assumptions stated there and the scale conditions \eqref{eq:ind-scales}, with $\rho_n=Rn^{-\lambda}$. \Cref{sec:references} is written for exponents satisfying \eqref{eq:ref-exponents} and an upper bound \eqref{eq:ref-mean-input} on the mean bias, and the static estimate \cref{b:lem:static-event} is stated for all scales satisfying the conditions \eqref{eq:st-S1}--\eqref{eq:st-S5}. In the proof below, the mean-bias input is the bound $0\le P_\beta-n^{-1}\E F_n\le C_bn^{-7/10}$ of \cref{inp:hb-rate}; the variance bound of \cref{inp:qp-rate} enters through \cref{sec:static}, and the trial bound of \cref{inp:qp-floor} through \cref{sec:fractional}.

\begin{proof}[Proof of \cref{thm:barrier-730}]
Fix $\beta>1$. We take the exponents $\lambda=7/30$ and $\xi=5/21$ in \eqref{eq:ref-exponents}, and the mean-bias input \eqref{eq:ref-mean-input} is the bound of \cref{inp:hb-rate}, with constant $C_b$. We fix the parameters of Steps 1--4 of \cref{tab:bp-order} as in \cref{sec:ref-parameters,sec:ref-families}, with $R^3>3C_b/(\varkappa g)$ as in \eqref{b:dyn:R-condition} and with an integer $B\ge H+2$ (for instance $B=H+2$). Then we fix $A_\beta$ as in Step 5, which is \eqref{b:dyn:coefficient} with $A=A_\beta$ and $E'=0$, and we set $L_n=A_\beta n^{7/30}$. The scales are $\rho_n=Rn^{-7/30}$, $\varepsilon_n=E\rho_n$, and $\eta_n=n^{-5/21}$. They satisfy \eqref{eq:ind-scales} with $E'=0$, since $\rho_n$, $\eta_n/\rho_n=n^{-1/210}/R$, and $1/(n\rho_n)=n^{-23/30}/R$ tend to zero.

We have $N_{\rm lev}\le t_0/(3\rho_n)=t_0n^{7/30}/(3R)\le e^{L_n}$ for all sufficiently large $n$. By \cref{rem:st-scales}(a), the scales satisfy \eqref{eq:st-S1}--\eqref{eq:st-S5}, so that \cref{b:lem:static-event} and therefore \cref{lem:ref-single-event} apply at these scales. Since $L_n=A_\beta n^{\lambda}$, the last assertion of \cref{lem:ref-families} shows that $L_n\to\infty$, $L_n/n\to0$, and \eqref{eq:ref-absorb} holds. By \cref{lem:ref-families}, applied with the time scale $L_n$, there exist measurable disorder events $\mathcal G_n$ with $\PP_J(\mathcal G_n)\to1$ such that, for all sufficiently large $n$ and every $J\in\mathcal G_n$, there exists a family of reference configurations with $N_{\rm lev}$ levels, with properties (i)--(iv) of that lemma, and with $\pi^J(V)\le3e^{-4L_n}$; as explained in \cref{sec:ind-setting}, it has properties (P1)--(P4). The family may depend on $J$ in an arbitrary way; it is not required to be a measurable function of $J$. Fix $\tau\in(0,1/2)$. By \cref{prop:bp-quenched}, for all sufficiently large $n$ and every $J\in\mathcal G_n$,
\[
 f_n(J)=\pi^J\bigl\{x:d^J(x,e^{L_n})\le\tau\bigr\}\le\frac{3(T_n+1)e^{-4L_n}+1/n}{1/2-\tau}.
\]
We write $e_n$ for the right side. The left side does not involve the family. By \cref{sec:model-joint}, $f_n$ is a Borel function of $J$, and $0\le f_n\le1$. By \eqref{eq:common-joint},
\[
 \PP_{J,X_0\sim\pi^J}\bigl\{d^J(X_0,e^{A_\beta n^{7/30}})\le\tau\bigr\}=\E_Jf_n(J)\le e_n+\PP_J(\mathcal G_n^c).
\]
Here $e_n\to0$ by \eqref{eq:bp-mixed-limit}, and $\PP_J(\mathcal G_n^c)\to0$. This proves \eqref{eq:barrier-730-continuous}. The same argument with \eqref{b:dyn:discrete-mixed} in place of \eqref{b:dyn:continuous-mixed} proves \eqref{eq:barrier-730-discrete}.

The coefficient $A_\beta$ was chosen in Step 5, before $\tau$, and the same coefficient serves both clocks. The threshold $\tau$ enters only through the factor $1/(1/2-\tau)$, which affects the rate of convergence and not the coefficient.
\end{proof}

The proof uses the two kinds of exceptional events of \cref{sec:static,sec:references} differently, as in \cref{rem:ref-two-errors}. The disorder probability $\PP_J(\mathcal G_n^c)$ is used once, additively, in the last display. It includes terms that decay only polynomially or as $e^{-cn\eta_n^4}$, and it is never multiplied by a number of references, levels, or time indices. At a fixed disorder in $\mathcal G_n$, the Gibbs mass $\pi^J(V)$ is multiplied by the $T_n+1$ time indices in \eqref{b:dyn:stationary-sign}, and the threshold probabilities are multiplied by the $T_n+1$ indices in the proof of \cref{prop:bp-obstruction}; both products are small.

\begin{proof}[Proof of \cref{b:cor:subcritical}]
Fix $\beta>1$, $\kappa\in(0,7/30)$, and $\tau\in(0,1/2)$, and let $A_\beta$ be the coefficient of \cref{thm:barrier-730}. Since $\kappa<7/30$, we have $n^\kappa\le A_\beta n^{7/30}$ for all sufficiently large $n$. By \cref{lem:model-dynamics}(d), the functions $t\mapsto d^J(x,t)$ and $k\mapsto d^J_{\rm disc}(x,k)$ are nonincreasing, and therefore, for all such $n$,
\[
 \bigl\{x:d^J(x,e^{A_\beta n^{7/30}})>\tau\bigr\}\subseteq\bigl\{x:d^J(x,e^{n^\kappa})>\tau\bigr\},
\]
and likewise in discrete time, with $\lfloor e^{n^\kappa}\rfloor\le\lfloor e^{A_\beta n^{7/30}}\rfloor$. By \eqref{eq:common-joint}, the realized-start probabilities are monotone in the same way, and \cref{thm:barrier-730} proves the corollary.
\end{proof}

\section{Quantitative and almost-sure forms}\label{sec:bp-strong}

The proof of \cref{thm:barrier-730} gives more than the convergence of the realized-start probabilities. In this section, we record four strengthenings. First, the disorder event can be chosen with a complement of stretched-exponentially small probability. Second, on this event there exists a set of initial states of Gibbs mass exponentially close to one from which the distance from equilibrium has a lower bound tending to $1/2$ up to the time horizon $e^{L_n}$. Third, the same holds for the Metropolis dynamics and for other single-site dynamics with bounded rates. Fourth, the relaxation time is at least of the order of the time horizon, up to a factor $n^{7/30}$. A consequence is the almost-sure form of the barrier along the full sequence of system sizes.

Three changes in the argument are needed. First, the lower bound on $F_n$ in the event of \cref{b:lem:static-event} is obtained from the small-deviation inequality of Paouris and Valettas for convex functions of Gaussian vectors \cite[Theorem~2.1 and Remark~2.4(3)]{paouris-valettas-2018}, in place of Chebyshev's inequality. Second, the tolerance $\eta_n$ is a fixed small multiple of $\rho_n$, so that the probability bounds for the fixed-scale constraints decay on the same scale $n\rho_n^4$ as the others. Third, the sign test of \cref{lem:bp-sign-test} is applied to each initial state separately, and the clock tail is bounded by a Chernoff bound instead of Markov's inequality. The relaxation-time bound uses in addition that the first reference takes few values. \Cref{thm:barrier-730} remains the main result; the coefficient in \cref{thm:barrier-strong} below is subject to a slightly smaller bound than in \eqref{b:dyn:coefficient} with $E'=0$, because of the second change.

\paragraph{Single-site dynamics.} Fix the disorder. For $x\in\Sigma_n$ and $1\le i\le n$, we write $x^{(i)}=x^{i,-x_i}$ for the configuration obtained by reversing the $i$th spin. For $C_{\rm rate}\ge1$, we let $\mathfrak L^J(C_{\rm rate})$ be the set of generators of the form
\begin{equation}\label{eq:bp-single-site}
 \mathcal Lf(x)=\sum_{i=1}^nc_i(x)\bigl(f(x^{(i)})-f(x)\bigr),\qquad f\colon\Sigma_n\to\R,
\end{equation}
with rates $c_i(x)\in[0,C_{\rm rate}]$, for which $\pi^J$ is stationary, that is, $\sum_x\pi^J(x)\mathcal Lf(x)=0$ for every $f$. We call $\mathcal L$ reversible if $\pi^J(x)c_i(x)=\pi^J(x^{(i)})c_i(x^{(i)})$ for all $x$ and $i$. By \eqref{eq:common-kernel}, the heat-bath generator $n(K^J-I)$ has the form \eqref{eq:bp-single-site} with $c_i(x)=p_i(-x_i\mid x)\le1$, and it is reversible by \cref{lem:model-dynamics}(a). The Metropolis kernel is
\[
 K_{\rm M}^J(x,x^{(i)})=\frac1n\min\bigl\{1,e^{\beta(H_n(x^{(i)})-H_n(x))}\bigr\},\qquad
 K_{\rm M}^J(x,x)=1-\sum_{i=1}^nK_{\rm M}^J(x,x^{(i)}),
\]
and $K_{\rm M}^J(x,y)=0$ otherwise. In one step, a uniformly chosen spin is reversed with the displayed acceptance probability. The identity $\pi^J(x)K^J_{\rm M}(x,x^{(i)})=(nZ_n)^{-1}\min\{e^{\beta H_n(x)},e^{\beta H_n(x^{(i)})}\}$, whose right side is symmetric in $x$ and $x^{(i)}$, shows that $K^J_{\rm M}$ is reversible with respect to $\pi^J$. Its continuous-time generator $n(K^J_{\rm M}-I)$ has the form \eqref{eq:bp-single-site} with $c_i(x)\le1$. Heat-bath and Metropolis dynamics belong to $\mathfrak L^J(1)$. For a reversible $\mathcal L$, the spectral gap $\operatorname{gap}(\mathcal L)$ is the infimum of $-\langle f,\mathcal Lf\rangle_{\pi^J}$ over $f$ with $\pi^J(f)=0$ and $\pi^J(f^2)=1$, where $\langle f,g\rangle_{\pi^J}=\sum_x\pi^J(x)f(x)g(x)$, and the relaxation time is $1/\operatorname{gap}(\mathcal L)$, with the convention $1/0=+\infty$.

\begin{theorem}[Quantitative and almost-sure forms]\label{thm:barrier-strong}
Fix $\beta>1$. There exist $A>0$, $c>0$, $C<\infty$, and Borel sets $\mathcal G_n^*$ of disorders, such that the following statements hold with $L_n=An^{7/30}$.
\begin{enumerate}[label=\textup{(\alph*)}]
\item For every $n\ge2$, we have $\PP_J\bigl((\mathcal G_n^*)^c\bigr)\le Ce^{-cn^{1/15}}$.
\item There exists $n_0$ such that, for every $n\ge n_0$ and every $J\in\mathcal G_n^*$, there exists a set $S\subseteq\Sigma_n$ with $\pi^J(S)\ge1-Ce^{-cn^{7/30}}$ such that every $x\in S$ satisfies
\[
 \inf_{0\le t\le e^{L_n}}d^J(x,t)\ge\frac12-Ce^{-cn^{7/30}},\qquad
 \min_{0\le k\le e^{L_n}}d^J_{\rm disc}(x,k)\ge\frac12-Ce^{-cn^{7/30}},
\]
and the same two inequalities hold for the Metropolis dynamics, that is, with $(K^J_{\rm M})^k$ in place of $(K^J)^k$ and $e^{nt(K^J_{\rm M}-I)}$ in place of $P^J_t$.
\item For every $C_{\rm rate}\ge1$ there exists $n_1$ such that, for every $n\ge n_1$, every $J\in\mathcal G_n^*$, and every $\mathcal L\in\mathfrak L^J(C_{\rm rate})$, there exists a set $S_{\mathcal L}\subseteq\Sigma_n$ with $\pi^J(S_{\mathcal L})\ge1-Ce^{-cn^{7/30}}$ such that
\[
 \norm{e^{t\mathcal L}(x,\cdot)-\pi^J}_\TV\ge\frac12-Ce^{-cn^{7/30}}\qquad\text{for every }x\in S_{\mathcal L}\text{ and }0\le t\le e^{L_n}.
\]
\item For every $C_{\rm rate}\ge1$, the threshold $n_1=n_1(\beta,C_{\rm rate})$ in part~(c) can be chosen so that, for every $n\ge n_1$, every $J\in\mathcal G_n^*$, and every reversible $\mathcal L\in\mathfrak L^J(C_{\rm rate})$, the relaxation time satisfies
\[
 \frac1{\operatorname{gap}(\mathcal L)}\ge\frac{e^{An^{7/30}}}{Cn^{7/30}}.
\]
In particular, this holds for the heat-bath and Metropolis generators $n(K^J-I)$ and $n(K^J_{\rm M}-I)$.
\item Let $(J^{(n)})_{n\ge2}$ be random disorders defined on one probability space, where $J^{(n)}$ has the law of the disorder of size $n$ and the dependence between different sizes is arbitrary. Then, almost surely, $J^{(n)}\in\mathcal G_n^*$ for all sufficiently large $n$. In particular, for every fixed $\tau\in(0,1/2)$, almost surely,
\[
 \pi^{J^{(n)}}\bigl\{x:d^{J^{(n)}}(x,e^{An^{7/30}})>\tau\bigr\}\longrightarrow1,\qquad
 \pi^{J^{(n)}}\bigl\{x:d^{J^{(n)}}_{\rm disc}(x,\lfloor e^{An^{7/30}}\rfloor)>\tau\bigr\}\longrightarrow1,
\]
and the same holds for the Metropolis dynamics.
\end{enumerate}
The constants $A$, $c$, $C$, and $n_0$ depend only on $\beta$; the threshold $n_1$ depends in addition on $C_{\rm rate}$.
\end{theorem}

Part~(b) contains the conclusions of \cref{thm:barrier-730}, with the coefficient $A$ of \cref{thm:barrier-strong} in place of $A_\beta$. It gives in addition the size of the exceptional sets, the uniformity over all earlier times, and a lower bound tending to $1/2$ on the distance. The threshold $1/2$ itself is not reached: the argument uses only that the half-space $\{y:R(v,y)\le0\}$ has Gibbs mass at least $1/2$. In part~(e), no independence between different system sizes is needed, because the first Borel--Cantelli lemma uses only the marginal laws.

The following table lists how the earlier statements enter the proof.

\begin{table}[ht]
\centering
\small
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.36\textwidth}>{\raggedright\arraybackslash}p{0.56\textwidth}@{}}
\toprule
Statement & Use in the proof of \cref{thm:barrier-strong} \\
\midrule
\cref{sec:induction}; \cref{lem:bp-product}; \cref{prop:bp-obstruction} & used as stated; they assume \eqref{eq:ind-scales}, which holds with $E'=1/1000$ \\
\cref{lem:bp-coefficient} & used as stated, with $E'=1/1000$, so that \eqref{b:dyn:coefficient} reads $A<t_0\log(250/11)/(3BR)$; it covers every $m\le n^3e^{L_n}$ \\
\cref{b:lem:static-event} & statement used as stated, at $\eta_n=\rho_n/1000$; its hypotheses are verified in \cref{lem:bp-strong-event} \\
Step~1 of the proof of \cref{b:lem:static-event} (Chebyshev) & replaced by the Paouris--Valettas inequality in \cref{lem:bp-strong-event} \\
\cref{b:lem:coverage-extension} & used as stated, including its bound on $\PP(\mathcal C_n(D_1)^c)$ \\
\cref{lem:ref-single-event} & used as stated, at $\eta_n=\rho_n/1000$, in Step~4 of the proof of \cref{lem:bp-strong-event} \\
\cref{lem:ref-families} & used as stated, with $q_n$ supplied by that Step~4; \eqref{eq:ref-absorb} is verified there \\
\cref{lem:bp-sign-test}, \cref{prop:bp-quenched}, \eqref{b:dyn:clock-tail} & replaced by \cref{lem:bp-per-start} and the Chernoff bound \eqref{eq:bp-chernoff} \\
new & \cref{lem:bp-gap} (relaxation time) \\
\bottomrule
\end{tabular}
\caption{The earlier statements in the proof of \cref{thm:barrier-strong}.}
\label{tab:bp-strong-uses}
\end{table}

\begin{lemma}[A disorder event with stretched-exponential failure]\label{lem:bp-strong-event}
Fix $\beta>1$ and let $\lambda=7/30$. Fix the parameters of Steps 1--4 of \cref{tab:bp-order}, with $C_b$ the constant of \cref{inp:hb-rate}, and set
\[
 \rho_n=Rn^{-\lambda},\qquad\varepsilon_n=E\rho_n,\qquad\eta_n=\frac{\rho_n}{1000},\qquad L_n=An^\lambda
\]
for a fixed $A>0$. Let $\mathcal A_n$ be the event of \cref{b:lem:static-event} at these scales, with the data of \cref{sec:references}, and let $\mathcal G_n^*=\mathcal A_n\cap\bigcap_{s<B}\mathcal C_n(C_s+5)$ be the event \eqref{eq:ref-good-event} at these scales. Then the following hold.
\begin{enumerate}[label=\textup{(\roman*)}]
\item There exist $C,c>0$ such that $\PP_J((\mathcal G_n^*)^c)\le Ce^{-cn^{1-4\lambda}}$ for all sufficiently large $n$.
\item For all sufficiently large $n$ and every $J\in\mathcal G_n^*$, there exists a family of reference configurations with $N_{\rm lev}$ levels, with properties (P1)--(P4), and with $\pi^J(V)\le3e^{-4L_n}$.
\end{enumerate}
\end{lemma}

\begin{proof}
Since $\lambda>0$, we have $\rho_n=Rn^{-\lambda}\to0$. We also verify the conditions \eqref{eq:st-S1}--\eqref{eq:st-S5} of \cref{b:lem:static-event}. For \eqref{eq:st-S1}, we have $\Delta_{n,\beta}/\rho_n^3\le C_b/R^3<\varkappa g/3$, by \cref{inp:hb-rate} and \eqref{b:dyn:R-condition}. Next, $n^{7/15}/(n^2\rho_n^6)=R^{-6}n^{7/15-2+6\lambda}=R^{-6}n^{-2/15}$, which is \eqref{eq:st-S2}. We have $n\rho_n^4=R^4n^{1-4\lambda}=R^4n^{1/15}$ and $n\eta_n^4=10^{-12}R^4n^{1/15}$, which gives \eqref{eq:st-S3} and \eqref{eq:st-S4}. Finally, $\rho_n^3/\eta_n^2=10^6\rho_n\to0$, which is \eqref{eq:st-S5}. Therefore \cref{b:lem:static-event} applies at these scales.

\paragraph{Step 1: the numerators.} Steps 2--4 of the proof of \cref{b:lem:static-event} bound the probability that one of the events (A2)--(A4) fails by
\[
 C(n+1)^3\bigl(e^{-cn\rho_n^4}+e^{-cn\eta_n^4}\bigr)+C(n+1)e^{-cn}\le C'e^{-c'n^{1-4\lambda}},
\]
for all large $n$, where we used $n\eta_n^4=10^{-12}n\rho_n^4$ and $n\rho_n^4=R^4n^{1-4\lambda}$, and absorbed the polynomial factors by decreasing the constant in the exponent.

\paragraph{Step 2: the denominator.} The function $F_n=\log Z_n$ of the standard Gaussian vector $J=(J_{ij})_{i<j}$ is convex. Indeed, its Hessian is $\beta^2n^{-1}$ times the covariance matrix of the vector $(x_ix_j)_{i<j}$ under $\pi^J$, which is positive semidefinite. It is Lipschitz by \cref{rem:inp-lipschitz}, so it is square integrable. By \cref{inp:qp-rate}, $\sigma_n^2=\Var F_n\le C_vn^{7/15}$ for large $n$, with $C_v$ the variance constant there. We write $a_0<c_0$ for the static denominator margins in \eqref{b:eq:denominator-margin}. For all sufficiently large $n$, $\Delta_{n,\beta}\le a_0\rho_n^3$, so failure of (A1) implies $F_n-\E F_n<-u_n$, where $u_n=(c_0-a_0)n\rho_n^3$. If $\sigma_n=0$, this event is empty. Otherwise, let $t_n=u_n/\sigma_n$. Then
\[
 t_n\ge\frac{(c_0-a_0)R^3}{C_v^{1/2}}\,n^{23/30-3\lambda},
\]
and the exponent $23/30-3\lambda=1/15$ is positive, so that $t_n\ge2$ for large $n$. By \cite[Theorem~2.1 and Remark~2.4(3), (2.4)]{paouris-valettas-2018}, a convex square-integrable function $f$ of a standard Gaussian vector $Z$ satisfies
\begin{equation}\label{eq:bp-pv}
 \PP\Bigl(f(Z)-\E f(Z)<-t\sqrt{\Var f(Z)}\Bigr)\le\frac12\exp\Bigl(-\frac{\pi}{1024}(t-1)^2\Bigr)\qquad(t>1).
\end{equation}
% CHECK: arXiv:1611.01723v2, Theorem 2.1 (median form) and Remark 2.4(3), eq. (2.4) (mean form), p. 6.
Applying \eqref{eq:bp-pv} with $t=t_n$ and using $(t_n-1)^2\ge t_n^2/4$, we obtain
\[
 \PP\bigl(F_n<n(P_\beta-c_0\rho_n^3)\bigr)\le\frac12\exp\Bigl(-\frac{\pi t_n^2}{4096}\Bigr)
 \le\frac12\exp\Bigl(-\frac{\pi(c_0-a_0)^2R^6}{4096\,C_v}\,n^{23/15-6\lambda}\Bigr).
\]
Since $23/15-6\lambda=2/15$ is larger than $1-4\lambda=1/15$, this is at most $e^{-cn^{1-4\lambda}}$ for large $n$. Only the full partition function enters here. For the constrained numerators, the upper tail is needed, and the bounds of Step~1 remain those of \cref{b:lem:static-event}.

\paragraph{Step 3: the overlap-neighborhood events.} Fix $D_1>0$. By \cref{b:lem:coverage-extension}, there exists $D_4>0$, depending only on $\beta$ and $D_1$, such that, for all sufficiently large $n$,
\[
 \PP\bigl(\mathcal C_n(D_1)^c\bigr)\le3e^{-D_4L_n}=3e^{-D_4An^\lambda}.
\]
Since $\lambda=7/30$ is larger than $1-4\lambda=1/15$, this is at most $e^{-cn^{1-4\lambda}}$ for large $n$. There are $B$ such events in the definition of $\mathcal G_n^*$, with $D_1=C_s+5$ for $0\le s<B$. Combining Steps~1--3, we obtain part~(i).

\paragraph{Step 4: the families.} The present sequences satisfy $\rho_n\to0$ and \eqref{eq:st-S1}--\eqref{eq:st-S5}, as we checked at the start of the proof. By \cref{lem:ref-single-event}, for all large $n$ and every $J\in\mathcal A_n$, we have $(\pi^J)^{\otimes3}(\mathcal B_n)\le q_n$ with $q_n=C(n+1)^3(e^{-cn\rho_n^3}+e^{-cn\eta_n^2}+e^{-cn})$, where $\mathcal B_n$ is defined by \eqref{eq:ref-exceptional} with the present $\eta_n$. Since $n\rho_n^3=R^3n^{1-3\lambda}$ and $n\eta_n^2=10^{-6}R^2n^{1-2\lambda}$, and $1-3\lambda=3/10$ and $1-2\lambda=8/15$ are larger than $\lambda=7/30$, we have $e^{CL_n}q_n\to0$ for every fixed $C$, which is \eqref{eq:ref-absorb}. Moreover $L_n\to\infty$, $L_n/n\to0$, and $N_{\rm lev}\le t_0n^\lambda/(3R)\le e^{L_n}$ for large $n$. \Cref{lem:ref-families} therefore provides, for all large $n$ and every $J\in\mathcal G_n^*$, a family with properties (i)--(iv) of that lemma and $\pi^J(V)\le3e^{-4L_n}$. As in \cref{sec:ind-setting}, these imply (P1)--(P4), with $\zeta=2\eta_n$. This proves part~(ii).
\end{proof}

The next lemma refines \cref{lem:bp-sign-test}: it bounds the distance from equilibrium for each initial state separately, in terms of the probability that the chain started there visits $V$.

\begin{lemma}[The sign test for each initial state]\label{lem:bp-per-start}
Let $Q$, $\pi$, $v$, $V$, and $m$ satisfy the hypotheses of \cref{lem:bp-sign-test}. Let $(X_a)_{a\ge0}$ be the Markov chain with kernel $Q$, write $\PP_x$ for its law when $X_0=x$, and set
\[
 \phi(x)=\PP_x\{X_a\in V\text{ for some }0\le a\le m\},\qquad G_x=\{y:R(v(x),y)\le0\}.
\]
Then the following hold.
\begin{enumerate}[label=\textup{(\alph*)}]
\item $\sum_x\pi(x)\phi(x)\le(m+1)\pi(V)$, and hence $\pi\{x:\phi(x)>u\}\le(m+1)\pi(V)/u$ for every $u>0$.
\item For every $x$ and every $0\le k\le m$, we have $Q^k(x,G_x)\le \phi(x)$ and $\norm{Q^k(x,\cdot)-\pi}_\TV\ge1/2-\phi(x)$.
\item For every $x$, every $q_{\rm u}>0$, and every $t\ge0$, let $\mathsf N$ have the Poisson law with mean $q_{\rm u}t$. Then
\[
 \begin{gathered}
 \norm{e^{q_{\rm u}t(Q-I)}(x,\cdot)-\pi}_\TV\ge\frac12-\phi(x)-\PP\{\mathsf N>m\},\\
 \sum_x\pi(x)e^{q_{\rm u}t(Q-I)}(x,G_x)\le(m+1)\pi(V)+\PP\{\mathsf N>m\}.
 \end{gathered}
\]
\end{enumerate}
\end{lemma}

\begin{proof}
For part~(a), we have $\sum_x\pi(x)\phi(x)=\PP_\pi\{X_a\in V\text{ for some }a\le m\}$, where $\PP_\pi$ is the law of the chain with $X_0\sim\pi$. Since $\pi Q=\pi$, the law of $X_a$ under $\PP_\pi$ is $\pi$ for every deterministic $a$, and a union bound over $0\le a\le m$ gives the first assertion, as in \eqref{b:dyn:stationary-sign}. The second assertion is Markov's inequality. For part~(b), almost surely under $\PP_x$ we have $Q(X_{a-1},X_a)>0$ for every $a\ge1$. By the hypothesis of \cref{lem:bp-sign-test}, on the event that $X_0,\dots,X_k$ avoid $V$ we have $R(v(x),X_k)>0$, that is, $X_k\notin G_x$. It follows that $Q^k(x,G_x)\le\PP_x\{X_a\in V\text{ for some }a\le k\}\le \phi(x)$. As in the proof of \cref{lem:bp-sign-test}, spin-flip symmetry gives $\pi(G_x)\ge1/2$, and therefore $\norm{Q^k(x,\cdot)-\pi}_\TV\ge\pi(G_x)-Q^k(x,G_x)\ge1/2-\phi(x)$. For part~(c), the matrices $q_{\rm u}tQ$ and $q_{\rm u}tI$ commute, so $e^{q_{\rm u}t(Q-I)}=\sum_{k\ge0}\PP\{\mathsf N=k\}Q^k$. By part~(b),
\[
 e^{q_{\rm u}t(Q-I)}(x,G_x)\le\sum_{k\le m}\PP\{\mathsf N=k\}\phi(x)+\PP\{\mathsf N>m\}\le \phi(x)+\PP\{\mathsf N>m\},
\]
which gives the first inequality as in part~(b). Multiplying by $\pi(x)$, summing over $x$, and using part~(a), we obtain the second.
\end{proof}

The next lemma combines the standard continuous-time $L^2$ spectral-gap
contraction \cite[Lemma~20.5]{levin-peres-2017} with a partition of the
initial states according to their chosen reference. This grouping gives a
bound in terms of the number of references.

\begin{lemma}[Relaxation time and a sign test with few references]\label{lem:bp-gap}
Let $\mathcal L$ be a generator on $\Sigma_n$ that is reversible with respect to a probability measure $\pi$ with $\pi(-x)=\pi(x)$, and let $\operatorname{gap}(\mathcal L)$ be its spectral gap. Let $v\colon\Sigma_n\to\Sigma_n$ take at most $N_v$ values, and let $G_x=\{y:R(v(x),y)\le0\}$. If $t>0$ and
\[
 \omega=\sum_x\pi(x)e^{t\mathcal L}(x,G_x)<\frac12,
\]
then $\operatorname{gap}(\mathcal L)\,t\le(\log N_v)/2-\log(1-2\omega)$.
\end{lemma}

\begin{proof}
For $f$ with $\pi(f)=0$, the function $f_t=e^{t\mathcal L}f$ also has mean zero, since $\pi$ is stationary, and the derivative of $\langle f_t,f_t\rangle_\pi$ in $t$ is $2\langle f_t,\mathcal Lf_t\rangle_\pi\le-2\operatorname{gap}(\mathcal L)\langle f_t,f_t\rangle_\pi$, by the definition of the spectral gap. Gronwall's inequality gives $\norm{e^{t\mathcal L}f}_{L^2(\pi)}\le e^{-\operatorname{gap}(\mathcal L)t}\norm f_{L^2(\pi)}$.

For each value ${\widetilde v}$ of $v$, let $U_{\widetilde v}=\{x:v(x)={\widetilde v}\}$ and $H_{\widetilde v}=\{y:R({\widetilde v},y)\le0\}$. The sets $U_{\widetilde v}$ partition $\Sigma_n$, and $G_x=H_{\widetilde v}$ for $x\in U_{\widetilde v}$. As in the proof of \cref{lem:bp-sign-test}, spin-flip symmetry gives $\pi(H_{\widetilde v})\ge1/2$, and therefore
\[
 \sum_{\widetilde v}\pi(U_{\widetilde v})\pi(H_{\widetilde v})\ge\frac12,\qquad
 \sum_{\widetilde v}\langle\mathbf 1_{U_{\widetilde v}},e^{t\mathcal L}\mathbf 1_{H_{\widetilde v}}\rangle_\pi=\sum_x\pi(x)e^{t\mathcal L}(x,G_x)=\omega.
\]
Since $e^{t\mathcal L}\mathbf 1=\mathbf 1$ and $\pi$ is stationary, $\langle\mathbf 1_{U},e^{t\mathcal L}\mathbf 1_{U'}\rangle_\pi-\pi(U)\pi(U')=\langle\mathbf 1_U-\pi(U),e^{t\mathcal L}(\mathbf 1_{U'}-\pi(U'))\rangle_\pi$ for all sets $U,U'\subseteq\Sigma_n$. Using the Cauchy--Schwarz inequality, the contraction just proved, $\norm{\mathbf 1_U-\pi(U)}_{L^2(\pi)}^2=\pi(U)(1-\pi(U))\le\pi(U)$, and $\norm{\mathbf 1_{U'}-\pi(U')}_{L^2(\pi)}\le1/2$ for $U=U_{\widetilde v}$ and $U'=H_{\widetilde v}$, we find
\[
 \frac12-\omega\le\sum_{\widetilde v}\bigl(\pi(U_{\widetilde v})\pi(H_{\widetilde v})-\langle\mathbf 1_{U_{\widetilde v}},e^{t\mathcal L}\mathbf 1_{H_{\widetilde v}}\rangle_\pi\bigr)
 \le\frac{e^{-\operatorname{gap}(\mathcal L)t}}2\sum_{\widetilde v}\sqrt{\pi(U_{\widetilde v})}\le\frac{e^{-\operatorname{gap}(\mathcal L)t}}2\sqrt{N_v}.
\]
In the last step, we used the Cauchy--Schwarz inequality and $\sum_{\widetilde v}\pi(U_{\widetilde v})=1$. Rearranging gives the claim.
\end{proof}

The bound depends on the number $N_v$ of values of $v$, not on the size of $\Sigma_n$. If $v$ took all $2^n$ values, the lemma would only give $\operatorname{gap}(\mathcal L)\,t\le Cn$. The first reference takes at most $e^{CL_n}$ values, and this gives the factor $n^{7/30}$, rather than $n$, in \cref{thm:barrier-strong}(d).

\begin{proof}[Proof of \cref{thm:barrier-strong}]
Fix $\beta>1$ and let $\lambda=7/30$. We fix the parameters of Steps 1--4 of \cref{tab:bp-order} as in \cref{sec:ref-parameters,sec:ref-families}, with $C_b$ the constant of \cref{inp:hb-rate}, with $R^3>3C_b/(\varkappa g)$, and with an integer $B\ge H+2$. We set $E'=1/1000=E/10$, $\eta_n=\rho_n/1000$, and fix $A$ with
\[
 0<A<\frac{t_0\log(250/11)}{3BR},
\]
which is \eqref{b:dyn:coefficient} with $E'=1/1000$, since $4(E+E')=11/250$. Let $L_n=An^\lambda$, let $\mathcal G_n^*$ be the event of \cref{lem:bp-strong-event}, and let $m_*=\lfloor n^3e^{L_n}\rfloor$, a budget of update attempts. Throughout, $n$ is large enough that the conclusions of \cref{lem:bp-strong-event,lem:ind-margins,lem:bp-coefficient} hold; this fixes $n_0$.

\paragraph{Part (a).} By \cref{lem:bp-strong-event}(i) and $1-4\lambda=1/15$, the bound holds for $n\ge n_0$, and we enlarge $C$ so that $Ce^{-cn^{1/15}}\ge1$ for $2\le n<n_0$.

\paragraph{The obstruction.} Fix $n\ge n_0$ and $J\in\mathcal G_n^*$, and fix a family as in \cref{lem:bp-strong-event}(ii). The scales satisfy \eqref{eq:ind-scales} with $E'=1/1000$. By \cref{lem:ind-margins}, the margins \eqref{b:dyn:finite-margins} hold, and by \cref{lem:bp-coefficient}, $C_n(m)<1$ for every $m\le m_*$. By \cref{prop:bp-obstruction}, every path $x_0,\dots,x_m$ with $m\le m_*$ that avoids $V$ satisfies $R(v(x_0),x_m)>0$.

\paragraph{Parts (b) and (c).} We use the uniformization of a bounded-rate generator into a discrete kernel and a Poisson clock \cite[Section~20.1]{levin-peres-2017}. Let $\mathcal L\in\mathfrak L^J(C_{\rm rate})$ and set $Q_{\mathcal L}=I+\mathcal L/(C_{\rm rate}n)$. Its off-diagonal entries are $Q_{\mathcal L}(x,x^{(i)})=c_i(x)/(C_{\rm rate}n)\ge0$, its diagonal entries are $1-\sum_ic_i(x)/(C_{\rm rate}n)\ge0$, and $Q_{\mathcal L}(x,y)=0$ unless $y=x$ or $y=x^{(i)}$ for some $i$. We see that $Q_{\mathcal L}$ is a Markov kernel, $\pi^J$ is stationary for it, every sequence with $Q_{\mathcal L}(x_{a-1},x_a)>0$ is a path in the sense of \eqref{eq:ind-jump}, and $e^{t\mathcal L}=e^{C_{\rm rate}nt(Q_{\mathcal L}-I)}$. For the heat-bath and Metropolis generators with $C_{\rm rate}=1$, we have $Q_{\mathcal L}=K^J$ and $Q_{\mathcal L}=K^J_{\rm M}$, respectively. Since $\pi^J(-x)=\pi^J(x)$, the obstruction shows that the hypotheses of \cref{lem:bp-sign-test} hold for $Q=Q_{\mathcal L}$, $\pi=\pi^J$, the map $v$, the set $V$ of the family, and $m=m_*$. Let $\phi_{\mathcal L}$ be the function $\phi$ of \cref{lem:bp-per-start} for this kernel, and let
\[
 S_{\mathcal L}=\{x:\phi_{\mathcal L}(x)\le e^{-L_n}\}.
\]
Since $m_*+1\le2n^3e^{L_n}$ and $\pi^J(V)\le3e^{-4L_n}$, by \cref{lem:bp-per-start}(a), we have
\begin{equation}\label{eq:bp-strong-starts}
 \pi^J(S_{\mathcal L}^c)\le e^{L_n}(m_*+1)\pi^J(V)\le6n^3e^{-2L_n}.
\end{equation}
For $x\in S_{\mathcal L}$ and $0\le k\le e^{L_n}\le m_*$, by \cref{lem:bp-per-start}(b), we have $\norm{Q_{\mathcal L}^k(x,\cdot)-\pi^J}_\TV\ge1/2-e^{-L_n}$. For continuous time, the bound at $t=0$ is immediate. Let $0<t\le e^{L_n}$ and let $\mathsf N$ have the Poisson law with mean $\mu_{\rm P}=C_{\rm rate}nt\le C_{\rm rate}ne^{L_n}$. For every integer $j\ge\mu_{\rm P}$, Markov's inequality applied to $z^{\mathsf N}$ with $z=j/\mu_{\rm P}\ge1$, together with $\E z^{\mathsf N}=e^{\mu_{\rm P}(z-1)}$, gives $\PP\{\mathsf N\ge j\}\le z^{-j}e^{\mu_{\rm P}(z-1)}=e^{-\mu_{\rm P}}(e\mu_{\rm P}/j)^j$. With $j=m_*+1\ge n^3e^{L_n}$, this implies
\begin{equation}\label{eq:bp-chernoff}
 \PP\{\mathsf N>m_*\}\le\Bigl(\frac{e\,C_{\rm rate}ne^{L_n}}{n^3e^{L_n}}\Bigr)^{m_*+1}=\Bigl(\frac{eC_{\rm rate}}{n^2}\Bigr)^{m_*+1}\le e^{-(m_*+1)}\le e^{-L_n}
\end{equation}
once $n^2\ge e^2C_{\rm rate}$. By \cref{lem:bp-per-start}(c) with $q_{\rm u}=C_{\rm rate}n$, every $x\in S_{\mathcal L}$ satisfies $\norm{e^{t\mathcal L}(x,\cdot)-\pi^J}_\TV\ge1/2-2e^{-L_n}$ for $0\le t\le e^{L_n}$. Taking $n_1\ge n_0$ so large that also $n_1^2\ge e^2C_{\rm rate}$ and $12n^3e^{-2L_n}\le e^{-L_n}$ for $n\ge n_1$, we obtain part~(c) with $c=A$ and $C=2$. For part~(b), we apply this with $C_{\rm rate}=1$ to the heat-bath and Metropolis generators, for which $Q_{\mathcal L}$ is $K^J$ and $K^J_{\rm M}$, and take $S=S_{n(K^J-I)}\cap S_{n(K^J_{\rm M}-I)}$. By \eqref{eq:bp-strong-starts}, $\pi^J(S^c)\le12n^3e^{-2L_n}\le e^{-L_n}$, and the four distance bounds hold on $S$, with $\lfloor e^{L_n}\rfloor$ discrete steps and $1/2-2e^{-L_n}$ as the lower bound. Enlarging $n_0$ to the value of $n_1$ for $C_{\rm rate}=1$ completes part~(b).

\paragraph{Part (d).} Let $\mathcal L\in\mathfrak L^J(C_{\rm rate})$ be reversible, and let $t=e^{L_n}$. By \cref{lem:bp-per-start}(c) with $q_{\rm u}=C_{\rm rate}n$, by \eqref{eq:bp-chernoff}, and by $\pi^J(V)\le3e^{-4L_n}$,
\[
 \omega=\sum_x\pi^J(x)e^{t\mathcal L}(x,G_x)\le6n^3e^{-3L_n}+e^{-L_n}\le\frac14
\]
for $n\ge n_1$, after enlarging the threshold $n_1$ of part~(c) by an amount depending only on $\beta$ and $C_{\rm rate}$. Write $\widehat c=c(5)>0$ for the first list-size constant in \eqref{eq:ref-schedule}. The map $v$ takes values among the signed entries of the list $\mathcal F_0$, which has $K_0=\lceil e^{(\widehat c+1)L_n}\rceil$ entries by \eqref{eq:ref-schedule}; hence it takes at most $N_v=2K_0\le4e^{(\widehat c+1)L_n}$ values, where $\widehat c$ depends only on $\beta$. By \cref{lem:bp-gap},
\[
 \operatorname{gap}(\mathcal L)e^{L_n}\le\frac12\log\bigl(4e^{(\widehat c+1)L_n}\bigr)+\log2\le(\widehat c+2)L_n
\]
for $n\ge n_1$, after enlarging $n_1$ once more by an amount depending only on $\beta$. If $\operatorname{gap}(\mathcal L)=0$, the conclusion of part~(d) is immediate by the convention $1/0=+\infty$. Otherwise we may divide by the positive gap, and we obtain $1/\operatorname{gap}(\mathcal L)\ge e^{L_n}/((\widehat c+2)L_n)=e^{An^\lambda}/((\widehat c+2)An^\lambda)$, which is part~(d) after enlarging $C$. The enlarged $n_1$ also serves in part~(c).

\paragraph{Part (e).} By part~(a), $\sum_n\PP(J^{(n)}\notin\mathcal G_n^*)\le\sum_nCe^{-cn^{1/15}}<\infty$. By the first Borel--Cantelli lemma, which uses only the marginal laws of the $J^{(n)}$, almost surely $J^{(n)}\in\mathcal G_n^*$ for all large $n$. On this event, part~(b) gives, for all large $n$, a set of Gibbs mass at least $1-Ce^{-cn^\lambda}$ on which $d^{J^{(n)}}(x,e^{An^\lambda})$ and $d^{J^{(n)}}_{\rm disc}(x,\lfloor e^{An^\lambda}\rfloor)$ are at least $1/2-Ce^{-cn^\lambda}$, which exceeds $\tau$ for large $n$, and likewise for the Metropolis dynamics. This completes the proof.
\end{proof}
\part{The upper bound}\label{part:upper}
\chapter{Primitive disorder, the scalar process beyond the support, and the parameters}\label{u:sec:setup}

Throughout \cref{part:upper}, we fix $\beta>1$. This section sets
the notation for that Part. We identify the
primitive Gaussian disorder with the completed model of \cref{sec:model},
record the Gaussian matrix estimates used in
\cref{u:sec:proposal,u:sec:connections,u:sec:proof}, and give the exponent
parameters. We then continue the scalar diffusion of \cref{sec:inputs}
beyond the support of the Parisi measure. The scalar profiles and the
one-row law defined below are used throughout the construction of the
proposal and its analysis. We follow the scalar construction of
\cite[Sections~2--3]{oai-typical-upper}.

The notation in \cref{u:tab:local-notation} is local to this Part.
All other notation retains the meaning fixed in \cref{sec:model,sec:inputs}.
In particular, $\PP$ denotes probability, $P_\beta$ is the limiting
pressure, and all exponents are written in sans serif.

\begin{table}[htbp]
\centering
\small
\caption{Notation local to \cref{part:upper}. The right column records
the meanings used here and any earlier meanings that they replace.}
\label{u:tab:local-notation}
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.30\textwidth}>{\raggedright\arraybackslash}p{0.66\textwidth}@{}}
\toprule
Symbols & Meaning in this Part\\
\midrule
$U,m$ & Scalar potential and its first derivative in variance time;
these replace $\Phi_{\alpha_\beta},u_{\alpha_\beta}$ after the time change.\\
$Y,M$ & Scalar diffusion and its Wiener driver in variance time;
$Y$ replaces the overlap-time diffusion of \cref{sec:inputs}.\\
$B,D$ & Reflected Wiener pair; $B$ replaces the driver denoted $B$
in overlap time.\\
$\delta,\epsilon_n$ & Smoothing variance $n^{-\ex D}$ and cut-balance
threshold $n^{-\ex d}$.\\
$h,h_j$ & Mesh lengths; fields involving the completed matrix are
written $(W(g)x)_i$.\\
$\rho,\sigma$ & Deterministic reflection coefficients, replacing the
overlap scale $\rho$ and the resampled spin symbol $\sigma$.\\
$\theta$ & Rotation angle, or an exponential-moment parameter in the
separate concentration arguments.\\
$p,r,e,j$ & Scalar profiles, with $e_h,b_j,s_j,w_j$ their mesh quantities;
the subscripted probabilities $p_i^J$ keep their meaning.\\
$R_V,R_h,R_n,R_D$ & Inverse-profile bound, mesh residual, enlargement
radius, and reference law, respectively; $R(x,y)$ remains the overlap.\\
$H,H_0,E_H$ & Query cross-moment matrix, its population value, and
its error; Hamiltonians carry their superscripts or subscripts.\\
$H_*$ & Maximum absolute spin energy in \cref{u:sec:fine}, and a
Bernstein scale in \cref{u:sec:concentration}.\\
$\mu^{\rm src}_0,\mu^{\rm rep}_0$ & Unlinked source and replay laws.\\
$\Lambda^{\rm src},\Lambda^{\rm rep}$ & Linked source and replay laws.\\
$Q_V,Q_F,\nu_V$ & Output law, fine output law, and one-row law.\\
$L$ & Spin likelihood in \cref{u:sec:proposal,u:sec:fine};
$L=\Upt(E_A)$ in \cref{u:sec:concentration,u:sec:coarse}.\\
$W_{\rm wt},W_0,W_2$ & Weighted error size, ambient GOE matrix, and
$2$-Wasserstein distance; $W(g)$ is the completed matrix.\\
$\mathcal C_n,\mathcal H_n,\mathcal W_n,\mathrm{Bad}$ & Events and spin sets defined
where they are used; $\mathcal H_n$ denotes the pressure event.\\
\bottomrule
\end{tabular}
\end{table}

\section{Primitive disorder and the completed Hamiltonian}\label{u:sec:primitive}

Let $\gamma_n$ be standard Gaussian measure on $\R^{n\times n}$.
For $g\in\R^{n\times n}$ and $x\in\Sigma_n$, define
\begin{equation}\label{u:eq:primitive}
\begin{gathered}
 W(g)=\frac{g+g^{\mathsf T}}{\sqrt{2n}},\qquad
 H_g^c(x)=\frac12x^{\mathsf T}W(g)x,\\
 F^c(g)=\log\sum_{x\in\Sigma_n}e^{\beta H_g^c(x)},\qquad
 \pi_g(x)=e^{\beta H_g^c(x)-F^c(g)}.
\end{gathered}
\end{equation}
The dictionary to \cref{sec:model-completion} is
\[
 J_{ij}(g)=\frac{g_{ij}+g_{ji}}{\sqrt2}\quad(i<j),\qquad
 J'_{ii}(g)=g_{ii},\qquad A_{ij}(g)=\frac{g_{ij}-g_{ji}}2\quad(i<j).
\]
We note that the variables $J_{ij}$ and $J'_{ii}$ are independent
standard Gaussians.
The antisymmetric coordinates $A_{ij}$ are independent $N(0,1/2)$
variables, independent of $W(g)$. Expanding the quadratic form, we find
\[
 H_g^c(x)=H_n^{J(g)}(x)+G(g),\qquad
 G(g)=\frac12\tr W(g)=\frac1{\sqrt{2n}}\sum_i g_{ii}\sim N(0,1/2).
\]
Thus $F^c(g)=F_n^c$, and \cref{lem:model-completion}(b) implies that
$\pi_g=\pi^{J(g)}$. We write $\mathcal G_n=W_*\gamma_n$ for the GOE
law with off-diagonal variance $1/n$ and diagonal variance $2/n$, and
$\Unif_n(x)=2^{-n}$ for the uniform law on $\Sigma_n$.

For the dynamics, let $P_t^g=P_t^{J(g)}$. The rate of a spin flip is
\begin{equation}\label{u:eq:flip-rate}
 p_i^{J(g)}(-x_i\mid x)
 =\frac{\pi_g(x^{i,-x_i})}{\pi_g(x)+\pi_g(x^{i,-x_i})}.
\end{equation}
The generator is $\mathcal L^{J(g)}$, with rate one per site as in
\eqref{eq:common-semigroup}. For a cut $S\subseteq\Sigma_n$, its
Dirichlet flow is
\begin{equation}\label{u:eq:cut-flow}
 \mathcal Q_g(S,S^c)=\sum_{x\in S}\sum_{i=1}^n
 \pi_g(x)p_i^{J(g)}(-x_i\mid x)
 \mathbf 1\{x^{i,-x_i}\notin S\}.
\end{equation}
The diagonal of $W(g)$ adds the same constant to all spin energies,
so it cancels from the Gibbs measure and the dynamics.
We compare proposals with the joint law
\begin{equation}\label{u:eq:target-law}
 \Pi_n(\dd g,x)=\gamma_n(\dd g)\pi_g(x).
\end{equation}
Its $(J(g),x)$-marginal is $\PP_{J,X_0\sim\pi^J}$ from
\cref{def:model-joint}. In the transition law, the spin sampled from
$\pi_g$ is held fixed.

The relative entropy is
$\KL(\mu\Vert\nu)=\int\log(\dd\mu/\dd\nu)\dd\mu$ when
$\mu\ll\nu$, where the integral takes values in the extended
nonnegative reals, and $+\infty$ otherwise. In finite sums, $0\log0=0$, and a positive numerator over
zero contributes $+\infty$. We use the asymptotic conventions of
\cref{sec:model-asymptotics}. In every eventual bound, we first fix
$\beta>1$ and all exponents, then choose finite constants and $n_0$,
and then assert the bound for every integer $n\ge n_0$. Constants $C_\beta,c_\beta>0$ may change between
inequalities. They depend only on $\beta$ and, when an exponent is
absorbed, on the displayed fixed exponents; they are independent of
$n$, the selected cut, and the rotation angle. The bounds are for
each fixed temperature and saving $\omega<1/4$.

\paragraph{Conventions.}\label{u:par:conventions}
Spaces of continuous paths carry the Borel sigma algebra of the
uniform topology, products carry the product sigma algebras, and
Wiener measures are supported on paths starting at zero. A bridge of duration $h>0$ has the law $\mathrm{Br}_h$
obtained from Wiener measure by $b(t)\mapsto b(t)-(t/h)b(h)$.
These spaces are standard Borel. We complete filtrations when
constructing stochastic integrals and then choose Borel versions on
the original spaces. All refinements defining integrals are taken at
fixed $n$, fixed horizon, and fixed query mesh, before $n\to\infty$.
Versions are identified outside fixed-$n$ null sets. For the terminal
spin likelihood $L$ defined in \eqref{u:eq:spin-link}, the bound
$L\le2^n$ is used only to transfer fixed-$n$ null sets and integral
versions. All probability estimates are proved under the law to which
they apply.

\section{Gaussian orthogonal ensemble facts}\label{u:sec:goe}

We record the Gaussian facts needed for the adaptive construction of
the matrix in \cref{u:lem:assembly} and for the exceptional events of
the disorder in \cref{u:sec:connections,u:sec:proof}. The successive
conditioning on orthogonal queries is the Gaussian-revelation method
used in \cite[Sections~2--3]{bolthausen-2014} and in the proposal of
\cite[Section~4.1]{oai-typical-upper}.

\begin{lemma}[Gaussian conditioning]\label{u:lem:goe-conditioning}
Let $W\sim\mathcal G_n$. Its law is invariant under orthogonal
conjugation. In an orthonormal basis, the entries on and above the
diagonal are independent, with variances $2/n$ on the diagonal and
$1/n$ off the diagonal. After revealing its columns in a fixed
subspace, the compression to the orthogonal complement is an
independent GOE block with the original variance normalization $1/n$.
The same conditional assertion holds for successive orthogonal unit
query directions chosen measurably from the previously revealed columns
and independent initial data.
\end{lemma}

\begin{proof}
For deterministic vectors $u,v,u',v'$, the covariance is
\[
 \E[(u^{\mathsf T}Wv)(u'^{\mathsf T}Wv')]
 =\frac1n\bigl[(u\cdot u')(v\cdot v')
                 +(u\cdot v')(v\cdot u')\bigr].
\]
This follows by expanding \eqref{u:eq:primitive}. Since $W$ is
Gaussian, this covariance implies that its law is invariant under
orthogonal conjugation and that its entries on and above the diagonal
are independent in any fixed orthonormal basis. In particular, the unrevealed block is independent
of the columns already read. Conditional on the past, the next query
direction is fixed in the remaining subspace. In a basis starting
with this direction, its diagonal entry is $N(0,2/n)$, its remaining
column entries are independent $N(0,1/n)$, and the smaller block is
independent of both. By induction, the assertion holds for
adaptive queries, including conditioning on independent initial data.
\end{proof}

\begin{lemma}[Operator norm]\label{u:lem:goe-norm}
For every $n\ge2$,
\begin{equation}\label{u:eq:goe-norm}
 \gamma_n\{g:\norm{W(g)}_{\mathrm{op}}>12\}\le2e^{-n}.
\end{equation}
\end{lemma}

\begin{proof}
We use the sphere-net argument of
\cite[Corollary~4.2.13 and Exercise~4.4.3(b)]{vershynin-2018}.
Let $\mathcal N$ be a $1/4$ net of the Euclidean unit sphere with
$|\mathcal N|\le9^n$, obtained by comparing the volumes of disjoint
balls. For a symmetric matrix, approximating a unit vector at which
the absolute value of the quadratic form is maximal by a point of the
net, we have
$\norm W_{\mathrm{op}}\le2\max_{u\in\mathcal N}|u^{\mathsf T}Wu|$.
For every fixed unit $u$, the quadratic form $u^{\mathsf T}Wu$ is
centered Gaussian with variance $2/n$. Using the Gaussian tail and a
union bound, we find that
\[
 \gamma_n\{\norm W_{\mathrm{op}}>12\}
 \le2\,9^n e^{-9n}
 =2e^{(\log9-9)n}\le2e^{-n}.
\]
\end{proof}

\begin{lemma}[Pressure tail]\label{u:lem:pressure-tail}
Fix $\beta>1$ and $0<\ex z_{\rm P}<1/2$. There exists $c_\beta>0$
such that
\begin{equation}\label{u:eq:pressure-tail}
 \gamma_n\{F^c(g)>nP_\beta+n^{1-\ex z_{\rm P}}\}
 \le e^{-c_\beta n^{1-2\ex z_{\rm P}}}=o(1).
\end{equation}
\end{lemma}

\begin{proof}
Differentiating with respect to the primitive entries, we have
\[
 \partial_{g_{ij}}F^c(g)
 =\frac\beta{\sqrt{2n}}\sum_x\pi_g(x)x_ix_j,\qquad
 \norm{\nabla F^c(g)}^2\le\frac{\beta^2n}{2}.
\]
By \cref{inp:parisi} and \cref{lem:model-completion}(c),
$\E_{\gamma_n}F^c=\E F_n\le nP_\beta$. Then \eqref{u:eq:pressure-tail}
follows from Gaussian concentration
\cite{tsirelson-ibragimov-sudakov-1976}, with the Lipschitz constant
given by the gradient bound.
\end{proof}

\section{The exponent parameters}\label{u:sec:parameters}

Fix $\omega\in(0,1/4)$ and define
\begin{equation}\label{u:eq:epsilon-omega}
 \varepsilon_\omega=\frac{1/4-\omega}{20}\in(0,1/80).
\end{equation}
We use the exponents in \eqref{u:eq:exponents} throughout this Part.
The strict inequalities between them are verified in
\cref{u:sec:budget}, where
we also give the instance $\omega=1/5$. The letters $\ex r$ and
$\ex w$ denote exponents, whereas $r(t)$ is a scalar profile and
$\gamma_n$ is Gaussian measure.

\begin{equation}\label{u:eq:exponents}
\begin{array}{@{}cc@{\qquad}cc@{}}
\toprule
\text{Exponent} & \text{Value} & \text{Exponent} & \text{Value}\\
\midrule
\ex a & 1/2 & \ex b & 1/2-14\varepsilon_\omega\\
\ex f & 1/2 & \ex z & 1/4-8\varepsilon_\omega\\
\ex k_{\rm f} & 1/2-2\varepsilon_\omega & \ex w & 1/8-3\varepsilon_\omega\\
\ex D & 1/4-\varepsilon_\omega & \ex k_{\rm b} & 1/4-9\varepsilon_\omega\\
\ex u_{\rm E} & 1/4-2\varepsilon_\omega & \ex s_{\rm b} & 1/4-5\varepsilon_\omega\\
\ex d & \varepsilon_\omega/2 & \ex s_{\rm seq} & 1/4-4\varepsilon_\omega\\
\ex e & 1/4-3\varepsilon_\omega & \ex r & 1/4-15\varepsilon_\omega/2\\
\ex u_{\rm F} & 1/4-39\varepsilon_\omega/4 & \ex q & 1/4-10\varepsilon_\omega\\
\ex z_{\rm P} & 3/10 & \ex g_{\rm rot} & 3/5\\
\bottomrule
\end{array}
\end{equation}

\section{The scalar Parisi process and its continuation}\label{u:sec:scalar}

We use \cref{inp:parisi,inp:unique,inp:full-rsb,inp:ob-scalar,cor:inp-identity}
and the scalar evolution identities of \cref{sec:scalar}.
Let $\mu_\beta$ be the unique minimizing Parisi measure and let
$q_\beta\in(0,1)$ be its maximal support point. In variance time
$v=\beta^2t$, define
\[
 v_*=\beta^2q_\beta,\qquad
 \bar\alpha(v)=\alpha_\beta(v/\beta^2),\qquad
 U(v,y)=\Phi_{\alpha_\beta}(v/\beta^2,y)
 \quad(0\le v\le\beta^2).
\]
Then $\partial_vU+U_{yy}/2+\bar\alpha U_y^2/2=0$, with terminal
value $\log(2\cosh y)$ at $\beta^2$. We continue the potential and the
distribution function by
\begin{equation}\label{u:eq:scalar-continuation}
 U(v,y)=\log(2\cosh y)+(\beta^2-v)/2,\qquad
 \bar\alpha(v)=1\quad(v\ge v_*).
\end{equation}
This agrees with the original solution on $[v_*,\beta^2]$ and
satisfies the same equation for all later times. Let $m=U_y$ and
$\chi=U_{yy}$. Up to $\beta^2$, these are
$u_{\alpha_\beta}(v/\beta^2,y)$ and
$\chi_{\alpha_\beta}(v/\beta^2,y)$, and for $v\ge v_*$ they are
$\tanh y$ and $\operatorname{sech}^2y$. We write
$b(v,y)=\bar\alpha(v)m(v,y)$ before $v_*$ and $b=m$ afterward.
The value of the distribution function at a time atom does not
affect its time integrals.

On canonical Wiener space with driver $M$, let
\begin{equation}\label{u:eq:scalar-process}
 Y_t=M_t+\int_0^t b(v,Y_v)\dd v,\qquad
 m_t=m(t,Y_t),\qquad \chi_t=\chi(t,Y_t).
\end{equation}
Here and below, the time argument of $Y$ is variance time. Up to
$\beta^2$, the dictionary to \eqref{b:eq:scalar-diffusion} is
$Y^{\rm IV}_v=Y_{v/\beta^2}$ and $M_v=\beta B_{v/\beta^2}$.
Since the drift is bounded and spatially $1$-Lipschitz, there exists
a unique continuous, measurable, causal Picard solution. We have
\begin{equation}\label{u:eq:scalar-bounds}
 m_0=0,\qquad |m(t,y)|<1,\qquad
 0\le\chi(t,y)\le1-m(t,y)^2\le1.
\end{equation}
The potential and $\chi$ are even in $y$, and $m$ and $b$ are odd.
These bounds follow from \cref{inp:ob-scalar} and the step-recursion
bound in \cite[Section~3]{oai-typical-upper}. For the latter bound, we note
that one Gaussian step at mass $\lambda\in[0,1]$ satisfies
$(Tf)'=\E_\lambda f'$ and
$(Tf)''=\E_\lambda f''+\lambda\Var_\lambda(f')$, where
$\E_\lambda$ is the normalized tilted expectation. If
$0<f''\le1-(f')^2$, then
$(Tf)''\le1-((Tf)')^2$.
By induction from $\log(2\cosh y)$ and scalar stability, we obtain
the inequality for $\chi$. Since $\chi>0$ by
\cref{inp:ob-scalar}(i), this implies that $|m|<1$. On the
continuation, the formulas are explicit.

\begin{proposition}[Scalar foundation]\label{u:prop:scalar}
Under the Wiener law in \eqref{u:eq:scalar-process}, $m_t$ is a bounded
martingale in the whole $Y$-past. Define
\begin{equation}\label{u:eq:scalar-profiles}
\begin{gathered}
 p(t)=\E\chi_t,\qquad r(t)=\E\chi_t^2,\qquad
 \varkappa(t)=\E\frac{\chi_t^2}{1-m_t^2},\\
 \rho(t)=\beta\sqrt{r(t)},\qquad
 \sigma(t)=\sqrt{1-\rho(t)^2}.
\end{gathered}
\end{equation}
All expectations are under the unlinked, zero-seed scalar Wiener law.
For $0\le s\le t$, we have
\begin{align}
 &\E(m_t-m_s)^2=\int_s^t r(v)\dd v,\qquad
   \int_0^t r(v)\dd v=\E m_t^2\le1,
   \label{u:eq:scalar-variance}\\
 &r=\beta^{-2},\quad\rho=1,\quad\sigma=0\quad(t\le v_*),
   \qquad 0<r(t)\le\beta^{-2},\quad p(t)^2\le r(t),
   \label{u:eq:scalar-stationarity}\\
 &r\text{ is nonincreasing},\quad |r'|\le2r,\qquad
   p\text{ is nonincreasing},\quad |p'|\le p
   \quad\text{almost everywhere}.
   \label{u:eq:profile-variation}
\end{align}
After $v_*$, we have $p=1-\E m_t^2$, $\varkappa=p$, and $p'=-r$.
The functions $\rho$ and $\sigma$ are continuous and of bounded
variation on every compact interval. Define
\begin{equation}\label{u:eq:scalar-energy-information}
 e(V)=\beta\int_0^V r(t)p(t)\dd t,\qquad
 j(V)=\frac12\int_0^V\rho(t)^2\varkappa(t)\dd t.
\end{equation}
For $V\ge v_*$,
\begin{equation}\label{u:eq:scalar-balance}
 j(V)-\log2+P_\beta-\beta e(V)=\frac{\beta^2}{4}p(V)^2,
 \qquad \E_{\gamma_n}F^c\le nP_\beta.
\end{equation}
For a fresh solution started at $Y_s=y$, we also have
\begin{equation}\label{u:eq:restart-coordinate}
 \E[Y_t m(t,Y_t)]-ym(s,y)
 =\int_s^t\E[\chi(v,Y_v)+m(v,Y_v)b(v,Y_v)]\dd v.
\end{equation}
This holds separately before and after $v_*$, and its integrand is
identically one after $v_*$.
\end{proposition}

\begin{proof}
We begin with the martingale and profile identities, which follow
from the scalar inputs. Then we identify the pressure normalization
in the entropy--energy identity and prove the restart formula.

By \cref{inp:ob-scalar}(iv), with $v=\beta^2t$, the process $m$ is a
bounded martingale before $v_*$. The explicit continuation has the
same property by It\^o's formula. The variance identity
\eqref{u:eq:scalar-variance} follows from the scalar evolution in
\cref{inp:ob-scalar}(iv) and the continuation. By
\cref{cor:inp-identity}, $\E m_v^2=v/\beta^2$ on $[0,v_*]$.
Continuity implies that $r(v)=\beta^{-2}$ on that closed interval.
Using Jensen's inequality and \eqref{u:eq:scalar-bounds}, we find
that $p^2\le r\le p$.

For $0\le v\le v_*$, the identity
\eqref{eq:inp-stationarity} and $\Gamma(v/\beta^2)=v/\beta^2$ imply
\begin{equation}\label{u:eq:mean-susceptibility}
 p(v)=1-\int\max(q,v/\beta^2)\,\mu_\beta(\dd q).
\end{equation}
Differentiating this Lipschitz integral almost everywhere, we obtain
$p'=-\bar\alpha/\beta^2=-\bar\alpha r$. This also follows directly
from \cref{inp:ob-scalar}(iv). After $v_*$, we have
$\chi_t=1-m_t^2$, so that $p=1-\E m_t^2$, $\varkappa=p$, and
$p'=-r$. In both intervals, $|p'|\le p$ because $r\le p$.
By the post-support estimate in \cite[Lemma~3]{oai-typical-upper},
$r$ is nonincreasing and $-2r\le r'\le0$; before
$v_*$ it is constant.
This proves \eqref{u:eq:scalar-stationarity} and
\eqref{u:eq:profile-variation}. Since $r$ is continuous and
monotone, the functions $\rho$ and $\sigma$ are continuous and have
bounded variation.

In variance time, \cref{inp:parisi,inp:unique} identify
\[
 P_\beta=U(0,0)-\frac1{2\beta^2}
                  \int_0^{\beta^2}v\bar\alpha(v)\dd v.
\]
This is the counting-measure normalization used in
\cite[Lemma~4]{oai-typical-upper}. The identity there reads
$\log2-j(V)+\beta e(V)=P_\beta-\beta^2p(V)^2/4$, which is
\eqref{u:eq:scalar-balance}; the endpoint $V=v_*$ follows by
continuity. The pressure inequality follows from \cref{inp:parisi}
and \cref{lem:model-completion}(c).

For the restart identity, the scalar evolution of
\cref{inp:ob-scalar}(v), after the time change, gives zero drift for
$m(v,Y_v)$. By the product rule \cite[Theorem~3.3.6]{karatzas-shreve-1991},
the drift of $Y_vm(v,Y_v)$ is $\chi(v,Y_v)+m(v,Y_v)b(v,Y_v)$. We use
localization and the bounded spatial derivatives of \cref{sec:scalar}
to justify the product rule under step approximation.
The stochastic integrand is $Y_v\chi(v,Y_v)+m(v,Y_v)$, which is
square integrable on every finite interval because $b$ is bounded.
The drift is bounded by two, so we may take expectations and pass
to the limit. This proves \eqref{u:eq:restart-coordinate}. On the
continuation, its
integrand is $\operatorname{sech}^2Y_v+\tanh^2Y_v=1$.
\end{proof}

\trilogyneedspace{12\baselineskip}
\begin{lemma}[Cross moments and an integral representation]\label{u:lem:cross}
For $0\le s\le t\le H$, a bounded measurable test $A$ of the
Brownian past through $s$, and deterministic $f\in L^2([s,t])$,
\begin{equation}\label{u:eq:cross-moment}
 \E\left[A m_H\int_s^t f(v)\dd M_v\right]
 =\int_s^t f(v)\E[A\chi_v]\dd v.
\end{equation}
Moreover, $m_t-m_s=\int_s^t\chi_v\dd M_v$, with the integral
constructed by predictable left sums in $L^2$ under the original
Wiener law. These assertions hold across $v_*$.
\end{lemma}

\begin{proof}
We begin with the tested identity for Brownian increments, which we
prove using the restart formula. We then pass to deterministic $L^2$ weights and
identify the integral by its second moment.

First let $A$ be a bounded Borel test of $Y_{[0,s]}$. The restart law
is the product of its $Y$-past law with a fresh Wiener driver,
followed by Picard restart. Multiplying
\eqref{u:eq:restart-coordinate} by $A$ and integrating against this
product law, we have
\[
 \E[A(Y_tm_t-Y_sm_s)]
 =\int_s^t\E[A(\chi_v+m_vb_v)]\dd v,
 \qquad b_v=b(v,Y_v).
\]
For unbounded endpoint observables, we clip the $y$ factor of
$ym(t,y)$ at $[-L,L]$ and take $L\to\infty$. The conditional first
moments are dominated by $|Y_s|+\sqrt{t-s}+t-s$, which is
integrable since $\E|Y_s|\le\sqrt s+s$. The drift
$\chi+mb$ is bounded by two, so Fubini's theorem applies to the signed
time integral. If the interval crosses $v_*$, we split there and
retain the original past test on the second interval. By continuity,
the split produces no endpoint term.

The martingale identity extends to integrable past tests by clipping,
with domination by twice their absolute value. In particular,
$\E[AY_s(m_t-m_s)]=0$, and for each $v\in[s,t]$,
$\E[Ab_vm_t]=\E[Ab_vm_v]$. Substituting
$M_t-M_s=Y_t-Y_s-\int_s^t b_v\dd v$, we obtain
\begin{equation}\label{u:eq:increment-cross-moment}
 \E[A m_t(M_t-M_s)]=\int_s^t\E[A\chi_v]\dd v.
\end{equation}
After completion, the sigma algebras of the Brownian past and of the
field past agree: the Picard solution is causal, and
$M_u=Y_u-\int_0^u b(v,Y_v)\dd v$ recovers the driver causally.
The recovery map into continuous paths is Borel by bounded joint
measurability and rational evaluations. Thus
\eqref{u:eq:increment-cross-moment} holds for Brownian-past tests.
Projecting $m_H$ to an intermediate time and using linearity, we
obtain \eqref{u:eq:cross-moment} for step functions $f$. By Brownian
isometry and Cauchy--Schwarz, we may pass to $L^2$ weights: the two
errors
are at most $\norm A_\infty\norm{f-f_\ell}_2$ and
$\norm A_\infty\sqrt{t-s}\norm{f-f_\ell}_2$.

For a refining deterministic partition $s=v_0<\dots<v_\ell=t$, let
$Z_\ell=\sum_i\chi_{v_i}(M_{v_{i+1}}-M_{v_i})$.
Predictable Gaussian isometry yields
$\E Z_\ell^2\to\int_s^t r(v)\dd v$.
Using \eqref{u:eq:increment-cross-moment} on each cell with past
test $\chi_{v_i}$, and projecting $m_t$ to that cell's endpoint,
we also have
$\E[(m_t-m_s)Z_\ell]\to\int_s^t r(v)\dd v$.
In these limits, we used that $\chi$ is continuous along the
paths and $\chi\le1$. By \eqref{u:eq:scalar-variance},
$\E|Z_\ell-(m_t-m_s)|^2\to0$, which proves the representation.
\end{proof}

On a pair $(M,\widetilde M)$ of independent Wiener processes, we define
\begin{equation}\label{u:eq:reflection}
 B=\int\rho\dd M+\int\sigma\dd\widetilde M,\qquad
 D=\int\sigma\dd M-\int\rho\dd\widetilde M.
\end{equation}
For a deterministic continuous function $f$ of bounded variation,
the integral is the path function
$I_f(b)(t)=f(t)b(t)-\int_{(0,t]}b(v)\dd f(v)$ on paths starting at
zero. By step approximation and Gaussian isometry, it agrees with the
$L^2$ integral. The reflection matrix
$\left(\begin{smallmatrix}\rho&\sigma\\\sigma&-\rho\end{smallmatrix}\right)$
is symmetric and orthogonal at each time. It follows that $(B,D)$
has the product Wiener law and has the causal inverse
\[
 M=\int\rho\dd B+\int\sigma\dd D,\qquad
 \widetilde M=\int\sigma\dd B-\int\rho\dd D.
\]
These identities hold for the laws of the full pairs of paths and for
the inverse maps.
For step weights, the reflection squares to the identity on every
independent Gaussian increment. Passing to uniform limits of paths,
with integration by parts for functions of bounded variation, we
find that the identities hold for $\rho$ and $\sigma$. Taking a countable
intersection over rational times, we identify the continuous versions.
The future increments of $(B,D)$ are independent of the pasts of both
original drivers, since the approximating deterministic integrals use
only fresh increments of $(M,\widetilde M)$.

Let $\mathcal W$ denote Wiener measure. For a terminal sign $x$,
we define the one-row law by
\begin{equation}\label{u:eq:row-law}
 \nu_V(\dd x,\dd M,\dd\widetilde M)
 =(1+x m_V)(\Unif_1\otimes\mathcal W\otimes\mathcal W)
                  (\dd x,\dd M,\dd\widetilde M).
\end{equation}
Averaging over the spin for each pair of paths, we see that this law is
normalized and that the law of the entire pair of paths is unchanged. By martingale projection, we conclude that
$\E_{\nu_V}[x\mid M_{[0,t]},\widetilde M_{[0,t]}]=m_t$ for
$t\le V$. We also use $\nu_V$ for its image on the history
$(x,Y,B,D)$ under the Picard solution map and \eqref{u:eq:reflection}.
For $0\le s\le t\le V$ and deterministic $f\in L^2([s,t])$,
by \cref{u:lem:cross} and the independence of the second original
driver, we have
\begin{equation}\label{u:eq:row-cross-moments}
\begin{aligned}
 \E_{\nu_V}\left[x\int_s^t f(v)\dd B_v\right]
 &=\int_s^t f(v)\rho(v)p(v)\dd v,\\
 \E_{\nu_V}\left[x\int_s^t f(v)\dd D_v\right]
 &=\int_s^t f(v)\sigma(v)p(v)\dd v.
\end{aligned}
\end{equation}
The tested conditional form of the first identity holds with any
bounded test $A$ of both pasts at $s$, with right side
$\int_s^t f(v)\rho(v)\E[A\chi_v]\dd v$.
For the $M$ term, we first integrate the independent
$\widetilde M$ past; for the $\widetilde M$ term, we first
integrate its centered fresh future. Unbounded Brownian tests are
handled by clipping in $L^2$. Under $\nu_V$, the path marginal remains
Wiener, but adjoining the terminal sign to the filtration need not
preserve the Brownian property. The filtrations used later are
recorded in \cref{u:sec:proposal-laws}.

\trilogyneedspace{12\baselineskip}
\begin{lemma}[Terminal and inverse-profile bounds]\label{u:lem:tails}
There exist constants depending only on $\beta$ such that, for all
$t\ge0$,
\begin{equation}\label{u:eq:scalar-tails}
\begin{gathered}
 p(t)\le C_\beta\frac{e^{-t/2}}{\sqrt{1+t}},\qquad
 r(t)\ge c_\beta\frac{e^{-t/2}}{\sqrt{1+t}},\\
 R_V=\left(\min_{0\le t\le V}r(t)\right)^{-1}
 \le C_\beta e^{V/2}\sqrt{1+V}.
\end{gathered}
\end{equation}
In particular, $p(V)^2\le C_\beta e^{-V}$.
\end{lemma}

\begin{proof}
By \cite[Lemma~3, proof]{oai-typical-upper}, the post-support
transition kernel is
\[
 K_t(y,z)=e^{-t/2}\varphi_t(z-y)\frac{\cosh z}{\cosh y},
\]
where $\varphi_t$ is the centered Gaussian density of variance $t$.
It integrates to one, and direct differentiation gives the generator
$\partial_{zz}/2+\tanh(z)\partial_z$. Since the drift is bounded and
Lipschitz, uniqueness of the solution of the Picard equation
identifies its law. Let $\lambda_*$ be
the endpoint law at $v_*$. For $k>1/2$, we have
\[
 \E\chi_{v_*+t}^k=e^{-t/2}\int\lambda_*(\dd y)\operatorname{sech}y
       \int\varphi_t(z-y)\operatorname{sech}^{2k-1}z\dd z.
\]
For $k=1$ and $t\ge1$, the bounds
$\varphi_t\le(2\pi t)^{-1/2}$ and
$\int\operatorname{sech}z\dd z=\pi$ imply the upper bound.
For $k=2$, let $C'_\beta<\infty$ be such that
$\lambda_*([-C'_\beta,C'_\beta])>0$, and restrict the inner integral
to $|z|\le1$. For $t\ge1$ and $|y|\le C'_\beta$, we have
\[
 \varphi_t(z-y)\ge
 \frac{e^{-(C'_\beta+1)^2/2}}{\sqrt{2\pi t}},
\]
which implies the lower bound. On $0\le t\le1$,
\eqref{u:eq:profile-variation} implies that
$r(v_*+t)\ge\beta^{-2}e^{-2t}$. The finite initial interval is
absorbed into the constants, which depend only on the fixed
temperature. The inverse bound follows from the monotonicity of $r$.
\end{proof}
\chapter{The Gaussian-query proposal on a growing mesh}\label{u:sec:proposal}

In this section, we construct the proposal used in
\cref{u:sec:fine,u:sec:coarse,u:sec:connections}. We first prove the
Gaussian assembly and reconstruction identities in
\cref{u:lem:assembly}. We then identify the law of the complete histories of the rows and prove the quadrature estimate in \cref{u:lem:quad}.
Throughout, we use the scalar inputs of \cref{u:sec:scalar}.

We follow the Gaussian-conditioning construction of
\cite[Section~4]{oai-typical-upper}. Successive Gaussian revelations along
orthogonalized directions occur in the iterative construction of
Bolthausen~\cite[Sections~2--3]{bolthausen-2014}. Montanari used related increments guided by the Parisi process in an
incremental algorithm for approximate message
passing~\cite[Sections~2 and~3.1]{montanari-2021}. Here the durations of the cells grow with
time, and the source and replay constructions retain all the auxiliary
coordinates needed for the row identities.

\section{Growing schedule and source law}\label{u:sec:proposal-schedule}

For $T>0$, let $\mathcal C_T=C([0,T],\R)$. We use the path-space
conventions of \cref{u:sec:setup}. We write $\mathrm{Br}_h$ for the law of a Brownian bridge of duration $h>0$,
which is the image of Wiener measure under
$b(u)\mapsto b(u)-(u/h)b(h)$.
Fix $n$, $h>0$, and an integer $N<n$. We define
\begin{equation}\label{u:eq:growing-schedule}
 t_0=0,\qquad h_j=h e^{t_j/4},\qquad
 t_{j+1}=t_j+h_j,\qquad V=t_{N+1}.
\end{equation}
Cell zero is auxiliary. Query $j\in\{1,\ldots,N\}$ uses
\begin{equation}\label{u:eq:query-targets}
 \begin{aligned}
 s_j^2&=\int_{t_{j-1}}^{t_j}r(v)\dd v,&
 \mathcal T_{j,i}&=\frac{m_i(t_j)-m_i(t_{j-1})}{s_j},\\
 b_j&=\frac1{\sqrt{h_j}}\int_{t_j}^{t_{j+1}}\rho(v)p(v)\dd v.
 \end{aligned}
\end{equation}
Here $m_i(t)=m(t,Y_i(t))$. The scalar reference $b_j$ should not be
confused with the drift $b(v,y)$ of the scalar equation. We use the normalized
inner product and norm
\[
 \ipn{u}{v}=n^{-1}u^{\mathsf T}v,\qquad
 \normn{u}^2=\ipn{u}{u}.
\]

We define the unlinked source law $\mu^{\rm src}_0$ as the independent product
of the laws of the following coordinates:
\begin{itemize}
\item $n$ Wiener paths $D_i$ through $V$ and auxiliary Wiener paths
 $B_i^0$ of duration $h_0$;
\item bridges $\mathrm{br}_{j,i}\sim\mathrm{Br}_{h_j}$ for
 $1\le j\le N$ and $1\le i\le n$;
\item independent $e_j,v_j\sim N(0,I_n)$ and $d_j\sim N(0,2/n)$
 for $1\le j\le N$;
\item a full primitive $g^0\sim\gamma_n$, with ambient matrix
 $W_0=W(g^0)$, and a spin $x\sim\Unif_n$.
\end{itemize}
Unused bridges may be retained as independent ancillary coordinates;
the estimates below do not depend on them. We write $\Omega^{\rm src}$
for this product space. When we use primitive disorder, we adjoin the
antisymmetric variable specified below, independently of the other inputs.

On the auxiliary cell, $B_i=B_i^0$. For $j\ge1$, we append cell $j$ by
\begin{equation}\label{u:eq:append-cell}
 B_i(t_j+u)=B_i(t_j)+\frac{u}{\sqrt{h_j}}e_{j,i}
                    +\mathrm{br}_{j,i}(u),\qquad 0\le u\le h_j.
\end{equation}
Almost surely, and simultaneously for all the cells and rows used here,
a bridge is unchanged when we subtract the line through its two endpoints. We define $M_i=\int\rho\dd B_i+\int\sigma\dd D_i$ and solve the
scalar equation of \cref{u:sec:scalar} in each coordinate. Since the weights in these integrals are deterministic and of bounded
variation, the integrals have the causal path interpretation given
there. In particular, we can evaluate the targets in
\eqref{u:eq:query-targets} before we append \eqref{u:eq:append-cell}.

Applying the Gram--Schmidt procedure to the $\mathcal T_j$ in chronological order,
we obtain directions $u_j$ such that $\normn{u_j}=1$; when a residual vanishes,
the choice of direction is deterministic and measurable. More precisely, let $E$ have
rows that are Euclidean orthonormal and span the remaining subspace. If
$E\mathcal T_j\ne0$, we normalize $E\mathcal T_j$; otherwise, we
normalize the first nonzero column of $E$ in the fixed coordinate
order. Let $q$ be the resulting vector, which has Euclidean norm one; the query
direction is then $u_j=\sqrt n E^{\mathsf T}q$. We complete $q$ to an orthonormal basis
by a fixed measurable rule, rotate $E$ by this basis, and retain all but
its first row as the next remaining frame. We do this also when a
residual vanishes.
We define
\[
 U_j=\frac1n\sum_{l\le j}u_lu_l^{\mathsf T},\qquad
 P_j=I-U_j,\qquad f_j=P_je_j.
\]
The assembled matrix and the completion vectors are defined by
\begin{equation}\label{u:eq:assembled-matrix}
 \begin{aligned}
 W&=\frac1n\sum_{j=1}^N
       (u_jf_j^{\mathsf T}+f_ju_j^{\mathsf T})
       +\frac1n\sum_{j=1}^N d_ju_ju_j^{\mathsf T}+P_NW_0P_N,\\
 C_j&=U_je_j+P_jv_j.
 \end{aligned}
\end{equation}
Note that the scalar recursion uses none of $d_j$, $v_j$, $W_0$, or $x$.

\begin{lemma}[Gaussian assembly, completion, and path reconstruction]
\label{u:lem:assembly}
Under $\mu^{\rm src}_0$, conditional on all initial data
$(D,B^0,\mathrm{br})$, the joint law of $(W,(C_j)_j)$ is
$\mathcal G_n\otimes N(0,I_n)^{\otimes N}$. This law does not depend
on the initial data. The fixed-matrix replay endpoint
\begin{equation}\label{u:eq:replay-endpoint}
 e_j^{\rm rep}=P_jWu_j+U_jC_j
\end{equation}
equals $e_j$ on the source realization. The reconstructed $(B_i,D_i)$
have the product of $n$ independent Wiener-pair laws through $V$.
\end{lemma}

\begin{proof}
We first prove the conditional Gaussian law by induction on the
remaining queries. We then expand the assembled matrix to identify the
replay endpoints. Finally, we compute the covariance of each appended
path to show that the reconstructed pairs have the Wiener law.

\paragraph{Conditional Gaussian law.}
For a fixed remaining frame, the column has Gaussian coordinates,
consisting of a diagonal $N(0,2/n)$ entry and independent off-diagonal $N(0,1/n)$
entries. The projection $P_je_j$ gives the off-diagonal column:
when we multiply a Euclidean unit direction by $\sqrt n$ to obtain $u_j$,
this cancels the $1/\sqrt n$ scale of the column. Conditional on the previously
read $e$'s and the initial data, the next frame is fixed.

We decompose the fresh $e_j,v_j$ along $U_j,P_j$. Since the projections are orthogonal, the four Gaussian components
are independent. It follows that $U_je_j+P_jv_j$ is standard Gaussian and independent of $P_je_j$
and $d_j$. By backward induction over the remaining queries, we find that the
unrevealed block is a GOE compression on the remaining subspace, with
the same $1/n$ normalization of the variance. It is independent of the
part already read and of the later completion outputs. The base case is given by the independent
ambient matrix $W_0$. Using orthogonal invariance as in \cref{u:sec:goe}, we assemble
each column with this block. This proves the assertion on the joint law, including
all later $C_j$, and that this law does not depend on the initial data.

\paragraph{Replay endpoints.}
Expanding \eqref{u:eq:assembled-matrix}, we find
$P_jWu_j=P_je_j$. Indeed, the earlier terms lie in the range of $U_j$, and the later terms
and the final compression annihilate $u_j$. We also have
$U_jC_j=U_je_j$, so \eqref{u:eq:replay-endpoint} equals $e_j$.
By induction, we conclude that every replay target and frame agrees
pointwise with the source, as does every initial segment of the
scalar paths.

\paragraph{Pair paths.}
On one cell, $(u/\sqrt h)e+\mathrm{br}(u)$ is a centered continuous
Gaussian path, and its covariance at $u,v$ is
\[
 \frac{uv}{h}+\min(u,v)-\frac{uv}{h}=\min(u,v).
\]
So the path has the Wiener law on this cell. Concatenating independently over
the deterministic cells, we obtain jointly Gaussian evaluations at
any finite set of times, with covariance $\min(s,t)$, including across
cells. Since evaluations at rational times determine a Borel law on
continuous paths, this identifies the law of the concatenation. The resulting
pairs are independent across rows, and the $B$ paths are independent
of the $D$ paths. Since \eqref{u:eq:append-cell} preserves every earlier
initial segment, we find by induction that the concatenation is the
reconstructed terminal state. The assertions also hold when unused bridges
are retained.
\end{proof}

\section{Spin laws, histories, and filtrations}\label{u:sec:proposal-laws}

\begin{definition}[The two spin laws and the replay space]\label{u:def:laws}
On the source, we define
\begin{equation}\label{u:eq:spin-link}
 L=\prod_{i=1}^n(1+x_i m_i(V)),\qquad
 \Lambda^{\rm src}=L\mu^{\rm src}_0.
\end{equation}
The unlinked replay law $\mu^{\rm rep}_0$ is the product of
$\mathcal G_n$, independent $C_j\sim N(0,I_n)$, all initial $B^0$
paths and bridges, the $n$ independent $D$ paths, and $\Unif_n$.
Its paths are calculated by \eqref{u:eq:replay-endpoint} and
\eqref{u:eq:append-cell}. We define
$\Lambda^{\rm rep}=L^{\rm rep}\mu^{\rm rep}_0$ using the terminal
fields in replay. We write $\Omega^{\rm rep}$ for this product space,
with the independent antisymmetric variable adjoined when using
primitive disorder. The output law $Q_V$ is the $(g,x)$ marginal of
$\Lambda^{\rm rep}$ on this primitive lift.
\end{definition}

For each fixed choice of all inputs except the spin, we have
$2^{-n}\sum_x L=1$. This implies that both linked laws are probability measures,
and that linking does not change the marginals obtained by forgetting the spin,
including the marginal of the Gaussian matrix. Conditional on the paths,
the terminal spins are independent with masses
$(1+x_i m_i(V))/2$. Equivalently, we may draw the spins using independent
uniform $[0,1]$ thresholds, with the same integrals in either
realization.

Let $\Omega^{\rm row}_V=\{-1,1\}\times\mathcal C_V^3$ be the
canonical Borel space of histories of $(x,Y,B,D)$ for one row through
$V$.
We continue to write $\nu_V$ for the one-row law of
\cref{u:sec:scalar} mapped into this space by the reflected pair and
the Picard solution. From $(\Omega^{\rm row}_V)^n$ and independent
ancillary coordinates, we reconstruct the source as follows. We
restrict $B$ to cell zero and recover, for $j\ge1$,
\begin{equation}\label{u:eq:history-reconstruction}
 \begin{aligned}
 e_{j,i}&=\frac{B_i(t_{j+1})-B_i(t_j)}{\sqrt{h_j}},\\
 \mathrm{br}_{j,i}(u)&=B_i(t_j+u)-B_i(t_j)
          -\frac{u}{h_j}\bigl(B_i(t_{j+1})-B_i(t_j)\bigr).
 \end{aligned}
\end{equation}
The ancillary coordinates are $d_j,v_j,g^0$, the antisymmetric variable
$A$ used in the final lift, and the unused bridges; they are independent, with law
\begin{equation}\label{u:eq:ancillary-law}
 \begin{aligned}
 \mu^{\rm anc}={}&N(0,2/n)^{\otimes N}\otimes N(0,I_n)^{\otimes N}
                  \otimes\gamma_n\otimes\gamma_n^{\rm anti}\\
 &\hspace{12mm}\otimes
       \bigotimes_{j\notin\{1,\ldots,N\}}
                     \mathrm{Br}_{h_j}^{\otimes n}.
 \end{aligned}
\end{equation}
The last factor is omitted if unused bridges are not retained, and
$\gamma_n^{\rm anti}$ is specified below. We write $\mathcal A$ for
the ancillary product space. Since $W_0=W(g^0)$, we retain the whole primitive matrix $g^0$,
including its unused antisymmetric coordinates.

The endpoints and bridges in \eqref{u:eq:history-reconstruction} are
independent, since their evaluations at any finite set of times are jointly
Gaussian with zero cross covariance. We have therefore identified
the product law on the source space. Using the causal
inverse of the reflected pair in \cref{u:sec:scalar}, we identify the
innovation and the scalar field in this reconstruction. All these maps are Borel
under the conventions of \cref{u:sec:setup}. Indeed, evaluations at finitely many times,
restrictions of paths, integration against weights of bounded variation, and
the Picard solution map are Borel,
and the frame rule uses finitely many measurable zero tests and
normalizations.

The fixed-matrix experiment and its independent coordinate-path law are
those of \cite[Section~4.1]{oai-typical-upper}. The source and replay
maps below retain the full histories and completion variables on the
growing mesh, making their equality of laws explicit.
We record the laws in the following diagram, in which every arrow except the
density arrow denotes an equality of laws under a measurable pushforward. The same
top arrows, applied before linking, identify the unlinked laws. The map $F$
retains the initial data and $D$ and replaces the source noises by
$(W,C)$ from \eqref{u:eq:assembled-matrix}.
\begin{equation}\label{u:eq:law-diagram}
 \begin{gathered}
 \bigl((\Omega^{\rm row}_V)^n\times\mathcal A,
       \nu_V^{\otimes n}\otimes\mu^{\rm anc}\bigr)
 \xrightarrow{\ \operatorname{Rec}\ }
       (\Omega^{\rm src},\Lambda^{\rm src})
 \xrightarrow{\ F\ }
       (\Omega^{\rm rep},\Lambda^{\rm rep})
 \\[4pt]
 \begin{array}{ccc}
 (\Omega^{\rm rep},R_D=E_D\mu^{\rm rep}_0)
 &\xrightarrow{\ d\Lambda^{\rm rep}/dR_D=L^{\rm rep}/E_D\ }&
 (\Omega^{\rm rep},\Lambda^{\rm rep})\\
 \downarrow\scriptstyle(g,x)&&\downarrow\scriptstyle(g,x)\\
 (\R^{n\times n}\times\Sigma_n,\gamma_n\otimes\Unif_n)&&Q_V.
 \end{array}
 \end{gathered}
\end{equation}
Here $\operatorname{Rec}$ is the reconstruction just described,
using \eqref{u:eq:history-reconstruction} and the reflected pair.
Along the density arrow, we change measure on the same replay space; $E_D$
and $R_D$ are constructed in \cref{u:sec:reference}. In particular,
the top row implies the identity for the complete histories
\begin{equation}\label{u:eq:row-history}
 \bigl((x_i,Y_i,B_i,D_i)_{i=1}^n\bigr)_*\Lambda^{\rm rep}
       =\nu_V^{\otimes n}.
\end{equation}

To check the linked identities, let $\nu_V^0$ be the canonical
history law before the one-row link. By the factorization into
endpoints and bridges and the inverse of the reflection, we have
\[
 \operatorname{Rec}_*\bigl((\nu_V^0)^{\otimes n}
                          \otimes\mu^{\rm anc}\bigr)
       =\mu^{\rm src}_0.
\]
On a supporting set of full measure, the recomputed field agrees with
the given $Y$. For every bounded Borel test function $f$ on the source space, we have
\[
 \begin{aligned}
 &\int f\bigl(\operatorname{Rec}(w,a)\bigr)
      \prod_{i=1}^n\bigl(1+x_i m(V,Y_i(V))\bigr)
      \dd\bigl((\nu_V^0)^{\otimes n}\otimes\mu^{\rm anc}\bigr)(w,a)\\
 &\hspace{35mm}=\int fL\dd\mu^{\rm src}_0.
 \end{aligned}
\]
This gives the law on the source space in the top row, with every
ancillary coordinate retained. Repeating this calculation of densities and maps for $F$,
and using the pointwise replay identity in \cref{u:lem:assembly}, we obtain the
pushforward identity for the linked replay law. The matrix and completion vectors are computed
from these rows and ancillary variables; they need not be independent
of the rows.

For the primitive $g$ in \eqref{u:eq:law-diagram}, we retain the
antisymmetric Gaussian variable $A$, independently of all the other
inputs.
Its entries $A_{ij}$, $i<j$, are independent $N(0,1/2)$, with
$A_{ji}=-A_{ij}$ and $A_{ii}=0$. We define
\[
 g=\sqrt{n/2}\,W+A.
\]
Then $g\sim\gamma_n$ and $W(g)=W$, as in \cref{u:sec:primitive}.
Since replay reads only $W$, this gives the proposal with primitive input.
The proposal and the target have the same conditional law of $A$ given
$(W,x)$. Their joint relative entropy on primitive matrices equals
their relative entropy on physical matrices. We use this lift whenever
we condition on $g$.

\paragraph{Filtrations.}
Whether a Brownian assertion may be used depends on the law and on the
information revealed at time zero. The following table records the
distinctions needed in the later proofs. In the last row,
$\xi_i(v)=x_i\chi(v,Y_i(v))/(1+x_i m(v,Y_i(v)))$, as in
\cref{u:sec:reference}.
\begin{center}
\small
\renewcommand{\arraystretch}{1.14}
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.29\textwidth}
 >{\raggedright\arraybackslash}p{0.39\textwidth}
 >{\raggedright\arraybackslash}p{0.25\textwidth}@{}}
\toprule
Law and filtration & Brownian or independence assertion & Use \\
\midrule
Scalar Wiener pair, with the spin-free pair past
& $(M,\widetilde M)$ and its deterministic orthogonal transform
  $(B,D)$ are Wiener pairs.
& Scalar martingales, brackets, and path estimates. \\[3pt]
Unlinked row, with an independent uniform sign revealed initially
& $(B,D)$ remains a Wiener pair in the filtration including this sign.
& Signed integrals and fixed-sign It\^o identities. \\[3pt]
Linked row $\nu_V$, with the sign revealed initially
& The entire path marginal is unchanged; $B,D$ need not be Brownian
  in the filtration including the sign.
& Posterior moments, path-only estimates, and the previously defined
  signed integrals. \\[3pt]
Product of complete linked rows and independent ancillary variables
& The complete rows are independent before conditioning on $W$.
& Independent-row concentration estimates. \\[3pt]
Unlinked replay, with $(W,C,B^0,\mathrm{br},x)$ revealed initially
& $D$ is Brownian and independent of the static inputs. No Brownian
  property for $B$ is asserted in this filtration.
& Construction and normalization of $R_D$. \\[3pt]
Linked source natural filtration, including $x$, excluding the final
matrix and unused matrix noises
& The compensated vectors
  $B-\int\rho\xi\dd v$ and $D-\int\sigma\xi\dd v$ are Brownian.
& The mean information identity by bounded-integrand Girsanov. \\
\bottomrule
\end{tabular}
\end{center}
Once the terminal sign or the assembled matrix is revealed, a Brownian
assertion must be verified separately in the resulting filtration.
We transport all scalar expectations at fixed $V$ and all Gram tails below
by \eqref{u:eq:row-history}. We use the bound $L\le2^n$ only for
null sets at fixed $n$ or for integral versions, as in \cref{u:sec:setup}.

\section{Population entries}\label{u:sec:proposal-entries}

These are the martingale and posterior-projection identities used in
\cite[Section~4, proof of Proposition~5]{oai-typical-upper}; we record
their form for the unequal neighboring cell durations.
For a single linked row, let
$e_j=(B(t_{j+1})-B(t_j))/\sqrt{h_j}$ and
$\mathcal T=(\mathcal T_j)_{j=1}^N$, $e=(e_j)_{j=1}^N$.
By \cref{u:lem:cross}, the identities for the reflected pair in
\cref{u:sec:scalar}, and posterior projection, we have
\begin{equation}\label{u:eq:population-entries}
 \begin{gathered}
 \E\mathcal T\mathcal T^{\mathsf T}=I_N,\qquad
 \E ee^{\mathsf T}=I_N,\qquad
 \E\mathcal T x=s,\qquad \E ex=b,\\
 H_0=\E\mathcal T e^{\mathsf T},\qquad
 (H_0)_{ij}=0\quad(i\le j).
 \end{gathered}
\end{equation}
Indeed, disjoint martingale increments are orthogonal.
The identity $\E\mathcal T_jx=s_j$ follows from the variance identity
in \cref{u:prop:scalar}, and the reflected cross moment gives
$\E e_jx=b_j$. For $i\le j$, $\mathcal T_i$ is measurable before
the fresh $B$ cell defining $e_j$, so the path-only moment is zero.
We retain the entire lower block $H_0$. By Bessel's inequality in the
$L^2$ space for one row, we have
\begin{equation}\label{u:eq:population-bounds}
 \norm{s},\norm{b},\norm{H_0}\le1,\qquad
 0\le e_h=s^{\mathsf T}b\le1.
\end{equation}
The row means vanish by zero-field symmetry. Note that these calculations do not use
independence of the entries within a row.

\begin{lemma}[Adjacent-cell quadrature and stopping]\label{u:lem:quad}
If $\max_{j\le N}h_j\le1/8$, then
\begin{equation}\label{u:eq:adjacent-quadrature}
 |e_h-e(V)|\le C_\beta h.
\end{equation}
For $h=n^{-\ex a}$ and $\ex L>0$, define
\[
 N_{\ex L}+1=\min\{j:t_j\ge\ex L\log n\},\qquad
 V_{\ex L}=t_{N_{\ex L}+1}.
\]
If $\ex a>\ex L/4$, then eventually
\begin{equation}\label{u:eq:growing-stopping}
 \begin{gathered}
 N_{\ex L}+1\le8n^{\ex a},\qquad
 \ex L\log n\le V_{\ex L}
                \le\ex L\log n+n^{-\ex a+\ex L/4},\\
 \max_{j<N_{\ex L}+1}h_j\le n^{-\ex a+\ex L/4}.
 \end{gathered}
\end{equation}
There are $N_{\ex L}+1$ cells and exactly $N_{\ex L}$ matrix queries.
If additionally $\ex a<1$, then eventually $N_{\ex L}+1\ge2$ and
$N_{\ex L}<n$.
\end{lemma}

\begin{proof}
We compare the normalization from the preceding cell with $r$ on the
next cell. Using the relative variation bounds of \cref{u:prop:scalar}, we
obtain an integrable quadrature error. For the stopping estimate, we
sum the decrements of $e^{-t_j/4}$.

For $v\in[t_j,t_{j+1}]$ and $u\in[t_{j-1},t_j]$, the relative
variation bounds, monotonicity, and
$h_j/h_{j-1}=e^{h_{j-1}/4}$ imply that
\begin{equation}\label{u:eq:adjacent-ratio}
 1-h_j/8\le\frac{s_j}{\sqrt{h_jr(v)}}\le1+4h_j.
\end{equation}
For the upper bound, we have $r(v)\ge(1-4h_j)r(u)$ and
$(1-4h_j)^{-1/2}\le1+4h_j$ for $h_j\le1/8$.
For the lower bound, we use $r(u)\ge r(v)$ and
$e^{-h_{j-1}/8}\ge1-h_j/8$. It follows that
\[
 \left|\frac{s_j}{\sqrt{h_j}}-\sqrt{r(v)}\right|
       \le4h_j\sqrt{r(v)}.
\]
With $q_e(v)=\beta r(v)p(v)$, we obtain
\begin{equation}\label{u:eq:quadrature-error}
 |e_h-e(V)|\le h+4h\int_0^V e^{v/4}q_e(v)\dd v.
\end{equation}
The first term $h$ bounds the contribution of the omitted auxiliary cell. The integral is
bounded uniformly in $V$. Indeed, on the finite interval before the endpoint of the support, we have
$q_e\le\beta$, and after it, using $r\le p$ and \cref{u:lem:tails},
we find that the integrand decays exponentially. This proves
\eqref{u:eq:adjacent-quadrature}, with, for example,
\[
 C_\beta=1+4\beta\int_0^\infty e^{v/4}p(v)r(v)\dd v<\infty.
\]

For $h_j\le1$, we have $1-e^{-h_j/4}\ge h_j/8$, and hence
\[
 e^{-t_j/4}-e^{-t_{j+1}/4}\ge h/8.
\]
Summing up to the first reaching time gives
$(N_{\ex L}+1)h\le8$. Every left endpoint that is used lies below
$\ex L\log n$, which proves the bounds on the number of cells and on the overshoot in
\eqref{u:eq:growing-stopping}. The reaching index exists because
$t_j\ge jh$. Since $\ex L\log n>0$, it is at least one.
If additionally $\ex a<1$, it is eventually at least two, because
$t_1=n^{-\ex a}\to0<\ex L\log n$ and
$\ex a>\ex L/4>0$. Further,
\eqref{u:eq:growing-stopping} implies that
\[
 N_{\ex L}<N_{\ex L}+1\le8n^{\ex a}<n
 \qquad\text{once }n^{1-\ex a}>8.
\]
In every fine or coarse use, the fine-energy conditions in
\cref{u:prop:fine-kl} supply $0<\ex k_{\rm f}<1-\ex a$, so
$\ex a<1$; in particular, $\ex a=1/2$ in the exponent table and
the range family of \cref{u:sec:parameters,u:sec:budget}.
We never use $\ex L=0$. All scalar references are
evaluated at $V_{\ex L}$, including its overshoot. The estimate needs neither an extra cell nor a quadrature error
bounded by the maximal duration of a cell.
\end{proof}

We use the same sequence \eqref{u:eq:growing-schedule} for both
$\ex L=\ex b$ and $\ex L=\ex f$, where $0<\ex b<\ex f$,
and the two replays use identical chronological inputs through
$V_{\ex b}$. By induction on the queries, every scalar
initial segment through $V_{\ex b}$ is the same in the two replays. The auxiliary cells agree. If the scalar paths agree through
the preceding cell, then the targets agree, the same remaining frame
and matrix give the same \eqref{u:eq:replay-endpoint}, and the same
bridge appends the same next cell. This argument covers the
deterministic fallback as well. At query $j$, the number of remaining queries
is $N_{\ex L}-j$; this does not change the chronological schedule.
The terminal spins drawn at the fine horizon are never fed back into
the continuation.
\chapter{Forward entropy at the fine horizon}\label{u:sec:fine}

The goal of this section is to prove \cref{u:prop:fine-kl}, which bounds
the relative entropy of the fine proposal with respect to $\Pi_n$.
We use the source and replay laws of \cref{u:def:laws}, the scalar
identities of \cref{u:prop:scalar,u:lem:cross}, and the quadrature estimate
of \cref{u:lem:quad}. We use this bound at the two ends of the
connecting sequence in \cref{u:sec:connections}.
For the reference measure and the comparison of likelihoods, we follow
\cite[Section~4, Proposition~5]{oai-typical-upper}. We then estimate the
expected energy directly, including the outcomes on which the query uses
the deterministic fallback.

Throughout this section, $V=t_{N+1}$ is the terminal time of a proposal
with $N<n$ queries. We write $m_i(t)=m(t,Y_i(t))$ and
$\chi_i(t)=\chi(t,Y_i(t))$. Expectations with subscript $\Lambda$
refer to the linked source or replay law, according to the space of
the integrand; these laws agree on the complete row histories and on
the output by \cref{u:sec:proposal-laws}.

\section{The reference measure}
\label{u:sec:reference}

On the unlinked replay space, we regard $(W,C,B^0,\mathrm{br},x)$ as static.
The $D_i$ are independent Brownian motions, independent of these
static data. The replay field at time $t$ is a Borel function of these
static data and $D_{[0,t]}$: each next query reads the field only at
the left endpoint of its new cell. We define
\begin{equation}\label{u:eq:reference-coefficient}
 \xi_i(t)=\frac{x_i \chi(t,Y_i(t))}{1+x_i m(t,Y_i(t))},\qquad
 |\xi_i(t)|\le2.
\end{equation}
The coefficient $\sigma(t)\xi_i(t)$ is continuous and predictable in this replay filtration.
It vanishes before $v_*$; afterward it is $\sigma(t)(x_i-m_i(t))$.
Under $\mu^{\rm rep}_0$, let
\begin{equation}\label{u:eq:reference-density}
 \begin{aligned}
 \mathscr M_D&=\sum_i\int_0^V\sigma(t)\xi_i(t)\dd D_i(t),\\
 \langle\mathscr M_D\rangle
   &=\sum_i\int_0^V\sigma(t)^2\xi_i(t)^2\dd t\le4nV,\\
 E_D&=e^{\mathscr M_D-\langle\mathscr M_D\rangle/2},\qquad
 R_D=E_D\mu^{\rm rep}_0.
 \end{aligned}
\end{equation}
We define $\mathscr M_D$ as the $L^2$ limit of the vector of
predictable sums at the left endpoints of dyadic partitions. This
definition, and not the normalization alone, is what identifies it
with the sum of the row integrals below. We choose versions according to the
conventions of \cref{u:sec:setup}.

\trilogyneedspace{12\baselineskip}
\begin{lemma}[The reference measure and the row integrals]
\label{u:lem:reference}
The measure $R_D$ is a probability measure, preserves the complete
static marginal, and has output marginal $\gamma_n\otimes \Unif_n$ on $(g,x)$.
For one unlinked row, let $\xi(t)=x\chi_t/(1+xm_t)$ and
\[
 \begin{aligned}
 \mathscr M_D^{\rm row}
   &=\int_0^V\sigma(t)\xi(t)\dd D(t),\\
 \langle\mathscr M_D^{\rm row}\rangle
   &=\int_0^V\sigma(t)^2\xi(t)^2\dd t.
 \end{aligned}
\]
The integral denotes its unlinked $L^2$ left-sum version;
we use the same Borel version under the equivalent linked row law.
Moreover,
\begin{equation}\label{u:eq:reference-ratio}
 \begin{aligned}
 \frac{\dd\Lambda^{\rm rep}}{\dd R_D}&=L/E_D,\qquad
 I^{(n)}=\log(L/E_D)=\sum_{i=1}^n I_{V,i},\\
 I_V&=\log(1+x m_V)-\mathscr M_D^{\rm row}
                   +\tfrac12\langle\mathscr M_D^{\rm row}\rangle,
 \end{aligned}
\end{equation}
Here $I_{V,i}$ is $I_V$ evaluated on row $i$.
The row histories are independent with law $\nu_V$ of
\cref{u:sec:scalar}, so the summands are independent with
common law $(I_V)_*\nu_V$.
Finally,
\begin{equation}\label{u:eq:reference-mean}
 \E_{\Lambda^{\rm rep}}I^{(n)}=n j(V).
\end{equation}
\end{lemma}
\begin{proof}
We first normalize the exponential conditional on the static data.
We then transport its defining left sums to the source row space and
compute the mean in the source natural filtration.

\paragraph{Normalization.}
This is the conditional stochastic-exponential normalization used in
\cite[Section~4, proof of Proposition~5]{oai-typical-upper}; the
bounded-integrand change of measure is Girsanov's theorem
\cite[Theorem~3.5.1]{karatzas-shreve-1991}.
Fix the static data and a deterministic grid for the left sums. The last
increment of $D$ is Gaussian and independent of the earlier ones, and
integrating it out shows that its factor in the exponential, together with
the compensator, has conditional mean one. Iterating over the increments, we find
that the exponential of the left sums has mean one. Since the predictable
coefficients are bounded and continuous, their left approximants
converge in $L^2(\dd t\,\dd\mu^{\rm rep}_0)$. By isometry, the sums
then converge to the integral defining $\mathscr M_D$. The exponentials are uniformly
integrable, because the same Gaussian identity at twice the coefficient
bounds their second moments by $e^{4nV}$. Testing against a bounded
function of the complete static data before taking the limit, we
conclude that the limiting exponential has conditional mean one given
these data.
This proves that $R_D$ is normalized and that it preserves the
complete static marginal, including the output marginal.

With the Borel versions fixed above, and after a modification on a null set,
$E_D$ is positive and finite everywhere.
Since $L>0$, multiplying the densities gives
$R_D\cdot(L/E_D)=\mu^{\rm rep}_0\cdot L=\Lambda^{\rm rep}$.

\paragraph{Identification of the row integrals.}
To identify the integral, we use the diagram of laws and the history
identity in \cref{u:sec:proposal-laws} for the unlinked laws. At every
finite dyadic level, the pullback of the vector sum on the replay
space equals the sum of the corresponding functions of the row
histories $(x,Y,B,D)$. On the unlinked
row space, $D$ is a Brownian motion in the filtration generated by the two original
drivers and by the independent uniform sign, which is revealed at time zero. The
coefficient $\sigma\xi_i$ from \eqref{u:eq:reference-coefficient} is bounded and
predictable, so the same sums converge in $L^2$ to its integral. The law identity and the
Cauchy--Schwarz inequality for finite sums imply that the sum of the row limits
converges in $L^2$ on the replay space. By uniqueness of the limit, $\mathscr M_D$ is
the sum of the row integrals. By Fubini's theorem in the time variable,
the brackets split in the same way.
This proves \eqref{u:eq:reference-ratio} almost everywhere under the
unlinked law. The linked and unlinked laws are equivalent, which implies that the
identity also holds almost everywhere under the linked law. Here the transported variables are the complete histories, and the
versions are fixed at the given $n,V$.

\paragraph{The mean.}
To compute the mean, we work in the natural filtration of the unlinked source,
$\mathcal F_t=\sigma(x,(B_i,D_i)_{[0,t]}:i\le n)$, which does not contain $W$,
$d_j$, and $W_0$. By \cref{u:lem:cross}, we have
$\dd m_i=\chi_i(\rho\dd B_i+\sigma\dd D_i)$. Using It\^o's formula,
localized where $1+x_i m_i\ge1/l$ and then letting $l\to\infty$, we find
\[
 \log L=\sum_i\int_0^V\xi_i(\rho\dd B_i+\sigma\dd D_i)
                         -\tfrac12\sum_i\int_0^V\xi_i^2\dd t.
\]
The localization can be removed because $|\xi_i|\le2$ and the denominator is
strictly positive on every finite continuous sample path.
The integral against $D$ is the pullback identified above, so
\begin{equation}\label{u:eq:information-brownian}
 I^{(n)}=\sum_i\int_0^V\rho\xi_i\dd B_i
                        -\tfrac12\sum_i\int_0^V\rho^2\xi_i^2\dd t.
\end{equation}
The process $L_t=\prod_i(1+x_i m_i(t))$ is the positive bounded
martingale that defines the terminal link, with $L_0=1$.
The expectation in Novikov's condition is at most $e^{2nV}$.
By Girsanov's theorem for bounded integrands
\cite[Theorem~3.5.1]{karatzas-shreve-1991}, under $\Lambda^{\rm src}$
the processes
\[
 B_i^\Lambda=B_i-\int_0^\cdot\rho\xi_i\dd t,\qquad
 D_i^\Lambda=D_i-\int_0^\cdot\sigma\xi_i\dd t
\]
form a Brownian vector in this natural filtration.
It follows from \eqref{u:eq:information-brownian} that
\[
 \E_\Lambda I^{(n)}
   =\frac12\sum_i\E_\Lambda\int_0^V\rho^2\xi_i^2\dd t.
\]
Given the past of the spin-free pair at time $t$, the linked spin probabilities
are $(1+x m_t)/2$, and therefore
\[
 \sum_{x=\pm1}\frac{1+x m_t}{2}
       \frac{\chi_t^2}{(1+x m_t)^2}=\frac{\chi_t^2}{1-m_t^2}.
\]
The spin-free paths keep their original law. Since the integrands
are bounded, we may exchange the expectation and the time integral. Using the definition of $j$ in \cref{u:prop:scalar}, we obtain
\eqref{u:eq:reference-mean}.

\paragraph{Integrability.}
All signed quantities used here are integrable under the indicated
laws. By isometry, we have $\mathscr M_D\in L^2(\mu^{\rm rep}_0)$; also
$\langle\mathscr M_D\rangle\le4nV$.
For fixed $n$, the bound $L\le2^n$ transfers this $L^2$ integrability to
$\Lambda$. Also, $\log L\in L^2(\Lambda)$, since $t(\log t)^2$ is bounded
on $0<t\le2$ and all other spin factors average to one.
After compensation in \eqref{u:eq:information-brownian}, we obtain the bound
$\E_\Lambda(I^{(n)})^2\le8nV+8n^2V^2$.
The calculation with conditional exponentials above also proves the case of
Girsanov's theorem with bounded integrands, by exponential tilting of
finitely many Gaussian steps and passage to the limit in $L^2$. Throughout,
the filtration does not contain the final matrix.
\end{proof}

\begin{lemma}[Forward relative entropy]\label{u:lem:forward-kl}
For the output $Q_V$ of \cref{u:def:laws},
\begin{equation}\label{u:eq:forward-kl}
 \KL(Q_V\Vert \Pi_n)\le n j(V)-n\log2+\E_{\gamma_n}F^c
                               -\beta\E_\Lambda H^c_g(x).
\end{equation}
\end{lemma}
\begin{proof}
We use the lifted-reference and entropy-contraction argument of
\cite[Section~4, proof of Proposition~5]{oai-typical-upper}.
On the replay space, we define
$\dd\widehat P=2^n e^{\beta H^c_g(x)-F^c(g)}\dd R_D$.
Its total mass is one, since $R_D$ projects to $\gamma_n\otimes\Unif_n$;
its output law is $\Pi_n$. Further,
\[
 \log\frac{\dd\Lambda}{\dd\widehat P}
      = I^{(n)}-n\log2+F^c(g)-\beta H^c_g(x).
\]
Averaging pointwise over the spins, we see that the matrix marginal
of the linked law is the GOE law. In particular, with $H_*(W)=\max_x|x^{\mathsf T}Wx/2|$,
\begin{equation}\label{u:eq:energy-integrability}
 \E_\Lambda H_*^2\le\frac{n^2}{4}\E_{\mathcal G_n}\|W\|_F^2
                =\frac{n^2(n+1)}4.
\end{equation}
Also $|F^c|\le n\log2+|\beta|H_*$. These bounds and \cref{u:lem:reference}
justify the finite KL calculation. The data processing inequality for the output
map \cite[Chapter~4, Exercise~4.10]{boucheron-lugosi-massart-2013}
and \eqref{u:eq:reference-mean} imply \eqref{u:eq:forward-kl}.
\end{proof}

\section{Fine energy}
\label{u:sec:fine-energy}

We now estimate the expected energy, one query at a time.
The decomposition into queried columns, diagonal innovations and an
unrevealed GOE block is the energy decomposition in
\cite[Section~4, proof of Proposition~5]{oai-typical-upper}. The
conditional expectation calculation below also controls the fallback
outcomes and the unequal cell durations.
Let $\mathcal F_{t_j}^{\rm paths}$ be the natural filtration of the
spin-free pair of drivers in the source, through time $t_j$.
Under the linked law, the marginal law of these paths is unchanged. The
conditional laws of the future rows and of the signs factor across
coordinates, because the unlinked rows factor, their pasts factor, and
the link is a product. We obtain their conditional densities by
integrating the terminal link of each row over its independent future.
In particular, the pairs $(e_{j,i},x_i)$
are conditionally independent across $i$ and satisfy
\begin{equation}\label{u:eq:fine-conditional-correlation}
 \begin{aligned}
 \E[e_{j,i}\mid\mathcal F_{t_j}^{\rm paths}]&=0,\qquad
 \E[x_i\mid\mathcal F_{t_j}^{\rm paths}]=m_i(t_j),\\
 \psi_i=\E[e_{j,i}x_i\mid\mathcal F_{t_j}^{\rm paths}]
   &=\frac1{\sqrt{h_j}}\int_{t_j}^{t_{j+1}}\rho(v)
            \E[\chi_i(v)\mid\mathcal F_{t_j}^{\rm paths}]\dd v.
 \end{aligned}
\end{equation}
The last identity is the cross-moment identity of \cref{u:sec:scalar}
tested against functions of the past. Here we condition on the past of
the original drivers, not on $W$. The Picard map with fresh future
drivers gives a Borel version of $\psi_i$ that depends only on the past of
its own row, in accordance with the conventions on versions in \cref{u:sec:setup}.
If $h_{j-1},h_j\le1/8$, then the relative variation
estimates of \cref{u:prop:scalar} and the growth of the schedule imply that
\begin{equation}\label{u:eq:fine-correlation-bound}
 0\le\psi_i\le\beta\sqrt{h_jr(t_j)}\le2\beta s_j,
 \qquad \E\overline\psi=b_j,\quad
 \overline\psi=n^{-1}\sum_i\psi_i.
\end{equation}
The factor two accounts for the different durations of the previous
and current cells.

In this section, we write $\varrho_j=\normn{P_{j-1}\mathcal T_j}$ for the
residual norm. Whether the query is normalized or uses the fallback, we have
$\ipn{u_j}{m(t_j)}=s_j\varrho_j$, because
$m(t_j)=\sum_{l\le j}s_l\mathcal T_l$ and each $\mathcal T_l$ belongs to the span
of the first $l$ queries. Also $P_jm(t_j)=0$.

\trilogyneedspace{12\baselineskip}
\begin{lemma}[Exact one-query conditional identity]\label{u:lem:onequery}
With $c_j=\ipn{u_j}x$,
\begin{equation}\label{u:eq:onequery}
 \begin{aligned}
 &\E[c_j\ipn{x}{P_je_j}\mid\mathcal F_{t_j}^{\rm paths}]\\
 &\qquad=s_j\varrho_j\frac1n\sum_i(P_j)_{ii}\psi_i
       -\frac2{n^2}\sum_i u_{j,i}(P_j)_{ii}\psi_i m_i(t_j).
 \end{aligned}
\end{equation}
\end{lemma}
\begin{proof}
We expand $(n^{-1}\sum_i u_i x_i)(n^{-1}\sum_{k,l}x_k P_{kl}e_l)$.
Since the row pairs are conditionally independent and $x_i^2=1$, the mixed
third moment is
\[
 \E[x_i x_k e_l\mid\mathrm{past}]
 =\mathbf1_{l=i}\psi_i m_k+\mathbf1_{l=k}m_i\psi_k
                  -2\mathbf1_{i=k=l}m_i\psi_i.
\]
For $i=k$ it is zero by Gaussian centering. For $i\ne k$, it follows
from independence, also when $l$ differs from both indices. The
first contraction contains $P_jm(t_j)=0$, and the second contains
$\ipn{u_j}{m(t_j)}=s_j\varrho_j$.
In more detail, given the past of a row, a fresh seed for the row consists
of the current $B$ bridge, the future $B$ path after that cell,
the future $D$ increments, and a terminal uniform threshold.
It is independent of the standard Gaussian endpoint $e_{j,i}$;
these seeds and endpoints are independent across coordinates.
Since the Picard solution is causal, the terminal sign is a Borel
function of its own endpoint and seed. These conditional product laws justify the
third-moment calculation above.
All terms are integrable, since the signs are bounded and Gaussian
first moments are finite.
\end{proof}

\begin{lemma}[Expected energy at the fine horizon]\label{u:lem:fine-energy}
For $V=t_{N+1}$, $\max h_j\le1/8$,
\begin{equation}\label{u:eq:fine-energy}
 \E_\Lambda\frac{H^c_g(x)}n\ge e(V)-C_\beta\left[
 h+\frac{(N+1)(1+V)+\sqrt N}{n}
                         +\frac{(N+1)^2VR_V}{n^2}\right].
\end{equation}
\end{lemma}
\begin{proof}
We first estimate the expectation in \cref{u:lem:onequery}, including
the residual norm and the projection diagonals. We then sum over the
queries and use the quadrature estimate of \cref{u:lem:quad}.

\paragraph{The residual norm.}
Since $\sum_i|u_{j,i}|\le n$, the correction in \eqref{u:eq:onequery} is at
most $4\beta s_j/n$ in absolute value. Removing the projection diagonals
from its first term costs at most $2\beta s_j^2j \varrho_j/n$, and
$\E \varrho_j\le\sqrt{\E\normn{\mathcal T_j}^2}=1$.
For this calculation, we write
$Q=\normn{\mathcal T_j}^2$, $Y=\normn{U_{j-1}\mathcal T_j}^2$, $X=\varrho_j^2=Q-Y$.
For $X\ge0$,
\begin{equation}\label{u:eq:residual-square-root}
 \sqrt X\ge1+(X-1)/2-(X-1)^2/2.
\end{equation}
Indeed, the remainder in the tangent-line identity for the square root
is $(\sqrt X-1)^2/2\le(X-1)^2/2$.
Since the row pairs $(\psi_i,\mathcal T_{j,i})$ are independent and identically distributed, we have
\begin{equation}\label{u:eq:residual-covariance}
 \E[\overline\psi(Q-1)]
       =n^{-1}\operatorname{Cov}(\psi_i,\mathcal T_{j,i}^2)
       \ge-2\beta s_j/n.
\end{equation}
Here we used that $\E \mathcal T_{j,i}^2=1$ and $\psi_i\ge0$.

Let $h_-=h_{j-1}$. Conditional on the past of the original pair up to time
$t_{j-1}$, the coordinates of the new vector $\mathcal T_j$
are independent, centered martingale increments, with conditional
second moment at most $h_-/s_j^2$ and fourth moment at most
$3h_-^2/s_j^4$. The bound on the fourth moment follows from It\^o's formula, applied to
the fourth power of an increment with coefficient $\chi\le1$, and the
bound on the second moment. The rank-$(j-1)$ projection $U_{j-1}$
is measurable at this time. Expanding its quadratic form using
conditional centering and independence, we obtain
\begin{equation}\label{u:eq:residual-projection-moments}
 \E Y\le\frac{j h_-}{ns_j^2},\qquad
 \E Y^2\le\frac{3(j^2+j)h_-^2}{n^2s_j^4}.
\end{equation}
For the unconditional fourth moment, we use the elementary bound
$\E Z_T^4\le144\E\langle Z\rangle_T^2$ for a martingale starting at
zero. Indeed, by It\^o's formula, we have
\[
 \E Z_T^4\le6(\E\sup|Z|^4)^{1/2}
                 (\E\langle Z\rangle_T^2)^{1/2}.
\]
Doob's fourth-moment inequality then implies the stated bound with
constant $144$.
Since $\langle\Delta m\rangle\le h_-$ and its expectation is $s_j^2$, we have
\begin{equation}\label{u:eq:residual-fourth-moments}
 \E \mathcal T_{j,i}^4\le144h_-/s_j^2,\qquad
 \E(Q-1)^2\le144h_-/(ns_j^2).
\end{equation}
All conditional statements here refer only to the pasts of the original drivers.

\paragraph{Summation of the query energies.}
We multiply \eqref{u:eq:residual-square-root} by $s_j\overline\psi\ge0$ and use \eqref{u:eq:fine-correlation-bound}, \eqref{u:eq:residual-covariance}--\eqref{u:eq:residual-fourth-moments},
and $(X-1)^2\le2(Q-1)^2+2Y^2$. Inserting the result into \eqref{u:eq:onequery}, we obtain
\begin{equation}\label{u:eq:onequery-energy-bound}
 \begin{aligned}
 \E[c_j\ipn{x}{P_je_j}]\ge s_jb_j-C_\beta\biggl[
 &\frac{(j+1)s_j^2+s_j+(j+1)h_-}{n}\\
 &+\frac{(j^2+j)h_-^2}{n^2s_j^2}\biggr].
 \end{aligned}
\end{equation}
This derivation also covers a vanishing residual, $\varrho_j=0$.
When summing over the queries, we use $\sum s_j^2\le1$, $\sum s_j\le\sqrt N$,
$s_j^2\ge h_-/R_V$, and $\sum h_-\le V$.

Expanding the assembled matrix of \cref{u:sec:proposal-schedule},
we have, on every outcome of the source,
\begin{equation}\label{u:eq:fine-energy-decomposition}
 \begin{aligned}
 \frac{H^c_g(x)}n&=E_{\rm qry}+Z,\qquad
 E_{\rm qry}=\sum_j c_j\ipn{x}{P_je_j},\\
 Z&=\frac12\sum_j c_j^2d_j
       +\frac1{2n}(P_Nx)^{\mathsf T}W_0(P_Nx).
 \end{aligned}
\end{equation}
Under the linked law, the unused variables $d_j,W_0$ are independent of
the complete scalar paths and the spins, because the link does not depend on them.
Thus $\E[Z\mid\text{complete scalar paths and spins}]=0$.
The required integrability follows from the Gaussian moments of these variables and \eqref{u:eq:energy-integrability}.
Summing \eqref{u:eq:onequery-energy-bound} and using
\cref{u:lem:quad} proves \eqref{u:eq:fine-energy}.
\end{proof}

\trilogyneedspace{12\baselineskip}
\begin{proposition}[Forward entropy at the fine horizon]\label{u:prop:fine-kl}
Let $h=n^{-\ex a}$, $V=V_{\ex f}$ as in \cref{u:lem:quad}, with
\begin{equation}\label{u:eq:fine-parameters}
 \ex a>\ex f/4,\quad \ex a+\ex f/2<1,\quad0<\ex k_{\rm f}<\min(\ex a,\ex f,1-\ex a).
\end{equation}
For the primitive fine output $Q_F$,
\begin{equation}\label{u:eq:fine-kl}
 \KL(Q_F\Vert \Pi_n)\le C_\beta n^{1-\ex k_{\rm f}}.
\end{equation}
The disorder and spin marginals of $Q_F$ are $\gamma_n$ and
$\Unif_n$, respectively.
\end{proposition}
\begin{proof}
Combining the entropy--energy identity of \cref{u:prop:scalar}, proved in
\cite[Lemma~4]{oai-typical-upper}, with
\eqref{u:eq:forward-kl} and \eqref{u:eq:fine-energy}, we find
\begin{equation}\label{u:eq:fine-kl-budget}
 \begin{aligned}
 \KL(Q_F\Vert\Pi_n)\le C_\beta\biggl[
 &nh+ne^{-V}+(N+1)(1+V)+\sqrt N\\
 &+\frac{(N+1)^2VR_V}{n}\biggr].
 \end{aligned}
\end{equation}
Here we also used $\E_{\gamma_n}F^c\le nP_\beta$ and
$p(V)^2\le C_\beta e^{-V}$. By \cref{u:lem:quad,u:lem:tails},
the powers of the five terms, up to fixed logarithmic factors, are
$1-\ex a$, $1-\ex f$, $\ex a$, $\ex a/2$, and $2\ex a+\ex f/2-1$.
The inequality $\ex a+\ex f/2<1$ implies that the last power is
smaller than $\ex a$.
Since the inequalities in \eqref{u:eq:fine-parameters} are strict, they absorb the constants and logarithmic factors.
For the affine family of \cref{u:sec:budget}, these powers are $1/2,1/2,1/2,1/4,1/4$, and the claimed
bound is $C_\beta n^{1/2+2\varepsilon_\omega}$. The energy estimate holds without restriction to a Gram event,
including every outcome on which a query uses the fallback. The disorder
marginal follows from averaging over the spins. The spin
marginal is $\Unif_n$ because the linked scalar rows are independent
and, by symmetry, $\E m_V=0$ in each row.
\end{proof}

\chapter{Concentration on the growing mesh}\label{u:sec:concentration}

In this section, we prove the concentration estimates used in
\cref{u:sec:coarse}. We use the population identities of
\cref{u:sec:proposal-entries} and the product law of the linked rows from
\cref{u:def:laws}. \Cref{u:lem:gram} controls the two blocks of the empirical
matrix. \Cref{u:lem:weighted} controls the weighted errors that enter the
energy expansion in \cref{u:lem:weighted-energy}.

Throughout, $V=V_{\ex{b}}$ and $N=N_{\ex{b}}$. All probability bounds are
under the linked row law, averaged over the disorder. The equality of
laws in \cref{u:sec:proposal-laws} transports them to the source and
replay experiments. Estimates involving only the paths are proved under
the original law of the Brownian paths, which is also their linked marginal.
Stochastic integrals use the spin-free filtration described there.

\section{Errors and dyadic depth}\label{u:sec:gram}

We first define the errors used in both concentration estimates.
For vectors in $\R^n$, we write
$\langle u,v\rangle_n=n^{-1}\sum_{i=1}^n u_iv_i$.
Let $\mathcal T_j$ and $e_j$ be the query vectors of
\cref{u:sec:proposal}, and define
\begin{equation}\label{u:eq:empirical-errors}
\begin{aligned}
 A_{\mathcal T}&=(\langle\mathcal T_i,\mathcal T_j\rangle_n),
 &q&=(\langle\mathcal T_j,x\rangle_n),
 &v&=(\langle e_j,x\rangle_n),
 &H&=(\langle\mathcal T_i,e_j\rangle_n),\\
 E_A&=A_{\mathcal T}-I,
 &E_q&=q-s,
 &E_v&=v-b,
 &E_H&=H-H_0.
\end{aligned}
\end{equation}
Here $v$ is an empirical vector. We use $\|\cdot\|$ for the Euclidean
norm of a vector and the operator norm of a matrix. Let
$D_*=1+\lceil\log_2N\rceil$ be the dyadic depth. We work in the case
$N\ge1$, which holds eventually; when there are no queries, the empty
matrix formulas are interpreted accordingly.

\begin{lemma}[Fixed Gram stability]\label{u:lem:gram}
For $n\ge1$ and $1\le N<n$, let $R=R_V\ge1$ and
$D_*=1+\lceil\log_2N\rceil$. For a linked row
$Z=(\mathcal T,e,x)$, define
$\widehat\Sigma_n=n^{-1}\sum_iZ_iZ_i^{\mathsf T}$ and
$\Sigma=\E ZZ^{\mathsf T}$. Then
\begin{equation}\label{u:eq:fixed-gram}
 \PP\left\{\|E_A\|\vee\|E_H\|>(16D_*^2)^{-1}\right\}
 \le 2\,9^{2N+1}
 \exp\left\{-\frac{n}{2^{24}RD_*^4\log(2+R)}\right\}.
\end{equation}
The blocks $E_A$ and $E_H$ are coordinate compressions of
$\widehat\Sigma_n-\Sigma$.
\end{lemma}

\begin{proof}
We bound the fourth moment of a row projection using its second moment
and its sub-Gaussian tail. We then use a net to control the empirical
covariance matrix, following the empirical-Gram argument of
\cite[Section~4, proof of Proposition~5]{oai-typical-upper}. The
variance-sensitive estimates below keep the dependence on $R$ needed
on the growing mesh.

Let $a=(a_{\mathcal T},a_e,a_x)$ be a deterministic unit vector, and
write $F=a^{\mathsf T}Z$. By the martingale representation, the
normalized $\mathcal T$ projections have sub-Gaussian proxy $R$ times
the squared coefficient norm. The $e$ projections are Gaussian, and
$x$ is a centered sign. By H\"older's inequality with exponents three,
which allows dependence within a row, we have that $F$ has proxy at
most $3R$. By the
population identities in \cref{u:sec:proposal-entries} and
$(u+v+w)^2\le3(u^2+v^2+w^2)$, we also have that $\E F^2\le3$.

Let $Y=F/\sqrt3$ and $Z'=F/\sqrt{3R}$. These have proxies $R$ and $1$,
respectively, and $\E Y^2\le1$. We split $\E(YZ')^2$ at
$(Z')^2=Q=16\log(2+R)$. The lower part is at most $Q$. For the upper
part, we have
\[
 \E Y^4\le16R^2,\qquad
 \E\bigl[(Z')^4\mathbf 1_{\{(Z')^2>Q\}}\bigr]
 \le2e^{-Q/2}(Q^2+4Q+8).
\]
By Cauchy--Schwarz, the upper part is at most
$4\sqrt2R(Q+3)/(R+2)^4<2$. Hence
$\E(YZ')^2\le20\log(2+R)$ and
$\E F^4\le180R\log(2+R)$. Integrating the Gaussian tail, we also have
$\|F^2\|_{\psi_1}\le18R$, where the Orlicz norm is defined below.
This implies that $X=F^2-\E F^2$ has variance at most
$M^2=180R\log(2+R)$ and Orlicz norm at most $S=36R$.

Let $H=4S\log(2+S)$. We split the $l$th moment at $H$. The lower
part is at most $M^2H^{l-2}$, and the upper part is at most
\[
 2H^l(2+S)^{-4}+2l!(2S)^l(2+S)^{-2}.
\]
For $M,S\ge1$, the two upper terms are bounded by $8H^{l-2}$ and
$8l!H^{l-2}$, respectively. Their sum with the lower part is at most
$13l!M^2H^{l-2}$. Expanding the centered exponential series, we find
\[
 \log\E e^{\theta X}
 \le\frac{26M^2\theta^2}{2(1-H|\theta|)}.
\]
Multiplying the moment generating functions of the rows, with
$\theta=t/(26M^2+Ht)$ and both signs, we obtain the two-sided tail
$2e^{-nt^2/[2(26M^2+Ht)]}$ for the empirical mean. The variance
parameter is $4680R\log(2+R)$, and the scale is
\[
 144R\log(2+36R)\le720R\log(2+R).
\]
At tolerance $\varepsilon_0/2$, where
$\varepsilon_0=(16D_*^2)^{-1}\le1$, the rate is at least
$n\varepsilon_0^2/[40320R\log(2+R)]$.

A deterministic $1/4$ net of the unit sphere in dimension $2N+1$ has
size at most $9^{2N+1}$. The operator norm of a symmetric matrix is at
most twice the largest absolute value of its quadratic form on this
net \cite[Corollary~4.2.13 and Exercise~4.4.3(b)]{vershynin-2018}.
Since $40320\cdot256<2^{24}$, we obtain \eqref{u:eq:fixed-gram}
from the union bound over the net. Here we used that the rows have the
product law of \cref{u:def:laws}, and that coordinate inclusions and
projections have norm one.
\end{proof}

\section{Weighted dyadic concentration}\label{u:sec:weighted}

We next prove the weighted estimate. Four elementary estimates will
control products, stochastic integrals with random brackets, weighted
squares, and empirical means. We use them in \cref{u:lem:weighted},
first for each dyadic node and then for the sum over the tree.
The dyadic triangular decomposition is used in
\cite[Section~4, proof of Proposition~5]{oai-typical-upper}. Here the
node estimates retain their durations before they are summed.

We write $\operatorname{SG}(s^2)$ for the centered bound
$\E e^{\theta X}\le e^{s^2\theta^2/2}$ for every real $\theta$, and let
\[
 \|X\|_{\psi_1}
 =\inf\{s>0:\E e^{|X|/s}\le2\}.
\]
The latter is a norm, so its triangle inequality holds also for
dependent random variables; see \cite[Section~2.7.1]{vershynin-2018}.

\begin{lemma}[Products of sub-Gaussian variables]
\label{u:lem:subgaussian-products}
Let $R\ge1$ and $u,s\ge0$. If $\E Y^2\le u^2$, $Y$ is
$\operatorname{SG}(Ru^2)$, and $Z$ is $\operatorname{SG}(s^2)$, then
\begin{equation}\label{u:eq:subgaussian-products}
\begin{aligned}
 \E(YZ)^2&\le20u^2s^2\log(2+R),\\
 \|YZ\|_{\psi_1}&\le10us\sqrt R,\qquad
 \|YZ-\E YZ\|_{\psi_1}\le20us\sqrt R.
\end{aligned}
\end{equation}
The variables $Y$ and $Z$ may be dependent.
\end{lemma}

\begin{proof}
The sub-exponential product estimate is standard
\cite[Lemma~2.7.7]{vershynin-2018}. We additionally retain the second
moment of $Y$ in the variance estimate.
If $us=0$, the assertions are immediate. Otherwise, we normalize to
$u=s=1$. By a union bound, we have that
$\PP(|YZ|>t)\le4e^{-t/(2\sqrt R)}$. Integrating this tail yields
the two Orlicz bounds. To bound the second moment, we split at
$Z^2=Q=16\log(2+R)$. The lower part is at most $Q\E Y^2$. For the
upper part, we use Cauchy--Schwarz, and the Gaussian tails give
\[
 \E Y^4\le16R^2,\qquad
 \E[Z^4\mathbf 1_{\{Z^2>Q\}}]
 \le2e^{-Q/2}(Q^2+4Q+8).
\]
The upper part is at most $4\sqrt2R(Q+3)/(R+2)^4<2$ for $R\ge1$.
In the last inequality, we used $Q+3\le9(R+2)$ and
$R/(R+2)^3\le1/27$. We note that the bound on the second moment uses
both the second moment of $Y$ and the two tails. This proves
\eqref{u:eq:subgaussian-products}.
\end{proof}

\begin{lemma}[A tail bound from the bracket]\label{u:lem:bracket-tail}
Let $\widetilde M$ be a continuous local martingale starting at zero,
and suppose that $\|\langle\widetilde M\rangle_V\|_{\psi_1}\le A$.
Then, for $t,u>0$,
\begin{equation}\label{u:eq:bracket-tail}
\begin{aligned}
 \PP(|\widetilde M_V|>t)
 &\le\PP(\langle\widetilde M\rangle_V>u)+2e^{-t^2/(2u)},\\
 \|\widetilde M_V\|_{\psi_1}&\le5\sqrt{2A}.
\end{aligned}
\end{equation}
\end{lemma}

\begin{proof}
We stop at bracket level $u$ and use the exponential supermartingale
with parameters $t/u$ and $-t/u$, the stopped stochastic-exponential
argument of \cite[Section~3.5]{karatzas-shreve-1991}. On
$\{\langle\widetilde M\rangle_V\le u\}$, we have the first inequality.
For the second, we take $u=t\sqrt{A/2}$ so that the resulting tail is
bounded by $4e^{-t/\sqrt{2A}}$. Integrating it, we find that
$\|\widetilde M_V\|_{\psi_1}\le5\sqrt{2A}$. If $A=0$, the martingale is
zero and the claim is immediate.
\end{proof}

\begin{lemma}[Weighted squares]\label{u:lem:jensen-weights}
Suppose that each $Y_j$ is $\operatorname{SG}(s^2)$, where $s>0$,
and that $p_j\ge0$ and $\sum_jp_j\le1$. Then
\begin{equation}\label{u:eq:jensen-weights}
 \E\exp\left\{\frac{\sum_jp_jY_j^2}{6s^2}\right\}\le2.
\end{equation}
The variables $Y_j$ may be dependent.
\end{lemma}

\begin{proof}
We use the sub-Gaussian-square estimate
\cite[Lemma~2.7.6]{vershynin-2018} and Jensen's inequality, keeping the
constants explicit.
For a single $Y_j$, we integrate its Gaussian tail and bound the
expectation by $1+2\int_1^\infty t^{-3}\dd t=2$. By Jensen's inequality, with the
unused weight assigned to zero, we obtain \eqref{u:eq:jensen-weights}.
\end{proof}

\begin{lemma}[Variance-sensitive Bernstein bound]\label{u:lem:bernstein}
Suppose that $\E X=0$, $\E X^2\le M^2$,
$\PP(|X|>t)\le2e^{-t/S}$, and $M,S\ge1$.
Let $H=4S\log(2+S)$. Then, for integers $l\ge2$,
\begin{equation}\label{u:eq:bernstein-moments}
 \E|X|^l\le13l!M^2H^{l-2}.
\end{equation}
If $X_1,\ldots,X_n$ are independent copies of $X$, then
\begin{equation}\label{u:eq:bernstein}
\begin{aligned}
 \log\E e^{\theta X}
 &\le\frac{26M^2\theta^2}{2(1-H|\theta|)},
 &&|\theta|<H^{-1},\\
 \PP\left(\left|\frac1n\sum_iX_i\right|>\eta\right)
 &\le2\exp\left\{-\frac{n\eta^2}{2(26M^2+H\eta)}\right\},
 &&\eta>0.
\end{aligned}
\end{equation}
\end{lemma}

\begin{proof}
This is the Bernstein inequality in the form of
\cite[Theorem~2.10 and Corollary~2.11]{boucheron-lugosi-massart-2013}; we include the
calculation to record the constants needed here. In passing from the
tail to the factorial moments, we keep the variance.

Splitting the $l$th moment at $H$ and integrating the upper tail, we have
\[
 \E|X|^l
 \le M^2H^{l-2}+2H^l(2+S)^{-4}
       +2l!(2S)^l(2+S)^{-2}
 \le13l!M^2H^{l-2}.
\]
The boundary term is at most $8H^{l-2}$, and the last term is at most
$8l!H^{l-2}$; here we used $H\ge2S$ and
$\log(2+S)\le(2+S)/2$. Expanding the exponential series, which is
absolutely convergent, and using centering, we conclude that the first
inequality in \eqref{u:eq:bernstein} holds. For the empirical mean, we use independence
to multiply the moment generating functions.
Taking $\theta=\eta/(26M^2+H\eta)$, with both
signs, gives the second inequality.
\end{proof}

\begin{lemma}[Duration-normalized weighted concentration]
\label{u:lem:weighted}
Let $m_0=2^{\lceil\log_2N\rceil}$ and $D_*=1+\log_2m_0$.
Pad the errors and $s,b$ by zeros, and $A_{\mathcal T}$ by an identity,
to dimension $m_0$. Dummy coordinates have duration zero. Let
$\mathcal D$ be the complete dyadic tree on $\{1,\ldots,m_0\}$, and
let $P_I$ denote coordinate projection onto $I$. Define
\begin{equation}\label{u:eq:weighted-definitions}
\begin{aligned}
 d_I&=\sum_{j\in I\cap[1,N]}h_{j-1},
 &U_{\rm dur}&=t_N,\\
 \mathcal N_w(X)^2&=\sum_{I\in\mathcal D}\|XP_Iw\|^2,
 &L&=\Upt(E_A),
\end{aligned}
\end{equation}
where $\Upt$ keeps the strict upper triangle and half its diagonal, and
let
\[
\begin{aligned}
 W_{\rm wt}^2={}&\|E_q\|^2+\|E_v\|^2
 +\mathcal N_s(L)^2+\mathcal N_s(L^{\mathsf T})^2\\
 &+\mathcal N_b(L)^2+\mathcal N_s(E_H)^2
 +\mathcal N_s(E_H^{\mathsf T})^2.
\end{aligned}
\]
The duration $d_I$ is that of the previous posterior cells; the query
endpoint $e_j$ has denominator $\sqrt{h_j}$ from the next cell.
Let $R=R_V\ge1$, $B_*=\max(1,\beta e^{1/8})$, and
\begin{equation}\label{u:eq:weighted-parameters}
\begin{aligned}
 \ell_R&=\log(2+R),
 &M_*^2&=25B_*^2D_*^2\ell_R,\\
 S_*&=20B_*D_*\sqrt R,
 &\mathsf v&=26M_*^2,
 &H_*&=4S_*\log(2+S_*).
\end{aligned}
\end{equation}
For every $n,N\ge1$, every deterministic grid with used cells at most
one and $h_j/h_{j-1}\le e^{1/4}$, and every $\eta>0$, under the
linked row product law,
\begin{equation}\label{u:eq:weighted-tail}
 \PP\left\{W_{\rm wt}>2\eta\sqrt{2+5D_*U_{\rm dur}}\right\}
 \le2(10m_0-3)5^{m_0}
 \exp\left\{-\frac{n\eta^2}{2(\mathsf v+H_*\eta)}\right\}.
\end{equation}
With this failure bound, the following estimates hold simultaneously:
\begin{equation}\label{u:eq:weighted-node-bounds}
\begin{aligned}
 &\|E_q\|,\|E_v\|\le2\eta,\\
 &\|LP_Is\|,\|L^{\mathsf T}P_Is\|,\|LP_Ib\|,
   \|E_HP_Is\|,\|E_H^{\mathsf T}P_Is\|
   \le2\eta\sqrt{d_I}\quad(I\in\mathcal D).
\end{aligned}
\end{equation}
Zero-duration nodes have zero error and are omitted when dividing by
$\sqrt{d_I}$. On the growing grid with $h=n^{-\ex{a}}$,
$V=V_{\ex{b}}$, and $\ex{a}>\ex{b}/4$, there exists a fixed finite $C$
such that
\begin{align}
 \PP\{W_{\rm wt}>C_\beta n^{-\ex{w}}\log^C n\}
 &\le2\exp\left\{C_\beta n^{\ex{a}}
 -\frac{c_\beta n^{1-\max\{2\ex{w},\ex{w}+\ex{b}/4\}}}
 {\log^C n}\right\},\label{u:eq:weighted-growing}\\
 \PP\{\|E_A\|\vee\|E_H\|>(16D_*^2)^{-1}\}
 &\le2\exp\left\{C_\beta n^{\ex{a}}
 -\frac{c_\beta n^{1-\ex{b}/2}}{\log^C n}\right\}.
 \label{u:eq:gram-growing}
\end{align}
Constants are independent of deterministic row directions, tree nodes,
the selected cut, and the rotation angle. The finite row estimate allows
all $N$; its transport to the $n$-spin proposal uses $N<n$, which holds
eventually on the parameter family since $\ex{a}<1$.
\end{lemma}

\begin{proof}
We first estimate projections of a single row. We decompose each
triangular projection by dyadic depth and separate its half diagonal.
After dividing each interval projection by $\sqrt{d_I}$, we use
\cref{u:lem:bernstein} and a sphere net. Summing the resulting squared
bounds over the tree gives \eqref{u:eq:weighted-tail}.

\paragraph{Row projections and the durations.}
For a deterministic $\alpha$, the population identities and martingale
representation in \cref{u:sec:proposal-entries} imply that
\begin{equation}\label{u:eq:row-projections}
\begin{aligned}
 \E(\alpha^{\mathsf T}\mathcal T)^2&=\|\alpha\|^2,\\
 \alpha^{\mathsf T}\mathcal T&\text{ is }
       \operatorname{SG}(R\|\alpha\|^2),\qquad
 \alpha^{\mathsf T}e\text{ is }
       \operatorname{SG}(\|\alpha\|^2).
\end{aligned}
\end{equation}
Indeed, $\alpha^{\mathsf T}\mathcal T$ is the integral with integrand
$\alpha_j\chi_t/s_j$ on previous cell $j$; its bracket is bounded by
$R\|\alpha\|^2$. For $u_j=w_js_j$ with $|w_j|\le B_*$, we have
\begin{equation}\label{u:eq:weighted-row-increments}
 Z_J=\sum_{j\in J}u_j\mathcal T_j
 =\sum_{j\in J}w_j\int_{t_{j-1}}^{t_j}\chi_t\dd M_t
 \text{ is }\operatorname{SG}(B_*^2d_I),\qquad J\subset I.
\end{equation}
The bracket bound is deterministic. This implies that the same estimate
holds conditionally before its supporting future cells. Further,
$\|P_Is\|^2\le d_I$, so $s^{\mathsf T}P_Ie$ is
$\operatorname{SG}(d_I)$. The weights $b_j$ satisfy
\begin{equation}\label{u:eq:adjacent-duration-weights}
 b_j\le\beta\sqrt{h_jr(t_j)},\qquad
 s_j\ge\sqrt{h_{j-1}r(t_j)},\qquad
 \frac{b_j}{s_j}\le\beta\sqrt{\frac{h_j}{h_{j-1}}}
 \le\beta e^{1/8}.
\end{equation}
Here we used the monotonicity of $r$ and $p\le1$. Note that the
previous and next durations enter this calculation separately.

\paragraph{The strict triangular part.}
Fix a deterministic unit direction $\alpha$ in the padded dimension
and one node $I$. At a dyadic depth, let $(A,B)$ range over the pairs of
disjoint left and right children of its parents. The projections of
the strict upper triangle at this level and of its transpose are,
respectively,
\[
\begin{aligned}
 N_+&=\sum_{(A,B)}Y_AZ_{B\cap I},
 &Y_A&=\sum_{j\in A}\alpha_j\mathcal T_j,\\
 N_-&=\sum_{(A,B)}Z_{A\cap I}Y_B,
 &Y_B&=\sum_{j\in B}\alpha_j\mathcal T_j.
\end{aligned}
\]
The first is the predictable integral with multiplier $Y_Aw_j\chi_t$
on future cells $j\in B\cap I$. The second has multiplier
$Z_{A\cap I}\alpha_j\chi_t/s_j$ on future cells $j\in B$.
The future supports at one depth are disjoint. By conditional centering
before each future support, the block products are orthogonal, although
their past and future factors need not be independent. The isometry for
$N_+$ and \cref{u:lem:subgaussian-products} for each product in $N_-$
give
\begin{equation}\label{u:eq:dyadic-level-variance}
 \E N_+^2\le B_*^2d_I\sum_A\|P_A\alpha\|^2\le B_*^2d_I,
 \qquad
 \E N_-^2\le20B_*^2d_I\ell_R.
\end{equation}
Their brackets satisfy
\begin{equation}\label{u:eq:dyadic-level-brackets}
 \langle N_+\rangle\le B_*^2d_I\sum_AY_A^2,\qquad
 \langle N_-\rangle
 \le R\sum_B\|P_B\alpha\|^2Z_{A\cap I}^2.
\end{equation}
Using \cref{u:lem:jensen-weights} with weights $\|P_A\alpha\|^2$
and $\|P_B\alpha\|^2$, respectively, we bound the Orlicz norms of
both brackets by $6B_*^2Rd_I$. \Cref{u:lem:bracket-tail} then implies
that the Orlicz norms of both projections are at most
$18B_*\sqrt{Rd_I}$. There are $D_*-1$ depths, and we sum the bounds
over them. By the triangle inequalities in $L^2$ and in the Orlicz
norm, this costs at most a factor equal to the number of depths.

\paragraph{The half diagonal.}
In either orientation, the half diagonal is
\begin{equation}\label{u:eq:half-diagonal}
 Q_I=\frac12\sum_{j\in I}\frac{\alpha_jw_j}{s_j}
       ((\Delta m_j)^2-s_j^2)=N_I+D_I,
\end{equation}
where
\[
\begin{aligned}
 N_I&=\sum_{j\in I}\frac{\alpha_jw_j}{s_j}
 \int_{t_{j-1}}^{t_j}(m_t-m_{t_{j-1}})\chi_t\dd M_t,\\
 D_I&=\frac12\sum_{j\in I}\frac{\alpha_jw_j}{s_j}
 \left(\int_{t_{j-1}}^{t_j}\chi_t^2\dd t-s_j^2\right),
 \qquad \Delta m_j=m_{t_j}-m_{t_{j-1}}.
\end{aligned}
\]
Both terms have mean zero, and $D_I$ is a centered random drift.
By isometry,
\[
 \E N_I^2\le B_*^2\sum_{j\in I}\alpha_j^2h_{j-1}
 \le B_*^2d_I.
\]
Let $S_j=\sup_{[t_{j-1},t_j]}|m_t-m_{t_{j-1}}|$. By the maximal
inequality for the exponential martingale, we have
$\PP(S_j>u)\le2e^{-u^2/(2h_{j-1})}$, so
$\|S_j^2\|_{\psi_1}\le6h_{j-1}$. The bracket of $N_I$ is bounded
by $B_*^2R\sum\alpha_j^2S_j^2$, whose Orlicz norm is at most
$6B_*^2Rd_I$. By \cref{u:lem:bracket-tail}, we have
$\|N_I\|_{\psi_1}\le18B_*\sqrt{Rd_I}$.
For $q_j=\int_{t_{j-1}}^{t_j}\chi_t^2\dd t$, we have
\[
 \|q_j-s_j^2\|_2\le\|q_j\|_2
 \le\int_{t_{j-1}}^{t_j}\sqrt{r(t)}\dd t
 \le\sqrt{h_{j-1}}s_j,
\]
where we used $\chi^4\le\chi^2$. It follows that
\begin{equation}\label{u:eq:half-diagonal-bounds}
\begin{aligned}
 \|D_I\|_2&\le\tfrac12B_*\sqrt{d_I},
 &|D_I|&\le\tfrac12B_*\sqrt{Rd_I},\\
 \E Q_I^2&\le\tfrac94B_*^2d_I,
 &\|Q_I\|_{\psi_1}&\le20B_*\sqrt{Rd_I}.
\end{aligned}
\end{equation}
These estimates keep track of the unequal durations of the cells. In
particular, the variance bound involves $d_I$ without a power of $R$,
which would be lost if each duration were replaced by the maximal one.

\paragraph{The remaining projections and normalization.}
The two projections of the full cross blocks are the centered products
$(\alpha^{\mathsf T}\mathcal T)(s^{\mathsf T}P_Ie)$ and
$(\alpha^{\mathsf T}e)(s^{\mathsf T}P_I\mathcal T)$. By
\eqref{u:eq:row-projections}--\eqref{u:eq:adjacent-duration-weights}
and \cref{u:lem:subgaussian-products}, their variances are at most
$20d_I\ell_R$, and their Orlicz norms are at most $20\sqrt{Rd_I}$.
Here the centering subtracts the block $H_0$.
The two projections without intervals,
$\alpha^{\mathsf T}(\mathcal T x-s)$ and
$\alpha^{\mathsf T}(ex-b)$, have variance at most one and Orlicz
norm at most $10\sqrt R$, since multiplication by the terminal sign
preserves squares and absolute values.

Combining \eqref{u:eq:dyadic-level-variance}--\eqref{u:eq:half-diagonal-bounds},
we find that each of the five interval projections, after division by
$\sqrt{d_I}$, and each of the two projections without intervals is
centered and satisfies
\begin{equation}\label{u:eq:normalized-moments}
 \E X^2\le M_*^2,\qquad \|X\|_{\psi_1}\le S_*.
\end{equation}
For a single row, the five vectors indexed by an interval are
\[
\begin{gathered}
 \Upt(\mathcal T\mathcal T^{\mathsf T}-I)P_Is,\qquad
 \Upt(\mathcal T\mathcal T^{\mathsf T}-I)^{\mathsf T}P_Is,\qquad
 \Upt(\mathcal T\mathcal T^{\mathsf T}-I)P_Ib,\\
 (\mathcal T e^{\mathsf T}-H_0)P_Is,\qquad
 (e\mathcal T^{\mathsf T}-H_0^{\mathsf T})P_Is.
\end{gathered}
\]
To pad these vectors, we extend their errors by zero on the dummy
coordinates. The identity matrix in their row formulas acts on the
first $N$ coordinates.

We have $M_*,S_*\ge1$ in \eqref{u:eq:normalized-moments}, even
for arbitrarily small $d_I$. \Cref{u:lem:bernstein} therefore applies
after the division by $\sqrt{d_I}$, although its hypotheses need not hold
before this normalization.

\paragraph{Nets and the growing grid.}
Fix a deterministic $1/2$ net of the unit sphere of size at most
$5^{m_0}$. For every vector $z$, we have
$\|z\|\le2\max|\alpha^{\mathsf T}z|$, where the maximum is over
this net \cite[Corollary~4.2.13 and Exercise~4.4.2]{vershynin-2018}.
There are $2+5(2m_0-1)=10m_0-3$ vectors to control.
Using \eqref{u:eq:bernstein} and the union bound, we conclude that
\eqref{u:eq:weighted-node-bounds} holds. At every depth, the durations
partition $t_N$, so $\sum_Id_I=D_*U_{\rm dur}$. Summing the squared
errors before normalization, we obtain
\[
 W_{\rm wt}^2\le4\eta^2(2+5D_*U_{\rm dur}),
\]
which proves \eqref{u:eq:weighted-tail}. The union bound is over
$(10m_0-3)5^{m_0}$ events, and the nets are in dimension $m_0$.

For the stopped growing grid, \cref{u:lem:quad} implies that
$N\le8n^{\ex{a}}$, $D_*=O(\log n)$, and
$U_{\rm dur}\le V=O(\log n)$. By \cref{u:lem:tails}, we have
$R\le C_\beta n^{\ex{b}/2}(\log n)^{1/2}$. Then
$M_*^2$ is bounded by a fixed power of $\log n$, and
$H_*\le C_\beta n^{\ex{b}/4}\log^C n$.
Taking $\eta=n^{-\ex{w}}$ in \eqref{u:eq:weighted-tail} and using
$x/(u+v)\ge(1/2)\min(x/u,x/v)$ yields
\eqref{u:eq:weighted-growing}.
To prove \eqref{u:eq:gram-growing}, we use \cref{u:lem:gram} for the
compressions $E_A,E_H$. Its rate is $n/R$ divided by a fixed power of
$\log n$, which gives the separate loss $\ex{b}/2$.
By the equality of row laws in \cref{u:sec:proposal-laws}, all these
bounds also hold under the linked source and replay laws.
\end{proof}
\chapter{The joint cancellation and coarse domination}\label{u:sec:coarse}

The goal of this section is to prove \cref{u:prop:coarse}, which bounds the restricted coarse output kernel by an unnormalized Gibbs weight. We use the Gram and weighted concentration estimates of \cref{u:lem:gram,u:lem:weighted}. The deterministic energy expansion and the joint information--energy identity proved below give the accuracy needed for this bound. The resulting kernel estimate is used in \cref{u:sec:connections} to control the connecting sequence. This sharpens the likelihood comparison of \cite[Section~4, Proposition~5, equation~(6)]{oai-typical-upper}, used for the coarse proposals in \cite[Section~6, proof of Proposition~7]{oai-typical-upper}.

Unless stated otherwise, $V=V_{\ex b}$ and $N=N_{\ex b}\ge1$. Throughout, we use the source and replay laws of \cref{u:def:laws}, and the empirical Gram quantities $A_{\mathcal T},q,v,H,E_A,E_q,E_v,E_H$ of \cref{u:sec:concentration}. Here $v$ is an empirical vector, distinct from the scalar time. We recall that $H_0$ is strictly lower triangular, $\|s\|,\|b\|,\|H_0\|\le1$, and $e_h=s^{\mathsf T}b$. Matrix norms are operator norms, and norms of coefficient vectors in $\R^N$ are Euclidean. For the query vectors in $\R^n$, we write $\langle u,v\rangle_n=n^{-1}u^{\mathsf T}v$.

\section{Dyadic algebra and the row telescope}\label{u:sec:dyadic}

We first record the deterministic estimates used in the energy expansion. Let $m_0=2^{\lceil\log_2N\rceil}$ and $D_*=1+\log_2m_0$. As in \cref{u:lem:weighted}, errors and coefficient vectors are padded by zeros to dimension $m_0$, and the Gram matrix is padded by an identity block. For the complete dyadic tree $\mathcal D$ and its coordinate projections $P_I$, we use
\[
 \mathcal N_w(X)^2=\sum_{I\in\mathcal D}\|XP_Iw\|^2,
 \qquad L=\Upt(E_A),
\]
where $\Upt$ keeps the strict upper triangle and half of the diagonal, and $\Up$ keeps the full upper triangle. The weighted error $W_{\rm wt}$ is defined in \cref{u:lem:weighted}. In this section $L$ denotes this triangular matrix. The dyadic decomposition of triangular truncations and the analysis of the Cholesky factor by half-diagonal truncation already appear in the proof of \cite[Section~4, Proposition~5]{oai-typical-upper}. Here we retain the weighted projections to obtain a quadratic remainder.

\trilogyneedspace{12\baselineskip}
\begin{lemma}[Deterministic dyadic estimates]\label{u:lem:dyadic}
For matrices $X,Y$ and vectors $a,b,w$ of the appropriate dimensions,
\begin{equation}\label{u:eq:dyadic-estimates}
\begin{aligned}
 |a^{\mathsf T}\Upt(X^{\mathsf T}Y)b|,\ 
 |a^{\mathsf T}\Up(X^{\mathsf T}Y)b|
 &\le2\mathcal N_a(X)\mathcal N_b(Y),\\
 \mathcal N_w(\Upt(X^{\mathsf T}Y))
 &\le2D_*\|X\|\mathcal N_w(Y),\\
 \mathcal N_w(\Upt(X^{\mathsf T}Y)^{\mathsf T})
 &\le2D_*\|Y\|\mathcal N_w(X),\\
 \mathcal N_w(X)&\le\sqrt{D_*}\|X\|\|w\|,\\
 \|\Upt(X)\|,\ \|\Up(X)\|&\le D_*\|X\|.
\end{aligned}
\end{equation}
Suppose $\|E_A\|\vee\|E_H\|\le\varepsilon_0=(16D_*^2)^{-1}$. Let $C$ be the upper triangular positive-diagonal Cholesky factor of $I+E_A$, and define $U=C-I$, $B=C^{-\mathsf T}$, and $c=B(s+E_q)$. Then
\begin{equation}\label{u:eq:cholesky-equation}
 U=L-\Upt(U^{\mathsf T}U),
\end{equation}
\begin{equation}\label{u:eq:cholesky-bounds}
 \|U\|\le2D_*\varepsilon_0,\qquad \|B\|\le2,\qquad
 2D_*\|U\|\le1/2,\qquad D_*\|E_H\|\le1.
\end{equation}
Further, $\mathcal N_w(U)\le2\mathcal N_w(L)$ for $w=s,b$, $\|U^{\mathsf T}s\|\le2W_{\rm wt}$, and, with $u=c-s$,
\begin{equation}\label{u:eq:coefficient-remainder}
\begin{aligned}
 u&=B(E_q-U^{\mathsf T}s),\qquad \|u\|\le6W_{\rm wt},\\
 u-E_q+L^{\mathsf T}s
 &=\Upt(U^{\mathsf T}U)^{\mathsf T}s-U^{\mathsf T}u.
\end{aligned}
\end{equation}
\end{lemma}

\begin{proof}
We decompose the strict upper triangle into dyadic rectangles, and then use the Cholesky equation to control the nonlinear term of the Cholesky factor. Each rectangle in the scalar estimates contributes
$\langle XP_{I_{\rm left}}a,YP_{I_{\rm right}}b\rangle$.
Cauchy--Schwarz over rectangles and then over singletons gives the factor two in \eqref{u:eq:dyadic-estimates}.

For the vector estimate, fix an input interval $I$. The external output $P_{<I}X^{\mathsf T}YP_Iw$ is orthogonal to the internal output and has squared norm at most $\|X\|^2\|YP_Iw\|^2$. At each internal depth, the outputs of the left children have disjoint supports. Cauchy--Schwarz over depths implies that the squared norm of the internal output is at most $D_*\|X\|^2$ times the sum of the squared norms of the right descendants and the singletons. When we sum over $I$, each descendant is counted at most $D_*$ times. The resulting factor $1+2D_*^2$ in squared norm is at most $(2D_*)^2$. Reversing left and right proves the transpose estimate. Partitioning at each depth gives the generic bound for $\mathcal N_w(X)$. Since the rectangle supports at one depth are disjoint, we also have the truncation bound.

The Cholesky equation $E_A=U+U^{\mathsf T}+U^{\mathsf T}U$ implies \eqref{u:eq:cholesky-equation}. Along $I+tE_A$, $0\le t\le1$, the Cholesky factor is continuous, as follows from the finite recursion that defines it by positive square roots. Writing $x(t)$ for the norm of this factor minus $I$, we have
\[
 x(t)\le D_*\varepsilon_0+D_*x(t)^2,\qquad x(0)=0.
\]
At $x=2D_*\varepsilon_0$, the right side is strictly smaller than $x$, since $4D_*^2\varepsilon_0=1/4$. By continuity, we have \eqref{u:eq:cholesky-bounds}.
Applying the first vector estimate in \eqref{u:eq:dyadic-estimates} to \eqref{u:eq:cholesky-equation} and absorbing the nonlinear term, we obtain $\mathcal N_w(U)\le2\mathcal N_w(L)$ for $w=s,b$. The transpose estimate at the root implies that $\|U^{\mathsf T}s\|\le2W_{\rm wt}$. Since $C^{\mathsf T}c=s+E_q$, we obtain both identities in \eqref{u:eq:coefficient-remainder}, and $\|u\|\le2(W_{\rm wt}+2W_{\rm wt})=6W_{\rm wt}$.
\end{proof}

\begin{lemma}[Weighted energy and discarded Gaussian variances]\label{u:lem:weighted-energy}
If $\|E_A\|\vee\|E_H\|\le(16D_*^2)^{-1}$, the queried energy is
\begin{equation}\label{u:eq:queried-energy}
\begin{aligned}
 E_{\rm qry}&=c^{\mathsf T}(b+E_v)
       -c^{\mathsf T}\Up\bigl(B(H_0+E_H)\bigr)c,\\
 C^{\mathsf T}C&=I+E_A,\qquad B=C^{-\mathsf T},\qquad c=B(s+E_q),
\end{aligned}
\end{equation}
where $C$ is upper triangular with positive diagonal. Define
\begin{equation}\label{u:eq:linear-energy}
 \ell(E)=E_q^{\mathsf T}b-s^{\mathsf T}\Upt(E_A)b
                  +s^{\mathsf T}E_v-s^{\mathsf T}\Up(E_H)s.
\end{equation}
Then
\begin{equation}\label{u:eq:weighted-energy-remainder}
 \|c-s\|\le6W_{\rm wt},\qquad
 |E_{\rm qry}-e_h-\ell(E)|\le200D_*W_{\rm wt}^2.
\end{equation}
Conditionally on the complete scalar paths and signs, the two unused energy terms in the source reconstruction are independent centered Gaussians with variances
\begin{equation}\label{u:eq:unused-gaussian-variances}
 \frac1{2n}\sum_jc_j^4,\qquad
 \frac{(1-\|c\|^2)^2}{2n}.
\end{equation}
If $\|c-s\|\le\tau$ and $t_N\ge v_*$, their sum $Z$ satisfies
\begin{equation}\label{u:eq:unused-energy-variance}
 \Var(Z\mid\text{paths, signs})
 \le\frac{C_\beta}{n}\{h+\tau^2+p(t_N)^2\}.
\end{equation}
These conditional assertions are taken before imposing any event that inspects the unused Gaussian variables.
\end{lemma}

\begin{proof}
We first expand the queried energy using \cref{u:lem:dyadic}, and then compute the two conditional Gaussian variances. Since the target Gram matrix is positive definite, no query uses the deterministic fallback. We write $\mathcal T^{\rm col}=(\mathcal T_1,\ldots,\mathcal T_N)$ and $\mathcal U=(u_1,\ldots,u_N)$ for the $n\times N$ matrices with these columns, so that $n^{-1}\mathcal U^{\mathsf T}\mathcal U=I_N$. The Gram--Schmidt factorization is $\mathcal T^{\rm col}=\mathcal U C$, with $C^{\mathsf T}C=A_{\mathcal T}$. Then $\mathcal U=\mathcal T^{\rm col}C^{-1}$ and $c=Bq$. The $(i,j)$ entry of $BH$ is $\langle u_i,e_j\rangle_n$. Subtracting the first $j$ query projections of $e_j$ in the energy gives \eqref{u:eq:queried-energy}, with truncation after multiplication. Since $B$ is lower triangular and $H_0$ is strictly lower triangular,
\begin{equation}\label{u:eq:population-triangle-cancellation}
 \Up(BH_0)=0.
\end{equation}
Indeed, a nonzero summand in an entry of the upper triangle would require $k\le i\le j<k$. Thus the population block cancels. If we pad the errors, $H_0$, and all vectors by zeros, and the Gram matrix by an identity block, then by uniqueness the Cholesky factor becomes $C\oplus I$. The padding preserves the norms, products, truncations, and energy.

Let $u=c-s$ as in \cref{u:lem:dyadic}. The remainder of $c^{\mathsf T}(b+E_v)$ after its linear terms is
\[
 s^{\mathsf T}\Upt(U^{\mathsf T}U)b-(Ub)^{\mathsf T}u+u^{\mathsf T}E_v.
\]
By \eqref{u:eq:dyadic-estimates}, its absolute value is at most $26W_{\rm wt}^2$. For the $E_H$ term, we first replace $c$ by $s$ on both sides. The three resulting errors are bounded by
\[
 2\sqrt{D_*}\|u\|\mathcal N_s(E_H),\qquad
 2\sqrt{D_*}\|u\|\mathcal N_s(E_H^{\mathsf T}),\qquad
 D_*\|E_H\|\|u\|^2.
\]
Their sum is at most $(24\sqrt{D_*}+36)W_{\rm wt}^2$. Finally, let $F_0=B-I$. Since $F_0^{\mathsf T}=-B^{\mathsf T}U$ and $\|F_0\|\le2\|U\|$, \cref{u:lem:dyadic} implies that
\[
 \mathcal N_c(F_0^{\mathsf T})\le7W_{\rm wt},\qquad
 \mathcal N_c(E_H)\le7W_{\rm wt}.
\]
By the scalar estimate, we have $|c^{\mathsf T}\Up(F_0E_H)c|\le98W_{\rm wt}^2$. Adding the bounds, we find that the total is
\[
 (160+24\sqrt{D_*})W_{\rm wt}^2\le200D_*W_{\rm wt}^2,
\]
which proves \eqref{u:eq:weighted-energy-remainder}.

In the source reconstruction of \cref{u:sec:proposal-schedule}, the variables $d_j\sim N(0,2/n)$ and $W_0$ are independent of all complete scalar paths and terminal signs. Neither the scalar update nor the terminal link reads them. The two terms of $Z$ in the energy decomposition of \cref{u:lem:fine-energy} have the conditional variances in \eqref{u:eq:unused-gaussian-variances}. On the scalar event in the statement, we have $\|c\|\le1$ by orthonormality, and
\[
 \sum_jc_j^4\le\max_jc_j^2
 \le2\max_js_j^2+2\tau^2\le C_\beta h+2\tau^2.
\]
Here we used that $s_j^2\le h_{j-1}r(t_{j-1})\le C_\beta h$, which follows from the exponential tail envelope and the growing durations. Also, $\|s\|^2=\int_0^{t_N}r=1-p(t_N)$ for $t_N\ge v_*$, so
\[
 1-\|c\|^2\le p(t_N)+2\tau.
\]
Adding \eqref{u:eq:unused-gaussian-variances} proves \eqref{u:eq:unused-energy-variance}. Since the unused primitives are independent of the complete scalar paths, these laws hold conditionally on the paths and terminal signs.
\end{proof}

The linear statistic \eqref{u:eq:linear-energy} can be written as an average over the scalar rows.

\begin{lemma}[Row telescope]\label{u:lem:telescope}
Let $A_t=xm_t-m_t^2/2$ and $w_j=b_j/s_j$, and define
\begin{equation}\label{u:eq:row-telescope-definition}
 \zeta_h=\sum_{j=1}^Nw_j(A_{t_j}-A_{t_{j-1}})
       +\sum_{j=1}^Ns_j(x-m_{t_j})e_j-\frac12e_h.
\end{equation}
Then
\begin{equation}\label{u:eq:row-telescope}
 e_h+\ell(E)=\frac1n\sum_{i=1}^n\zeta_{h,i},\qquad
 \E_{\nu_V}\zeta_h=e_h.
\end{equation}
\end{lemma}

\begin{proof}
Expanding the triangles and using $\sum_{l\le j}s_l\mathcal T_l=m_{t_j}$ gives the first identity. To check the constant, we note that the empirical expression is the row average of
\begin{align*}
 &xb^{\mathsf T}\mathcal T+xs^{\mathsf T}e
 -s^{\mathsf T}\Upt(\mathcal T\mathcal T^{\mathsf T})b\\
 &\hspace{3em}-s^{\mathsf T}\Up(\mathcal T e^{\mathsf T})s-e_h/2.
\end{align*}
The first and third terms combine into $\sum_jw_j\Delta A_j$, and the other pair gives the second sum in \eqref{u:eq:row-telescope-definition}. The population expectation of $\Upt(\mathcal T\mathcal T^{\mathsf T})$ is $I/2$, while that of $\Up(\mathcal T e^{\mathsf T})$ is zero. Since $\E x\mathcal T=s$ and $\E xe=b$, the expectation is $e_h+e_h-e_h/2-e_h/2=e_h$. This verifies \eqref{u:eq:row-telescope}, including the term $-e_h/2$.
\end{proof}

\section{The joint information--energy cancellation}\label{u:sec:joint}

We next compare the row telescope with the continuous scalar energy. The It\^o identities below and the information integral of \cref{u:lem:reference} imply an identity at the terminal time. We estimate its terminal fluctuation and its mesh residual separately.
The scalar entropy--energy balance is
\cite[Lemma~4]{oai-typical-upper}, and the auxiliary-reference
likelihood is from \cite[Section~4, proof of Proposition~5]{oai-typical-upper}.
The joint terminal identity below compares the fluctuations of these
quantities on the same row history. Its terminal correction and the
growing-mesh residual are derived here.

\begin{lemma}[It\^o formula for the time-dependent field]\label{u:lem:ito}
On every finite horizon, under the original scalar Wiener law,
\begin{equation}\label{u:eq:field-ito}
 \dd U(t,Y_t)=m_t\dd M_t+\frac12\bar\alpha(t)m_t^2\dd t,
 \qquad \dd m_t=\chi_t\dd M_t.
\end{equation}
The identities hold across CDF atoms and the support endpoint. They continue to hold after adjoining an independent initial sign, and their fixed-horizon almost-sure identities transfer to the linked row law.
\end{lemma}

\begin{proof}
By \cref{inp:ob-scalar} and the scalar estimates of \cref{sec:scalar}, the Parisi solution used in \cref{u:prop:scalar} is continuous in time with its spatial derivatives, smooth in space on compact rectangles, and absolutely continuous in time there. Its time derivative is $-(U_{yy}+\bar\alpha U_y^2)/2$ almost everywhere, and its spatial derivative satisfies the differentiated equation.

To justify It\^o's formula directly, we stop $Y$ on leaving $[-A,A]$, extend the time fields slightly at the endpoints, and mollify in time. The fields and spatial derivatives converge uniformly on compact subrectangles. The time derivatives converge at Lebesgue-almost-every time, uniformly on compact spatial sets: the CDF has only countably many discontinuities, and the other factors are continuous. The derivatives are bounded on the stopped rectangle. The classical It\^o formula \cite[Theorem~3.3.6]{karatzas-shreve-1991} holds for the mollifications, and we pass to the limit using dominated convergence for the drift integrals and the isometry for the stochastic integrals. The drift for $U$ is $\bar\alpha m^2/2$, and the drift for $m$ is $-(m_{yy}+2\bar\alpha m\chi)/2+\bar\alpha m\chi+m_{yy}/2=0$.
Passing to an almost-sure subsequence and then using rational times and continuous versions, we obtain the identities at all times. As $A\to\infty$, every continuous path on a finite horizon is eventually unstopped. Since $m,\chi,\bar\alpha$ are bounded, the integrands displayed above satisfy the required integrability conditions. There is no integral against $\dd\bar\alpha$, so the atoms produce no jump term. By uniqueness, the second identity agrees with \cref{u:lem:cross}. Since the terminal link is positive and bounded, these identities hold under the linked law, with the integrals still defined in the original Brownian filtration.
\end{proof}

\begin{lemma}[Joint information--energy terminal cancellation]\label{u:lem:joint}
For a deterministic finite $V\ge v_*$, define
\begin{equation}\label{u:eq:continuous-row-energy}
\begin{aligned}
 A_t&=xm_t-m_t^2/2,\\
 Z_V&=\beta\int_0^Vp(t)\dd A_t+
                         \int_0^V\sqrt{r(t)}(x-m_t)\dd B_t,\\
 \zeta_\infty&=Z_V-e(V)/2.
\end{aligned}
\end{equation}
All stochastic integrals here and in the following lemma are first defined under the unlinked Brownian pair law with an independent initial uniform sign, using the measurable left-sum versions of \cref{u:sec:reference}. Then, under $\nu_V$,
\begin{equation}\label{u:eq:joint-cancellation}
 I_V-\beta\zeta_\infty
 =\log2-P_\beta-\frac{\beta^2p(V)^2}{4}
                      +\frac{\beta^2p(V)}2(x-m_V)^2.
\end{equation}
The centered terminal fluctuation
\[
 T_V=\frac{\beta^2p(V)}2\{(x-m_V)^2-p(V)\}
\]
satisfies $|T_V|\le2\beta^2p(V)$ and
\begin{equation}\label{u:eq:terminal-variance}
\begin{aligned}
 \Var_{\nu_V}(T_V)
 &=\frac{\beta^4p(V)^2}{4}\{4p(V)-3r(V)-p(V)^2\}\\
 &\le\beta^4p(V)^3.
\end{aligned}
\end{equation}
The identity \eqref{u:eq:joint-cancellation} holds at the endpoint $V\ge v_*$.
\end{lemma}

\begin{proof}
We first match the information and energy at the support endpoint, and then extend the identity beyond support and compute the terminal variance. Before support, we have $\rho=1$ and $\sigma=0$, so the query driver $B$ equals $M$. By \eqref{u:eq:field-ito}, $\beta^2p'=-\bar\alpha$, and $A_0=Y_0=0$, we have
\[
 \dd\{\beta^2pA+xY-U(t,Y)\}
 =\beta^2p\dd A+(x-m)\dd B=\beta\dd Z.
\]
Indeed, the extra drift is $\bar\alpha(-A+xm-m^2/2)=0$. Integrating up to $v_*$ yields
\[
 \beta Z_{v_*}=\beta^2p_*A_*+xY_*-U(v_*,Y_*)+U(0,0).
\]
At the support endpoint,
\[
 m_*=\tanh Y_*,\qquad \beta^2p_*=\beta^2-v_*,\qquad
 U(v_*,y)=\log(2\cosh y)+(\beta^2-v_*)/2.
\]
Since $I_{v_*}=\log(1+xm_*)$, $\log(1+x\tanh y)=xy-\log\cosh y$, and $A+(x-m)^2/2=1/2$, we obtain
\begin{equation}\label{u:eq:support-cancellation}
 I_{v_*}-\beta Z_{v_*}
 =\log2-U(0,0)+\frac12\beta^2p_*(x-m_*)^2.
\end{equation}
After support, $\chi=1-m^2$ and $\xi=x-m$. By the information integral of \cref{u:lem:reference}, we have
\[
\begin{aligned}
 \dd(I-\beta Z)
 &=-\beta^2p\dd A-\frac12\beta^2r(x-m)^2\dd t\\
 &=\dd\bigl\{\tfrac12\beta^2p(x-m)^2\bigr\},
\end{aligned}
\]
where we used $p'=-r$. This extends \eqref{u:eq:support-cancellation} to every $V\ge v_*$.

With the time change of \cref{u:prop:scalar}, the normalization of the pressure in \eqref{b:eq:Parisi-functional} reads
\[
 P_\beta=U(0,0)-\frac1{2\beta^2}
                  \int_0^{\beta^2}t\bar\alpha(t)\dd t.
\]
This implies the identity
\begin{equation}\label{u:eq:energy-pressure-conversion}
 \beta e(V)+\frac12\beta^2p(V)^2
 =2\{U(0,0)-P_\beta\},\qquad V\ge v_*.
\end{equation}
To check it, note that the left side is constant after support. At support it equals $\int_0^{v_*}p+\beta^2p_*^2/2$. Using integration by parts and $p'=-\bar\alpha/\beta^2$, we find that
\[
 \int_0^{v_*}p=v_*p_*+\beta^{-2}\int_0^{v_*}t\bar\alpha(t)\dd t.
\]
The remaining expression is
\[
 v_*p_*+\beta^2p_*^2/2
 =\frac{\beta^4-v_*^2}{2\beta^2}
 =\beta^{-2}\int_{v_*}^{\beta^2}t\dd t,
\]
which proves \eqref{u:eq:energy-pressure-conversion}. Substituting $\zeta_\infty=Z_V-e(V)/2$ proves \eqref{u:eq:joint-cancellation}.

Given the terminal path, the sign has mean $m_V$. Averaging over this sign, we find that
\[
\begin{aligned}
 \E[(x-m_V)^2\mid\text{path}]&=1-m_V^2,\\
 \E[(x-m_V)^4\mid\text{path}]&=(1-m_V^2)(1+3m_V^2).
\end{aligned}
\]
For $V\ge v_*$, their expectations are $p(V)$ and $4p(V)-3r(V)$. This proves \eqref{u:eq:terminal-variance}, including the centering. At a fixed finite such $V$, the variance is positive: the conditional sign variance of $(x-m_V)^2$ is $4m_V^2(1-m_V^2)$, which is positive with positive probability since the scalar diffusion is nondegenerate. Thus the centered row as a whole cannot have variance $O(h)$ uniformly at fixed $V$ as $h\to0$.

The restriction to the terminal endpoint is necessary. If the identity extended to times before support, then $I_t-\beta Z_t-\beta^2p(t)(x-m_t)^2/2$ would be constant. The coefficient of $\dd B$ in its differential is $\xi_t-(x-m_t)$. At time zero, $\chi_0=p(0)=1/\beta$, which implies that this coefficient is $x(1/\beta-1)\ne0$ and has positive bracket on a short interval. It is the matching at $v_*$ in \eqref{u:eq:support-cancellation} that makes the terminal identity possible.
\end{proof}

\begin{lemma}[Growing-mesh residual MGF]\label{u:lem:lag}
On the growing mesh of \cref{u:sec:proposal-schedule}, suppose $0<h\le1$, all used cells are at most one, and $V=t_{N+1}\ge v_*$. Define
\[
 R_h=Z_V-(\zeta_h+e_h/2).
\]
There exist $C_\beta,c_\beta>0$, independent of $h,N,V$, such that
\begin{equation}\label{u:eq:residual-mgf}
\begin{aligned}
 \log\E_{\nu_V}e^{\theta(R_h-\E R_h)}
 &\le C_\beta\theta^2h,\qquad |\theta|\le c_\beta/h,\\
 \E R_h&=\frac32(e(V)-e_h).
\end{aligned}
\end{equation}
For every $n\ge1$ and $0<\eta\le1$, independent identically distributed linked rows obey
\begin{equation}\label{u:eq:residual-average-tail}
 \PP\left\{\left|\frac1n\sum_i(R_{h,i}-\E R_h)\right|>\eta\right\}
 \le2e^{-c_\beta n\eta^2/h}.
\end{equation}
\end{lemma}

\begin{proof}
We decompose the residual into four terms, estimate their centered MGFs, and then compute the mean of the residual. Both omitted endpoint cells are retained in the decomposition.

\paragraph{The residual decomposition.}
On previous cells $[t_{j-1},t_j)$, $1\le j\le N$, let $C(t)=\beta p(t)-w_j$; on the final cell let $C(t)=\beta p(t)$. On current cells $[t_j,t_{j+1})$, $j\ge1$, let $G(t)=\sqrt{r(t)}-s_j/\sqrt{h_j}$; on the auxiliary cell zero, let $G(t)=\sqrt{r(t)}$. We write $m_{\lfloor t\rfloor}=m_{t_j}$ on cell $j$, including $m_0=0$, and define
\begin{align*}
 \mathcal M&=\int_0^VC\chi\dd M+\int_0^VG\dd B,\\
 \mathcal N&=-\int_0^VCm\chi\dd M-
                  \int_0^VGm_{\lfloor t\rfloor}\dd B,\\
 \mathcal Q&=-\frac12\int_0^VC\chi^2\dd t,\qquad
 \mathscr L_j=\int_{t_j}^{t_{j+1}}\sqrt{r(t)}(m_t-m_{t_j})\dd B_t.
\end{align*}
Then
\begin{equation}\label{u:eq:residual-decomposition}
 R_h=x\mathcal M+\mathcal N+\mathcal Q-\sum_{j=0}^N\mathscr L_j.
\end{equation}
This follows from $\dd A=(x-m)\chi\dd M-\chi^2\dd t/2$ and
\[
\begin{aligned}
 &\sqrt r(x-m_t)-\frac{s_j}{\sqrt{h_j}}(x-m_{t_j})\\
 &\hspace{2em}=G(t)(x-m_{t_j})-\sqrt r(m_t-m_{t_j}).
\end{aligned}
\]
The initial Brownian cell and the final posterior increment, which is not queried, are two distinct omissions, and both occur in \eqref{u:eq:residual-decomposition}. The current cell of the last query remains included in $e_h$.

\paragraph{Uniform bounds on the growing mesh.}
By the continuation kernel of \cref{u:lem:tails}, we have the envelope $p(t),r(t)\le C_\beta e^{-t/2}$ for all $t\ge0$, after increasing $C_\beta$. In particular, $\int_0^\infty e^{t/4}(p+r)\dd t<\infty$. For this mesh, we have
\begin{equation}\label{u:eq:mesh-summability}
\begin{gathered}
 (N+1)h\le8,\qquad
 \max_jh_j\sqrt{r(t_j)}\le\sqrt{C_\beta}h,\\
 \sum_jh_j^2r(t_j)\le8C_\beta h.
\end{gathered}
\end{equation}
Indeed, $e^{-t_j/4}-e^{-t_{j+1}/4}\ge h/8$ for $h_j\le1$, and $h_j^2e^{-t_j/2}=h^2$. These two facts also bound the number of queries.

The logarithmic profile bounds of \cref{u:prop:scalar}, with $h_j/h_{j-1}=e^{h_{j-1}/4}$, imply, for $t$ in previous cell $j$ and $u$ in current cell $j$, respectively,
\begin{equation}\label{u:eq:adjacent-profile-ratios}
\begin{aligned}
 \left|\log\frac{w_j}{\beta p(t)}\right|&\le(33/8)h_j,\\
 -h_{j-1}/8\le\log\frac{s_j}{\sqrt{h_jr(u)}}&\le(15/8)h_j.
\end{aligned}
\end{equation}
For the first bound, we compare the integrand $p\sqrt r$ throughout the current cell to its value on the previous cell and use $\sqrt{h_j/h_{j-1}}=e^{h_{j-1}/8}$. Each logarithm varies by at most the elapsed time. For the second, we compare the average of $r$ on the previous cell to $r(u)$ and use the same duration ratio. These comparisons imply that
\begin{equation}\label{u:eq:residual-coefficient-bounds}
\begin{aligned}
 |C(t)|&\le C_\beta h_jp(t),&&t\in[t_{j-1},t_j),\\
 |G(t)|&\le C_\beta h_j\sqrt{r(t)},&&t\in[t_j,t_{j+1}),\ j\ge1.
\end{aligned}
\end{equation}
Indeed, the inequality $|e^x-1|\le e^L|x|$ for $|x|\le L$ proves this with $L=33/8$ and $15/8$. On a previous cell, $h_j\le e^{1/4}he^{t/4}$, and on a current cell, $h_j\le he^{t/4}$. These bounds imply
\begin{equation}\label{u:eq:residual-bracket-bounds}
 \int C^2\dd t\le C_\beta h,\qquad
 \int G^2\dd t\le C_\beta h,\qquad
 |\mathcal Q|\le C_\beta h.
\end{equation}
In detail, the integral of $C^2$ over the interior cells is bounded by $C_\beta h^2\int e^{t/2}p(t)^2\dd t=O_\beta(h^2)$, and the final cell contributes at most $C_\beta h_Ne^{-t_N}\le C_\beta h$. For $G^2$, the auxiliary cell zero contributes $O_\beta(h)$; elsewhere we use $h_j^2\le h_j\le he^{t/4}$ and integrate $e^{t/4}r$. To bound $\mathcal Q$, we integrate $|C|$, using $h\int e^{t/4}p$ and the bound $h_Np(t_N)\le C_\beta h$ on the final cell. All constants are uniform in the horizon. Since $|\chi|,|m|,|m_{\lfloor t\rfloor}|\le1$ and $|\rho|\le1$, both path martingales $\mathcal M,\mathcal N$ have deterministic bracket at most $C_\beta h$.

\paragraph{The lag integrals.}
For cell $j$ of length $h_j$, we condition only on the past $\mathcal F_{t_j}^{\rm path}$ of the original spin-free Brownian pair. The increment $m_t-m_{t_j}$ has bracket at most $h_j$. We stop it at its first crossing of $\sqrt{uh_j}$. The outer lag integral, stopped at this time, has bracket at most $uh_j^2r(t_j)$, since $r$ is nonincreasing. By the stopped exponential-martingale argument
\cite[Section~3.5]{karatzas-shreve-1991} and a union bound, we have, for $u>0$,
\begin{equation}\label{u:eq:lag-cell-tail}
\begin{aligned}
 \PP\bigl\{|\mathscr L_j|>uh_j\sqrt{r(t_j)}
                  \mid\mathcal F_{t_j}^{\rm path}\bigr\}&\le4e^{-u/2},\\
 \E[\mathscr L_j\mid\mathcal F_{t_j}^{\rm path}]&=0.
\end{aligned}
\end{equation}
Integrating this tail and expanding the centered series, we obtain
\begin{equation}\label{u:eq:lag-cell-mgf}
 \E[e^{\theta\mathscr L_j}\mid\mathcal F_{t_j}^{\rm path}]
 \le e^{32\theta^2h_j^2r(t_j)},\qquad
 |\theta|\le[4h_j\sqrt{r(t_j)}]^{-1}.
\end{equation}
Indeed, for every integer $l\ge2$, the conditional $l$th absolute moment is at most $4l![2h_j\sqrt{r(t_j)}]^l$. Beyond the linear term, which vanishes, the series of the conditional MGF is at most $16\theta^2h_j^2r(t_j)/(1-2|\theta|h_j\sqrt{r(t_j)})$.
Iterating over cells and using \eqref{u:eq:mesh-summability}, we find that the centered log MGF is bounded by $C_\beta\theta^2h$ on $|\theta|\le c_\beta/h$ for $\sum_j\mathscr L_j$. The iteration uses only the conditional bounds and needs no independence between cells. The isometry also gives the variance identity $\E\mathscr L_j^2=(1/2)(\int_{t_j}^{t_{j+1}}r)^2$.
The tail and MGF estimates above are conditional on the past of the original pair; this variance identity is unconditional.

\paragraph{The centered MGF and the mean.}
Linking does not change the law of the paths. In particular, $\mathcal N$ has its usual sub-Gaussian bound under $\nu_V$. Multiplying $\mathcal M$ by $x$ preserves its absolute tail, but the product need not be a martingale. We use the standard centering principle for sub-Gaussian variables
\cite[Lemma~2.6.8]{vershynin-2018}, verifying an MGF constant by symmetrization.
To obtain a centered MGF without a prefactor, let $X'$ be an independent copy of $X=x\mathcal M$. By Jensen's inequality, $\E e^{\theta(X-\E X)}\le\E e^{\theta(X-X')}$.
If $X$ has tail $2e^{-u^2/(2v)}$, the symmetric difference has tail $4e^{-u^2/(8v)}$ and even moments at most $4(8v)^ll!$. Since $(2l)!\ge(l!)^2$, its series is at most $4e^{8v\theta^2}-3\le e^{32v\theta^2}$.
This applies with $v=C_\beta h$. By Hoeffding's lemma \cite[Lemma~2.2]{boucheron-lugosi-massart-2013}, we have a centered log MGF bounded by $C_\beta\theta^2h^2$ for $\mathcal Q$, using \eqref{u:eq:residual-bracket-bounds}. We combine these bounds and \eqref{u:eq:lag-cell-mgf} in \eqref{u:eq:residual-decomposition} by H\"older's inequality with four factors. Since the number of factors is fixed, this proves the MGF bound in \eqref{u:eq:residual-mgf}.

For the mean, we have, on every deterministic grid, the identities
\begin{equation}\label{u:eq:residual-mean-integrals}
 \int_0^VCr=e(V)-e_h,\qquad
 \int_0^VG\rho p=e(V)-e_h.
\end{equation}
The first uses the previous-cell value $s_j^2$, and the second the current-cell value $\sqrt{h_j}b_j$. By \cref{u:lem:cross} and posterior averaging, $\E[x\mathcal M]=\int Cr+\int G\rho p$, $\E\mathcal N=\E\mathscr L_j=0$, and $\E\mathcal Q=-(1/2)\int Cr$.
We conclude that $\E R_h=3(e(V)-e_h)/2$. The same integrals of the coefficients show that $|e(V)-e_h|\le C_\beta h$, in agreement with \cref{u:lem:quad}. For independent identically distributed rows, the centered MGFs multiply. The exponential Markov inequality with $\theta=c\eta/h$, for a sufficiently small fixed $c>0$, proves \eqref{u:eq:residual-average-tail} for all $0<\eta\le1$.

\paragraph{Integral versions and replay.}
At fixed finite $V$ and fixed query cells, we refine within each cell by dyadic left sums, retaining its endpoints. For the lag sums, the isometry and $\E(m_t-m_u)^2\le t-u$ bound the squared error by $8V\omega_{\sqrt r}(\varepsilon')^2+V\varepsilon'\to0$, where $\omega_{\sqrt r}$ is the modulus of continuity of $\sqrt r$ on this fixed horizon. The integrals with deterministic weights and the likelihood integral against $D$ are identified with limits of left sums in the same way. Passing to an almost-sure subsequence and using Fatou's lemma, we obtain the same MGF bound for the limit as for the finite sums. With bounded test functions of the past, we obtain the conditional version of Fatou's lemma in the same way.

The query driver is $B=\int\rho\dd M+\int\sigma\dd\widetilde M$, with the independent orthogonal driver of \cref{u:sec:scalar}. We retain both terms. Freezing $\rho,\sigma$ at the endpoints of the refinement produces a squared error at most $4\int[(\rho_{\varepsilon'}-\rho)^2+(\sigma_{\varepsilon'}-\sigma)^2]\dd t\to0$.
Before support, we use $\dd m=\chi\dd M$. We do not split a cell that crosses the support endpoint, since splitting it and restarting its lag baseline would omit a term. The identity of the full histories in \cref{u:def:laws} and the uniqueness of these sums identify the replay versions, including the reference of \cref{u:lem:reference}. For fixed $n$, the linking transfers null sets. The MGF and tail estimates follow from the unchanged path law and sign symmetrization, without a factor $2^n$. The conditioning in \eqref{u:eq:lag-cell-tail} and \eqref{u:eq:lag-cell-mgf} never involves the terminal sign, the final matrix, the futures of the other rows, or any crossing event.
\end{proof}

\section{Coarse domination}\label{u:sec:coarse-domination}

We combine the terminal cancellation and the residual estimate to bound the information minus the queried energy. The weighted remainder and the unused Gaussian energy then give the bound on the restricted kernel in \cref{u:prop:coarse}.

\begin{lemma}[Concentration of the row $I-\beta\zeta_h$]\label{u:lem:row}
For $V\ge v_*$, we have the exact identity
\begin{equation}\label{u:eq:finite-grid-cancellation}
\begin{aligned}
 I_V-\beta\zeta_h
 ={}&\log2-P_\beta-\frac{\beta^2p(V)^2}{4}
       +\frac{\beta^2p(V)}2(x-m_V)^2\\
 &+\beta R_h+\frac\beta2(e_h-e(V)).
\end{aligned}
\end{equation}
Its mean is
\begin{equation}\label{u:eq:finite-grid-mean}
 j(V)-\beta e_h=\log2-P_\beta+\frac{\beta^2p(V)^2}{4}
                                      +\beta(e(V)-e_h),
\end{equation}
and its centered random part is $T_V+\beta(R_h-\E R_h)$, with variance at most $C_\beta\{p(V)^3+h\}$. In particular, for $V=V_{\ex b}$, $h=n^{-\ex a}$, $\ex z<\min(\ex a,\ex b/2)$, and all sufficiently large $n$,
\begin{equation}\label{u:eq:row-concentration}
 \PP\left\{\frac1n\sum_i(I_{V,i}-\beta\zeta_{h,i})
                    >\log2-P_\beta+C_\beta n^{-\ex z}\right\}
 \le2e^{-c_\beta n^{1+\ex a-2\ex z}}.
\end{equation}
\end{lemma}

\begin{proof}
By \eqref{u:eq:continuous-row-energy} and the definition of $R_h$, we have $\zeta_h=\zeta_\infty-R_h+(e(V)-e_h)/2$.
Inserting this into \eqref{u:eq:joint-cancellation} yields \eqref{u:eq:finite-grid-cancellation}. Using \eqref{u:eq:residual-mgf} and $\E(x-m_V)^2=p(V)$, we obtain \eqref{u:eq:finite-grid-mean}. This also agrees with the information mean of \cref{u:lem:reference} and with \eqref{u:eq:row-telescope}. Combining the centered decomposition with \eqref{u:eq:terminal-variance} and \eqref{u:eq:residual-mgf}, we conclude that the variance has the stated bound; this does not require the two terms to be independent.

The terminal correction is deterministically at most $2\beta^2p(V)=O_\beta(n^{-\ex b/2})=o(n^{-\ex z})$. Its mean and the deterministic mesh error $O_\beta(h)$ fit within the same accuracy. By \eqref{u:eq:residual-average-tail} with $\eta=n^{-\ex z}$, absorbing the fixed coefficient $\beta$ into its constants, we obtain \eqref{u:eq:row-concentration}. The terminal variance remains $O(p(V)^3)$. Bernstein's inequality for bounded variables
\cite[Theorem~2.10 and Corollary~2.11]{boucheron-lugosi-massart-2013}
would give it the rate $n\min(\eta^2/p(V)^3,\eta/p(V))$; here we use only its deterministic amplitude.
\end{proof}

\begin{proposition}[Intrinsic coarse domination]\label{u:prop:coarse}
Suppose the fine conditions of \cref{u:prop:fine-kl} hold and, in addition,
\begin{equation}\label{u:eq:coarse-conditions}
\begin{gathered}
 \ex a>\ex b/4,\qquad
 0<\ex{k}_{\rm b}<\ex z<\min(\ex a,\ex b/2,2\ex w),\\
 \max\{2\ex w,\ex w+\ex b/4,\ex b/2,
       2\ex z-\ex a,2\ex z-2\ex w,2\ex z-\ex b\}<\ex{s}_{\rm b},\\
 \ex a<1-\ex{s}_{\rm b}.
\end{gathered}
\end{equation}
Let $\mu^{\rm aux}_n$ be the independent product law of the spin-free replay inputs $(D,B^0,\mathrm{br},(C_j)_j)$ from \cref{u:def:laws}, and let $U$ be independent and uniform on $[0,1]^n$. Only the inputs through $V_{\ex b}$ are read here; unused coordinates extending the experiment to the fine horizon may be retained. We write $m_i^{g,a}(V_{\ex b})$ for the resulting replay terminal magnetizations and define
\[
 X_B(g,a,U)_i
 =2\mathbf 1\!\left\{U_i\le
    \frac{1+m_i^{g,a}(V_{\ex b})}{2}\right\}-1.
\]
There exists a Borel replay event $E_B$ such that, eventually,
\begin{equation}\label{u:eq:coarse-failure}
 \PP_{\gamma_n\otimes\mu^{\rm aux}_n\otimes\dd U}(E_B^c)
 \le e^{-n^{1-\ex{s}_{\rm b}}}.
\end{equation}
The event includes $\|A_{\mathcal T}-I\|\le3/4$. Its restricted output kernel satisfies, for $\gamma_n$-almost every $g$, simultaneously for all $x\in\Sigma_n$,
\begin{equation}\label{u:eq:coarse-kernel}
\begin{aligned}
 q_B^{\rm good}(g,x)
 &=\int\mathbf 1_{E_B}\mathbf 1_{\{X_B(g,a,U)=x\}}
                      \dd\mu^{\rm aux}_n(a)\dd U\\
 &\le e^{-nP_\beta+n^{1-\ex{k}_{\rm b}}}e^{\beta H^c_g(x)}.
\end{aligned}
\end{equation}
The event $E_B$ contains no pressure event, and \eqref{u:eq:coarse-failure} is an annealed failure bound.
\end{proposition}

\begin{proof}
We intersect the weighted and Gram events with bounds on the unused Gaussian energy and the row residual. The row identity gives a likelihood bound on this event, which we then disintegrate over the disorder.

\paragraph{The energy errors.}
At $V=V_{\ex b}$, we take the simultaneous weighted event of \cref{u:lem:weighted} at $\eta=n^{-\ex w}$ and the fixed Gram event $\|E_A\|\vee\|E_H\|\le(16D_*^2)^{-1}$. On their intersection, by \cref{u:lem:weighted,u:lem:weighted-energy}, we have
\begin{equation}\label{u:eq:coarse-weighted-errors}
\begin{aligned}
 \|c-s\|&\le\tau_n=C_\beta n^{-\ex w}\log^Cn,\\
 |E_{\rm qry}-e_h-\ell(E)|&\le C_\beta n^{-2\ex w}\log^Cn.
\end{aligned}
\end{equation}
The failure probabilities of the two events have the respective raw rates $n^{1-\max\{2\ex w,\ex w+\ex b/4\}}/\operatorname{polylog}n$ and $n^{1-\ex b/2}/\operatorname{polylog}n$, less the cost $C_\beta n^{\ex a}$ of the net. By the strict inequalities in \eqref{u:eq:coarse-conditions}, these rates eventually dominate both the cost of the net and any fixed multiple of $n^{1-\ex{s}_{\rm b}}$.

For the discarded energy, we use its conditional law before restricting it by its own tail. The endpoint of the last previous cell satisfies $t_N=\ex b\log n+o(1)$, since $V_{\ex b}-t_N=h_N=o(1)$. Thus $p(t_N)^2\le C_\beta n^{-\ex b}$. On the event in \eqref{u:eq:coarse-weighted-errors}, we see from \eqref{u:eq:unused-energy-variance} that its conditional variance is at most
\[
 C_\beta n^{-1}\{n^{-\ex a}+n^{-2\ex w}\log^Cn+n^{-\ex b}\}.
\]
The conditional probability that $|Z|>n^{-\ex z}$ is at most
\begin{equation}\label{u:eq:coarse-gaussian-tail}
 2\exp\left\{-\frac{c_\beta
 n^{1-\max\{2\ex z-\ex a,\,2\ex z-2\ex w,\,2\ex z-\ex b\}}}
 {\log^Cn}\right\}.
\end{equation}
If the conditional variance is zero, this probability is zero. Integrating over the event on paths and signs in \eqref{u:eq:coarse-weighted-errors} gives the same bound for the probability of its intersection with this failure. Here we used the sum of the two unused Gaussian terms, with the conditional variance computed above. The coarser bound $1/(2n)$ would not give the required rate at these exponents.

\paragraph{The replay event.}
We intersect the preceding events with $|Z|\le n^{-\ex z}$ and
\[
 \left|\frac1n\sum_i(R_{h,i}-\E R_h)\right|\le n^{-\ex z}.
\]
The latter fails with probability at most $2e^{-c_\beta n^{1+\ex a-2\ex z}}$ by \eqref{u:eq:residual-average-tail}, with loss $2\ex z-\ex a$. All these events are replay events: the Gram blocks and scalar histories are replay functions, the residual uses the integral versions of \cref{u:lem:reference}, and $Z=H^c_g(x)/n-E_{\rm qry}$. We use the Borel versions fixed by the conventions of \cref{u:sec:setup}. The identity between the source and replay laws in \cref{u:def:laws} transports their probabilities. The sum of the finitely many failure probabilities is eventually at most $e^{-n^{1-\ex{s}_{\rm b}}}$, because each raw exponent exceeds $1-\ex{s}_{\rm b}$ and the exponent of the net by a strict margin. We denote the resulting event by $E_B$.

On $E_B$, \eqref{u:eq:row-telescope}, \eqref{u:eq:finite-grid-cancellation}, and the mean in \eqref{u:eq:residual-mgf} imply that
\begin{align*}
 \frac1nI^{(n)}-\log2-\frac\beta nH^c_g(x)
 &\le-P_\beta+C_\beta\bigl\{p(V_{\ex b})+p(V_{\ex b})^2+h\\
 &\hspace{5em}+n^{-\ex z}+n^{-2\ex w}\log^Cn\bigr\}\\
 &\le-P_\beta+n^{-\ex{k}_{\rm b}}
\end{align*}
eventually. Here the terminal correction has amplitude $O_\beta(p(V_{\ex b}))=O_\beta(n^{-\ex b/2})$, the deterministic mean of $R_h$ is $O_\beta(h)$, and the weighted remainder is controlled by \eqref{u:eq:coarse-weighted-errors}. Since $\ex{k}_{\rm b}<\ex z<\min(\ex a,\ex b/2,2\ex w)$, all constants and logarithms are absorbed. The terminal variance \eqref{u:eq:terminal-variance} is controlled here by its deterministic amplitude.

\paragraph{Disintegration.}
On this replay space, $\Lambda=e^{I^{(n)}}R_D$ and $(g,x)_*R_D=\gamma_n\otimes\Unif_n$. For every Borel set $A$ of primitive matrices and each spin,
\[
 \Lambda(E_B,\ g\in A,\ x)
 \le\int_A e^{-nP_\beta+n^{1-\ex{k}_{\rm b}}+\beta H^c_g(x)}\,
                   \gamma_n(\dd g).
\]
The factor $2^n$ from the logarithmic inequality cancels the factor $2^{-n}$ of the reference law of the spin. Realizing the linked spin by terminal thresholds does not change the restricted law. Disintegration over the disorder, whose marginal is unchanged, gives \eqref{u:eq:coarse-kernel}. Since the spin space is finite, there exists one common measurable null set for all spins. We redefine the restricted kernel to be zero there. This does not affect its later use, since the nondegenerate Gaussian regressions used later are absolutely continuous with respect to $\gamma_n$. This proves the bound on the subprobability kernel and, separately, the annealed failure estimate. In the coupled experiment of \cref{u:sec:sequence}, the auxiliary inputs are independent, and this will justify applying the proposition after the outer target is adjoined.
\end{proof}
\chapter{Smoothing, coverage, and the connecting sequence}\label{u:sec:connections}

The goal of this section is to prove \cref{u:prop:witness}, which excludes
balanced cuts with small stationary flow on the event
$\|W(g)\|_{\mathrm{op}}\le12$. We use the fine entropy estimate of
\cref{u:prop:fine-kl} to obtain coverage of both sides of a cut, and the
coarse domination estimate of \cref{u:prop:coarse} to construct a sequence
that avoids its enlarged boundary. These are the smoothing and connecting
sequence arguments of \cite[Sections~5--6, Propositions~6--7]{oai-typical-upper},
with the estimates at the fine and coarse horizons established in
\cref{u:sec:fine,u:sec:coarse}. In \cref{u:sec:proof}, the witness estimate
and \cref{u:lem:goe-norm} will imply \cref{thm:upper}.

Throughout, we fix $\beta>1$ and the exponent parameters before taking
$n\to\infty$, as in \cref{u:sec:parameters}. We retain the primitive disorder,
completed Hamiltonian, and joint law $\Pi_n$ of \cref{u:sec:primitive},
and the source and replay constructions of \cref{u:sec:proposal}.

\section{Smoothing and coverage}\label{u:sec:smoothing}

We first record the transport inequality for disorder conditioned on a
spin. Under $\Pi_n$, the spin marginal is $\Unif_n$: this follows from the
gauge transformation $g_{ij}\mapsto x_ix_jg_{ij}$ and summation over spins.
Conditional on $x$, primitive disorder has density
\begin{equation}\label{u:eq:conditional-density}
 p_x(g)=2^n(2\pi)^{-n^2/2}
       \exp\{-\|g\|_F^2/2+\beta H^c_g(x)-F^c(g)\}.
\end{equation}
Since $F^c$ is a log-sum-exp of linear functions and $H^c_g(x)$ is
linear in $g$, we have that the negative logarithm of $p_x$ is
$1$-strongly convex.
We write $W_2$ for the quadratic Wasserstein distance for the Euclidean
norm on the primitive matrix space, that is, the Frobenius norm.

\begin{lemma}\label{u:lem:transport}
For every $x\in\Sigma_n$ and every probability measure $\eta$ on the
primitive matrix space,
\begin{equation}\label{u:eq:conditional-transport}
 W_2(\eta,p_x)^2\le2\KL(\eta\Vert p_x).
\end{equation}
\end{lemma}

\begin{proof}
This is the transport inequality for measures with a strongly convex potential; see
\cite[Corollary~3.10]{bogachev-kolesnikov-medvedev-2005} and its use in
\cite[Section~5]{oai-typical-upper}. We recall the finite-dimensional
argument. For a smooth positive density $\eta$, let $T$ be the optimal
transport map from $p_x$ to $\eta$, which is the gradient of a convex function
by the convex-gradient transport theorem of Brenier~\cite{brenier-1991}
in the full Euclidean generality of \cite[Main Theorem]{mccann-1995},
and let $V_x$ be the
negative logarithm of $p_x$. By the change-of-variables formula, we have
\[
 \KL(\eta\Vert p_x)
 =\int\bigl(V_x(T(g))-V_x(g)-\log\det DT(g)\bigr)p_x(g)\dd g.
\]
By strong convexity and $-\log\det DT\ge\tr(I-DT)$, we have
\[
 \KL(\eta\Vert p_x)
 \ge\frac12\int\|T(g)-g\|_F^2p_x(g)\dd g.
\]
In the last step, we used integration by parts to cancel the linear terms.
For a general measure with finite entropy, we obtain the inequality by smooth
truncation and regularization, followed by lower semicontinuity. If the
entropy is infinite, there is nothing to prove.
\end{proof}

For every $L$-Lipschitz linear function $f$ and every $t\in\R$, the variational formula for relative entropy
\cite[Corollary~4.14]{boucheron-lugosi-massart-2013} and \cref{u:lem:transport} imply that
\begin{equation}\label{u:eq:conditional-linear-mgf}
 \E_{p_x}\exp\{t(f-\E_{p_x}f)\}\le e^{t^2L^2/2}.
\end{equation}
Indeed, $\E_\eta f-\E_{p_x}f\le L W_2(\eta,p_x)$, and maximizing
$|t|L\sqrt{2K}-K$ over $K\ge0$ proves
\eqref{u:eq:conditional-linear-mgf}.

Let $\delta=n^{-\ex{D}}$, and write the fine output as
$Q_F(\dd g,x)=\gamma_n(\dd g)q_F(g,x)$. We take the version of $q_F$
given by the replay integral, with the conventions of
\cref{u:sec:primitive}, and define
\begin{equation}\label{u:eq:smoothed-kernel}
 \nu^{\rm sm}_g(x)=\int
 q_F(\sqrt{1-\delta}\,g+\sqrt\delta\,z,x)\,\gamma_n(\dd z).
\end{equation}
For every $g$, we have that $\nu^{\rm sm}_g$ is a probability measure
on the spins.

\begin{lemma}\label{u:lem:smoothing}
Let $g_\star,g_\circ$ be independent with law $\gamma_n$, and set
$g_{\rm base}=\sqrt{1-\delta}\,g_\star+\sqrt\delta\,g_\circ$. Then
\begin{equation}\label{u:eq:reverse-entropy}
 \E_{g_\star,g_\circ}\KL(\pi_{g_\star}\Vert\nu^{\rm sm}_{g_{\rm base}})
 \le\frac{\beta^2(n-1)}2\delta+\delta^{-1}\KL(Q_F\Vert\Pi_n).
\end{equation}
In particular, if $\ex{u}_{\rm E}<\min(\ex{D},\ex{k}_{\rm f}-\ex{D})$,
the outer event
\begin{equation}\label{u:eq:coverage-event}
 \mathcal C_n=\bigl\{(g_\star,g_\circ):
 \KL(\pi_{g_\star}\Vert\nu^{\rm sm}_{g_{\rm base}})
                         \le n^{1-\ex{u}_{\rm E}}\bigr\}
\end{equation}
has complementary probability at most
$C_\beta(n^{\ex{u}_{\rm E}-\ex{D}}+
 n^{\ex{u}_{\rm E}-\ex{k}_{\rm f}+\ex{D}})=o(1)$.
\end{lemma}

\begin{proof}
We first compare the joint laws after smoothing, conditioning on the spin.
We then compare the Gibbs laws at the target and its smoothing base.
This is the reverse comparison of \cite[Section~5, Proposition~6]{oai-typical-upper};
we use Gaussian integration by parts for the second comparison.

We disintegrate $\Pi_n$ and $Q_F$ over the spin, whose marginal is uniform
under both laws. For each spin, by \eqref{u:eq:conditional-transport},
we can couple the conditional laws of the disorder with squared cost at
most twice their forward relative entropy. Under the channel
$g\mapsto\sqrt{1-\delta}\,g+\sqrt\delta\,Z$, the relative entropy between
the Gaussian output laws at two inputs is $(1-\delta)\|g-g'\|_F^2/(2\delta)$ in either
direction. By convexity of relative entropy, applied to the coupled mixture and
averaged over the common spin, we have
\begin{equation}\label{u:eq:smoothed-joint-entropy}
 \KL(\Pi_n^{\rm sm}\Vert Q_F^{\rm sm})
                 \le\delta^{-1}\KL(Q_F\Vert\Pi_n).
\end{equation}
This comparison for Gaussian convolutions also appears in
\cite[Lemma~11]{lugosi-neu-2022}. Both smoothed measures have disorder
marginal $\gamma_n$. By Gaussian reversibility, we identify the
conditional spin law of $Q_F^{\rm sm}$ with \eqref{u:eq:smoothed-kernel},
and we denote that of $\Pi_n^{\rm sm}$ by $\bar\pi_g$. Under the joint law of $(g_\star,g_{\rm base})$, expanding the logarithms
of the probabilities on the finite spin space gives the entropy chain
rule \cite[Chapter~4]{boucheron-lugosi-massart-2013}, in the form used in
\cite[Section~5, proof of Proposition~6]{oai-typical-upper}:
\begin{equation}\label{u:eq:entropy-decomposition}
 \begin{split}
 \E\KL(\pi_{g_\star}\Vert\nu^{\rm sm}_{g_{\rm base}})
 &=\E\KL(\pi_{g_\star}\Vert\bar\pi_{g_{\rm base}})\\
 &\quad+\E\KL(\bar\pi_{g_{\rm base}}\Vert\nu^{\rm sm}_{g_{\rm base}}).
 \end{split}
\end{equation}
Using the same decomposition with any alternative kernel at $g_{\rm base}$,
we find that the first term is at most
$\E\KL(\pi_{g_\star}\Vert\pi_{g_{\rm base}})$. When intermediate expressions may be infinite, we justify these
identities by finite approximations, or by the variational formula for relative entropy and monotone
approximation. By \cref{u:prop:fine-kl} and the following calculation,
the upper bounds are finite.

Since $g_\star$ and $g_{\rm base}$ have the same marginal law, the expected
difference of log partition functions vanishes. Using the independence of
$g_\circ$, we obtain
\begin{equation}\label{u:eq:gibbs-smoothing-cost}
 \begin{split}
 \E\KL(\pi_{g_\star}\Vert\pi_{g_{\rm base}})
 &=\beta(1-\sqrt{1-\delta})
   \E_{g_\star}\sum_x\pi_{g_\star}(x)H^c_{g_\star}(x)\\
 &\le\tfrac12\beta^2(n-1)(1-\sqrt{1-\delta})
 \le\tfrac12\beta^2(n-1)\delta.
 \end{split}
\end{equation}
For the middle inequality, Gaussian integration by parts gives the
off-diagonal contribution
$(\beta/n)\sum_{i<j}(1-\langle x_ix_j\rangle_{g_\star}^2)$;
the spin-independent diagonal has mean zero. Since the derivatives that occur here are bounded spin moments and
Gaussian first moments are finite, the integrations are justified. Combining \eqref{u:eq:smoothed-joint-entropy}--\eqref{u:eq:gibbs-smoothing-cost}
proves \eqref{u:eq:reverse-entropy}. The final assertion follows from
\cref{u:prop:fine-kl} and Markov's inequality.
\end{proof}

\begin{lemma}\label{u:lem:coverage}
On $\mathcal C_n$, every spin has positive
$\nu^{\rm sm}_{g_{\rm base}}$ probability, and, simultaneously for every
$S\subseteq\Sigma_n$,
\begin{equation}\label{u:eq:all-set-coverage}
 \pi_{g_\star}(S)\ge\epsilon_n=n^{-\ex{d}}
 \quad\Longrightarrow\quad
 \nu^{\rm sm}_{g_{\rm base}}(S)
 \ge\exp\{-n^{\ex{d}}(n^{1-\ex{u}_{\rm E}}+\log2)\}.
\end{equation}
If $\ex{e}+\ex{d}<\ex{u}_{\rm E}$, two independent draws from
$\nu^{\rm sm}_{g_{\rm base}}$ cross every $\epsilon_n$-balanced cut with
probability at least
\begin{equation}\label{u:eq:crossing-probability}
 \varpi_n=\exp\{-2n^{\ex{d}}(n^{1-\ex{u}_{\rm E}}+\log2)\}
                       \ge e^{-2n^{1-\ex{e}}}
\end{equation}
eventually.
\end{lemma}

\begin{proof}
This is the two-point entropy-contraction argument of
\cite[Section~5, Proposition~6]{oai-typical-upper}; the partition bound
is \cite[Theorem~4.20]{boucheron-lugosi-massart-2013}.
Finite relative entropy and the strict positivity of $\pi_{g_\star}(x)$
imply that $\nu^{\rm sm}_{g_{\rm base}}(x)>0$ for every spin. We pass to the partition $\{S,S^c\}$, which does not increase relative
entropy. If $a=\pi_{g_\star}(S)$ and
$b=\nu^{\rm sm}_{g_{\rm base}}(S)$, the binary relative entropy is at least
$a\log(1/b)-\log2$, since binary entropy is at most $\log2$.
We have proved \eqref{u:eq:all-set-coverage}. Applying it to both sides
of a balanced cut and multiplying the independent probabilities gives
\eqref{u:eq:crossing-probability}; the strict power gap also absorbs the
$\log2$ term. The implication holds for every set once the outer pair is
fixed, including a cut selected from this pair.
\end{proof}

\section{The witness event and the enlarged boundary}\label{u:sec:witness}

We now select a balanced cut with small flow, whenever one exists, and
bound the Gibbs mass of its enlarged boundary. The selection and the
comparison of boundary weights follow
\cite[Section~6, Proposition~7]{oai-typical-upper}. Here the enlargement
cost is controlled by concentration of sums of local fields over small
sets of sites under the conditional disorder law. We derive this
sparse-field estimate below from \cref{u:lem:transport} and its
linear-function MGF consequence; the transport input is
\cite[Corollary~3.10]{bogachev-kolesnikov-medvedev-2005}.

For $\epsilon>0$ and $\phi>0$, we define the witness event
\begin{equation}\label{u:eq:witness-event}
 \begin{split}
 \mathcal W_n(\epsilon,\phi)=\{g:\ &\|W(g)\|_{\mathrm{op}}\le12,\\
 &\exists S\subseteq\Sigma_n:\epsilon\le\pi_g(S)\le1-\epsilon,
                         \ \mathcal Q_g(S,S^c)\le\phi\}.
 \end{split}
\end{equation}
Fix an ordering of the subsets of $\Sigma_n$, and let $S_g$ be the first
cut satisfying these conditions. The cut $S_g$ depends only on $g$, and any fixed order
works for the argument. The mass and flow of each cut are continuous
finite sums in $g$, so this selection is measurable, as in the conventions
of \cref{u:sec:primitive}. The selected sets defined below are empty off
the witness event.

Let
\[
 \epsilon_n=n^{-\ex{d}},\qquad \phi_n=e^{-n^{1-\ex{q}}},\qquad
 R_n=\lceil n^{1-\ex{r}}\rceil,\qquad
 A_n^{\rm fld}=n^{1-\ex{u}_{\rm F}}.
\]
We write $d_H(x,y)=\#\{i\in[n]:x_i\ne y_i\}=\|x-y\|_2^2/4$
for the Hamming distance between $x,y\in\Sigma_n$, and
$d_H(x,A)=\min_{y\in A}d_H(x,y)$, with
$d_H(x,\varnothing)=+\infty$.
We write $\partial S$ for the vertex boundary that contains both endpoints
of every edge across the cut, and define
$\partial_R S=\{x:d_H(x,\partial S)\le R\}$. Let $\mathcal V_R$ be the family of differences of the indicator vectors
of two subsets of $[n]$ with at most $R$ elements each. Then $v_i\in\{-1,0,1\}$ and
$\|v\|_1=\|v\|_2^2\le2R$. The family contains both signs of every
indicator of at most $R$ sites, and
\begin{equation}\label{u:eq:field-direction-count}
 |\mathcal V_R|\le[(R+1)(n+1)^R]^2.
\end{equation}
Define
\[
 \mathrm{Bad}_{R,A}(g)=\Bigl\{x:\exists v\in\mathcal V_R,
                         \ \sum_i v_i(W(g)x)_i\ge A\Bigr\}.
\]
The excluded set is
\begin{equation}\label{u:eq:excluded-set}
 \mathcal E_n(g)=\partial_{R_n}S_g
           \ \cup\ \mathrm{Bad}_{R_n,A_n^{\rm fld}}(g)
 \quad\text{on }\mathcal W_n(\epsilon_n,\phi_n),
\end{equation}
and is empty off that event. It contains the enlarged boundary of the selected cut and the spins with large sums of local fields.

\begin{lemma}\label{u:lem:boundary}
Suppose that
\begin{equation}\label{u:eq:boundary-conditions}
 0<\ex{q}<\ex{u}_{\rm F}<\ex{r}<1,\qquad
 2\ex{u}_{\rm F}-\ex{r}<\ex{q},\qquad \ex{q}<1/2.
\end{equation}
Then, eventually,
\begin{equation}\label{u:eq:annealed-boundary-mass}
 \E_{\gamma_n}\pi_g(\mathcal E_n(g))
                         \le e^{-\frac12 n^{1-\ex{q}}}.
\end{equation}
\end{lemma}

\begin{proof}
We first bound the annealed mass of $\mathrm{Bad}_{R,A}$ by conditioning
on the spin. We then compare Gibbs weights along a short path from an
enlarged boundary vertex to the boundary.

Fix a spin $x$. Recall that, under the joint law $\Pi_n$, the conditional
density of the disorder given this spin is
\eqref{u:eq:conditional-density}. For fixed $v$, the linear functional
$f_v(g)=\sum_i v_i(W(g)x)_i$ has coefficients
$a_{ij}=(v_ix_j+v_jx_i)/\sqrt{2n}$, so
\[
 \|a\|_2^2=\|v\|_2^2+(\textstyle\sum_i v_ix_i)^2/n
                         \le2\|v\|_2^2\le4R.
\]
By integration by parts under \eqref{u:eq:conditional-density}, we have
$|\E_{p_x}f_v|\le2\beta\|v\|_1\le4\beta R$. Indeed, the derivative of
$\beta H^c_g(x)-F^c(g)$ in the $(i,j)$ entry is
$\beta(x_ix_j-\langle y_iy_j\rangle_g)/\sqrt{2n}$, whose absolute value
is at most $2\beta/\sqrt{2n}$. The density has Gaussian tails up to a linear exponential factor; by
truncation, this justifies the integration by parts, including its
boundary terms. Using
\eqref{u:eq:conditional-linear-mgf} and a union bound over
$\mathcal V_R$, we find that, for $A\ge4\beta R$,
\begin{equation}\label{u:eq:sparse-field-tail}
 \E_{\gamma_n}\pi_g(\mathrm{Bad}_{R,A}(g))
 \le[(R+1)(n+1)^R]^2
                    \exp\{-(A-4\beta R)^2/(8R)\}.
\end{equation}
When $R=R_n$ and $A=A_n^{\rm fld}$, the inequality
$\ex{u}_{\rm F}<\ex{r}$ implies that $A\ge8\beta R$ eventually. We also
have that the logarithm of \eqref{u:eq:field-direction-count} is $o(A^2/R)$.
Then \eqref{u:eq:sparse-field-tail} implies the upper bound
$\exp(-c_\beta n^{1+\ex{r}-2\ex{u}_{\rm F}})$ eventually.
By \eqref{u:eq:boundary-conditions}, the power of $n$ in this exponent is
strictly larger than $1-\ex{q}$.

Next, fix a witness matrix and its selected cut. On
$\|W(g)\|_{\mathrm{op}}\le12$, we find by polarization that the change
in energy when one spin is flipped is at most $24\sqrt n$. Then every
flip rate is at least
$(1/2)e^{-24\beta\sqrt n}$. By reversibility, the flow in each direction
across the selected cut is at most $\phi_n$. We conclude that its two-sided
boundary has Gibbs mass at most $4\phi_n e^{24\beta\sqrt n}$.
If $x\notin\mathrm{Bad}_{R,A}(g)$ and $d_H(x,y)\le R$, expansion about
$x$ gives
\begin{equation}\label{u:eq:sparse-energy-change}
 |H^c_g(y)-H^c_g(x)|\le2A+24R.
\end{equation}
To see this, let $I$ be the set of flipped sites. Both $-x\mathbf 1_I$
and its negative belong to $\mathcal V_R$, since $|I|\le R$. Their linear
fields are bounded by $A$ in both signs, and the quadratic correction is
at most $2\|W(g)\|_{\mathrm{op}}R$.
For each vertex of the enlarged boundary, we select a nearest boundary
vertex, using a fixed order to break ties. By the bound on the size of a
Hamming ball, each boundary vertex is selected at most
$(R+1)(n+1)^R$ times. Comparing Gibbs weights by
\eqref{u:eq:sparse-energy-change}, we obtain
\begin{equation}\label{u:eq:boundary-weight-comparison}
 \begin{split}
 \pi_g(\mathcal E_n(g))
 &\le4(R_n+1)(n+1)^{R_n}\phi_n
       e^{\beta(2A_n^{\rm fld}+24R_n+24\sqrt n)}\\
 &\quad+\pi_g(\mathrm{Bad}_{R_n,A_n^{\rm fld}}(g)).
 \end{split}
\end{equation}
The inequality also holds off the witness event, since its left side
vanishes there. In the first term, the logarithms of the positive factors other than
$\phi_n$ have powers $1-\ex{r}$ (with a factor $\log n$), $1-\ex{u}_{\rm F}$,
and $1/2$, all smaller than $1-\ex{q}$. Averaging
\eqref{u:eq:boundary-weight-comparison} and using
\eqref{u:eq:sparse-field-tail}, we obtain
\eqref{u:eq:annealed-boundary-mass} after absorbing the sum and constants.
\end{proof}

We impose a pressure cutoff only at the outer target. We define
\begin{equation}\label{u:eq:target-pressure-event}
 \begin{split}
 \mathcal H_n&=\{g:F^c(g)\le nP_\beta+n^{1-\ex{z}_{\rm P}}\},\\
 \overline{\mathcal E}_n(g)&=
 \begin{cases}
 \mathcal E_n(g),&g\in\mathcal H_n,\\
 \varnothing,&g\notin\mathcal H_n.
 \end{cases}
 \end{split}
\end{equation}
The gradient of $F^c$ has norm at most $\beta\sqrt{n/2}$.
By Gaussian concentration and $\E F^c\le nP_\beta$
(\cref{inp:ob-concentration,inp:parisi,lem:model-completion}), we have
\begin{equation}\label{u:eq:target-pressure-tail}
 \gamma_n(\mathcal H_n^c)
 \le e^{-c_\beta n^{1-2\ex{z}_{\rm P}}}=o(1)
                         \qquad(0<\ex{z}_{\rm P}<1/2).
\end{equation}
This is also \cref{u:lem:pressure-tail}. Each set is measurable by the
conventions of \cref{u:sec:primitive}; the disorder used in each coarse
proposal is not restricted by $\mathcal H_n$.

\section{The connecting sequence}\label{u:sec:sequence}

We construct the sequence on one product space, following
\cite[Section~6, Proposition~7]{oai-typical-upper}. At the two endpoints
we continue the coarse proposal to the fine horizon. At every other
vertex we use only the coarse proposal. The simultaneous Gaussian
rotation, continuity of the query recursion, and pinning of spins to
their magnetizations are the connecting-sequence method of that proof;
we retain the source and replay histories on the growing mesh.

For the common fine schedule, let $\mu^{\rm aux}_n$ be the independent
product law of all $D_i$ Wiener paths to $V_{\ex{f}}$, auxiliary $B_i^0$
paths, used bridge paths through $N_{\ex{f}}$, and full completion vectors
$C_j\sim N(0,I_n)$ through $N_{\ex{f}}$. We may retain the unused bridge coordinates independently; this gives
the same finite experiment. At both horizons, the matrix has dimension $n$ and the initial frame is
the identity. By
\cref{u:lem:assembly}, these are the replay inputs.

Under a single product probability $\PP_n$, we sample independently
\[
 g_\star,g_\circ,g_1,g_2\sim\gamma_n,\qquad
 a_1,a_2\sim\mu^{\rm aux}_n,
\]
and independent arrays of uniform thresholds at each grid point, together
with two arrays for the fine endpoints. For $0\le\theta\le\pi/2$, we define
\begin{equation}\label{u:eq:coupled-rotation}
 \begin{split}
 g_{\rm base}&=\sqrt{1-\delta}\,g_\star+\sqrt\delta\,g_\circ,\\
 g_\theta&=\sqrt{1-\delta}\,g_{\rm base}
                +\sqrt\delta(\cos\theta\,g_1+\sin\theta\,g_2),\\
 a_\theta&=\cos\theta\,a_1+\sin\theta\,a_2.
 \end{split}
\end{equation}
The rotation of the auxiliary inputs is defined by blockwise addition of
vectors and paths, which preserves the zero endpoints of bridges. The uniform thresholds are not
rotated. We use the coarse replay at every angle and continue it to the fine
horizon at the two endpoints. Let
\[
 \ell_n=\lceil(\pi/2)e^{n^{\ex{g}_{\rm rot}}}\rceil,
 \qquad \theta_j=\frac{\pi j}{2\ell_n}\quad(0\le j\le\ell_n).
\]
We denote the coarse spins by $X_j^B$, their coarse magnetizations by
$M_j$, and the fine endpoint spins by $X_0^F,X_1^F$.
For a grid angle $\theta=\theta_j$, we write $X_\theta^B=X_j^B$,
and let $U_j^B$ be its coarse threshold array. We define
\[
 E_{B,\theta_j}
   =\{(g_{\theta_j},a_{\theta_j}|_B,U_j^B)\in E_B\},
\]
where $E_B$ is the event of \cref{u:prop:coarse} and
$a|_B$ restricts the $D$ paths to $V_{\ex{b}}$ and retains only
the auxiliary cell, bridges, and completion vectors used by the
coarse replay.

The outer pair $(g_\star,g_\circ)$ has law $\gamma_n^{\otimes2}$.
The internal law $\mu^{\rm int}_n$ is the fixed product law of
$(g_1,g_2,a_1,a_2,\text{thresholds})$, so
$\PP_n=\gamma_n^{\otimes2}\otimes\mu^{\rm int}_n$. After an outer pair is
fixed, the internal law remains the same. For each fixed angle,
$g_\theta$ is standard Gaussian, and $a_\theta\sim\mu^{\rm aux}_n$
independently of all matrix drivers. For the complete paths, we note that every finite collection of path
evaluations is jointly Gaussian, that the covariances within and
between blocks are unchanged by the rotation, and that cylinders at
rational times generate the path sigma algebra. After fixing $(g_\star,g_\circ)$, the spin $X_0^F$ depends only on
$(g_1,a_1,U_0^F)$, and $X_1^F$ depends only on $(g_2,a_2,U_1^F)$. We have
\begin{equation}\label{u:eq:endpoint-product-law}
 \Law_{\mu^{\rm int}_n}(X_0^F,X_1^F)
        =\nu^{\rm sm}_{g_{\rm base}}\otimes\nu^{\rm sm}_{g_{\rm base}}.
\end{equation}
This is the law of the endpoints under the internal measure, before any
Gram, pinning, or crossing event is imposed.

\begin{lemma}\label{u:lem:transfer}
If $\ex{q}<\min(\ex{k}_{\rm b},\ex{z}_{\rm P},\ex{D})$, then at each fixed
grid angle, eventually,
\begin{equation}\label{u:eq:boundary-transfer}
 \PP_n(E_{B,\theta},\ X_\theta^B\in\overline{\mathcal E}_n(g_\star))
                   \le e^{-\frac14 n^{1-\ex{q}}}.
\end{equation}
\end{lemma}

\begin{proof}
We refine the target-to-proposal comparison in
\cite[Section~6, proof of Proposition~7]{oai-typical-upper} by integrating
the conditional Gaussian energy at each fixed spin. The calculation
below gives an $O_\beta(n\delta)$ regression cost while retaining the
target Gibbs normalization.
We condition only on $g_\star$. From \eqref{u:eq:coupled-rotation}, we have
\begin{equation}\label{u:eq:gaussian-regression}
 \begin{split}
 g_\theta&=(1-\delta)g_\star+\sqrt{\delta(2-\delta)}\,Z_\theta,\\
 Z_\theta&=\frac{\sqrt{1-\delta}\,g_\circ
                    +\cos\theta\,g_1+\sin\theta\,g_2}{\sqrt{2-\delta}},
 \end{split}
\end{equation}
where $Z_\theta\sim\gamma_n$ is independent of $g_\star$. For a fixed
spin, $H^c_Z(x)$ is Gaussian with variance $n/2$, including the completed
diagonal. It follows that
\begin{equation}\label{u:eq:regression-energy-moment}
 \E[e^{\beta H^c_{g_\theta}(x)}\mid g_\star]
 =e^{\beta(1-\delta)H^c_{g_\star}(x)+\beta^2n\delta(2-\delta)/4}.
\end{equation}
At a fixed angle, the auxiliary input and thresholds are independent of
the entire matrix tuple. The restricted coarse kernel of
\cref{u:prop:coarse} therefore still applies after adjoining
$g_\star,g_\circ,g_\theta$. The conditional Gaussian in
\eqref{u:eq:gaussian-regression} has positive covariance, so it does not
charge the common $\gamma_n$-null exceptional set of that proposition.

Recall that on the support of $\overline{\mathcal E}_n(g_\star)$, we have
$\|W(g_\star)\|_{\mathrm{op}}\le12$ and $|H^c_{g_\star}(x)|\le6n$.
Using \cref{u:prop:coarse} and \eqref{u:eq:regression-energy-moment}, then
summing over the set $\overline{\mathcal E}_n(g_\star)$ fixed by the target,
we bound the conditional probability of hitting this set by
\[
 \exp\bigl\{n^{1-\ex{k}_{\rm b}}+F^c(g_\star)-nP_\beta
                    +(6\beta+\beta^2/2)n\delta\bigr\}
             \pi_{g_\star}(\overline{\mathcal E}_n(g_\star)).
\]
Here we retained the target Gibbs normalization. By
\eqref{u:eq:target-pressure-event} and
\eqref{u:eq:annealed-boundary-mass}, its average in $g_\star$ is at most
\[
 \exp\{n^{1-\ex{k}_{\rm b}}+n^{1-\ex{z}_{\rm P}}
                   +C_\beta n^{1-\ex{D}}\}
                         e^{-\frac12 n^{1-\ex{q}}}.
\]
By the strict inequalities between the exponents, this proves
\eqref{u:eq:boundary-transfer}. The cost of the regression is of order
$n\delta$; the exceptional event \eqref{u:eq:target-pressure-tail} for the
target pressure is included only after the crossing
argument in \cref{u:sec:exclusion}.
\end{proof}

\begin{lemma}\label{u:lem:stability}
There exists a measurable cap event, with failure probability at most
$C(N_{\ex{f}}+2)e^{-cn}$, that controls all angles. On this event and all
coarse Gram events, neighboring grid points satisfy
\begin{equation}\label{u:eq:coarse-continuity}
 \normn{M_{j+1}-M_j}
 \le e^{C_\beta n^{\ex{a}}\log n}|\theta_{j+1}-\theta_j|
 \le e^{-n^{\ex{g}_{\rm rot}}/2}
\end{equation}
eventually, provided $\ex{a}<\ex{g}_{\rm rot}$.
\end{lemma}

\begin{proof}
This is the input-cap and recursive-continuity argument of
\cite[Section~6, proof of Proposition~7]{oai-typical-upper}, with the
number of steps and the inverse scalar profiles controlled by the
growing schedule.
We first bound the primitive inputs uniformly in the angle. We then
compare the two coarse recursions at neighboring grid points.

For a Wiener path of duration $T$, the reflection bound gives
$\PP(\sup|B|>u\sqrt T)\le4e^{-u^2/2}$. Then, for some fixed small parameter, the exponential moment of
$\sup|B|^2/T$ is bounded by a universal constant. By Chernoff's inequality
over the independent rows, we have
$\sum_i\sup|B_i|^2\le CTn$ except with probability $e^{-cn}$ when $C$
is sufficiently large. A bridge has supremum at most twice that of its
generating Wiener path. These bounds cover the two $D$ blocks, the
auxiliary cells, and the finitely many used bridge blocks. We also bound
every coordinate of the four primitive matrices and of the two streams of
completion vectors by $n$; this adds failure probability at most
$C(n^2+nN_{\ex{f}})e^{-n^2/2}$. All required path norms, completion norms,
and matrix operator norms are then at most $n^{C_0}$ for a fixed $C_0$;
for example, an enlarged $n^4$ cap suffices. Rotated inputs satisfy the
same cap up to a factor of two, and their differences are at most a
polynomial times $|\theta-\phi|$. This bounds the inputs for all angles.

For the recursion, we use the path norm
$(n^{-1}\sum_i\sup_t|b_i(t)|^2)^{1/2}$. On the target Gram events at the two grid points
$\|A_{\mathcal T}-I\|\le3/4$, every Gram--Schmidt residual has normalized
norm at least $1/2$. Indeed, the squared norm of each residual is a Schur complement
and is at least the minimum eigenvalue of the relevant Gram matrix.
If $u_l,u_l'$ are the earlier unit queries, the operator norm of their
projection difference is at most $2\sum_l\normn{u_l-u_l'}$.
Normalization is $4$-Lipschitz on vectors of norm at least $1/2$.
The next query difference is bounded by a fixed multiple of the target
difference and this sum of earlier differences.

Each target difference is bounded by $2s_j^{-1}$ times the scalar field
difference on the preceding path. Using Picard's construction, Gronwall's inequality, and integration by
parts for bounded-variation coefficients, we bound that difference by
$Ce^{V_{\ex{f}}}$ times the gap between the inputs to the preceding drivers. All $s_j^{-1}$ are polynomially bounded, since
$s_j^2\ge h/R_{V_{\ex{f}}}$ and \cref{u:lem:tails} applies. By the query
endpoint formula in \cref{u:lem:assembly}, the next endpoint gap is bounded
by a polynomial times the primitive matrix and completion gap plus the
sum of preceding query gaps. Appending the next cell adds at most the
old path gap, $\sqrt{h_j}$ times this endpoint gap, and the bridge gap.
Here $\sqrt{h_j}\le1$. The recursion implies that one chronological step multiplies the accumulated gap by
at most $n^{C_\beta}$ and adds at most that factor times the primitive
input gap. After $N_{\ex{b}}\le8n^{\ex{a}}$ steps, the
amplification is at most $e^{C_\beta n^{\ex{a}}\log n}$.
The final map $m$ is spatially $1$-Lipschitz. The grid spacing is at most
$e^{-n^{\ex{g}_{\rm rot}}}$, proving \eqref{u:eq:coarse-continuity}.
This comparison uses only coarse queries and the Gram events at the
two endpoints of each grid edge.
\end{proof}

\trilogyneedspace{12\baselineskip}
\begin{lemma}\label{u:lem:pinning}
If $\ex{r}<\ex{b}/2$, each of the $\ell_n+3$ pairs
\[
 \begin{gathered}
 (X,M_B)=(X_j^B,M_j)\quad(0\le j\le\ell_n),\\
 (X_0^F,M_0),\qquad (X_1^F,M_{\ell_n})
 \end{gathered}
\]
satisfies
\begin{equation}\label{u:eq:pinning-tail}
 \PP_n(\|X-M_B\|_2^2>R_n/4)\le2e^{-R_n/64}
\end{equation}
eventually.
\end{lemma}

\begin{proof}
We use posterior projection and the independent-row pinning argument of
\cite[Section~6, proof of Proposition~7]{oai-typical-upper}.
Under the unconditioned law at a single angle, the source identification
in \cref{u:sec:proposal} gives the law of the complete coordinate rows,
which are independent under this law.
For $T=V_{\ex{b}}$ or $V_{\ex{f}}$, posterior projection at $S=V_{\ex{b}}$ gives
\[
 \E[(X_T-m_S)^2]=1-\E m_S^2=p(S)\le C_\beta n^{-\ex{b}/2}.
\]
The squared distances lie in $[0,4]$ and are independent across rows.
Using $e^{z/4}\le1+(e-1)z/4$ for $0\le z\le4$ and Chernoff's inequality
with parameter $1/4$, we find that, once $np(S)\le R_n/16$, their sum
exceeds $R_n/4$ with probability at most $e^{-R_n/32}$. This implies \eqref{u:eq:pinning-tail}.
For the fine endpoints, the equality of the common initial segment after
\cref{u:lem:quad} gives the same $m_S$. We have proved the bounds under the unconditioned law of each run.
\end{proof}

Define the sequence
\begin{equation}\label{u:eq:connecting-sequence}
 f_0=X_0^F,\qquad f_{j+1}=X_j^B\quad(0\le j\le\ell_n),
 \qquad f_{\ell_n+2}=X_1^F.
\end{equation}
Every step has a coarse endpoint. Let $\mathrm{Good}_n$ be the intersection
of the cap event, all $E_{B,\theta_j}$, all pinning events in
\eqref{u:eq:pinning-tail}, and the avoidance events
$X_j^B\notin\overline{\mathcal E}_n(g_\star)$ for every coarse vertex.
This is a finite intersection of measurable events on the product space.

\begin{proposition}\label{u:prop:sequence}
Assume the hypotheses of
\cref{u:prop:coarse,u:lem:boundary,u:lem:transfer,u:lem:stability,u:lem:pinning}.
On $\mathrm{Good}_n$, every step of \eqref{u:eq:connecting-sequence} has
Hamming length at most $R_n$. Moreover,
\begin{equation}\label{u:eq:sequence-union-bound}
 \begin{split}
 \PP_n(\mathrm{Good}_n^c)
 &\le C(N_{\ex{f}}+2)e^{-cn}
       +(\ell_n+1)e^{-n^{1-\ex{s}_{\rm b}}}\\
 &\quad+2(\ell_n+3)e^{-R_n/64}
       +(\ell_n+1)e^{-\frac14 n^{1-\ex{q}}}.
 \end{split}
\end{equation}
If, in addition,
\begin{equation}\label{u:eq:sequence-conditions}
 \begin{gathered}
 \ex{r}<\ex{s}_{\rm b},\qquad \ex{q}<\ex{s}_{\rm b},\qquad
 \ex{a}<\ex{g}_{\rm rot}<1-\ex{s}_{\rm b},\\
 \ex{s}_{\rm b}<\ex{s}_{\rm seq}<\ex{e},
 \end{gathered}
\end{equation}
then, eventually,
\begin{equation}\label{u:eq:sequence-failure}
 \PP_n(\mathrm{Good}_n^c)\le e^{-n^{1-\ex{s}_{\rm seq}}}.
\end{equation}
\end{proposition}

\begin{proof}
For spins $x,y$ and real vectors $m,m'$, we have
\[
 4d_H(x,y)\le3\{\|x-m\|_2^2+\|m-m'\|_2^2+\|m'-y\|_2^2\}.
\]
Each term is at most $R_n/4$ on $\mathrm{Good}_n$, eventually, by
\eqref{u:eq:coarse-continuity} and \eqref{u:eq:pinning-tail}. For the steps between fine and coarse spins, we have that the middle
term is zero, since the common initial segments agree.

A union bound using \cref{u:prop:coarse},
\eqref{u:eq:boundary-transfer}, the cap bound, and
\eqref{u:eq:pinning-tail} proves \eqref{u:eq:sequence-union-bound}.
The strict gaps in \eqref{u:eq:sequence-conditions} absorb its constants
and $\log(\ell_n+1)=O(n^{\ex{g}_{\rm rot}})$, giving
\eqref{u:eq:sequence-failure}. The fine endpoints contribute their pinning
failures to this union. The Gram and likelihood events concern only coarse proposals. Each
bound holds under the unconditioned law of the corresponding run. This union bound does not use independence between different angles.
\end{proof}

\section{Exclusion of low-flow witnesses}\label{u:sec:exclusion}

We now combine coverage with the connecting sequence. The crossing
argument is the one in \cite[Section~6, Proposition~7]{oai-typical-upper}.

\begin{proposition}\label{u:prop:witness}
Assume the hypotheses of
\cref{u:lem:smoothing,u:lem:coverage,u:prop:sequence}, and let
$0<\ex{z}_{\rm P}<1/2$. Then
\begin{equation}\label{u:eq:witness-decay}
 \begin{split}
 \gamma_n(\mathcal W_n(\epsilon_n,\phi_n))
 &\le C_\beta\bigl(n^{\ex{u}_{\rm E}-\ex{D}}
                 +n^{\ex{u}_{\rm E}-\ex{k}_{\rm f}+\ex{D}}\bigr)
        +e^{-c_\beta n^{1-2\ex{z}_{\rm P}}}\\
 &\quad+e^{2n^{1-\ex{e}}-n^{1-\ex{s}_{\rm seq}}}=o(1).
 \end{split}
\end{equation}
\end{proposition}

\begin{proof}
Fix an outer pair $(g_\star,g_\circ)\in\mathcal C_n$ with
$g_\star\in\mathcal H_n\cap\mathcal W_n(\epsilon_n,\phi_n)$.
The selected cut $S_{g_\star}$ is $\epsilon_n$-balanced. By
\eqref{u:eq:endpoint-product-law} and \cref{u:lem:coverage}, the event
$f_0\in S_{g_\star}$, $f_{\ell_n+2}\notin S_{g_\star}$ has internal
probability at least $\varpi_n$.

If a sequence joins opposite sides with steps of Hamming length at most
$R_n$, then some step has endpoints on opposite sides.
A shortest path in the hypercube between the endpoints of that step
contains an edge across the cut, and both endpoints of this edge lie in the
boundary.
Both endpoints of the step are within $R_n$ of that boundary. One endpoint
is coarse by \eqref{u:eq:connecting-sequence}, so it belongs to
$\partial_{R_n}S_{g_\star}$, which is contained in
$\mathcal E_n(g_\star)=\overline{\mathcal E}_n(g_\star)$ on this target.
Crossing is therefore impossible on $\mathrm{Good}_n$, and
\begin{equation}\label{u:eq:conditional-crossing-obstruction}
 \varpi_n\le
 \mu^{\rm int}_n\bigl(\mathrm{Good}_n(g_\star,g_\circ)^c\bigr).
\end{equation}
The crossing event may depend on the fixed outer pair, while \eqref{u:eq:endpoint-product-law} gives the law of the endpoints
under the unconditioned internal measure.

Integrating \eqref{u:eq:conditional-crossing-obstruction} against
$\gamma_n^{\otimes2}$ and using Fubini's theorem for the product measure, we obtain
\[
 \begin{split}
 \gamma_n(\mathcal W_n(\epsilon_n,\phi_n))
 &\le C_\beta\bigl(n^{\ex{u}_{\rm E}-\ex{D}}
                +n^{\ex{u}_{\rm E}-\ex{k}_{\rm f}+\ex{D}}\bigr)
        +e^{-c_\beta n^{1-2\ex{z}_{\rm P}}}\\
 &\quad+\varpi_n^{-1}\PP_n(\mathrm{Good}_n^c).
 \end{split}
\]
Here we used \cref{u:lem:smoothing} and
\eqref{u:eq:target-pressure-tail} for the outer exceptions. By
\eqref{u:eq:crossing-probability} and \eqref{u:eq:sequence-failure}, this
proves \eqref{u:eq:witness-decay}. The exceptional events for the pressure and for coverage each enter
once, before we divide by the crossing probability.
\end{proof}
\chapter{Proof of the upper bound and the parameter system}\label{u:sec:proof}

We prove \cref{thm:upper,cor:upper-discrete,cor:two-sided} from
\cref{u:prop:witness,u:lem:goe-norm}. The finite-state argument of
\cite[Section~7, Lemma~8]{oai-typical-upper} converts the exclusion of
balanced cuts with small flow into a bound on the average distance from
equilibrium. We give its proof, then establish the estimate for the
discrete chain. In \cref{u:sec:budget}, we check the parameter
comparisons used throughout \cref{part:upper}.

Throughout, the disorder, Gibbs measure, and continuous-time semigroup
are those of \cref{u:sec:primitive}. In particular,
$\pi_g=\pi^{J(g)}$ and $P_t^g=P_t^{J(g)}$. The exponents are fixed before
$n$ tends to infinity.

\section{From balanced-cut flow to mixing}\label{u:sec:pruning}

The next lemma is \cite[Lemma~8]{oai-typical-upper}, specialized to the
heat-bath chain. Its proof uses the coarea argument for the reversible
Cheeger inequality of Lawler and Sokal~\cite[Theorem~3.5]{lawler-sokal-1988} and
entropy dissipation as in Diaconis and Saloff-Coste
\cite[Lemmas~2.5 and~2.7]{diaconis-saloff-coste-1996}. We keep track of the estimates on
the Dirichlet form, since they are also used in
\cref{u:sec:proof-discrete}.

\begin{lemma}[Pruning and averaged mixing]\label{u:lem:pruning}
Fix $n\ge2$, $\beta>0$, and $g$. Let $0<\epsilon<1/3$ and $\phi>0$.
Suppose that
\[
 \mathcal Q_g(S,S^c)>\phi
 \qquad\text{whenever }\epsilon\le\pi_g(S)\le1-\epsilon.
\]
Then, for every $t>0$,
\begin{equation}\label{u:eq:averaged-mixing}
 \sum_{x\in\Sigma_n}\pi_g(x)
       \norm{P_t^g(x,\cdot)-\pi_g}_{\TV}
 \le\frac{n\sqrt{2\log2}}{\phi\sqrt t}+2\epsilon.
\end{equation}
\end{lemma}

\begin{proof}
We first remove sets that fail an internal expansion inequality. On the
remaining set, we prove a Poincar\'e inequality by integrating over level
sets. Entropy dissipation for the full chain then gives a time at which
the average distance is small; contraction gives the estimate at $t$.

In this proof, we write $\pi=\pi_g$. For distinct configurations $x,y$,
let $q(x,y)=\pi(x)p_i^{J(g)}(-x_i\mid x)$ if $y=x^{i,-x_i}$, and let
$q(x,y)=0$ otherwise. By reversibility, $q(x,y)=q(y,x)$, and
\[
 \sum_{y\ne x}q(x,y)\le n\pi(x).
\]
We use the Dirichlet form
\begin{equation}\label{u:eq:pruning-dirichlet}
 \mathfrak D(u)=\sum_{\{x,y\}}q(x,y)(u(x)-u(y))^2
              =-\langle u,\mathcal L^{J(g)}u\rangle_\pi,
\end{equation}
where each unordered pair of distinct configurations occurs once and
$\langle u,v\rangle_\pi=\sum_x\pi(x)u(x)v(x)$.

For disjoint sets $A,B\subseteq\Sigma_n$, we write
\[
 \mathcal Q_g(A,B)=\sum_{x\in A}\sum_{y\in B}q(x,y).
\]
This agrees with \eqref{u:eq:cut-flow} when $B=A^c$.

\paragraph{Removing sets.}
We use the pruning procedure of
\cite[Section~7, proof of Lemma~8]{oai-typical-upper}.
We start with $C=\Sigma_n$. Whenever a nonempty set $U\subset C$ satisfies
\[
 \pi(U)\le\pi(C)/2,
 \qquad \mathcal Q_g(U,C\setminus U)<\phi\pi(U),
\]
we remove $U$ from $C$. The procedure terminates after finitely many
removals. Let $A$ be the union of the sets removed up to any
stage. Every edge from $A$ to its current complement was counted when
its first endpoint was removed. Summing the inequalities at those
stages, we have, whenever $A$ is nonempty,
\[
 \mathcal Q_g(A,\Sigma_n\setminus A)<\phi\pi(A).
\]
Suppose that the removed mass reaches $\epsilon$, and consider the first
removal at which this happens. Before this removal, the removed mass
was some $a<\epsilon$, and the newly removed set has mass at most
$(1-a)/2$. After it, the removed mass is at most
$(1+a)/2<(1+\epsilon)/2<1-\epsilon$. This contradicts the hypothesis on balanced
cuts, because $\phi\pi(A)\le\phi$. We have proved that
the terminal set satisfies $\pi(C)>1-\epsilon$ and
\begin{equation}\label{u:eq:internal-expansion}
 \mathcal Q_g(U,C\setminus U)\ge\phi\pi(U)
 \quad\text{if }U\subset C\text{ and }\pi(U)\le\pi(C)/2.
\end{equation}

\paragraph{A Poincar\'e inequality on $C$.}
We apply the coarea proof of the reversible Cheeger bound
\cite[Theorem~3.5]{lawler-sokal-1988}, with unnormalized weights on the
retained set as in \cite[Section~7, proof of Lemma~8]{oai-typical-upper}.
We claim that, for every real function $u$ on $\Sigma_n$,
\begin{equation}\label{u:eq:retained-poincare}
 \sum_{y\in C}\pi(y)(u(y)-\bar u_C)^2
 \le\frac{2n}{\phi^2}\mathfrak D(u),
 \qquad
 \bar u_C=\frac1{\pi(C)}\sum_{y\in C}\pi(y)u(y).
\end{equation}
Let $m$ be a weighted median of $u$ on $C$. Each of
$w=(u-m)_+$ and $w=(m-u)_+$ has support of mass at most $\pi(C)/2$.
Integrating \eqref{u:eq:internal-expansion} over the level sets of $w^2$,
we find
\[
 \phi\sum_C\pi w^2
 \le\sum_{\{x,y\}\subset C}q(x,y)|w(x)^2-w(y)^2|.
\]
By Cauchy--Schwarz, the right side is at most
\[
 \left(\sum_{\{x,y\}\subset C}q(x,y)(w(x)-w(y))^2\right)^{1/2}
 \left(2n\sum_C\pi w^2\right)^{1/2}.
\]
Here we used $(w(x)+w(y))^2\le2(w(x)^2+w(y)^2)$ and the bound on the
outgoing rates. Dividing and squaring, we find the desired estimate for $w$ when
$\sum_C\pi w^2>0$; when this sum vanishes, the estimate is
immediate. The sum of the energies of the two parts is at most
$\mathfrak D(u)$. Since the weighted mean minimizes the squared error,
\eqref{u:eq:retained-poincare} follows.

\paragraph{Entropy dissipation.}
The derivative formula and square-root Dirichlet bound are
\cite[Lemmas~2.5 and~2.7]{diaconis-saloff-coste-1996}, with the rates
of \cref{u:sec:primitive}.
For a fixed starting configuration $x$, define
$f_s^x(y)=P_s^g(x,y)/\pi(y)$. We write
\[
 \Ent_\pi(f)=\sum_y\pi(y)f(y)\log f(y),\qquad
 H(\pi)=-\sum_x\pi(x)\log\pi(x),
\]
where $\pi(f)=1$ and $0\log0=0$. Reversibility implies
\begin{align*}
 \frac{\dd}{\dd s}\Ent_\pi(f_s^x)
 &=-\sum_{\{y,z\}}q(y,z)
       (f_s^x(y)-f_s^x(z))(\log f_s^x(y)-\log f_s^x(z))\\
 &\le-4\mathfrak D(\sqrt{f_s^x}).
\end{align*}
In the last step, we used
$(a-b)(\log a-\log b)\ge4(\sqrt a-\sqrt b)^2$ for $a,b>0$.
The initial entropy is $\log(1/\pi(x))$, and the final entropy is
nonnegative. Integrating and averaging over $x\sim\pi$, we find
\begin{equation}\label{u:eq:continuous-dissipation}
 \int_0^t\sum_x\pi(x)\mathfrak D(\sqrt{f_s^x})\dd s
 \le\frac14H(\pi).
\end{equation}
To justify the calculation when the density has zero entries, we start
with $(1-\eta)f_0^x+\eta$ and let $\eta\downarrow0$. Since the state
space is finite and entropy is continuous at zero, the integrated
inequality passes to this limit.

\paragraph{Averaging the distances.}
By \eqref{u:eq:retained-poincare} and
\eqref{u:eq:continuous-dissipation}, there
exists $s\in(0,t]$ such that
\begin{equation}\label{u:eq:continuous-variance}
 \begin{split}
  \sum_x\pi(x)r_x^2&\le\frac{nH(\pi)}{2t\phi^2},\\
  r_x^2&=\sum_{y\in C}\pi(y)(\sqrt{f_s^x(y)}-a_x)^2,
  \qquad
  a_x=\frac1{\pi(C)}\sum_{y\in C}\pi(y)\sqrt{f_s^x(y)}.
 \end{split}
\end{equation}
Using Cauchy--Schwarz and the fact that $a_x^2\pi(C)\le\sum_C\pi f_s^x\le1$, we have
\[
 \sum_C\pi|f_s^x-a_x^2|\le2r_x.
\]
Further, the normalization of the density gives
$|\sum_C\pi(f_s^x-1)|\le\sum_{C^c}\pi(f_s^x+1)$.
Comparing $f_s^x$ first with $a_x^2$ and then with $1$ on $C$,
and adding the complement, we have
\begin{equation}\label{u:eq:pruning-distance}
 \sum_{\Sigma_n}\pi|f_s^x-1|
 \le4r_x+2\sum_{C^c}\pi(f_s^x+1).
\end{equation}
By stationarity, we have $\sum_x\pi(x)f_s^x(y)=1$. It follows that the
last sum in the preceding display, without its factor two, has
average $2\pi(C^c)$ over $x\sim\pi$. Dividing
\eqref{u:eq:pruning-distance} by two and using Cauchy--Schwarz in $x$,
we obtain
\[
 \sum_x\pi(x)\norm{P_s^g(x,\cdot)-\pi}_{\TV}
 \le\frac{\sqrt{2nH(\pi)}}{\phi\sqrt t}+2\epsilon.
\]
Since $|\Sigma_n|=2^n$, we have $H(\pi)\le n\log2$. Total variation
contracts under $P_{t-s}^g$ by \cref{lem:model-dynamics}, which gives
\eqref{u:eq:averaged-mixing} at time $t$.
\end{proof}

\section{Proof of the main theorem}\label{u:sec:proof-upper}

We now combine the witness estimate with \cref{u:lem:pruning}. Recall
that $\epsilon_n=n^{-\ex d}$, $\phi_n=e^{-n^{1-\ex q}}$, and
$\mathcal W_n(\epsilon_n,\phi_n)$ is the event that $\norm{W(g)}_{\mathrm{op}}
\le12$ and some $\epsilon_n$-balanced cut has flow at most $\phi_n$.

\begin{proof}[Proof of \cref{thm:upper}]
Fix $\beta>1$ and $0<\omega<1/4$, and take the affine family of
\cref{u:sec:parameters} with $\varepsilon_\omega=(1/4-\omega)/20$.
The comparisons verified in \cref{u:sec:budget} imply all hypotheses
of the preceding estimates. By \cref{u:prop:witness}, we have
\begin{equation}\label{u:eq:witness-family}
 \gamma_n(\mathcal W_n(\epsilon_n,\phi_n))
 \le C_\beta n^{-\varepsilon_\omega}+e^{-c_\beta n^{2/5}}
       +\exp\{2n^{3/4+3\varepsilon_\omega}
                    -n^{3/4+4\varepsilon_\omega}\}=o(1).
\end{equation}
Outside this event and $\{\norm{W(g)}_{\mathrm{op}}>12\}$, every
$\epsilon_n$-balanced cut has flow greater than $\phi_n$. For all
sufficiently large $n$, we have $0<\epsilon_n<1/3$. We use
\eqref{u:eq:averaged-mixing} with
\[
 \epsilon=\epsilon_n,\qquad \phi=\phi_n,
 \qquad t=e^{n^{1-\omega}}.
\]
Using Markov's inequality in the Gibbs draw and
\cref{u:lem:goe-norm}, we find
\begin{equation}\label{u:eq:joint-bad-start}
 \begin{split}
 &\Pi_n\bigl\{(g,x):
       \norm{P_{e^{n^{1-\omega}}}^g(x,\cdot)-\pi_g}_{\TV}>1/4\bigr\}\\
 &\quad\le\gamma_n(\mathcal W_n(\epsilon_n,\phi_n))+2e^{-n}
       +4n\sqrt{2\log2}\,
          e^{n^{1-\ex q}-\frac12n^{1-\omega}}+8n^{-\ex d}.
 \end{split}
\end{equation}
The right side tends to zero, since $\ex d>0$ and
$\ex q-\omega=10\varepsilon_\omega>0$. By the dictionary in
\cref{u:sec:primitive}, the $(J(g),x)$-marginal of $\Pi_n$ is
$\PP_{J,X_0\sim\pi^J}$. This proves the assertion under the joint law.
Here the Gibbs draw is held fixed when the distance is formed, and
the semigroup has rate one per site.
\end{proof}

\section{Proof of the discrete-time corollary}\label{u:sec:proof-discrete}

We use the retained set and the Poincar\'e inequality from the proof of
\cref{u:lem:pruning}. What changes is the entropy dissipation
inequality, which we prove for one step of the discrete heat-bath chain. The subsequent averaging argument
is that of \cite[Section~7, Lemma~8]{oai-typical-upper}.

\begin{proof}[Proof of \cref{cor:upper-discrete}]
Fix $\beta>1$ and $0<\omega<1/4$, and choose the same exponents as
in the proof of \cref{thm:upper}. We first fix a disorder $g$
outside the exceptional events of that proof. We abbreviate
$\pi=\pi_g$ and $K=K^{J(g)}$. For a function $u$ on $\Sigma_n$, define
\[
 Q_i u(x)=\sum_{\sigma\in\{-1,1\}}
                  p_i^{J(g)}(\sigma\mid x)u(x^{i,\sigma}).
\]
Each $Q_i$ is conditional expectation given all spins except the
$i$th, and $K=n^{-1}\sum_iQ_i$. In $L^2(\pi)$, the operators $Q_i$
are orthogonal projections. By \eqref{u:eq:pruning-dirichlet},
\begin{equation}\label{u:eq:discrete-dirichlet}
 \mathfrak D(u)=\sum_{i=1}^n\langle u,(I-Q_i)u\rangle_\pi.
\end{equation}

Let $f\ge0$ satisfy $\pi(f)=1$. The convexity of $s\log s$ implies that
$\Ent_\pi(Kf)\le n^{-1}\sum_i\Ent_\pi(Q_i f)$.
We claim that
\[
 \Ent_\pi(f)-\Ent_\pi(Q_i f)
 \ge\langle\sqrt f,(I-Q_i)\sqrt f\rangle_\pi.
\]
Indeed, on each fiber obtained by fixing all spins except the $i$th,
write $a=Q_i f$. For $s\ge0$ and $a>0$,
\[
 s\log(s/a)-s+a\ge(\sqrt s-\sqrt a)^2.
\]
This follows by writing the difference as
$2a\sqrt{s/a}(\sqrt{s/a}\log\sqrt{s/a}-\sqrt{s/a}+1)\ge0$,
with its continuous interpretation at $s=0$. Averaging on the fiber
eliminates $-s+a$. The resulting squared error is at least the squared
error about the fiber mean $Q_i\sqrt f$. Fibers with $a=0$ have $f=0$
throughout and contribute zero. Averaging over the other spins proves
the claim. Combining it with \eqref{u:eq:discrete-dirichlet}, we have
\begin{equation}\label{u:eq:discrete-entropy-step}
 \Ent_\pi(f)-\Ent_\pi(Kf)
 \ge\frac1n\mathfrak D(\sqrt f).
\end{equation}

For $f_k^x(y)=K^k(x,y)/\pi(y)$, reversibility gives
$f_{k+1}^x=Kf_k^x$. Telescoping \eqref{u:eq:discrete-entropy-step}
and averaging over $x\sim\pi$, we obtain, for every integer $m\ge1$,
\begin{equation}\label{u:eq:discrete-dissipation}
 \sum_{k=0}^{m-1}\sum_x\pi(x)\mathfrak D(\sqrt{f_k^x})
 \le nH(\pi)\le n^2\log2.
\end{equation}
Since the disorder lies outside the exceptional events, the hypothesis
on balanced cuts holds with $\phi=\phi_n$ and $\epsilon=\epsilon_n$, and
it gives the same set $C$ and inequality
\eqref{u:eq:retained-poincare}. Then there exists
$k\in\{0,\ldots,m-1\}$ such that
\[
 \sum_x\pi(x)\sum_{y\in C}\pi(y)
       (\sqrt{f_k^x(y)}-a_x)^2
 \le\frac{2n^2H(\pi)}{m\phi_n^2},\qquad
 a_x=\frac1{\pi(C)}\sum_{y\in C}\pi(y)\sqrt{f_k^x(y)}.
\]
Using \eqref{u:eq:pruning-distance}, stationarity, and
Cauchy--Schwarz, we find
\[
 \sum_x\pi(x)d^{J(g)}_{\mathrm{disc}}(x,k)
 \le\frac{2n\sqrt{2H(\pi)}}{\phi_n\sqrt m}+2\epsilon_n.
\]
Using that the distance contracts under $K^{m-k}$, we find
\begin{equation}\label{u:eq:discrete-averaged-mixing}
 \sum_x\pi_g(x)d^{J(g)}_{\mathrm{disc}}(x,m)
 \le\frac{2\sqrt{2\log2}\,n^{3/2}e^{n^{1-\ex q}}}{\sqrt m}
       +2n^{-\ex d}.
\end{equation}
Let $m=\lfloor e^{n^{1-\omega}}\rfloor$. For all sufficiently large
$n$, we have $m\ge e^{n^{1-\omega}}/2$. The first term in
\eqref{u:eq:discrete-averaged-mixing} is at most
\[
 Cn^{3/2}\exp\{n^{1-\ex q}-\tfrac12n^{1-\omega}\},
\]
which tends to zero because $\omega<\ex q$. By Markov's inequality in
the Gibbs draw, we obtain
\begin{equation}\label{u:eq:discrete-bad-start}
 \begin{split}
 &\Pi_n\{(g,x):d^{J(g)}_{\mathrm{disc}}(x,m)>1/4\}\\
 &\quad\le\gamma_n(\mathcal W_n(\epsilon_n,\phi_n))+2e^{-n}
       +Cn^{3/2}e^{n^{1-\ex q}-\frac12n^{1-\omega}}
       +8n^{-\ex d}=o(1).
 \end{split}
\end{equation}
Using the marginal identity for $\Pi_n$, we conclude that \cref{cor:upper-discrete} holds.
\end{proof}

\section{Proof of the two-sided corollary}\label{u:sec:proof-two-sided}

The two-sided statement uses the lower bound at threshold $1/4$ and
the two upper bounds just proved.

\begin{proof}[Proof of \cref{cor:two-sided}]
Fix $\beta>1$ and $0<\omega<1/4$, and let $A_\beta$ be the coefficient
in \cref{thm:barrier-730}. With $\tau=1/4$, that theorem implies
\[
 \PP_{J,X_0\sim\pi^J}
   \{d^J(X_0,e^{A_\beta n^{7/30}})>1/4\}\longrightarrow1.
\]
By the definition of the mixing time in \eqref{eq:model-mixing-time} and
\cref{lem:model-dynamics}, this event is
$\{T^J(X_0)>e^{A_\beta n^{7/30}}\}$. Intersecting it with the event
of probability tending to one in \cref{thm:upper} proves the
continuous-time assertion.

For the discrete chain, by \eqref{eq:barrier-730-discrete} at $\tau=1/4$,
we have, with probability tending to one,
\[
 d^J_{\mathrm{disc}}(X_0,\lfloor e^{A_\beta n^{7/30}}\rfloor)>1/4.
\]
Since $k\mapsto d^J_{\mathrm{disc}}(X_0,k)$ is nonincreasing, this
implies $T^J_{\mathrm{disc}}(X_0)>\lfloor e^{A_\beta n^{7/30}}\rfloor$.
On the event in \cref{cor:upper-discrete}, we also have
$T^J_{\mathrm{disc}}(X_0)\le\lfloor e^{n^{1-\omega}}\rfloor$.
The probability of their intersection tends to one, as desired.
\end{proof}

\section{The parameter system}\label{u:sec:budget}

We verify all comparisons used in the preceding chapters. Fix
$0<\omega<1/4$ and define
$\varepsilon_\omega=(1/4-\omega)/20$. Then
$0<\varepsilon_\omega<1/80$ and
$\omega=1/4-20\varepsilon_\omega$.
The affine family from \cref{u:sec:parameters} is
\begin{equation}\label{u:eq:parameter-family}
 \begin{aligned}
 \ex a&=\tfrac12,&
 \ex f&=\tfrac12,&
 \ex k_{\rm f}&=\tfrac12-2\varepsilon_\omega,\\
 \ex D&=\tfrac14-\varepsilon_\omega,&
 \ex u_{\rm E}&=\tfrac14-2\varepsilon_\omega,&
 \ex d&=\tfrac12\varepsilon_\omega,\\
 \ex e&=\tfrac14-3\varepsilon_\omega,&
 \ex b&=\tfrac12-14\varepsilon_\omega,&
 \ex z&=\tfrac14-8\varepsilon_\omega,\\
 \ex w&=\tfrac18-3\varepsilon_\omega,&
 \ex k_{\rm b}&=\tfrac14-9\varepsilon_\omega,&
 \ex s_{\rm b}&=\tfrac14-5\varepsilon_\omega,\\
 \ex s_{\rm seq}&=\tfrac14-4\varepsilon_\omega,&
 \ex r&=\tfrac14-\tfrac{15}2\varepsilon_\omega,&
 \ex u_{\rm F}&=\tfrac14-\tfrac{39}4\varepsilon_\omega,\\
 \ex q&=\tfrac14-10\varepsilon_\omega,&
 \ex z_{\rm P}&=\tfrac3{10},&
 \ex g_{\rm rot}&=\tfrac35.
 \end{aligned}
\end{equation}
An optional intermediate loss $\ex s_{\rm F}$ can be chosen to satisfy
$2\ex u_{\rm F}-\ex r<\ex s_{\rm F}<\ex q$. We leave it unspecified;
only the strict comparison $2\ex u_{\rm F}-\ex r<\ex q$ enters the
probability estimates.

The sufficient system consists of the positivity of the exponents and
\begin{equation}\label{u:eq:comparison-system}
 \begin{gathered}
  \ex a>\ex f/4,\qquad \ex a>\ex b/4,\qquad
  \ex a+\ex f/2<1,\\
  \ex k_{\rm f}<\min(\ex a,\ex f,1-\ex a),\qquad \ex b<\ex f,\\
  0<\ex k_{\rm b}<\ex z<\min(\ex a,\ex b/2,2\ex w),\\
  \max\{2\ex w,\ex w+\ex b/4,\ex b/2,
          2\ex z-\ex a,2\ex z-2\ex w,2\ex z-\ex b\}<\ex s_{\rm b},\\
  \ex a<1-\ex s_{\rm b},\qquad
  \ex e+\ex d<\ex u_{\rm E}<\min(\ex D,\ex k_{\rm f}-\ex D),\\
  \ex q<\ex u_{\rm F}<\ex r<\ex b/2,\qquad
  2\ex u_{\rm F}-\ex r<\ex q,\qquad \ex q<1/2,\\
  \ex q<\min(\ex k_{\rm b},\ex D,\ex z_{\rm P}),\qquad
  0<\ex z_{\rm P}<1/2,\\
  \ex q<\ex s_{\rm b},\qquad \ex r<\ex s_{\rm b},\qquad
  \ex a<\ex g_{\rm rot}<1-\ex s_{\rm b},\\
  \ex s_{\rm b}<\ex s_{\rm seq}<\ex e,\qquad \omega<\ex q.
 \end{gathered}
\end{equation}
All exponents are below one whenever a growing cutoff or a tail
estimate requires this. The scalar, Gram, concentration, regression,
and finite-chain arguments show that these comparisons suffice.

We list the powers used in those arguments in
\cref{u:tab:budget}. A rate is the power in a negative logarithm.
Polylogarithmic denominators in rates are retained until a strict
comparison absorbs them. Query counts and expectation bounds may
have fixed polylogarithmic numerators. The last column gives the
instance $\omega=1/5$, $\varepsilon_\omega=1/400$.

\begingroup
\small
\setlength{\tabcolsep}{4pt}
\begin{longtable}{@{}>{\raggedright\arraybackslash}p{.43\textwidth}ll@{}}
\caption{Powers in the parameter system.}\label{u:tab:budget}\\
\toprule
Quantity & Family power & $\omega=1/5$\\
\midrule
\endfirsthead
\toprule
Quantity & Family power & $\omega=1/5$\\
\midrule
\endhead
\bottomrule
\endfoot
Query count $N_{\ex f},N_{\ex b}$ & $1/2$ & $1/2$\\
Fine maximum-cell saving & $3/8$ & $3/8$\\
Coarse maximum-cell saving & $3/8+7\varepsilon_\omega/2$ & $307/800$\\
Fine $nh$, $ne^{-V_{\ex f}}$, $N_{\ex f}(1+V_{\ex f})$ & $1/2$ & $1/2$\\
Fine $\sqrt{N_{\ex f}}$, $N_{\ex f}^2V_{\ex f}R_{V_{\ex f}}/n$ & $1/4$ & $1/4$\\
Fine KL cutoff $1-\ex k_{\rm f}$ & $1/2+2\varepsilon_\omega$ & $101/200$\\
Weighted variance loss $2\ex w$ & $1/4-6\varepsilon_\omega$ & $47/200$\\
Weighted scale loss $\ex w+\ex b/4$ & $1/4-13\varepsilon_\omega/2$ & $187/800$\\
Fixed Gram loss $\ex b/2$ & $1/4-7\varepsilon_\omega$ & $93/400$\\
Weighted raw rate $1-2\ex w$ & $3/4+6\varepsilon_\omega$ & $153/200$\\
Weighted scale raw rate $1-\ex w-\ex b/4$ & $3/4+13\varepsilon_\omega/2$ & $613/800$\\
Fixed Gram raw rate $1-\ex b/2$ & $3/4+7\varepsilon_\omega$ & $307/400$\\
Joint lag / diagonal base rate $1+\ex a-2\ex z$ & $1+16\varepsilon_\omega$ & $26/25$\\
Gaussian coefficient rate $1+2\ex w-2\ex z$ & $3/4+10\varepsilon_\omega$ & $31/40$\\
Gaussian terminal rate $1+\ex b-2\ex z$ & $1+2\varepsilon_\omega$ & $201/200$\\
Coarse one-run rate $1-\ex s_{\rm b}$ & $3/4+5\varepsilon_\omega$ & $61/80$\\
Weighted quadratic energy saving $2\ex w$ & $1/4-6\varepsilon_\omega$ & $47/200$\\
Coarse log-density cost $1-\ex k_{\rm b}$ & $3/4+9\varepsilon_\omega$ & $309/400$\\
Both expected reverse-KL costs & $3/4+\varepsilon_\omega$ & $301/400$\\
Outer reverse-KL cutoff $1-\ex u_{\rm E}$ & $3/4+2\varepsilon_\omega$ & $151/200$\\
Both outer Markov decay savings & $\varepsilon_\omega$ & $1/400$\\
Raw crossing exponent $1-\ex u_{\rm E}+\ex d$ & $3/4+5\varepsilon_\omega/2$ & $121/160$\\
Crossing envelope $1-\ex e$ & $3/4+3\varepsilon_\omega$ & $303/400$\\
Total sequence failure rate $1-\ex s_{\rm seq}$ & $3/4+4\varepsilon_\omega$ & $19/25$\\
Pinning/radius power $1-\ex r$ & $3/4+15\varepsilon_\omega/2$ & $123/160$\\
Flow / good-output hit rate $1-\ex q$ & $3/4+10\varepsilon_\omega$ & $31/40$\\
Sparse-field threshold $1-\ex u_{\rm F}$ & $3/4+39\varepsilon_\omega/4$ & $1239/1600$\\
Hamming-ball entropy power $1-\ex r$ & $3/4+15\varepsilon_\omega/2$ & $123/160$\\
Bad-field rate $1+\ex r-2\ex u_{\rm F}$ & $3/4+12\varepsilon_\omega$ & $39/50$\\
Intermediate bad-field rate (optional) & see text & $(31/40,39/50)$\\
Target pressure cutoff / rate & $7/10,\ 2/5$ & $7/10,\ 2/5$\\
Regression log-cost $1-\ex D$ & $3/4+\varepsilon_\omega$ & $301/400$\\
Rotation logarithmic grid size & $3/5$ & $3/5$\\
Physical time exponent $1-\omega$ & $3/4+20\varepsilon_\omega$ & $4/5$\\
\end{longtable}
\endgroup
In the optional row, the last entry is the interval of allowed values.
In the family, that interval is
$(3/4+10\varepsilon_\omega,3/4+12\varepsilon_\omega)$.

For example, $\max\{2\ex w,\ex w+\ex b/4\}=2\ex w$, since the
difference of the two entries is $\varepsilon_\omega/2$. The weighted
variance rate and the fixed-Gram rate exceed the one-run rate by
$\varepsilon_\omega$ and $2\varepsilon_\omega$, respectively. Each of
these rates exceeds the net exponent $\ex a=1/2$ with a fixed positive
gap. We first establish the coarse event with rate $1-\ex s_{\rm b}$,
then take a union over the rotation grid. Since
$\ex s_{\rm seq}-\ex s_{\rm b}=\varepsilon_\omega$ and
$\ex e-\ex s_{\rm seq}=\varepsilon_\omega$, the grid union and the
comparison with the rare crossing each retain a strict gap. The
target pressure failure, whose rate is $2/5$, is charged once outside
that comparison.

The parameter instance at $\omega=1/5$ is
\begin{equation}\label{u:eq:parameter-instance}
 \begin{aligned}
 \omega&=\tfrac15,& \ex q&=\tfrac9{40},&
 \ex a&=\tfrac12,& \ex f&=\tfrac12,\\
 \ex k_{\rm f}&=\tfrac{99}{200},& \ex D&=\tfrac{99}{400},&
 \ex u_{\rm E}&=\tfrac{49}{200},& \ex d&=\tfrac1{800},\\
 \ex e&=\tfrac{97}{400},& \ex b&=\tfrac{93}{200},&
 \ex z&=\tfrac{23}{100},& \ex w&=\tfrac{47}{400},\\
 \ex k_{\rm b}&=\tfrac{91}{400},& \ex s_{\rm b}&=\tfrac{19}{80},&
 \ex s_{\rm seq}&=\tfrac6{25},& \ex r&=\tfrac{37}{160},\\
 \ex u_{\rm F}&=\tfrac{361}{1600},& \ex z_{\rm P}&=\tfrac3{10},&
 \ex g_{\rm rot}&=\tfrac35.
 \end{aligned}
\end{equation}
Some of the tight comparisons are listed below; the slack is the
right side minus the left side.
\begin{equation}\label{u:eq:representative-slacks}
\begin{array}{@{}lcc@{}}
\toprule
\text{Comparison}&\text{Family slack}&\omega=1/5\\
\midrule
 \ex k_{\rm f}<\ex a&2\varepsilon_\omega&1/200\\
 \ex u_{\rm E}<\ex D,\ \ex u_{\rm E}<\ex k_{\rm f}-\ex D
     &\varepsilon_\omega&1/400\\
 \ex e+\ex d<\ex u_{\rm E}&\varepsilon_\omega/2&1/800\\
 \ex z<\ex b/2&\varepsilon_\omega&1/400\\
 \ex z<2\ex w&2\varepsilon_\omega&1/200\\
 2\ex w<\ex s_{\rm b}&\varepsilon_\omega&1/400\\
 \ex w+\ex b/4<\ex s_{\rm b}&3\varepsilon_\omega/2&3/800\\
 \ex b/2<\ex s_{\rm b}&2\varepsilon_\omega&1/200\\
 2\ex z-2\ex w<\ex s_{\rm b}&5\varepsilon_\omega&1/80\\
 \ex q<\ex k_{\rm b}&\varepsilon_\omega&1/400\\
 \ex q<\ex u_{\rm F}&\varepsilon_\omega/4&1/1600\\
 \ex u_{\rm F}<\ex r&9\varepsilon_\omega/4&9/1600\\
 \ex r<\ex b/2&\varepsilon_\omega/2&1/800\\
 2\ex u_{\rm F}-\ex r<\ex q&2\varepsilon_\omega&1/200\\
 \ex s_{\rm b}<\ex s_{\rm seq},\ \ex s_{\rm seq}<\ex e
     &\varepsilon_\omega&1/400\\
 \omega<\ex q&10\varepsilon_\omega&1/40\\
\bottomrule
\end{array}
\end{equation}
The least slack in the full comparison list at this instance is
$1/1600$. The witness estimate specializes to
\begin{equation}\label{u:eq:witness-instance}
 \gamma_n(\mathcal W_n(\epsilon_n,\phi_n))
 \le C_\beta n^{-1/400}+e^{-c_\beta n^{2/5}}
       +\exp\{2n^{303/400}-n^{19/25}\}=o(1).
\end{equation}
The bound \eqref{u:eq:joint-bad-start} further adds the norm exception
$2e^{-n}$, the term
$4n\sqrt{2\log2}\exp\{n^{31/40}-n^{4/5}/2\}$, and
$8n^{-1/800}$.

\begin{proposition}[Every fixed saving below one quarter]\label{u:prop:range}
The family \eqref{u:eq:parameter-family} satisfies every sufficient
comparison in \eqref{u:eq:comparison-system} for every
$0<\varepsilon_\omega<1/80$. Hence \cref{thm:upper} holds for every
fixed $0<\omega<1/4$. These estimates do not reach $\omega=1/4$.
\end{proposition}

\begin{proof}
All parameters and all differences between the right and left sides
of \eqref{u:eq:comparison-system} are affine in $\varepsilon_\omega$,
after expanding each maximum and minimum into its constituent
comparisons. Using \eqref{u:eq:parameter-family}, we find that all endpoint values
at $0$ and $1/80$ are nonnegative and that no difference is identically
zero. Each interior value is a strict convex combination of these two
endpoint values, at least one of which is positive. Every comparison
is therefore strict. This also proves that
$\omega=1/4-20\varepsilon_\omega$ is positive; zero values at the excluded
endpoints are allowed. The optional $\ex s_{\rm F}$ can be chosen
between $2\ex u_{\rm F}-\ex r$ and $\ex q$, whose difference is
$2\varepsilon_\omega$.

In particular, $\ex q-\omega=10\varepsilon_\omega>0$.
By \cref{u:prop:witness} and \eqref{u:eq:joint-bad-start}, we conclude
that the asserted range holds. All constants
are chosen after $\omega$ and $\beta$ are fixed.

We now explain the obstruction in these estimates. Fine entropy requires
$\ex k_{\rm f}<\min(\ex a,1-\ex a)\le1/2$.
Reverse smoothing and coverage require
\[
 \ex e+\ex d<\ex u_{\rm E}
 <\min(\ex D,\ex k_{\rm f}-\ex D)
 \le\ex k_{\rm f}/2<1/4.
\]
The connecting sequence needs
$\ex q<\ex s_{\rm b}<\ex s_{\rm seq}<\ex e$, and mixing needs
$\omega<\ex q$. For any parameters satisfying these strict estimates,
we have $\omega<1/4$, independently of the affine family.
At $\varepsilon_\omega=0$, the fine, coverage, concentration, and
crossing gaps vanish. Constants and logarithms cannot be absorbed by
the arguments above. This is an obstruction to these estimates; it
does not give a lower bound or an optimality statement for the SK
dynamics.
\end{proof}

A calculation in exact rational arithmetic checks the comparisons; it
lists each affine slack and its values at $0$, $1/80$, and $1/400$.
With an optional intermediate loss chosen inside the stated interval,
the calculation checks 114 strict comparisons and 37 identities for
the powers in \cref{u:tab:budget}. The proof of \cref{u:prop:range}
does not rely on this calculation: the endpoint argument there gives
the comparisons on the whole open interval.
\part{Fluctuations and mean bias}\label{part:fluct}
\chapter{The fluctuation scale}\label{sec:fluct}

This section concerns \cref{f:main}, the fluctuation theorem of OpenAI \cite[Theorem~1.1]{oai-fluctuations}. Its proof in \cite{oai-fluctuations} has two parts of very different lengths. The main part, which occupies Sections~2--7.4 of that paper, is a two-sided bound on a positive logarithmic moment of the completed log partition function at a tilt slightly larger than $n^{-1/6}$. We state this bound in our normalization as \cref{f:window}, and in \cref{sec:fl-route} we summarize how it is proved in \cite{oai-fluctuations}; we do not reprove it. The second part converts \cref{f:window} into the three assertions of \cref{f:main}. It rests on an exponential augmentation argument of Aronow and Lopatto \cite[Proposition~5.1]{quantitative-parisi}, in the form given in \cite[Lemmas~7.6 and~7.7 and Section~7.6]{oai-fluctuations}. We give complete proofs of this part: the log-concavity lemma (\cref{f:logconcave}) and elementary bounds for one-dimensional log-concave laws (\cref{f:one-dim}) in \cref{sec:fl-logconcave}, and the deduction of \cref{f:main} from \cref{f:window} in \cref{sec:fl-endgame}. \Cref{sec:bias} uses the lower bound on the variance in \cref{f:main}.

As recalled in \cref{sec:intro}, OpenAI's companion preprint \cite{oai-limiting-law} claims more than \cref{f:main}: a positive limit of $n^{-1/3}\Var(F_n)$ and a nondegenerate limit law for $(F_n-\E F_n)/\Var(F_n)^{1/2}$ \cite[Theorems~1.1 and~1.2]{oai-limiting-law}. We do not use these claims; their consequence for the mean bias is recalled in \cref{rem:bias-prefactor}.

Throughout this section, $\beta>1$ is fixed, and constants may depend on $\beta$. Recall from \cref{sec:model} that the completed log partition function is $F_n^c=F_n+\beta G$, where $G$ is a centered Gaussian variable of variance $1/2$, independent of the disorder $J$. For $t\in\R$, we define the logarithmic moment generating function of the centered variable $F_n^c-\E F_n^c$,
\begin{equation}\label{eq:fl-transform}
 K_n(t)=\log\E\exp\bigl\{t\bigl(F_n^c-\E F_n^c\bigr)\bigr\}.
\end{equation}
By \cref{lem:model-completion}, $K_n(t)$ is finite for every $t\in\R$, $F_n^c$ has mean $\E F_n$ and variance $\Var(F_n)+\beta^2/2$, and $K_n(t)=\log\E e^{t(F_n-\E F_n)}+\beta^2t^2/4$ by \eqref{eq:model-completion-laplace}. The proof of that lemma also shows that $F_n$ has finite exponential moments, so all its moments are finite.

\section{The two-sided bound on positive logarithmic moments}\label{sec:fl-window}

\begin{proposition}[Two-sided bound on positive logarithmic moments]\label{f:window}
Fix $\beta>1$ and $0<m<1/300$, and set $s_n=n^{-1/6+5m}$. There exist constants $c,C>0$ and $n_0\in\N$, depending only on $\beta$ and $m$, such that
\begin{equation}\label{f:window-eq}
 c\,n^{20m}\le K_n(s_n)\le C\,n^{100m}\qquad(n\ge n_0).
\end{equation}
\end{proposition}

This is \cite[Proposition~7.5]{oai-fluctuations}. We call it the two-sided bound on positive logarithmic moments; in \cite{oai-fluctuations} it is called a positive Laplace window. The normalization of \cite{oai-fluctuations} differs from ours in two ways, and neither affects \eqref{f:window-eq}. First, the Hamiltonian there is $\beta H_n$ and the partition function carries the factor $2^{-n}$, so that the log partition function there is $F_n-n\log 2$. Second, the completion there adds a common centered Gaussian variable $D$ of variance $\beta^2/2$, which has the law of $\beta G$. The completed log partition function of \cite{oai-fluctuations} therefore has the law of $F_n^c-n\log2$, the centered logarithmic moment in \cite[Proposition~7.5]{oai-fluctuations} is $K_n$, and the tilt $s=n^{-1/6+5m}$, the exponents $20m$ and $100m$, and the range $0<m<1/300$ are the same.

We use only two features of \eqref{f:window-eq} below: for each fixed small $m$, the lower bound tends to infinity, and the upper bound grows at most like a fixed small power of $n$. The particular exponents $20m$ and $100m$ play no further role.

\section{OpenAI's proof of the two-sided bound}\label{sec:fl-route}

We now summarize the proof of \cref{f:window} in \cite{oai-fluctuations}. The summary is a description of that proof, not a proof, and no other part of this paper depends on it. We follow the notation of \cite{oai-fluctuations} where possible, writing $\bar u$, $\widehat\lambda$, and $\widehat h$ for their mass $u$, strength $\lambda$, and field covariance path $h$. Throughout, $m\in(0,1/300)$ is fixed and $\delta=m/10000$, as in \cite[Section~3.1]{oai-fluctuations}, and we abbreviate $s=s_n$.

\paragraph{The two-level pressure.} Let $H^{(0)}$ and $H^{(1)}$ be independent copies of the completed Hamiltonian $H_n^c$, that is, independent centered Gaussian processes indexed by $\Sigma_n$, each with covariance $nR(x,y)^2/2$. For $0\le\eps\le\beta^2$, we set
\begin{equation}\label{eq:fl-two-level}
 \Psi_n(\eps)=\frac{1}{ns}\,\E_{(0)}\log\E_{(1)}\exp\Bigl\{s\log\sum_{x\in\Sigma_n}
 e^{\sqrt{\beta^2-\eps}\,H^{(0)}(x)+\sqrt\eps\,H^{(1)}(x)}\Bigr\},
\end{equation}
where $\E_{(i)}$ is the expectation over $H^{(i)}$. At $\eps=0$ the inner expectation is trivial, and $\Psi_n(0)=n^{-1}\E F_n^c$. At $\eps=\beta^2$ the outer expectation is trivial, and $\Psi_n(\beta^2)=(ns)^{-1}\log\E e^{sF_n^c}$. We conclude that
\begin{equation}\label{eq:fl-two-level-endpoints}
 K_n(s)=ns\bigl(\Psi_n(\beta^2)-\Psi_n(0)\bigr).
\end{equation}
Because $ns=n^{5/6+5m}$, the estimate \eqref{f:window-eq} is equivalent to
\begin{equation}\label{eq:fl-pressure-window}
 c\,n^{-5/6+15m}\le\Psi_n(\beta^2)-\Psi_n(0)\le C\,n^{-5/6+95m}.
\end{equation}

\paragraph{Gaussian hierarchies.} To prove \eqref{eq:fl-pressure-window}, OpenAI compares $\Psi_n$ with the pressures of Gaussian hierarchies \cite[Section~2.1]{oai-fluctuations}. Let $0=m_0<m_1<\dots<m_{L+1}=1$, and let $b$ and ${\widehat h}$ be nonnegative nondecreasing functions of the \emph{mass} ${\bar u}\in[0,1)$ that are constant on each interval $[m_j,m_{j+1})$. We call $b$ and ${\widehat h}$ the matrix and the field covariance paths. Their increments at $m_j$ define independent centered Gaussian processes on $\Sigma_n$, with covariances $(b(m_j)-b(m_j-))\,nR(x,y)^2/2$ and $({\widehat h}(m_j)-{\widehat h}(m_j-))\,nR(x,y)$, where $b(0-)={\widehat h}(0-)=0$. Together they form the stage at mass $m_j$. Let $\Xi$ be the sum of all stages. Starting from $\log(2^{-n}\sum_xe^{\Xi(x)})$, we integrate the stages in the order $j=L,\dots,0$. The stage at mass $m_j$ is integrated by the Gaussian integration step $X\mapsto m_j^{-1}\log\E_je^{m_jX}$ of \cref{def:inp-step}, where $\E_j$ is the expectation over that stage, and by $X\mapsto\E_jX$ if $m_j=0$. The pressure $f_n(b,{\widehat h})$ is the resulting number divided by $n$, with $b(1-)/4$ and ${\widehat h}(1-)/2$ subtracted. It extends to bounded nondecreasing paths by approximation \cite[Proposition~2.14]{oai-fluctuations}. The associated \emph{path law} is the law of the stages and of a configuration $x$ obtained as follows. Given the earlier stages, the stage at mass $m_j$ is drawn from the tilted Gaussian kernel \eqref{eq:inp-kernel} of this recursion; then $x$ is drawn from the Gibbs measure $e^{\Xi(x)}/\sum_ye^{\Xi(y)}$. For ${\bar u}\in[0,1)$, let $x^1$ and $x^2$ be two configurations that share the stages up to mass ${\bar u}$ and are conditionally independent given them. We set $S_{\bar u}=\E R(x^1,x^2)$, $B_{\bar u}=\E R(x^1,x^2)^2$, and $D_{\bar u}^2=B_{\bar u}-S_{\bar u}^2$ \cite[Lemma~2.1]{oai-fluctuations}. By \cite[Lemma~2.2]{oai-fluctuations}, a variation of the paths changes $f_n$ by
\[
 \dd f_n=-\frac14\int_0^1B_{\bar u}\,\dd b({\bar u})\,\dd {\bar u}-\frac12\int_0^1S_{\bar u}\,\dd {\widehat h}({\bar u})\,\dd {\bar u}.
\]
Such hierarchies descend from the Parisi recursion and Guerra's interpolation \cite{parisi-1979,guerra-2003}. The path law and the variance bound with deterministic weights, on which their moment estimates rest, are from \cite[Section~2 and Proposition~2.2]{quantitative-parisi}. The other basic tools are moment bounds under the path law \cite[Lemma~2.3]{oai-fluctuations} and a differentiation formula for expectations over finite trees of replicas \cite[Proposition~2.5]{oai-fluctuations}.

In this notation, $\Psi_n(\eps)=f_n(\beta^2-\eps\mathbf 1_{[0,s)},0)+\beta^2/4+\log2$: the matrix covariance $\beta^2-\eps$ is integrated at mass zero, and the remaining covariance $\eps$ at mass $s$ \cite[Section~7, opening]{oai-fluctuations}. By the variation formula, the derivative of $\Psi_n$ is $\frac14\int_0^sB_{\bar u}\,\dd {\bar u}\in[0,s/4]$ at each point of differentiability, and $0\le\Psi_n(\eps')-\Psi_n(\eps)\le s(\eps'-\eps)/4$ for $\eps\le\eps'$. The function $\Psi_n$ is nondecreasing.

\paragraph{Comparisons.} A comparison \cite[Section~3.1, (3)--(6)]{oai-fluctuations} has parameters $d$ and $E$ with $n^{-2/3+m}d\lesssim E\le n^{-1/2}d^2$. We write $\underline x=\sqrt nE/d$; this is the smallest scale at which the profile of its minimizers is examined. The comparison acts on a mass interval $[{\bar u}_0,1]$, where ${\bar u}_0=0$ or ${\bar u}_0=o(\underline x)$. It starts from unperturbed paths $(b_0,{\widehat h}_0)$ that are bounded and nondecreasing on $[{\bar u}_0,1]$ and satisfy $b_0({\bar u})=\beta^2+o({\bar u}^2)$ and ${\widehat h}_0({\bar u})=o({\bar u}^3)$ for ${\bar u}\gtrsim\underline x$ \cite[(4)]{oai-fluctuations}. Let $f$ be a fixed smooth positive function that equals $1$ near $0$ and is comparable to $(1+\log t)/t$ for large $t$, with logarithmic slope $-tf'(t)/f(t)$ nondecreasing from $0$ to $1$. We set $e(r)={\widehat\lambda} f({\widehat\lambda} r/d^3)$, where $0\le{\widehat\lambda}\le d^2$ is the \emph{strength}. For a nondecreasing overlap quantile $r$ on $[{\bar u}_0,1]$, the perturbed paths are $b_0-e(r)$ and ${\widehat h}_0+e(r)r$ on $[{\bar u}_0,1]$, and $b_0$ and ${\widehat h}_0$ below ${\bar u}_0$; here $e(r)$ is the function ${\bar u}\mapsto e(r({\bar u}))$. The comparison minimizes
\[
 \mathcal F_{\widehat\lambda}(r)=f_n\bigl(b_0-e(r),{\widehat h}_0+e(r)r\bigr)+\frac14\int_{{\bar u}_0}^1e(r({\bar u}))\,r({\bar u})^2\,\dd {\bar u}
\]
over a compact class of quantiles \cite[(5)]{oai-fluctuations}. This class is defined by two-sided bounds on the derivative of the inverse function of $r$ and by a prescribed initial value $r({\bar u}_0)=r_{\min}\ge0$, which we call the lower radius; $r_{\min}=0$ when ${\bar u}_0=0$. The minimum $\mathcal M_{\widehat\lambda}$ is attained. The comparison is \emph{admissible} at strength ${\widehat\lambda}$ if the perturbed paths are nonnegative and nondecreasing on $[0,1)$ for every quantile in the class. For a comparison admissible at every ${\widehat\lambda}\in[0,d^2]$, including ${\widehat\lambda}=0$, where the perturbed paths are $(b_0,{\widehat h}_0)$, we have $\mathcal M_0=f_n(b_0,{\widehat h}_0)$, and $\mathcal M_{\widehat\lambda}$ is locally Lipschitz in ${\widehat\lambda}$. At almost every ${\widehat\lambda}>0$, its derivative in $\log{\widehat\lambda}$ is at most
\[
 \frac14\int_{{\bar u}_0}^1e(r({\bar u}))\bigl[D_{\bar u}^2+(S_{\bar u}-r({\bar u}))^2\bigr]\,\dd {\bar u}
\]
for every minimizer $r$ at strength ${\widehat\lambda}$, where the moments are those of the perturbed paths \cite[Lemma~3.1]{oai-fluctuations}. Now let ${\widehat\lambda}\ge n^{-10}$, and suppose that a minimizer at strength ${\widehat\lambda}$ satisfies the \emph{two-sided linear bound} $c\bar u\le r(\bar u)\le C\bar u$, with fixed constants $c,C>0$, for all masses ${\bar u}$ above $A_{\widehat\lambda}=n^\delta\sqrt nE/\sqrt{\widehat\lambda}$; this assumption is not needed when $A_{\widehat\lambda}$ stays bounded below by a fixed positive constant. The last integral is then at most $n^{10\delta}E$. If the comparison is admissible at every strength and this holds at every strength ${\widehat\lambda}\ge n^{-10}$, integration in $\log{\widehat\lambda}$ gives the comparison bound
\begin{equation}\label{eq:fl-comparison-bound}
 0\le\mathcal M_{d^2}-f_n(b_0,{\widehat h}_0)\le n^{20\delta}E
\end{equation}
\cite[Propositions~3.9 and~3.10]{oai-fluctuations}. The next two steps establish the two-sided linear bound for the minimizers.

\paragraph{The one-site cavity estimate and the linear profile.} Consider a minimizer at a positive strength and a scale $x\to0$. Suppose that the minimizer satisfies the two-sided linear bound above a fixed multiple of $x$, and that $x\gtrsim\underline x$, $e(x)x^2\gtrsim nE^2$, and $e(x)\gtrsim x^2$ \cite[Section~4, opening]{oai-fluctuations}. The last condition says that the strength is comparable to the scale. We write $K={\widehat h}+bS$, where $b$ and ${\widehat h}$ are the perturbed paths and $S$ is the function ${\bar u}\mapsto S_{\bar u}$. Let $\Gamma_K({\bar u})$ be the second moment, under the path law, of the conditional mean of the spin given the stages up to mass ${\bar u}$, in the hierarchy with $n=1$, field covariance path $K$, and no matrix covariance. A one-site cavity estimate \cite[Proposition~4.1]{oai-fluctuations} shows that a mass ${\bar u}$ with $r({\bar u})\le Cx$, $cx\le {\bar u}\le Cx$, and $D_{\bar u}=o(x)$ satisfies $S_{\bar u}=\Gamma_K({\bar u})+o(x^3)$. When only $D_{\bar u}=O(x)$ is known, the remainder is $O(x^3)$, uniformly over such masses. The proposition also gives integrated estimates with remainder $o(x^3)$ over these ranges, in mass and in overlap. Its proof removes one spin and controls the resulting susceptibility correction through identities for a fixed number of additional replicas \cite[Sections~4.1--4.5]{oai-fluctuations}. Using this estimate and the first variation of the comparison, OpenAI proves that, under the same hypotheses, along a subsequence
\[
 \frac{r(xv)}{x}\longrightarrow\frac v\gamma\qquad\text{locally uniformly in }v>0,
\]
where $\gamma$ lies between positive constants that depend only on $\beta$ and on the bounds on the paths \cite[Theorem~5.1]{oai-fluctuations}. The proof identifies the limits of the rescaled scalar and overlap distributions \cite[Lemma~5.4]{oai-fluctuations} and derives a limiting scalar equation \cite[Lemmas~5.3 and~5.5]{oai-fluctuations}. It then excludes gaps in the limiting support by a first-variation argument \cite[Lemma~5.6]{oai-fluctuations}. The scalar diffusion identities used there go back to Auffinger and Chen \cite[Proposition~3]{auffinger-chen-2015} and to Jagannath and Tobasco \cite[Lemma~8.4]{jagannath-tobasco-2017}. At macroscopic scales, let $c_*>0$ be fixed. If $x\ge c_*$, $e(x)/x^2\ge c_*$, and $d^2\le\beta^2-c_*$, then the minimizers converge, in $L^1$ and locally uniformly on $(0,1)$, to the quantile function of the Parisi measure \cite[Proposition~5.9]{oai-fluctuations}. This uses the Parisi formula \cite{parisi-formula}, the uniqueness of the Parisi measure \cite{auffinger-chen-uniqueness-2015}, and the interval support theorem \cite[Theorem~1.1]{full-rsb}, through \cite[Lemma~5.8]{oai-fluctuations}. Both statements allow an omitted interval $[0,{\bar u}_0)$ and a positive lower radius $r_{\min}$, provided that the perturbed paths and the competitors used in the first variation are admissible.

\paragraph{Finite propagation.} The assumption that the linear bound holds above a multiple of $x$ is removed in \cite[Section~6]{oai-fluctuations}. There, in addition, $d^2\le\beta^2-c_*$. Either the comparison acts on all masses and $b_0$ is bounded below by a fixed positive constant, or it acts on $[{\bar u}_0,1]$, where $b_0$ is bounded below by a fixed positive constant, and $b_0$ has a fixed positive jump at ${\bar u}_0$ \cite[Definition~6.1]{oai-fluctuations}. The selected alternative must retain a fixed positive margin after the initial comparison: a lower bound on the matrix covariance on $[{\bar u}_0,1]$ and, when the lower interval is omitted, a positive jump in that covariance from $[0,{\bar u}_0)$ to $[{\bar u}_0,1]$. A comparison on an omitted lower interval may instead be followed by comparisons on all masses if its perturbed matrix path has a global lower bound by a fixed positive constant. Subsequent comparisons on the omitted lower interval have lower radius zero. Suppose that the linear bound fails for some minimizer, at some strength ${\widehat\lambda}$, at some mass at least $A_{\widehat\lambda}$. Let $x$ be such a mass within a factor two of the supremum of these masses, so that the bound holds above $2x$. If $e(x)/x^2$ stays bounded below along a subsequence, \cite[Theorem~5.1]{oai-fluctuations} (when $x\to0$) or \cite[Proposition~5.9]{oai-fluctuations} (when $x$ stays bounded below) contradicts the failure, since $c$ and $C$ are chosen outside the bounds of the limiting profiles. Otherwise $e(x)/x^2\to0$. In this case OpenAI builds a new comparison, whose unperturbed paths are the perturbed paths of this minimizer, with
\[
 d'=c_0x,\qquad E'=\min\bigl\{n^{-1/2}(c_0x)^2,\;n^{-100\delta}e(x)x^3\bigr\},\qquad E'\ge n^{\delta/4}E
\]
\cite[Lemma~6.2]{oai-fluctuations}. Suppose that the new comparison satisfies its comparison bound \eqref{eq:fl-comparison-bound}, with $E'$ in place of $E$, and let $r'$ be its minimizer at its final strength. The two minimizers are then close at every scale $y\gtrsim x$ \cite[Lemmas~6.3--6.5]{oai-fluctuations}. More precisely, for fixed $[t_1,t_2]\subset(0,\infty)$ and $\eta>0$, with $[t_1y,t_2y]\subset(0,1)$ when $y$ is bounded below, the set of masses ${\bar u}\in[t_1y,t_2y]$ with $\abs{r({\bar u})-r'({\bar u})}>\eta y$ has Lebesgue measure $o(y)$. It follows that $r'$ satisfies the linear bound above a multiple of $x$. We then transfer the linear profile of $r'$ at scale $x$, given by \cite[Theorem~5.1]{oai-fluctuations} or \cite[Proposition~5.9]{oai-fluctuations}, to $r$, contradicting the failure. Hence the new comparison violates its comparison bound. By \cite[Proposition~3.10]{oai-fluctuations}, one of its minimizers then violates the linear bound, and the argument repeats with $E'$ in place of $E$. The parameter range confines $E$ between fixed multiples of $n^{-5/6+2m}$ and $n^{-1/2}$, and each repetition multiplies it by at least $n^{\delta/4}$. Since $N\delta/4\ge5/2>1/3-2m$ for $N=\lceil10/\delta\rceil$, at most $N$ repetitions are possible. Let $\kappa_0>0$ be a fixed lower bound for the residual matrix covariance and, when required, the residual matrix jump after the initial comparison. Before $n\to\infty$, we fix $c_0>0$ so that $Nc_0^2<\min\{1,\kappa_0/2\}$ and $c_0^2\le\beta^2-c_*$. The total matrix covariance removed by the subsequent comparisons is then less than $\kappa_0/2$, and the total field covariance added is less than $1$. Together with the zero lower radius for subsequent comparisons on the omitted lower interval, these bounds preserve nonnegativity, monotonicity, and the fixed bounds on the total covariance throughout the chain \cite[Proposition~6.6, proof]{oai-fluctuations}. The conclusion is \cite[Proposition~6.6]{oai-fluctuations}: every minimizer at a positive strength satisfies the linear bound above $A_{\widehat\lambda}$, and \eqref{eq:fl-comparison-bound} holds when the comparison is admissible at every strength. The linear bound also holds, uniformly in the parameters, for the comparisons with a low-field perturbation used below, which are admissible only at their final strength \cite[Corollary~6.7]{oai-fluctuations}.

\paragraph{The lower bound.} Let $\eps=n^{-1/3}$, $d=s$, and $E=n^{-2/3+2m}s$. OpenAI applies a comparison on all masses whose unperturbed paths $(\beta^2-\eps\mathbf 1_{[0,s)},0)$ are those of $\Psi_n(\eps)$. It combines \eqref{eq:fl-comparison-bound} with the chord inequality
\[
 f_n(b',{\widehat h}')\le f_n(b,{\widehat h})+\frac14\int_0^1\frac{({\widehat h}({\bar u})-{\widehat h}'({\bar u}))^2}{b'({\bar u})-b({\bar u})}\,\dd {\bar u},
\]
valid for bounded nonnegative nondecreasing paths with $b'\ge b$, ${\widehat h}'={\widehat h}$ wherever $b'=b$, and a finite right side \cite[Lemma~7.1]{oai-fluctuations}, and with \cite[Proposition~6.6]{oai-fluctuations}, which gives $r\asymp s$ on $[s/2,s]$ at a minimizer. The result is
\[
 \Psi_n(\eps)-\Psi_n(0)\ge c\,\eps s^3-n^{20\delta}E=c\,n^{-5/6+15m}-n^{-5/6+7m+20\delta}
\]
\cite[Section~7.1, (30)]{oai-fluctuations}. The second term is smaller than the first by the factor $n^{-8m+20\delta}$. Since $\Psi_n$ is nondecreasing, we have $\Psi_n(\beta^2)-\Psi_n(0)\ge\Psi_n(\eps)-\Psi_n(0)$, and the lower bound in \eqref{eq:fl-pressure-window} follows.

\paragraph{The upper bound.} An initial comparison on all masses, with $d_N=n^{-1/6+25m}$, bounds $\Psi_n(\eps_0)-\Psi_n(0)$, where $\eps_0=n^{-1/3+40m}$ \cite[Section~7.2]{oai-fluctuations}. Starting at $\eps_0$, OpenAI then iterates
\[
 \eps\longmapsto\eps+q_\eps,\qquad q_\eps=\tfrac14\min\{\eps,\beta^2-\eps\},
\]
until $\beta^2-\eps\le n^{-5}$. Below $\beta^2/2$ the parameter $\eps$ grows by the factor $5/4$, and above it $\beta^2-\eps$ shrinks by the factor $3/4$, so $O(\log n)$ steps suffice. Each step uses a comparison on $[s,1]$ with $d=n^{-1/6+15m}$ and $E=n^{-2/3+10m}d$. Its lower radius is $r_{\min}={\bar u}_*/(2M(s))\asymp n^{-1/6-55m-\delta}$, where ${\bar u}_*=(nE)^{-1}$ and $M({\bar u})$ is the upper bound on the derivative of the inverse quantile in \cite[(5)]{oai-fluctuations}. The comparison also includes a low-field perturbation of strength $\theta\in[0,1]$ on the lower interval $[0,s)$, which lowers the matrix covariance on that interval by $\theta c_L$ and raises the field covariance by $\theta c_Lr_0\varphi$ \cite[Section~7.3]{oai-fluctuations}. Here $c_L=\frac12\min\{\eps,\beta^2-\eps\}$, $r_0=d^2r_{\min}/(2\eps)$, and $\varphi$ increases from $0$ to $1$ on $[0,s]$. These comparisons are admissible at their final strength $d^2$, and their minimizers satisfy the linear bound above the scale $d$, uniformly in $\eps$ and $\theta$ \cite[Lemma~7.2]{oai-fluctuations}. This uses \cite[Corollary~6.7]{oai-fluctuations} at a fixed positive strength and requires no integration from strength zero. On the lower interval, $S_{\bar u}\le Cd^3/\eps$ for $0\le {\bar u}<s$ \cite[Lemma~7.3]{oai-fluctuations}. Integrating in $\theta$ from $0$ to $1$, and comparing with $\Psi_n(\eps+q_\eps)$ by the chord inequality, OpenAI obtains \cite[Lemma~7.4]{oai-fluctuations}
\[
 \Psi_n(\eps+q_\eps)-\Psi_n(\eps)\le n^{20\delta}E
 +C\Bigl[\frac{sd^6}{\eps_0}+\frac{d}{nsr_{\min}}(\log n)^3+n^{-10}\Bigr].
\]
By the bound on the derivative of $\Psi_n$, the last interval, of length at most $n^{-5}$, costs at most $sn^{-5}/4$. Summing over the steps, we have \cite[Section~7.4]{oai-fluctuations}
\begin{equation}\label{eq:fl-upper-sum}
 \Psi_n(\beta^2)-\Psi_n(0)\le C\bigl[n^{-5/6+95m}+n^{-5/6+65m+\delta}(\log n)^4
 +n^{-5/6+27m+20\delta}\log n+n^{-5}s\bigr].
\end{equation}
\Cref{t:fl-powers} lists the powers of $n$ that enter the two bounds. In \eqref{eq:fl-upper-sum}, the first term dominates for large $n$, because $65m+\delta$ and $27m+20\delta$ are both smaller than $95m$ when $\delta=m/10000$; the logarithmic factors are absorbed by the remaining powers. Multiplying \eqref{eq:fl-pressure-window} by $ns=n^{5/6+5m}$ turns the exponents $-5/6+15m$ and $-5/6+95m$ into $20m$ and $100m$, and we obtain \eqref{f:window-eq}.

\begin{table}[ht]
\centering
\caption{The terms in the proof of \eqref{eq:fl-pressure-window} in \cite[Section~7]{oai-fluctuations}, with logarithmic factors and constants suppressed. Here $s=n^{-1/6+5m}$, $d_N=n^{-1/6+25m}$, $\eps_0=n^{-1/3+40m}$, $d=n^{-1/6+15m}$, $E=n^{-2/3+10m}d$, and $r_{\min}\asymp n^{-1/6-55m-\delta}$. The middle column refers to the sections of \cite{oai-fluctuations}.}
\label{t:fl-powers}
\begin{tabular}{@{}llc@{}}
\toprule
Term & In \cite{oai-fluctuations} & Power of $n$\\
\midrule
main term $\eps s^3$ of the lower bound, $\eps=n^{-1/3}$ & Section~7.1 & $-5/6+15m$\\
error $n^{20\delta}E$ of the lower comparison & Section~7.1 & $-5/6+7m+20\delta$\\
chord term $s\eps_0d_N^2$ of the initial comparison & Section~7.2 & $-5/6+95m$\\
error of the initial comparison & Section~7.2 & $-5/6+27m+20\delta$\\
error $n^{20\delta}E$ of one step & Section~7.4 & $-5/6+25m+20\delta$\\
$sd^6/\eps_0$ & Section~7.4 & $-5/6+55m$\\
$d/(nsr_{\min})$ & Section~7.4 & $-5/6+65m+\delta$\\
\bottomrule
\end{tabular}
\end{table}

The order in which the parameters are chosen matters. In \cite{oai-fluctuations}, $m$ is fixed first and $\delta=m/10000$. Then the constants in the two-sided linear bound, the depth $N$, and the coefficient $c_0$ are fixed, and only then does $n$ tend to infinity. The resulting threshold in $n$ is uniform in $\eps$, $\theta$, the strength, and the minimizer. This uniformity is needed because the number and the locations of the steps depend on $n$.

\section{Log-concave augmentation}\label{sec:fl-logconcave}

A two-sided bound on $K_n$ at a single tilt does not by itself control $\Var(F_n)$ or the typical size of $F_n-\E F_n$: a rare large positive deviation can dominate an exponential moment, and a rare negative deviation can dominate the variance without changing the positive exponential moment \cite[Section~7.5]{oai-fluctuations}. The additional input is that $F_n^c$ has a log-concave law after an independent exponential variable is added. This observation is due to Aronow and Lopatto \cite[Proposition~5.1, proof, Steps~1 and~2]{quantitative-parisi}, and \cite[Lemma~7.6]{oai-fluctuations} states it in the present setting. Let $\mathcal E$ be an exponential random variable with mean one, independent of $J$ and $G$. We set
\begin{equation}\label{f:smoothing}
 U_n=F_n^c+\mathcal E=F_n+\beta G+\mathcal E,\qquad \sigma_n=\bigl(\Var(F_n)+\beta^2/2+1\bigr)^{1/2}.
\end{equation}

\begin{lemma}[Log-concave augmentation]\label{f:logconcave}
Let $n\ge2$ and $\beta\ne0$. Then $U_n$ has a density $\varrho_n\colon\R\to(0,\infty)$ such that $\log \varrho_n$ is concave. Moreover, $\E U_n=\E F_n+1$, $\Var(U_n)=\sigma_n^2$, and for $0\le t<1$,
\begin{equation}\label{f:smooth-transform}
 \log\E e^{t(U_n-\E U_n)}=K_n(t)-t-\log(1-t).
\end{equation}
\end{lemma}

\begin{proof}
We follow \cite[Proposition~5.1, proof]{quantitative-parisi} and \cite[Lemma~7.6]{oai-fluctuations}. The proof has three steps: we write $F_n^c$ as a convex function of a standard Gaussian vector, use the gauge symmetry of the model to express $\varrho_n$ through a single configuration, and then apply Pr\'ekopa's theorem.

\paragraph{Step 1: a Gaussian representation.} Let $\bar n=n(n-1)/2+1$, and let $g=((J_{ij})_{i<j},g_0)$ be a standard Gaussian vector in $\R^{\bar n}$ with $G=g_0/\sqrt2$. For $x\in\Sigma_n$, we define $a_x\in\R^{\bar n}$ by $(a_x)_{ij}=\beta x_ix_j/\sqrt n$ for $i<j$ and $(a_x)_0=\beta/\sqrt2$. Then $\beta(H_n(x)+G)=a_x\cdot g$, and
\[
 F_n^c=F(g),\qquad F(g)=\log\sum_{x\in\Sigma_n}e^{a_x\cdot g}.
\]
The function $F$ is convex, as the logarithm of a sum of exponentials of linear functions.

\paragraph{Step 2: gauge symmetry.} For $y\in\Sigma_n$, let $O_y\colon\R^{\bar n}\to\R^{\bar n}$ be the linear map with $(O_yg)_{ij}=y_iy_jg_{ij}$ for $i<j$ and $(O_yg)_0=g_0$. It is diagonal with entries $\pm1$. It is therefore orthogonal, and it preserves the law of $g$. Writing $yx=(y_ix_i)_{i\le n}$, we have $a_x\cdot O_yg=a_{yx}\cdot g$. Summing over $x$ and using that $x\mapsto yx$ is a bijection of $\Sigma_n$, we obtain $F(O_yg)=F(g)$.

The density of $U_n=F(g)+\mathcal E$ is the convolution of the law of $F(g)$ with the density $e^{-y}\mathbf 1_{\{y\ge0\}}$, namely
\[
 \varrho_n(z)=\E\bigl[e^{F(g)-z}\mathbf 1_{\{F(g)\le z\}}\bigr]
 =e^{-z}\sum_{x\in\Sigma_n}\E\bigl[e^{a_x\cdot g}\mathbf 1_{\{F(g)\le z\}}\bigr],
 \qquad z\in\R.
\]
This expectation is finite, since its integrand is at most $1$. Fix $x_*\in\Sigma_n$ and write $a_*=a_{x_*}$. For $x\in\Sigma_n$, let $y\in\Sigma_n$ be given by $y_i=x_ix_{*,i}$, so that $yx_*=x$. Using $a_x\cdot g=a_*\cdot O_yg$, $F(g)=F(O_yg)$, and the invariance of the law of $g$ under $O_y$, we find that
\[
 \E\bigl[e^{a_x\cdot g}\mathbf 1_{\{F(g)\le z\}}\bigr]
 =\E\bigl[e^{a_*\cdot O_yg}\mathbf 1_{\{F(O_yg)\le z\}}\bigr]
 =\E\bigl[e^{a_*\cdot g}\mathbf 1_{\{F(g)\le z\}}\bigr].
\]
All $2^n$ terms of the sum are therefore equal, and
\begin{equation}\label{eq:fl-pinned-density}
 \varrho_n(z)=2^n(2\pi)^{-\bar n/2}\int_{\R^{\bar n}}\Theta(g,z)\,\dd g,\qquad
 \Theta(g,z)=\exp\Bigl\{-\frac{|g|^2}{2}+a_*\cdot g-z\Bigr\}\mathbf 1_{\{F(g)\le z\}}.
\end{equation}

\paragraph{Step 3: log-concavity and positivity.} The set $\{(g,z)\in\R^{\bar n+1}:F(g)\le z\}$ is convex, since it is the epigraph of the convex function $F$. On this set, $\log\Theta(g,z)=-|g|^2/2+a_*\cdot g-z$ is concave, and $\Theta$ vanishes off it; that is, $\Theta$ is a log-concave function on $\R^{\bar n+1}$. Since $F(g)\ge a_*\cdot g$, we have $a_*\cdot g-z\le0$ wherever $F(g)\le z$, and hence $0\le\Theta(g,z)\le e^{-|g|^2/2}$ for every $(g,z)$. Moreover,
\[
 \int_\R\Theta(g,z)\,\dd z=e^{-|g|^2/2+a_*\cdot g-F(g)}\le e^{-|g|^2/2}.
\]
Every slice $g\mapsto\Theta(g,z)$ is therefore integrable, and $\Theta$ is integrable on $\R^{\bar n+1}$ by Tonelli's theorem. By Pr\'ekopa's theorem \cite[Theorems~6 and~8]{prekopa-1973}, the integral of a log-concave function over some of its variables is log-concave in the remaining ones. Then \eqref{eq:fl-pinned-density} implies that $\log \varrho_n$ is concave. To see that $\varrho_n(z)>0$ for every $z$, note that $F(g)=\log Z_n+\beta g_0/\sqrt2$. For fixed $J$, the set of $g_0$ with $F(g)\le z$ is a half-line, since $\beta\ne0$. By Fubini's theorem, $\{g:F(g)\le z\}$ has positive Lebesgue measure, and the integrand in \eqref{eq:fl-pinned-density} is positive on it.

Finally, $J$, $G$, and $\mathcal E$ are independent, $\E G=0$, $\E\mathcal E=\Var(\mathcal E)=1$, and $\Var(\beta G)=\beta^2/2$. It follows that $\E U_n=\E F_n+1$ and $\Var(U_n)=\sigma_n^2$. For $0\le t<1$, we have $\E e^{t(\mathcal E-1)}=e^{-t}/(1-t)$, and \eqref{f:smooth-transform} follows from the independence of $F_n^c$ and $\mathcal E$.
\end{proof}

The next lemma collects the estimates for one-dimensional log-concave laws that we need. It is a variant of \cite[Lemma~7.7]{oai-fluctuations}; the tail bound in its proof is also in \cite[Proposition~5.1, proof, Step~3]{quantitative-parisi}.

\begin{lemma}[One-dimensional log-concave bounds]\label{f:one-dim}
There exist universal constants $c_1,c_2,p_*,C_1>0$ with the following property. Let $Z$ be a real random variable with mean $\mu$ and finite variance $\sigma^2>0$, and suppose that $Z$ has a density $\varrho\colon\R\to(0,\infty)$ such that $\log\varrho$ is concave. Then
\begin{align}
 \log\E e^{t(Z-\mu)}&\le\log3 &&\Bigl(|t|\le\frac{c_1}{2\sigma}\Bigr),\label{f:small-mgf}\\
 \log\E e^{t(Z-\mu)}&\ge tc_2\sigma+\log p_* &&(t\ge0),\label{f:large-mgf}\\
 \PP\{z-r\le Z\le z+r\}&\le\frac{2C_1r}{\sigma} &&(z\in\R,\ r\ge0).\label{f:interval}
\end{align}
In \eqref{f:large-mgf}, the left side may be $+\infty$.
\end{lemma}

\begin{proof}
We show that the lemma holds with
\[
 c_1=\frac{\log3}{4},\qquad C_2=\frac{48}{c_1^4},\qquad c_2=\frac{1}{4\sqrt{C_2}},\qquad p_*=c_2^2,
\]
and
\[
 C_1=\Bigl(16+\frac{4}{c_3^3}\Bigr)^{1/2},\qquad c_3=\frac{\log 2}{4}.
\]
We first prove a tail bound and then the three assertions in order.

\paragraph{Step 1: an exponential tail bound.} The function $(x,y)\mapsto\mathbf 1_{\{y\ge x\}}\varrho(y)$ is log-concave on $\R^2$. By Pr\'ekopa's theorem \cite[Theorems~6 and~8]{prekopa-1973}, the survival function $Q(x)=\PP(Z\ge x)$ is log-concave. It is positive, because $\varrho>0$. By Chebyshev's inequality,
\[
 Q(\mu-2\sigma)\ge1-\PP(|Z-\mu|\ge2\sigma)\ge\frac34,\qquad Q(\mu+2\sigma)\le\PP(|Z-\mu|\ge2\sigma)\le\frac14.
\]
Let $x\ge\mu+2\sigma$. Since $\log Q$ is concave, the slope of its secant over $[\mu+2\sigma,x]$ is at most its slope over $[\mu-2\sigma,\mu+2\sigma]$, which is at most $(\log\frac14-\log\frac34)/(4\sigma)=-c_1/\sigma$. This gives
\[
 Q(x)\le\frac14e^{-c_1(x-\mu-2\sigma)/\sigma}=\frac{\sqrt3}{4}e^{-c_1(x-\mu)/\sigma},
\]
where we used $e^{2c_1}=\sqrt3$. The variable $-Z$ satisfies the same hypotheses, with mean $-\mu$. Applying the same bound to it and adding, we find $\PP(|Z-\mu|\ge u)\le(\sqrt3/2)e^{-c_1u/\sigma}$ for $u\ge2\sigma$. For $0\le u<2\sigma$, we have $2e^{-c_1u/\sigma}\ge2/\sqrt3>1$. We conclude that
\begin{equation}\label{eq:fl-tail}
 \PP(|Z-\mu|\ge u)\le2e^{-c_1u/\sigma}\qquad(u\ge0).
\end{equation}

\paragraph{Step 2: small tilts.} Let $|t|\le c_1/(2\sigma)$ and $\ell=|t|$. Writing $e^{\ell|Z-\mu|}=1+\int_0^\infty\ell e^{\ell u}\mathbf 1_{\{|Z-\mu|>u\}}\,\dd u$ and using \eqref{eq:fl-tail}, we obtain
\[
 \E e^{t(Z-\mu)}\le\E e^{\ell|Z-\mu|}\le1+2\ell\int_0^\infty e^{(\ell-c_1/\sigma)u}\,\dd u
 =1+\frac{2\ell}{c_1/\sigma-\ell}\le3.
\]
In the last inequality, we used $c_1/\sigma-\ell\ge c_1/(2\sigma)\ge\ell$. This proves \eqref{f:small-mgf}.

\paragraph{Step 3: positive tilts.} We use the fourth-moment and positive-part argument of \cite[Lemma~7.7, proof]{oai-fluctuations}. Let $\hat Z=(Z-\mu)/\sigma$, so that $\E\hat Z=0$ and $\E\hat Z^2=1$. By \eqref{eq:fl-tail},
\[
 \E\hat Z^4=\int_0^\infty4u^3\,\PP(|\hat Z|\ge u)\,\dd u\le8\int_0^\infty u^3e^{-c_1u}\,\dd u=\frac{48}{c_1^4}=C_2.
\]
By H\"older's inequality with exponents $3/2$ and $3$, we have $1=\E\bigl[|\hat Z|^{2/3}|\hat Z|^{4/3}\bigr]\le(\E|\hat Z|)^{2/3}(\E\hat Z^4)^{1/3}$, which implies that $\E|\hat Z|\ge C_2^{-1/2}$. The positive part $\hat Z_+$ satisfies $\E\hat Z_+=\E|\hat Z|/2\ge C_2^{-1/2}/2=2c_2$, because $\E\hat Z=0$. We have $\hat Z_+<c_2$ on $\{\hat Z<c_2\}$. Using this, the Cauchy--Schwarz inequality, and $\E\hat Z_+^2\le1$, we obtain
\[
 2c_2\le\E\hat Z_+\le c_2+\E\bigl[\hat Z_+\mathbf 1_{\{\hat Z\ge c_2\}}\bigr]\le c_2+\PP(\hat Z\ge c_2)^{1/2}.
\]
Rearranging, we find $\PP(\hat Z\ge c_2)\ge c_2^2=p_*$. For $t\ge0$, restricting the expectation to $\{\hat Z\ge c_2\}$, we obtain $\E e^{t(Z-\mu)}=\E e^{t\sigma\hat Z}\ge e^{tc_2\sigma}p_*$, which is \eqref{f:large-mgf}.

\paragraph{Step 4: the density.} We adapt the log-density secant argument of \cite[Lemma~7.7, proof]{oai-fluctuations}, using an arbitrary point in place of a mode. Fix $z\in\R$, and set $\ell=4/\varrho(z)$. Suppose that $\varrho(z+\ell)\ge\varrho(z)/2$. Since $\log\varrho$ is concave, for $0\le y\le\ell$ we have
\[
 \log\varrho(z+y)\ge\Bigl(1-\frac y\ell\Bigr)\log\varrho(z)+\frac y\ell\log\varrho(z+\ell)\ge\log\frac{\varrho(z)}2,
\]
so that $\int_z^{z+\ell}\varrho\ge\ell\varrho(z)/2=2$, which is impossible. We conclude that $\varrho(z+\ell)<\varrho(z)/2$. For $y\ge\ell$, the slope of the secant of $\log\varrho$ over $[z,z+y]$ is at most its slope over $[z,z+\ell]$, which is less than $-(\log2)/\ell=-c_3\varrho(z)$. Exponentiating, we find that $\varrho(z+y)\le\varrho(z)e^{-c_3\varrho(z)y}$ for $y\ge\ell$, and by the same argument applied to $y\mapsto\varrho(z-y)$, also $\varrho(z-y)\le\varrho(z)e^{-c_3\varrho(z)y}$ for $y\ge\ell$. Since the mean minimizes the mean square distance, we have
\[
\begin{aligned}
 \sigma^2\le\E(Z-z)^2&\le\ell^2\,\PP(|Z-z|<\ell)+2\int_\ell^\infty y^2\varrho(z)e^{-c_3\varrho(z)y}\,\dd y\\
 &\le\frac{16}{\varrho(z)^2}+\frac{4}{c_3^3\varrho(z)^2}=\frac{C_1^2}{\varrho(z)^2}.
\end{aligned}
\]
In the second inequality, we bounded $(Z-z)^2$ by $\ell^2$ on $\{|Z-z|<\ell\}$ and used the two exponential bounds on the complement. This shows that $\varrho(z)\le C_1/\sigma$ for every $z\in\R$, and integrating over $[z-r,z+r]$ proves \eqref{f:interval}.
\end{proof}

\section{Proof of the fluctuation theorem}\label{sec:fl-endgame}

The argument below is that of \cite[Section~7.6]{oai-fluctuations}, with two elementary changes: we use the constant bound \eqref{f:small-mgf} for small tilts in place of the quadratic bound of \cite[Lemma~7.7]{oai-fluctuations}, and we remove the added noise through the inequality \eqref{f:transfer} below.

\begin{proof}[Proof of \cref{f:main} from \cref{f:window}]
The proof has three steps. We first bound $\sigma_n$ above and below, using the two halves of \eqref{f:window-eq} together with \cref{f:logconcave,f:one-dim}. We then subtract the variance of the added noise to obtain the bounds on $\Var(F_n)$. Finally, we use the bound \eqref{f:interval} on the density of $U_n$ to show that $|F_n-\E F_n|$ is rarely smaller than $n^{1/6-\eta}$. Throughout, $c_1,c_2,p_*,C_1$ are the universal constants of \cref{f:one-dim}, and we write
\[
 d_\beta=\frac{\beta^2}{2}+1,\qquad\text{so that}\qquad \Var(F_n)=\sigma_n^2-d_\beta.
\]
By \cref{f:logconcave}, the variable $Z=U_n$ satisfies the hypotheses of \cref{f:one-dim}, with $\mu=\E U_n$ and $\sigma=\sigma_n\ge1$.

\paragraph{Step 1: bounds on $\sigma_n$.} Fix $0<m<1/300$, and let $c,C,n_0$ be as in \cref{f:window}. Let $\omega(t)=-t-\log(1-t)$ for $0\le t<1$. Since $\omega(t)=\sum_{k\ge2}t^k/k$, we have
\[
 0\le\omega(t)\le\frac{t^2}{2(1-t)}\le t^2\qquad\Bigl(0\le t\le\frac12\Bigr).
\]
By \eqref{f:smooth-transform}, $\log\E e^{t(U_n-\E U_n)}=K_n(t)+\omega(t)$. Let $n_1\ge n_0$ be such that $s_n\le1/2$ and $c\,n^{20m}>\log3$ for all $n\ge n_1$, and let $n\ge n_1$. If $s_n\le c_1/(2\sigma_n)$, then \eqref{f:small-mgf} and $\omega\ge0$ imply that $K_n(s_n)\le\log3$, which contradicts the lower bound in \eqref{f:window-eq}. It follows that $\sigma_n>c_1/(2s_n)$. On the other hand, \eqref{f:large-mgf} and the upper bound in \eqref{f:window-eq} give
\[
 s_nc_2\sigma_n+\log p_*\le K_n(s_n)+\omega(s_n)\le C\,n^{100m}+1.
\]
Using $p_*<1$ and $n^{100m}\ge1$, we find $\sigma_n\le C'n^{100m}/s_n$ with $C'=(C+1+\log(1/p_*))/c_2$. We have shown that
\begin{equation}\label{f:augmented-sd}
 \frac{c_1}{2}\,n^{1/6-5m}\le\sigma_n\le C'n^{1/6+95m}\qquad(n\ge n_1).
\end{equation}
Now fix $\eta>0$, and let $m=\min\{1/600,\eta/200\}$. Then $0<m<1/300$, $5m<\eta$, and $95m<\eta$. Since $n^{\eta-5m}$ and $n^{\eta-95m}$ tend to infinity, \eqref{f:augmented-sd} implies that there exists $n_2(\beta,\eta)$ such that
\begin{equation}\label{eq:fl-sigma-eta}
 n^{1/6-\eta}\le\sigma_n\le n^{1/6+\eta}\qquad(n\ge n_2(\beta,\eta)).
\end{equation}
Here $m$ depends only on $\eta$ and is chosen before $n\to\infty$; the threshold $n_2$ depends only on $\beta$ and $\eta$.

\paragraph{Step 2: the standard deviation and the variance.} Fix $\eta>0$. By \eqref{eq:fl-sigma-eta} and $\Var(F_n)\le\sigma_n^2$, we have $\Var(F_n)^{1/2}\le n^{1/6+\eta}$ for $n\ge n_2(\beta,\eta)$. For the lower bound, let $\eta'=\min\{\eta/2,1/12\}$. By \eqref{eq:fl-sigma-eta} with $\eta'$ in place of $\eta$, we have $\sigma_n^2\ge n^{1/3-2\eta'}$ for $n\ge n_2(\beta,\eta')$. Since $1/3-2\eta'\ge1/6>0$, we have $d_\beta\le n^{1/3-2\eta'}/2$ for all sufficiently large $n$, and then
\[
 \Var(F_n)=\sigma_n^2-d_\beta\ge\frac12n^{1/3-2\eta'}\ge n^{1/3-2\eta}.
\]
The last inequality holds once $n^{2\eta-2\eta'}\ge2$, which is true eventually because $2\eta-2\eta'\ge\eta>0$. Taking square roots, we conclude that $n^{1/6-\eta}\le\Var(F_n)^{1/2}\le n^{1/6+\eta}$ for all sufficiently large $n$, for every fixed $\eta>0$. This is \eqref{f:sd}. Applying it with $\eta/2$ in place of $\eta$ and squaring, we obtain $n^{1/3-\eta}\le\Var(F_n)\le n^{1/3+\eta}$ for all sufficiently large $n$, which is \eqref{f:variance}.

\paragraph{Step 3: the typical magnitude.} We have $U_n-\E U_n=(F_n-\E F_n)+\beta G+(\mathcal E-1)$. Let $r\ge0$ and $v>0$. On the event $\{|F_n-\E F_n|\le r\}\cap\{|\beta G|\le v\}\cap\{|\mathcal E-1|\le v\}$, we have $|U_n-\E U_n|\le r+2v$. A union bound, \eqref{f:interval} with $z=\E U_n$, and Chebyshev's inequality for $\beta G$ and $\mathcal E$, whose variances are $\beta^2/2$ and $1$, give
\begin{equation}\label{f:transfer}
 \PP(|F_n-\E F_n|\le r)\le\frac{2C_1(r+2v)}{\sigma_n}+\frac{d_\beta}{v^2}.
\end{equation}
Fix $\eta>0$, and set $\eta_1=\min\{\eta/4,1/48\}$, $r=n^{1/6-\eta}$, and $v=n^{\eta_1}$. By \eqref{eq:fl-sigma-eta} with $\eta_1$ in place of $\eta$, we have $\sigma_n\ge n^{1/6-\eta_1}$ for $n\ge n_2(\beta,\eta_1)$, and then \eqref{f:transfer} gives
\[
 \PP\bigl(|F_n-\E F_n|\le n^{1/6-\eta}\bigr)\le2C_1\bigl(n^{\eta_1-\eta}+2n^{2\eta_1-1/6}\bigr)+d_\beta n^{-2\eta_1}.
\]
Since $\eta_1-\eta\le-3\eta/4$ and $2\eta_1-1/6\le-1/8$, the right side tends to zero. For the upper tail, Chebyshev's inequality and the bound on the variance from Step~2 give
\[
 \PP\bigl(|F_n-\E F_n|>n^{1/6+\eta}\bigr)\le\frac{\Var(F_n)}{n^{1/3+2\eta}}\le n^{-\eta}
\]
for all sufficiently large $n$. Combining the two bounds, we find that $\PP\{n^{1/6-\eta}\le|F_n-\E F_n|\le n^{1/6+\eta}\}\to1$ for every fixed $\eta>0$, which is \eqref{f:typical}. This completes the proof.
\end{proof}
\chapter{The finite-size mean bias}\label{sec:bias}

In this section, we prove \cref{cor:mean-bias}, the lower bound with exponent $5/6$ on the mean bias $\Delta_{n,\beta}=P_\beta-p_n(\beta)$ stated in the introduction. Combined with the upper bound of Aronow and Lopatto (\cref{inp:hb-rate}), it shows that, for every fixed $\beta>1$ and $\eps>0$, we have $c_{\beta,\eps}n^{-5/6-\eps}\le\Delta_{n,\beta}\le C_b n^{-7/10}$ for all sufficiently large $n$. If $\Delta_{n,\beta}=n^{-a+o(1)}$ for some exponent $a$, then $7/10\le a\le5/6$ (\cref{rem:bias-exponent}). \Cref{cor:mean-bias} follows from the lower bound on the variance in \cref{f:main} and from a variance--bias inequality of Aronow and Lopatto (\cref{prop:bias-variance-deficit}). The deduction itself is not new: Aronow and Lopatto record it in \cite[Section~1.6, p.~11]{hexagon-bias}, as a consequence of the variance exponent stated in \cite[Theorem~1.1]{oai-fluctuations}. Here we formally verify this deduction with a Lean proof that follows the argument of \cref{sec:bias-proof} and composes the formal proof of \cref{f:main} with a formal proof of \cref{prop:bias-variance-deficit} (see \cref{sec:formal-statements}). We include the short written proof for completeness, and in \cref{rem:bias-prefactor,rem:bias-exponent} we discuss what it does not determine. \Cref{sec:open} returns to the exact exponent of the mean bias.

Recall from \cref{sec:model} that $p_n(\beta)=n^{-1}\E F_n$, with $F_n=\log Z_n$, that $P_\beta=\lim_np_n(\beta)$, and that $\Delta_{n,\beta}=P_\beta-p_n(\beta)\ge0$ for every $n\ge2$ (see \eqref{eq:model-bias}).

\section{The variance--bias inequality}\label{sec:bias-inequality}

\begin{proposition}[Variance--bias inequality]\label{prop:bias-variance-deficit}
Fix $\beta>1$. There exists a constant $c_\beta>0$ such that, for every $n\ge2$,
\begin{equation}\label{eq:mean-bias-variance-deficit}
 \Delta_{n,\beta}\ge c_\beta\Bigl(\frac{\Var(F_n)+\beta^2/2}{n}\Bigr)^{5/4}.
\end{equation}
\end{proposition}

This is \cite[Remark~7.14, (7.68)]{quantitative-parisi}, which restates \cite[Lemma~7.4]{quantitative-parisi} in the normalization of the log partition function. The normalization of \cite[(1.1)]{quantitative-parisi} is ours: the Hamiltonian is $n^{-1/2}\sum_{i<j}J_{ij}x_ix_j$, the partition function is a sum over $\Sigma_n$ with counting measure, $\beta$ multiplies the Hamiltonian, and the Parisi value has the terminal condition $\log(2\cosh z)$. The inequality \eqref{eq:mean-bias-variance-deficit} is \cite[(7.68)]{quantitative-parisi} verbatim; there it is stated for every $n\ge1$. The term $\beta^2/2$ is the variance of the completion $\beta G$ (\cref{lem:model-completion}), and $\Var(F_n)+\beta^2/2=\Var(F_n^c)$ for the completed log partition function $F_n^c=F_n+\beta G$.

We recall the argument of \cite[Lemma~7.4]{quantitative-parisi}, which works with the variable $F_n^c/\beta$. For $t>0$, let $\Lambda(t)=(nt)^{-1}\log\E e^{tF_n^c/\beta}$, and let $\Lambda(0)=\E F_n^c/(\beta n)=p_n(\beta)/\beta$. Since $F_n^c/\beta$ is a convex Lipschitz function of a standard Gaussian vector, \cite[Lemma~5.2]{quantitative-parisi}, which rests on W.-K.~Chen's convexity theorem for Gaussian logarithmic moments \cite[Theorem~1]{chen-gaussian-convexity}, shows that $\Lambda$ is convex with $\Lambda'(0)=V/2$, where $V=\Var(F_n^c)/(\beta^2n)$. By convexity, $\Lambda(t)\ge\Lambda(0)+tV/2$: a gain linear in $t$ and proportional to the variance. On the other hand, the finite-size comparison \cite[(7.1)]{quantitative-parisi} and the fifth-order bound for a scalar trial profile \cite[Lemma~7.2, (7.3)]{quantitative-parisi} give $\Lambda(t)\le P_\beta/\beta+Ct^5$ for small $t>0$. Here $C$ and the range $0<t\le t_*$ of the trial bound depend only on $\beta$. Subtracting, we see that $\Delta_{n,\beta}/\beta$ is at least $tV/2-Ct^5$. The gradient of $F_n^c/\beta$ is a convex combination of vectors of squared norm $n/2$, and the Gaussian Poincar\'e inequality \cite[Theorem~3.20]{boucheron-lugosi-massart-2013}, which is the Gaussian case of the Brascamp--Lieb bound in \cref{inp:ob-concentration} \cite{brascamp-lieb-1976}, gives $\Var(F_n^c)\le\beta^2n/2$; also $\Var(F_n^c)\ge\Var(\beta G)=\beta^2/2$. We conclude that $0<V\le1/2$. With $c'=\min\{1,t_*,(4C)^{-1/4}\}$ and $t=c'V^{1/4}$, we have $t\le t_*$ and
\[
 \frac{\Delta_{n,\beta}}{\beta}\ge\frac{c'}{2}V^{5/4}-C(c')^5V^{5/4}\ge\frac{c'}{4}V^{5/4}.
\]
Since $V=\beta^{-2}(\Var(F_n)+\beta^2/2)/n$, multiplying by $\beta$ gives \eqref{eq:mean-bias-variance-deficit} with $c_\beta=(c'/4)\beta^{-3/2}$, independent of $n$.

\section{Proof of the mean-bias bound}\label{sec:bias-proof}

\begin{proof}[Proof of \cref{cor:mean-bias}]
Fix $\beta>1$ and $\eps>0$. By the lower bound in \eqref{f:variance}, applied with $\eta=4\eps/5$, there exists $n_0(\beta,\eps)\ge2$ such that
\[
 \Var(F_n)\ge n^{1/3-4\eps/5}\qquad(n\ge n_0(\beta,\eps)).
\]
Since $t\mapsto t^{5/4}$ is increasing on $[0,\infty)$, we may drop the term $\beta^2/2$ in \eqref{eq:mean-bias-variance-deficit}. Inserting the variance bound, we obtain, for $n\ge n_0(\beta,\eps)$,
\[
 \Delta_{n,\beta}\ge c_\beta\bigl(n^{1/3-4\eps/5-1}\bigr)^{5/4}=c_\beta\,n^{-5/6-\eps},
\]
where we used $\frac54\bigl(-\frac23-\frac{4\eps}{5}\bigr)=-\frac56-\eps$. Multiplying by $n$, we find $nP_\beta-\E F_n\ge c_\beta n^{1/6-\eps}$. This proves \cref{cor:mean-bias} with $c_{\beta,\eps}=c_\beta$ and this threshold $n_0(\beta,\eps)$; only the threshold depends on $\eps$.
\end{proof}

If we use instead the earlier lower bound $\Var(F_n)\ge c\,n^{4/15}$ of \cite[Theorem~1.4]{quantitative-parisi}, the same computation yields $\Delta_{n,\beta}\ge c\,n^{-11/12}$, which is how the lower bound on the mean bias in \cite[Theorem~1.4]{quantitative-parisi} is obtained there. Using the variance exponent $1/3$ gives $5/6+\eps$ in place of $11/12$, as already observed in \cite[Section~1.6, p.~11]{hexagon-bias}.

\section{What the bound does not determine}\label{sec:bias-scope}

\begin{remark}[A positive variance prefactor]\label{rem:bias-prefactor}
If there were a constant $k_\beta>0$ such that $\Var(F_n)\ge k_\beta n^{1/3}$ for all sufficiently large $n$, the same computation would give
\begin{equation}\label{eq:mean-bias-prefactor-target}
 \Delta_{n,\beta}\ge c_\beta k_\beta^{5/4}\,n^{-5/6}
\end{equation}
for all sufficiently large $n$. This is \cite[(1.4), p.~11]{hexagon-bias}, where Aronow and Lopatto note that it would follow from the positive limit of $n^{-1/3}\Var(F_n)$ claimed in \cite[Theorem~1.2]{oai-limiting-law}. \Cref{f:main} does not give such a prefactor: a variance equal to $n^{1/3}/\log n$, for example, is of the form $n^{1/3+o(1)}$, and inserting it in \eqref{eq:mean-bias-variance-deficit}, we obtain \eqref{eq:mean-bias-prefactor-target} only up to the factor $(\log n)^{-5/4}$. We therefore prove the bound with exponent $5/6+\eps$, not \eqref{eq:mean-bias-prefactor-target}.
\end{remark}

\begin{remark}[The exponent of the mean bias]\label{rem:bias-exponent}
For every fixed $\beta>1$, Aronow and Lopatto prove that $\Delta_{n,\beta}\le C_b n^{-7/10}$ for $n\ge2$ \cite[Theorem~1.1]{hexagon-bias}. Combined with \cref{cor:mean-bias}, this bound implies that if $\Delta_{n,\beta}=n^{-a+o(1)}$ for some exponent $a$, then $7/10\le a\le5/6$ \cite[Section~1.6, p.~11]{hexagon-bias}. The exact exponent is not determined by these bounds. Franz, Parisi, and Ricci-Tersenghi predict $a=3/4$, possibly with logarithmic corrections \cite[(13)--(17)]{franz-parisi-ricci-2026}; this prediction is consistent with both bounds.
\end{remark}
\part{Formal verification and open problems}\label{part:formal}
\chapter{Formal verification}\label{sec:formal}

The assembled Palomar submission
\url{https://github.com/pmaronow/fluctuations-palomar} exposes five
claims through \path{Solution.lean}, selected by \path{comparator.json}; its
independent \path{Challenge.lean} contains those five claims. The combined
package passed the recorded checks on 19 October 2026 (20 October UTC),
for proof-source commit
\path{5551e62264a270fd5dcd396f36389eaf77eddebc}.
The default build succeeded using unchanged compiled objects, and a
strict axiom audit accepted all five exports with only
\path{propext}, \path{Quot.sound}, and \path{Classical.choice}.
A fresh protected Comparator run compiled the independent Challenge,
compared the five statements and their complete reachable helper
dependencies, and replayed the Solution export through Lean's kernel,
NanoDa, and con-ron. All three accepted it. The records are published at
\url{https://github.com/pmaronow/fluctuations-palomar/tree/351cc102439ff1d741d240391dfb62258605b4d2/verification}.
All Lean sources and build/comparison controls are unchanged from the
verified proof-source commit. These checks concern the combined
five-claim package; the source-package checks below, including the
historical 47-claim upper-bound check, retain their original scope.

In this section, we describe formal proofs of the four results
\cref{thm:barrier-730,b:cor:subcritical,f:main,cor:mean-bias} of the lower bound
and the fluctuation part in the Lean~4 proof assistant \cite{lean4}, built on the Mathlib library \cite{mathlib}. No argument elsewhere in the paper depends on this section. In \cref{sec:formal-oai}, we recall what OpenAI's public formalization proves. In \cref{sec:formal-statements}, we state the four formal theorems, explain how they correspond to the statements of this paper, and describe how they were checked. \Cref{sec:formal-barrier} compares the formal proof of \cref{thm:barrier-730} with the proof in this paper, and \cref{sec:formal-window} describes the mathematics that the formal proof of \cref{f:main} had to supply beyond OpenAI's formalization. \Cref{sec:formal-upper} describes the upper-bound formalization, and
\cref{sec:formal-scope} lists what is not formally verified.

\section{OpenAI's formalization}\label{sec:formal-oai}

OpenAI's public repository \cite{oai-math-lean} contains Lean formalizations accompanying \cite{oai-barriers,oai-fluctuations}. For the barrier, the theorem \path{OAI.SK.main}, in the file \path{lean/OAI/Probability/SKBarriers/Main.lean}, states that for every $\beta>1$ two quantities tend to one: the Gibbs mass, averaged over the disorder, of the initial states whose total variation distance from $\pi^J$ at continuous time $\exp(n^{1/10000})$ is greater than $1/4$, and the same mass with the distance after $\lfloor\exp(n^{1/10000})\rfloor$ discrete attempts. This is the formal counterpart of \cite[Theorem~1.1]{oai-barriers}; the repository's scope document \path{lean/docs/227.md} describes it.

For the fluctuations, the theorem \path{main_of_positiveLaplaceWindow}, at line~294 of \path{lean/OAI/Probability/SKFluctuations/EventuallyNatPos.lean}, has the hypothesis \path{hw : PositiveLaplaceWindow} and the conclusion \path{MainFluctuationStatement}. Both propositions are defined in the file \path{Spin.lean} of the same directory, at lines~453 and~531. The proposition \path{PositiveLaplaceWindow} is the statement of \cref{f:window}, for every $\beta>1$ and every $0<m<1/300$, and \path{MainFluctuationStatement} is the conjunction of the three assertions of \cref{f:main}, for every $\beta>1$ and every $\eta>0$. \path{MainFluctuationStatement} uses the log partition function before completion, normalized as $\log(2^{-n}\sum_xe^{\beta H_n(x)})=F_n-n\log2$, which has the same variance and the same centered fluctuations as $F_n$. \path{PositiveLaplaceWindow} uses its Gaussian completion $F_n^c-n\log2$, whose logarithmic moment generating function after centering is $K_n$. OpenAI's formalization proves the deduction of \cref{f:main} from \cref{f:window} given in \cite[Sections~7.5 and~7.6]{oai-fluctuations}, of which \cref{sec:fl-logconcave,sec:fl-endgame} give a variant, but not \cref{f:window} itself. Aronow and Lopatto inspected this directory and found no proof of the hypothesis there \cite[Section~1.6, footnote~1]{hexagon-bias}. In the paper \cite{oai-fluctuations}, by contrast, \cref{f:window} is proved, as summarized in \cref{sec:fl-route}.

\section{Statements of the formal results}\label{sec:formal-statements}

We originally formalized the four results in two source packages, one for \cref{thm:barrier-730,b:cor:subcritical} and one for \cref{f:main,cor:mean-bias}. Both use Lean version \texttt{v4.35.0-rc2} and Mathlib at commit \path{065356127b1dc0016f66b7283ce0ce2c4055aa55}. Both include OpenAI's Lean sources at commit \path{fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb} as a dependency, adapted to the module system of this version of Lean. The second package also depends on the first, which contains the formal proof of the variance--bias inequality.

\paragraph{The barrier theorem.} The theorem \path{SKBarriers2.mainStatementSevenThirty} has type \path{OAI.SK.Strengthened.MainStatementAt (7 / 30)}. This proposition states that for every $\beta>1$ there exists $A>0$ with the following property: for every $\tau\in(0,1/2)$, the two quantities
\[
 \int\sum_{x\in\Sigma_n}\pi^J(x)\,\mathbf 1_{\{d^J(x,\exp(An^{7/30}))>\tau\}}\,\PP(\dd J),\qquad
 \int\sum_{x\in\Sigma_n}\pi^J(x)\,\mathbf 1_{\{d^J_{\rm disc}(x,\lfloor\exp(An^{7/30})\rfloor)>\tau\}}\,\PP(\dd J)
\]
tend to one as $n\to\infty$. These are the two probabilities in \cref{thm:barrier-730}, written as in \eqref{eq:common-joint}, and $A$ plays the role of $A_\beta$. As in \cref{thm:barrier-730}, the coefficient is chosen before the threshold $\tau$, and the same coefficient serves both clocks. The disorder law, the Hamiltonian, the Gibbs measure, the two heat-bath kernels, and the total variation distances are those of OpenAI's development. They agree with the definitions in \cref{sec:model}: the partition function $Z_n$ is a sum over $\Sigma_n$ with counting measure, one discrete step is an update attempt at a uniformly chosen site, an attempt that leaves the configuration unchanged is counted, and the continuous-time kernel is given by \eqref{eq:model-poisson}. Both displayed quantities are instances of a definition in the development that takes the coefficient, the exponent of $n$, and the threshold as arguments. The development proves that the instance with coefficient $1$, exponent $1/10000$, and threshold $1/4$ is the corresponding quantity of OpenAI's theorem. The theorem \path{SKBarriers2.subcriticalStatementSevenThirty}, of type \path{OAI.SK.Strengthened.SubcriticalStatementBelow (7 / 30)}, is \cref{b:cor:subcritical}: the same conclusion with coefficient one, at the time horizons $\exp(n^\kappa)$ and $\lfloor\exp(n^\kappa)\rfloor$, for every $0<\kappa<7/30$. The two theorems have no hypotheses.

\paragraph{The fluctuation theorem.} The fluctuation theorem is the declaration
\path{OAI.SKFluctuations.LaplaceWindowFormalization.main_fluctuation_statement_unconditional_original_q}
in the module \path{SKProof.OriginalQUnconditionalMain}. Its type is \path{OAI.SKFluctuations.MainFluctuationStatement}, the definition of OpenAI's development described in \cref{sec:formal-oai}, and it has no hypotheses. The identifier is only a locator; its parts are not used as terminology in this paper. In \path{MainFluctuationStatement}, the bounds on the standard deviation and on the variance come with separate thresholds in $n$, and their maximum serves as the threshold $n_0(\beta,\eta)$ of \cref{f:main}. The proof of the theorem establishes \path{PositiveLaplaceWindow} and then applies OpenAI's theorem \path{main_of_positiveLaplaceWindow}. In other words, our formal proof of \cref{f:main} consists of a formal proof of \cref{f:window} together with OpenAI's formal proof of the deduction.

\paragraph{The mean-bias corollary.} The declaration
\path{OAI.SKFluctuations.LaplaceWindowFormalization.mean_bias_lower_bound_unconditional},
in the module \path{SKProof.MeanBiasLowerBound}, has type \path{OAI.SKFluctuations.MeanBias.LowerBoundStatement}. This proposition states that for every $\beta>1$ and every $\eps>0$, there exist $c>0$ and $n_0$ such that
\[
 c\,n^{-5/6-\eps}\le\lim_{m\to\infty}p_m(\beta)-p_n(\beta)\qquad(n\ge n_0),
\]
where we write $p_n(\beta)=n^{-1}\E\log\sum_{x\in\Sigma_n}e^{\beta H_n(x)}$, with OpenAI's disorder law and Hamiltonian, which are those of \cref{sec:model}. Since $P_\beta=\lim_mp_m(\beta)$ by definition, this is \cref{cor:mean-bias}. The development proves that this limit exists and equals the infimum of the Parisi functional, as in \cref{inp:parisi}.

To prove this bound, we follow the proof of \cref{cor:mean-bias} in \cref{sec:bias-proof}. We combine our formal proof of \cref{f:main} with the theorem \path{QuantitativePaper.short_variance_mean} of the Lean development \cite{hexagon-bias-lean} accompanying \cite{hexagon-bias}, which is a formal version of \cref{prop:bias-variance-deficit}. That theorem concerns the variable $F_n/\beta$: for every $\beta_0>1$, there exists $c>0$ such that
\[
 c\Bigl(\frac{\Var(F_n/\beta)+1/2}{n}\Bigr)^{5/4}\le\frac{\Delta_{n,\beta}}{\beta}
\]
for every $n\ge1$ and every $\beta\ge\beta_0$. Since $\Var(F_n/\beta)+1/2=\beta^{-2}(\Var(F_n)+\beta^2/2)$, at a fixed $\beta$ this is \eqref{eq:mean-bias-variance-deficit} with $c_\beta=c\beta^{-3/2}$. The module \path{SKProof.MeanBiasBridge} proves the identities that connect the two developments: their disorder laws and Hamiltonians coincide, their expected pressures agree, and OpenAI's log partition function is $F_n-n\log2$, so that its variance is $\Var(F_n)$. As in \cref{sec:bias-proof}, the lower bound on the variance in \cref{f:main}, with $\eta=4\eps/5$, then gives the exponent $-5/6-\eps$.

\paragraph{Axioms and the independent check.} The four theorems use only the three standard axioms \path{propext}, \path{Classical.choice}, and \path{Quot.sound} of Lean's logic. Since they also have no hypotheses, every result that their proofs use is proved in Lean, in the two packages or in their dependencies. The upper bounds on the mean bias and on the variance in \cref{inp:hb-rate,inp:qp-rate} and the variance--bias inequality enter the formal proofs as Lean theorems. The first and the last of these come from \cite{hexagon-bias-lean}.

In their source packages, the four theorems also passed an independent check, which has three parts. First, we wrote out their statements a second time, with every definition that they involve, in separate files that import only Mathlib and none of our proofs. For the barrier, these are the definitions of the disorder law, the Hamiltonian, the Gibbs measure, the heat-bath kernels, the total variation distances, and the two quantities displayed above. For the fluctuations and the mean bias, they are the definitions on which \path{MainFluctuationStatement} rests, copied from OpenAI's sources at the commit cited above, and the definitions of the pressures $p_n(\beta)$ and of their limit. A comparison confirmed that each proved theorem has exactly the statement written out in this way. Second, the same comparison confirmed that the proofs use no axioms other than the three above. Third, the statements and proofs of the theorems were exported from the compiled packages, with every declaration on which they depend, including the proofs of all lemmas that they use. This export was checked again by three type checkers for Lean's logic: Lean's own kernel and the external checkers nanoda and con-ron. All three accepted the barrier statements and, in a separate comparison, both the fluctuation theorem and the mean-bias corollary. Declarations of the packages that these proofs do not use are not part of the check.

\section{The formal and the written barrier proofs}\label{sec:formal-barrier}

The formal proof of \cref{thm:barrier-730} does not follow the proof in this paper in every step. The paper proves the local three-replica expansion of \cref{b:prop:os-expansion} with a remainder $O(h^4k^2)$, where $h$ is the width parameter and $k$ the relative imbalance of \eqref{b:eq:os-parameters}, by estimates for smooth limiting flows. The formal proof instead works with finite chains of Gaussian integrations, using the hierarchical recursion of \cite[Section~2]{oai-barriers} and its accompanying Lean development \cite{oai-math-lean}. It proves the upper bound contained in \eqref{b:eq:finiteE-polynomial} for these chains, at the ratio $\vartheta=1/2$ used in the dynamical argument and on a smaller range of the relative imbalance $k$. Its remainder is of order $h^{7/2}k^2+h^3|k|^3$ instead of $h^4k^2$, up to an additional error that tends to zero as the length of the chains tends to infinity. Only this upper bound enters the proof of the uniform cubic deficit in \cref{prop:cub-band}, where the remainder must be small compared with $h^3k^2$. The weaker remainder has this property when $h$ and $|k|$ are small. Either estimate suffices for \cref{thm:barrier-730}; they lead to different constants, not to a different exponent.

In the final counting step, the formal proof and \cref{sec:barrier-proof} use the same kind of count. Only the first reference configuration of each block is counted, and the sign obstruction follows, as in \cref{prop:bp-obstruction}, once a count of the form \eqref{b:dyn:prefix-count} is less than one. The formal proof passes through finite grids of threshold values, while \cref{sec:barrier-proof} uses the uniform law of the thresholds directly. We conclude that the formal proof certifies \cref{thm:barrier-730} and \cref{b:cor:subcritical}, not every intermediate estimate in the paper's proof of them.

\section{The formal proof of the two-sided bound}\label{sec:formal-window}

Our formal proof of \cref{f:window} follows the route of \cite{oai-fluctuations} summarized in \cref{sec:fl-route}, and its mathematics is that of \cite{oai-fluctuations}. We do not claim a new proof of \cref{f:window} or of \cref{f:main}, and we do not claim to have found or repaired a gap in the argument of \cite{oai-fluctuations}. What the formalization supplies is a complete treatment of the particular variational problems to which \cite[Section~7]{oai-fluctuations} applies the general results of \cite[Sections~3--6]{oai-fluctuations}. A paper proof can handle these problems by checking that they satisfy the hypotheses of general statements. In the formal proof, each such check is carried out in full detail. We now describe these problems and the cases that arise, in the notation of \cref{sec:fl-route}.

\paragraph{The comparisons and their minimizers.} The upper bound in \cref{f:window} rests on the comparisons of \cite[Section~7.3]{oai-fluctuations}. For each $\eps\in[\eps_0,\beta^2)$ and $\theta\in[0,1]$, we minimize a functional of the form \cite[(6)]{oai-fluctuations} over overlap quantiles $r$ on the mass interval $[s,1]$. The lower radius is $r(s)=r_{\min}={\bar u}_*/(2M(s))$, and the low-field perturbation of strength $\theta$ acts on $[0,s)$. The general results of \cite{oai-fluctuations} are stated for the comparison families of \cite[Definition~6.1]{oai-fluctuations}, which allow a positive lower radius $r_{\min}$ and an omitted lower interval. All overlap moments in them refer to minimizers of the perturbed paths. In the formal proof, we establish their hypotheses and conclusions for the minimizers of these particular functionals and for the finitely many successive comparisons built from them. For $\theta=0$, and for the successive comparisons, whose unperturbed paths are nonnegative and nondecreasing, we do this throughout the strength interval $0<{\widehat\lambda}\le d^2$. For $\theta>0$, we use the two-sided linear bound only at the fixed final strength ${\widehat\lambda}=d^2$, uniformly in $\eps$, $\theta$, and the minimizer. We use neither a pressure at the unperturbed paths nor an integration in the strength from zero. This restriction is necessary. Let $e_{\widehat\lambda}$ be the function $e$ of \cref{sec:fl-route} at strength ${\widehat\lambda}$. Since $e_{\widehat\lambda}(r_{\min})={\widehat\lambda}$ for large $n$, the perturbed field path is nondecreasing at $s$ only if ${\widehat\lambda} r_{\min}\ge\theta c_Lr_0$, that is, only if ${\widehat\lambda}\ge\theta c_Ld^2/(2\eps)$.

In the formal proof, we use the strength profile $e_{\widehat\lambda}(r)={\widehat\lambda}\tilde f({\widehat\lambda} r/d^3)$, where $\tilde f(t)=1$ for $t\le1$ and $\tilde f(t)=(1+\log t)/t$ for $t>1$. This function has the monotonicity properties required of the profile $f$ in \cite[(6)]{oai-fluctuations}: its logarithmic slope $-t\tilde f'(t)/\tilde f(t)$ equals $0$ for $t\le1$ and $\log t/(1+\log t)$ for $t>1$. It is continuously differentiable but not smooth at $t=1$, and the parts of the argument that use the profile were carried out for this choice.

\paragraph{The three cases.} Consider a minimizer at strength ${\widehat\lambda}$ and a mass $x\ge A_{\widehat\lambda}$ at which the two-sided linear bound fails, while it holds above $2x$, as in the propagation argument of \cref{sec:fl-route}. There are three cases.
\begin{enumerate}
\item \emph{Comparable strength at a small scale}: $x\to0$ and $e_{\widehat\lambda}(x)/x^2$ is bounded below. This case is excluded by the linear limit of \cite[Theorem~5.1]{oai-fluctuations}, which has to be proved for minimizers with the positive lower radius $r_{\min}$. Terms involving $r_{\min}$ enter as multiples of ${\widehat\lambda} r_{\min}$, and they are negligible at the cubic scale. To see this, recall from \cite[(3) and (5)]{oai-fluctuations} that
\[
 \underline x=\frac{\sqrt nE}{d},\qquad {\bar u}_*=\frac1{nE},\qquad L=\min\Bigl\{{\bar u}_*,\frac{\underline x^3}{d^2}\Bigr\},
\]
and that the upper bound on the derivative of the inverse quantile is
\[
 M({\bar u})=n^\delta\Bigl[\frac{{\bar u}+{\bar u}_*}{L}+\frac{d^3}{E}({\bar u}+{\bar u}_*)^2\Bigr].
\]
Since $M(s)\ge n^\delta {\bar u}_*/L$ and $r_{\min}={\bar u}_*/(2M(s))$, and since $x\ge A_{\widehat\lambda}\ge n^\delta\underline x\ge\underline x$ for ${\widehat\lambda}\le d^2$, we have
\[
 {\widehat\lambda} r_{\min}\le\frac{{\widehat\lambda} L}{2n^\delta}\le\frac{d^2L}{2n^\delta}\le\frac{\underline x^3}{2n^\delta}\le\frac{x^3}{2n^\delta}.
\]
We use the one-site cavity estimate of \cite[Proposition~4.1]{oai-fluctuations} at the two scales $x$ and $\bar x=\max\{x,\sqrt{e_{\widehat\lambda}(x)}\}$, and $\bar x/x$ may tend to infinity; for example, $\bar x/x=n^{5m-\delta}$ when ${\widehat\lambda}=d^2$ and $x=A_{d^2}$. The identification of the limiting scalar and overlap distributions \cite[Lemma~5.4]{oai-fluctuations} and the first variations on gaps of the limiting support \cite[Lemma~5.6]{oai-fluctuations} are carried out for these minimizers and their admissible competitors.
\item \emph{Vanishing strength}: $e_{\widehat\lambda}(x)/x^2\to0$, either with $x\to0$ or with $x$ bounded below. The next comparison is built from the perturbed paths of this minimizer, as in \cite[Lemma~6.2]{oai-fluctuations}. The alternative described in \cref{sec:fl-route} is then followed for a fixed number $N=\lceil10/\delta\rceil$ of successive comparisons \cite[Lemmas~6.3--6.5 and Proposition~6.6]{oai-fluctuations}. At each stage, we exclude a violation with strength comparable to the scale as in the first or the third case.
\item \emph{Macroscopic scale}: $x$ and $e_{\widehat\lambda}(x)/x^2$ are bounded below, and $d^2\le\beta^2-c_*$ for a fixed $c_*>0$. We exclude this case using the convergence of the minimizers to the quantile function of the Parisi measure \cite[Proposition~5.9]{oai-fluctuations}, applied to the comparison at hand.
\end{enumerate}
The successive comparisons follow one of the two covariance alternatives of \cite[Definition~6.1 and Corollary~6.7]{oai-fluctuations}. If $\eps<\beta^2/4$, the perturbed matrix covariance is bounded below by a fixed positive constant, and they act on all masses. If $\eps\ge\beta^2/4$, they act only on $[s,1]$, with lower radius zero, and the matrix covariance has a jump of at least $\beta^2/4-d^2$ at $s$. We then choose the coefficient $c_0$ so that the total matrix covariance $Nc_0^2$ removed by the successive comparisons is less than half of a fixed positive constant that bounds this jump from below; their perturbed matrix paths therefore remain nondecreasing at $s$. For $\theta>0$, we establish the two-sided linear bound at the fixed final strength $d^2$ \cite[Lemma~7.2]{oai-fluctuations} and the bound $S_{\bar u}\le Cd^3/\eps$ on $[0,s)$ \cite[Lemma~7.3]{oai-fluctuations}, uniformly in $\eps$ and $\theta$. These two estimates enter the integration in $\theta$ of \cite[Lemma~7.4]{oai-fluctuations}. As in \cite{oai-fluctuations}, the constants in the two-sided linear bound, the depth $N$, and the coefficient $c_0$ are fixed before $n\to\infty$. The final threshold in $n$ is uniform in $\eps$, $\theta$, and the minimizer, and in ${\widehat\lambda}$ over the strengths at which each comparison is used.

\section{The upper bound}\label{sec:formal-upper}
\statusFormalUpper

\section{Scope and availability}\label{sec:formal-scope}

The combined check concerns the five final statements described above; the source-package checks of \cref{sec:formal-statements} concern the four lower-bound and fluctuation results described there. Intermediate results in their proofs, such as \cref{f:window}, are checked only as parts of these proofs. The quantitative and almost-sure forms in \cref{thm:barrier-strong} are not formally verified. These are the bounds on the probability of the exceptional disorders and on the Gibbs mass of the exceptional initial states, the lower bounds on the distance that tend to $1/2$, the extension to the Metropolis dynamics and to other single-site dynamics with bounded rates, the bound on the relaxation time, and the almost-sure statements for arbitrary couplings of the disorders across system sizes. The sharper local expansion of \cref{b:prop:os-expansion} is not formally verified either; see \cref{sec:formal-barrier}. The proposition \path{MainFluctuationStatement} concerns exponents only, and the limit law and the positive variance prefactor claimed in \cite{oai-limiting-law} are not part of it. The proposition \path{LowerBoundStatement} contains the loss $n^{-\eps}$ for an arbitrary $\eps>0$, and the bound \eqref{eq:mean-bias-prefactor-target}, which would follow from a positive variance prefactor, is not part of it.

\statusFormalUpperScope

OpenAI's Lean sources \cite{oai-math-lean} and the development
\cite{hexagon-bias-lean} are public. The combined sources, independent
\path{Challenge.lean}, and verification records are available in
\url{https://github.com/pmaronow/fluctuations-palomar}; the records
identify the checked proof-source revision and the scope of each check.
The original two source packages and the historical upper-bound package
are maintained in the private repository
\url{https://github.com/pmaronow/fluctuations-temp}. The latter was checked
at commit \path{352426b9007b6bbb31a3c8cfe6887caa0badf3fd}, in its
\path{glauber/lean} directory. The historical records remain available
from the authors.
\chapter{Open problems}\label{sec:open}

We collect some questions that this paper leaves open. The first four concern
the dynamics, and the last two concern the law of $F_n$ and the mean bias.
We use the mixing times from a fixed configuration with threshold $1/4$
defined in \eqref{eq:model-mixing-time}.

\paragraph{The exponent of the typical mixing time.}
By \cref{cor:two-sided}, for every fixed $\beta>1$ and every fixed
$\omega\in(0,1/4)$, the realized initial configuration $X_0\sim\pi^J$
satisfies $e^{A_\beta n^{7/30}}<T^J(X_0)\le e^{n^{1-\omega}}$ with
probability tending to one. The same bounds hold in discrete time, with
the integer parts of the two time horizons. We do not know whether there
exists an exponent $\zeta$ such that $\log T^J(X_0)=n^{\zeta+o_\PP(1)}$, in
the sense of \cref{sec:model-asymptotics}. If it exists, then
$7/30\le\zeta\le3/4$. The numerical and analytical work cited in
\cref{sec:intro-dynamics} suggests that $\zeta=1/3$
\cite{kinzelbach-horner-1991,billoire-marinari-2001,monthus-garel-2009,billoire-2010}.
Billoire also studied the distribution of the logarithm of the relaxation
time over samples of the disorder~\cite{billoire-2010}. In the same way, one
may ask for the fluctuations of $\log T^J(X_0)$ over $J$ and $X_0$.

\paragraph{The limits of the present argument.}
In the proof of \cref{thm:barrier-730}, the upper bound on the mean bias is
used to make the cubic deficit, of order $\rho^3$ at the scale
$\rho=Rn^{-\lambda}$, dominate $\Delta_{n,\beta}$. Suppose more generally that
$\Delta_{n,\beta}\le C_\beta n^{-\gamma}$ for some fixed $\gamma>0$, and retain the
variance bound $\Var(F_n)\le C_v n^{7/15}$. We set the tolerance of the
overlap constraints at fixed scales to $\eta_n=n^{-\xi}$, where the exponents
$(\lambda,\xi)$ satisfy \eqref{eq:ref-exponents} and $3\lambda\le \gamma$; that is,
\[
 0<\lambda\le \gamma/3,\qquad \lambda<1/4,\qquad \lambda<\xi<\min\{1/4,3\lambda/2\}.
\]
We check that the conditions on the scales in the proof of
\cref{thm:barrier-730} still hold. Indeed, the
interval for $\xi$ is nonempty; it ensures that $\eta_n=o(\rho)$,
$\rho^3=o(\eta_n^2)$, and $n\rho^4/\log n,\,n\eta_n^4/\log n\to\infty$; and the
Chebyshev bound for the denominator vanishes since $\lambda<1/4<23/90$.
Choosing $R$ with a fixed margin when $3\lambda=\gamma$, we obtain the time
$\exp(An^{\lambda})$ for some $A>0$. The barrier exponent is therefore one third of
the exponent of the upper bound on the mean bias, as long as this third is
below $1/4$: the case $0<\gamma<3/4$ permits $\lambda=\gamma/3$, while $\gamma\ge3/4$ permits
every fixed $\lambda<1/4$. In the latter case, choosing a slightly larger
admissible power and using the contraction of total variation also gives the
time $\exp(n^\lambda)$ with coefficient one. The upper bound $n^{-7/10}$
of~\cite[Theorem~1.1]{hexagon-bias} is the case $\gamma=7/10$, and we obtain the
exponent $\lambda=7/30$ of \cref{thm:barrier-730}. The exponent $1/4$ itself
is not reached: at $\rho=Rn^{-1/4}$, the quantity $n\rho^4$ is bounded, and
the union bound over polynomially many constrained partition functions in the
disorder event fails. The prediction $\Delta_{n,\beta}\approx n^{-3/4}$ of
Franz, Parisi, and Ricci-Tersenghi~\cite{franz-parisi-ricci-2026} lies exactly
at this threshold. We expect that the exponent $1/4$, and the value $1/3$
suggested by physics, can be reached only with new ideas, such as overlap
constraints with a larger pressure deficit or bounds on constrained partition
functions that hold simultaneously without a union over constraints.

For the upper bound, the sufficient conditions of \cref{u:prop:range}
require
\[
 \omega<\ex{q}<\ex{s}_{\rm b}<\ex{s}_{\rm seq}<\ex{e}
 <\ex{u}_{\rm E}<\ex{k}_{\rm f}/2\le1/4.
\]
Both arguments stop at the exponent $1/4$. We do not know whether this
coincidence has a common explanation.

\paragraph{Dependence on the temperature.}
The coefficient $A_\beta$ depends on $\beta$ through the scalar Parisi
problem, through the constant in the upper bound on the mean bias
of~\cite{hexagon-bias}, and through the constant in the upper bound on the
variance of~\cite{quantitative-parisi}. The estimates used here do not control
these constants uniformly as $\beta\downarrow1$ or $\beta\to\infty$.
At $\beta=1$, the mixing time is
$n^{2/3+o(1)}$~\cite[Theorem~1.1]{oai-critical-mixing}. Billoire's
simulations with binary couplings suggest an approximate collapse of the
centered variable $\log\tau_J-\overline{\log\tau}$ after division by
$n^{1/3}(\beta-1)$, with visible corrections near the
transition~\cite[equation~(6), Figure~3 and the following discussion]{billoire-2010}.
Here $\tau_J$ is an equilibrium autocorrelation time that depends on the
disorder sample, and $\overline{\log\tau}$ is the disorder average of
$\log\tau_J$.
We ask for a description of the crossover between the two regimes in a window
of temperatures around $\beta=1$, and for bounds that remain effective as
$\beta\to\infty$.

\paragraph{Worst-case and typical initial configurations.}
Sellke proved that the mixing time from the worst initial configuration is
exponential in $n$ when $\beta$ exceeds an absolute
constant~\cite[Theorem~1.1]{sellke-2025}. By \cref{thm:upper}, the mixing
time from a typical equilibrium configuration is at most
$\exp(n^{1-\omega})$ for every fixed $\omega\in(0,1/4)$, with joint
probability tending to one, whereas Sellke's worst-start bound is
exponential in $n$ at sufficiently large $\beta$. For every $\beta>1$,
\cref{thm:barrier-strong} implies that, with probability tending to one, the
relaxation time of the heat-bath and Metropolis dynamics is at least
$e^{An^{7/30}}/(Cn^{7/30})$, so that their spectral gaps are at most
$Cn^{7/30}e^{-An^{7/30}}$. Is the spectral gap exponentially small in $n$ for
every $\beta>1$, and is the mixing time from the worst initial configuration
exponential in $n$? The sufficient hypotheses
of~\cite[Theorem~1.9]{ben-arous-jagannath-2018} include the vanishing of the
second derivative at zero of the covariance function, and exclude
the quadratic covariance of SK. Their general overlap-barrier
criterion~\cite[Theorem~1.3]{ben-arous-jagannath-2018} remains available if
its barrier hypothesis can be verified. A second question concerns the size
of the distance. By \cref{thm:barrier-strong}, the distance from equilibrium
of the chain started from most initial configurations is at least
$1/2-Ce^{-cn^{7/30}}$ at every time $t\le e^{An^{7/30}}$. The value $1/2$
comes from spin-flip symmetry, which gives Gibbs mass at least $1/2$ to every
half-space $\{y:R(v,y)\le0\}$, and it is the limit of the argument. Does the
conclusion of \cref{thm:barrier-730} hold for every fixed $\tau<1$?

\paragraph{The limiting law of $F_n$.}
\Cref{f:main} identifies the exponent of the fluctuations but not their law.
OpenAI states that $n^{-1/3}\Var(F_n)$ converges to a positive constant
$a_2(\beta)$, and that $(F_n-\E F_n)/\sqrt{\Var(F_n)}$ converges in law to a
nondegenerate limit $\nu_\beta$~\cite[Theorems~1.1 and~1.2]{oai-limiting-law}.
We do not use these statements. The limit $\nu_\beta$ is not identified
with a known distribution. Parisi and Rizzo predicted upward deviations of
$F_n/n$ with speed $n$ and power $6/5$, and downward deviations with speed
$n^2$ and power $12/5$~\cite[equations~(12)--(15)]{parisi-rizzo-2010}. If
these tails match the typical fluctuations, the right and left tails of
$\nu_\beta$ should be of the forms $\exp(-cx^{6/5})$ and $\exp(-cx^{12/5})$.
This conditional passage from large-deviation tails to a typical scale follows
the matching argument of \cite[Section~VI]{parisi-rizzo-2009}.
On the rigorous side, \cite[Theorem~1.5]{quantitative-parisi} gives two-sided
bounds with the power $6/5$ on the probability that $F_n/n$ exceeds
$P_\beta+x$, for $x$ in a range $C_\beta n^{-2/3}\le x\le x_\beta$. The lower
end of this range is of the order of the upper bound on the mean bias
in~\cite[Theorem~1.4]{quantitative-parisi}. Bounds of this type for
deviations from $\E F_n/n$, down to the fluctuation scale $n^{-5/6}$, would
make the matching argument of Parisi and Rizzo rigorous for upward
deviations.

\paragraph{The exponent of the mean bias.}
By \cref{cor:mean-bias}, we have
$\Delta_{n,\beta}\ge c_{\beta,\varepsilon}n^{-5/6-\varepsilon}$ for every
fixed $\varepsilon>0$, and by~\cite[Theorem~1.1]{hexagon-bias}, we have
$\Delta_{n,\beta}\le C_b n^{-7/10}$. If
$\Delta_{n,\beta}=n^{-a+o(1)}$ for some $a$, then $7/10\le a\le5/6$. Franz,
Parisi, and Ricci-Tersenghi predict $a=3/4$, possibly with logarithmic
corrections~\cite{franz-parisi-ricci-2026}. The lower bound
cannot be improved by the variance--bias inequality
\eqref{eq:intro-variance-deficit} alone. Since $\Var(F_n)=n^{1/3+o(1)}$, this
inequality gives no lower bound with an exponent smaller than $5/6$, and such
a bound would require a different argument. On the upper side, the
discussion in~\cite[Section~1.4]{hexagon-bias} explains which features of the
proof of~\cite[Theorem~1.1]{hexagon-bias} account for the difference between
$7/10$ and $3/4$. The estimates there do not control the
constants uniformly as $\beta\to\infty$, so the corresponding question for the
ground-state energy is also open.

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