\chapter{Deterministic covariance evolution and heat estimates}\label{sec:heat}

This section collects deterministic facts about the exponential of a diagonal matrix perturbed by a matrix of rank one. For positive precisions $p_1,\dots,p_N$ with $\sum_ip_i^{-1}=N$, we study the heat matrix $H=-2\diag(p)+(2/N)\mathbf 1\mathbf 1^\T$, the heat vector $h(t)=e^{tH}\mathbf 1$, and its mean $\bar h(t)$. These objects describe the evolution of Gaussian variances that we use at high temperature to approximate the time-$t$ law, and they also describe the second moments of the unnormalized process of \cref{sec:process}. In \cref{sec:heat-algebra}, we prove the algebraic facts once (\cref{lem:heat-algebra}): positivity of $e^{tH}$, the kernel of $H$ and the weight of its zero eigenvalue, a representation of $\bar h$ by a positive spectral measure, the coordinate Duhamel formula, and the resolvent of $H$. We also prove an exact formula for the information in terms of $\bar h$ (\cref{mc-exact-info}) and rescale the spectral measure at the scale of the smallest precision (\cref{lem:heat-rescaled}). In \cref{sec:heat-covariance}, we define the covariance evolution and the Bingham comparison law $\mu_t$ of a symmetric matrix $A$, and prove their basic properties (\cref{ht-covariance-system,ht-effective-normalization}). The next two subsections prove estimates under two different hypotheses on the precisions. In \cref{sec:heat-hot}, the smallest precision $d$ is at most one and the other precisions exceed $d$ by an amount of order $((i-1)/N)^{2/3}$, which is the situation at high temperature (\cref{hi3-finite-heat,iw-all-time-heat,hi3-information-envelopes,hmr-heat-derivatives}). In \cref{sec:heat-critical}, the half-precisions $r_i=p_i/2$ are of order $(i/N)^{2/3}$ for all $i$, including $i=1$, which is the situation in the critical window (\cref{cd-heat-profiles}). Finally, in \cref{pr-sec:real-measures}, we show that convergence of the real squared resolvents of positive measures implies their convergence against a large class of test functions (\cref{pr-real-measure-transfer}).

We use only the definitions of \cref{sec:model}, namely the operators \eqref{eq:model-general} and the Lagrange multiplier of \cref{def:model-multiplier}, and the identification of Bingham laws with conditioned Gaussian laws in \cref{prop:cond-bingham}. All statements are deterministic, except for the random version of \cref{pr-real-measure-transfer}. The covariance evolution and the Bingham comparison law are used in \cref{sec:comparison} and in \cref{part:hot}. The algebra of \cref{lem:heat-algebra} is used in \cref{sec:process,sec:hot-meso}, and the estimates of \cref{sec:heat-hot} are used in \cref{sec:process,sec:hot-meso,sec:outer-hot}. \Cref{cd-heat-profiles} is used in \cref{sec:inequalities,sec:process,sec:density,sec:crit-process,sec:crit-profile}, and \cref{mc-exact-info,pr-real-measure-transfer} are used in \cref{sec:hot-meso,sec:outer-hot}.

\paragraph{Notation.}
Throughout this section, $N\ge2$ and $p=(p_1,\dots,p_N)$ is a sequence of positive numbers such that
\begin{equation}\label{eq:heat-trace}
\sum_{i=1}^N\frac1{p_i}=N .
\end{equation}
Recall from \cref{def:model-multiplier} that the precisions $p_i=z-\alpha_i$ of a real symmetric matrix $A$ with eigenvalues $\alpha_1\ge\dots\ge\alpha_N$ satisfy \eqref{eq:heat-trace}. Conversely, every sequence $p$ satisfying \eqref{eq:heat-trace} arises in this way up to a permutation of the eigencoordinates, since the Lagrange multiplier of $A=-\diag(p)$ is $z=0$. We write
\begin{equation}\label{ht-effective-normalization-definition}
q_i=\frac1{p_i},\qquad d=\min_ip_i,\qquad s=\max_ip_i,\qquad \vartheta=\frac N{\sum_{i=1}^Nq_i^2} .
\end{equation}
For the matrix $A$, we have $\vartheta=N/\Tr Q^2$ with $Q=(zI-A)^{-1}$. We call $\vartheta$ the weight of the zero eigenvalue; \cref{lem:heat-algebra} below explains the name. We distinguish this weight from the low-temperature top fraction $\theta_N$. Let $\mathbf 1=(1,\dots,1)^\T$. We define the heat matrix, the heat vector, and its mean by
\begin{equation}\label{eq:heat-objects}
H=-2\diag(p)+\frac2N\mathbf 1\mathbf 1^\T,\qquad h(t)=e^{tH}\mathbf 1,\qquad \bar h(t)=\frac1N\sum_{i=1}^Nh_i(t),\qquad t\ge0 .
\end{equation}
The heat vector $h(t)$ has positive coordinates (\cref{lem:heat-algebra} below). We define the variances and the relative errors, and recall the information $\cI_N(t)=\cI(q,v(t))$ of $v(t)$ relative to $q$ from \cref{def:model-information}:
\begin{equation}\label{eq:heat-errors}
v_i(t)=\frac{h_i(t)}{\bar h(t)},\qquad e_i(t)=\frac{v_i(t)}{q_i}-1=p_iv_i(t)-1,\qquad \cI_N(t)=\frac12\sum_{i=1}^Ne_i(t)^2 .
\end{equation}
For a matrix $A$, we show in \cref{sec:heat-covariance} that $v(t)$ is the covariance evolution of $A$; in particular, \eqref{eq:heat-errors} agrees with \eqref{eq:intro-information} and with \cref{def:model-information}. Finally, for complex $\zeta$ with $\zeta\ne-2p_i$ for all $i$, let
\begin{equation}\label{eq:heat-GS}
\mathsf G(\zeta)=\frac1N\sum_{i=1}^N\frac1{p_i+\zeta/2},\qquad
\mathsf S(\zeta)=\frac1N\sum_{i=1}^N\frac1{p_i(p_i+\zeta/2)},
\end{equation}
and for a locally integrable function $f$ on $[0,\infty)$ of at most exponential growth, let $\widehat f(\zeta)=\int_0^\infty e^{-\zeta t}f(t)\,dt$ be its Laplace transform, for real $\zeta$ large enough.

\section{The heat matrix}\label{sec:heat-algebra}

The following lemma contains the algebra used in all later estimates. A real square matrix whose off-diagonal entries are nonnegative is called a Metzler matrix, after \cite{Metzler1945}. The positivity of its exponential, which we use below, follows from the series argument in the proof of the first part; for background on positive linear systems, see \cite[Chapter~2, pp.~7--15]{FarinaRinaldi2000}.

\begin{lemma}[The heat matrix]\label{lem:heat-algebra}
Let $p$ satisfy \eqref{eq:heat-trace}.
\begin{enumerate}
\item The matrix $H$ is symmetric, and its off-diagonal entries are positive. For every $t\ge0$, the matrix $e^{tH}$ has nonnegative entries, and $h_i(t)\ge e^{-2p_it}>0$ for all $i$.
\item We have $d\le1\le s$ and $d\le\vartheta\le1$. The matrix $H$ is negative semidefinite, and its kernel is spanned by the vector $q=(q_1,\dots,q_N)^\T$. Every eigenvalue of the restriction of $-H$ to the orthogonal complement of $q$ lies in $[2d,2s]$.
\item There exists a positive measure $\rho$ on $[2d,2s]$ with total mass $1-\vartheta$ such that
\begin{equation}\label{eq:heat-spectral}
\bar h(t)=\vartheta+\int e^{-\lambda t}\,\rho(d\lambda),\qquad t\ge0 .
\end{equation}
In particular, $\bar h$ is completely monotone, $\bar h(0)=1$, $\vartheta\le \bar h(t)\le1$, and $\bar h(t)$ decreases to $\vartheta$ as $t\to\infty$. For every integer $j\ge1$ and all $0\le t_0\le t$ with $t>0$,
\begin{equation}\label{eq:heat-derivatives}
\begin{gathered}
0\le(-1)^j\bar h^{(j)}(t)\le(2s)^j\big(\bar h(t)-\vartheta\big),\qquad
(-1)^j\bar h^{(j)}(t)\le\Big(\frac{2j}{et}\Big)^j\big(\bar h(t/2)-\vartheta\big),\\
(-1)^j\bar h^{(j)}(t)\le e^{-2d(t-t_0)}(-1)^j\bar h^{(j)}(t_0).
\end{gathered}
\end{equation}
\item For all $t\ge0$ and $1\le i,j\le N$,
\begin{equation}\label{eq:heat-entries}
(e^{tH})_{ij}=e^{-2p_it}\delta_{ij}+\frac2N\int_0^te^{-2p_i(t-u)}h_j(u)\,du,
\end{equation}
and in particular
\begin{equation}\label{eq:heat-duhamel}
h_i(t)=e^{-2p_it}+2\int_0^te^{-2p_i(t-u)}\bar h(u)\,du .
\end{equation}
With $k(t)=N^{-1}\sum_ie^{-2p_it}$, we have $\bar h(t)=k(t)+2\int_0^tk(t-u)\bar h(u)\,du$.
\item For all $t\ge0$, we have $\sum_iq_ih_i(t)=N$.
\item For every complex $\zeta\notin\{0\}\cup[-2s,-2d]$, the matrix $\zeta I-H$ is invertible, $\mathsf G(\zeta)\ne1$, and
\begin{equation}\label{hi3-hot-renewal-transform}
\frac1N\mathbf 1^\T(\zeta I-H)^{-1}\mathbf 1=\frac\vartheta\zeta+\int\frac{\rho(d\lambda)}{\zeta+\lambda}=\frac{\mathsf G(\zeta)}{2\big(1-\mathsf G(\zeta)\big)},\qquad
1-\mathsf G(\zeta)=\frac\zeta2\,\mathsf S(\zeta).
\end{equation}
For real $\zeta>0$, these quantities are equal to $\widehat{\bar h}(\zeta)$.
\item For all $t\ge0$ and all $i$, we have $\sum_iv_i(t)=N$, $\sum_iq_ie_i(t)=0$, and
\begin{equation}\label{eq:heat-derivative-identity}
h_i'(t)=-2\bar h(t)e_i(t),\qquad \bar h'(t)=-\frac{2\bar h(t)}N\sum_{j=1}^Ne_j(t).
\end{equation}
\end{enumerate}
\end{lemma}

\begin{proof}
We prove the parts in order, except that the lower bound on $h_i(t)$ in the first part uses \eqref{eq:heat-duhamel}, whose proof does not use it.

\paragraph{Part 1.}
The off-diagonal entries of $H$ are equal to $2/N$. Since the diagonal entries of $H+2sI$ are $2(s-p_i)+2/N\ge0$, the matrix $H+2sI$ and all its powers have nonnegative entries. The expansion
\[
e^{tH}=e^{-2st}\sum_{n=0}^\infty\frac{t^n(H+2sI)^n}{n!}
\]
shows that $e^{tH}$ has nonnegative entries for $t\ge0$. In particular, $h(t)$ has nonnegative coordinates and $\bar h(t)\ge0$. The integral in \eqref{eq:heat-duhamel} is nonnegative, and $h_i(t)\ge e^{-2p_it}$.

\paragraph{Part 2.}
The numbers $q_i$ lie in $[1/s,1/d]$, and their average is one by \eqref{eq:heat-trace}. Hence $d\le1\le s$. By the Cauchy--Schwarz inequality, we have $N^2=(\sum_iq_i)^2\le N\sum_iq_i^2$, that is, $\vartheta\le1$. Also $\sum_iq_i^2\le d^{-1}\sum_iq_i=N/d$, that is, $\vartheta\ge d$. For $y\in\R^N$, we have
\begin{equation}\label{eq:heat-form}
-y^\T Hy=2\sum_ip_iy_i^2-\frac2N\Big(\sum_iy_i\Big)^2 .
\end{equation}
Writing $y_i=\sqrt{q_i}\cdot\sqrt{p_i}\,y_i$ and using the Cauchy--Schwarz inequality and \eqref{eq:heat-trace}, we find $(\sum_iy_i)^2\le\sum_iq_i\sum_ip_iy_i^2=N\sum_ip_iy_i^2$, with equality if and only if $(\sqrt{p_i}\,y_i)_i$ is proportional to $(\sqrt{q_i})_i$, that is, if and only if $y$ is proportional to $q$. We conclude that $-H$ is positive semidefinite and that $y^\T Hy=0$ only for $y\in\operatorname{span}(q)$. Every vector $y$ in the kernel satisfies $y^\T Hy=0$, and $(Hq)_i=-2+2N^{-1}\sum_jq_j=0$. Therefore the kernel is spanned by $q$. Since $H$ is symmetric and $Hq=0$, the orthogonal complement $q^\perp$ is invariant under $H$. Let $y\in q^\perp$. Using $p_iq_i=1$ and $\sum_iq_iy_i=0$, we find $\sum_iy_i=\sum_i(p_i-d)q_iy_i$, where the numbers $p_i-d$ are nonnegative. Then the Cauchy--Schwarz inequality implies that
\[
\Big(\sum_iy_i\Big)^2\le\sum_i(p_i-d)q_i^2\sum_i(p_i-d)y_i^2\le N\sum_i(p_i-d)y_i^2,
\]
where in the last step we used $\sum_i(p_i-d)q_i^2=N-dN/\vartheta\le N$. Inserting this bound into \eqref{eq:heat-form} gives $-y^\T Hy\ge2d|y|^2$. Also $-y^\T Hy\le2\sum_ip_iy_i^2\le2s|y|^2$. This shows that the eigenvalues of $-H$ on $q^\perp$ lie in $[2d,2s]$.

\paragraph{Part 3.}
By the second part, there exists an orthonormal basis $u_0,u_1,\dots,u_{N-1}$ of $\R^N$ with $u_0=q/|q|$ and $Hu_k=-\lambda_ku_k$, where $\lambda_0=0$ and $\lambda_k\in[2d,2s]$ for $k\ge1$. Then
\[
\bar h(t)=\frac1N\mathbf 1^\T e^{tH}\mathbf 1=\sum_{k=0}^{N-1}e^{-\lambda_kt}\,\frac{(\mathbf 1^\T u_k)^2}N .
\]
By \eqref{eq:heat-trace}, the term $k=0$ is $(\sum_iq_i)^2/(N\sum_iq_i^2)=\vartheta$. Let $\rho=\sum_{k\ge1}N^{-1}(\mathbf 1^\T u_k)^2\delta_{\lambda_k}$. Its mass is $N^{-1}|\mathbf 1|^2-\vartheta=1-\vartheta$ by Parseval's identity, and \eqref{eq:heat-spectral} holds. Differentiating the finite sum, we have $(-1)^j\bar h^{(j)}(t)=\int\lambda^je^{-\lambda t}\rho(d\lambda)\ge0$. In particular, $\bar h$ is completely monotone and decreases from $\bar h(0)=1$ to its limit $\vartheta$. For $\lambda\in[2d,2s]$, we have $\lambda^je^{-\lambda t}\le(2s)^je^{-\lambda t}$ and $\lambda^je^{-\lambda t}\le e^{-2d(t-t_0)}\lambda^je^{-\lambda t_0}$. Since $\sup_{x>0}x^je^{-xt/2}=(2j/(et))^j$, we also have $\lambda^je^{-\lambda t}\le(2j/(et))^je^{-\lambda t/2}$. Integrating these three inequalities against $\rho$ and recalling that $\int e^{-\lambda t}\rho(d\lambda)=\bar h(t)-\vartheta$, we obtain \eqref{eq:heat-derivatives}.

\paragraph{Part 4.}
Since $e^{tH}$ is symmetric, we have $\sum_k(e^{tH})_{kj}=(e^{tH}\mathbf 1)_j=h_j(t)$. The equation $\frac d{dt}e^{tH}=He^{tH}$ reads, entrywise,
\[
\frac d{dt}(e^{tH})_{ij}=-2p_i(e^{tH})_{ij}+\frac2N\sum_k(e^{tH})_{kj}=-2p_i(e^{tH})_{ij}+\frac2Nh_j(t),
\]
and the variation of constants formula with $(e^{0H})_{ij}=\delta_{ij}$ yields \eqref{eq:heat-entries}. Summing \eqref{eq:heat-entries} over $j$ gives \eqref{eq:heat-duhamel}, and averaging \eqref{eq:heat-duhamel} over $i$ gives the equation for $\bar h$.

\paragraph{Part 5.}
Since $Hq=0$, we have $e^{tH}q=q$. By the symmetry of $e^{tH}$, we find $\sum_iq_ih_i(t)=q^\T e^{tH}\mathbf 1=(e^{tH}q)^\T\mathbf 1=\sum_iq_i=N$.

\paragraph{Part 6.}
The eigenvalues of $\zeta I-H$ are $\zeta$ and $\zeta+\lambda_k$, $k\ge1$, which do not vanish for $\zeta\notin\{0\}\cup[-2s,-2d]$. Expanding $\mathbf 1$ in the basis of the third part gives the first identity in \eqref{hi3-hot-renewal-transform}. Let $y=(\zeta I-H)^{-1}\mathbf 1$ and $\bar y=N^{-1}\sum_iy_i$. The $i$th coordinate of $(\zeta I-H)y=\mathbf 1$ reads $(\zeta+2p_i)y_i-2\bar y=1$. Since $\zeta+2p_i\ne0$, we obtain $y_i=(1+2\bar y)/(\zeta+2p_i)$. Averaging over $i$, we find $\bar y=(1+2\bar y)\,\mathsf G(\zeta)/2$, that is,
\[
\bar y\big(1-\mathsf G(\zeta)\big)=\mathsf G(\zeta)/2 .
\]
If $\mathsf G(\zeta)=1$, this identity would read $0=1/2$. Therefore $\mathsf G(\zeta)\ne1$ and $\bar y=\mathsf G(\zeta)/(2(1-\mathsf G(\zeta)))$. By \eqref{eq:heat-trace}, $1-\mathsf G(\zeta)=N^{-1}\sum_i(q_i-(p_i+\zeta/2)^{-1})=(\zeta/2)\,\mathsf S(\zeta)$. For real $\zeta>0$, we have $\widehat{\bar h}(\zeta)=\vartheta/\zeta+\int\rho(d\lambda)/(\zeta+\lambda)$ by \eqref{eq:heat-spectral} and Fubini's theorem.

\paragraph{Part 7.}
We have $\sum_iv_i=\sum_ih_i/\bar h=N$ by \eqref{eq:heat-objects}, and $\sum_iq_ie_i=\sum_iv_i-\sum_iq_i=0$ by \eqref{eq:heat-trace}. The equation $h'=Hh$ reads $h_i'=-2p_ih_i+2\bar h=-2\bar h(p_iv_i-1)=-2\bar h\,e_i$. Averaging over $i$ gives the formula for $\bar h'$.
\end{proof}

The positivity in the first part holds more generally for the solutions of the linear system $y'=My$ for every real matrix $M$ with nonnegative off-diagonal entries: if $y(0)$ has nonnegative coordinates, then $y(t)=e^{tM}y(0)$ has nonnegative coordinates for all $t\ge0$. The proof is the same, with $H+2sI$ replaced by $M+\tau I$ for $\tau$ large. We use this in the proof of \cref{ht-covariance-system}.

\begin{remark}[Precisions and half-precisions]\label{rem:heat-conventions}
The critical sections use the half-precisions $r_i=p_i/2$. In terms of these, the trace condition \eqref{eq:heat-trace} and the heat matrix read
\begin{equation}\label{cd-heat-matrix}
\sum_{i=1}^N\frac1{2r_i}=N,\qquad H=-4\diag(r)+\frac2N\mathbf 1\mathbf 1^\T,\qquad h(t)=e^{tH}\mathbf 1,\qquad \sum_{j=1}^Nh_j(t)=N\bar h(t),
\end{equation}
and the matrix $-4\diag(r)+(2/N)\mathbf 1\mathbf 1^\T$ of the critical sections is the same matrix $H$. With $k(t)=N^{-1}\sum_ie^{-4r_it}$, the fourth and sixth parts of \cref{lem:heat-algebra} read $N\bar h=Nk+2k*(N\bar h)$ and, for $\zeta>0$,
\begin{equation}\label{cd-renewal-resolvent}
\widehat{N\bar h}(\zeta)=\frac{N\widehat k(\zeta)}{1-2\widehat k(\zeta)},\qquad
\widehat k(\zeta)=\frac1N\sum_{i=1}^N\frac1{\zeta+4r_i},\qquad
1-2\widehat k(\zeta)=\frac\zeta{2N}\sum_{i=1}^N\frac1{r_i(\zeta+4r_i)} .
\end{equation}
Indeed $2\widehat k(\zeta)=\mathsf G(\zeta)$ and $\mathsf S(\zeta)=N^{-1}\sum_i[r_i(\zeta+4r_i)]^{-1}$. \Cref{tab:heat-conventions} lists the two conventions side by side.

\begin{table}[ht]
\centering
\begin{tabular}{@{}lll@{}}
\toprule
 & precisions $p_i$ & half-precisions $r_i=p_i/2$\\
\midrule
trace condition & $\sum_ip_i^{-1}=N$ & $\sum_i(2r_i)^{-1}=N$\\
heat matrix $H$ & $-2\diag(p)+(2/N)\mathbf 1\mathbf 1^\T$ & $-4\diag(r)+(2/N)\mathbf 1\mathbf 1^\T$\\
kernel of $H$ & spanned by $(p_i^{-1})_i$ & spanned by $(r_i^{-1})_i$\\
decay in \eqref{eq:heat-entries} & $e^{-2p_it}$ & $e^{-4r_it}$\\
limit of $\sum_jh_j(t)$ & $N\vartheta=N^2/\sum_ip_i^{-2}$ & $4N^2/\sum_ir_i^{-2}$\\
\bottomrule
\end{tabular}
\caption{The heat matrix in terms of precisions and of half-precisions.}\label{tab:heat-conventions}
\end{table}

At low temperature the potential is written with the scaled eigenvalue gaps $a_i=c(\lambda_1-\lambda_i)/2$ of \eqref{eq:model-raw}, and $a_i=r_i-d/2$. More generally, for a real number $\tau$, the matrix built from the coefficients $r_i-\tau/2$ is $-4\diag(r_i-\tau/2)+(2/N)\mathbf 1\mathbf 1^\T=H+2\tau I$, and its exponential is $e^{2\tau t}e^{tH}$. For $\tau=d$, the zero eigenvalue of $H$ becomes the eigenvalue $2d$, and the exponential grows at rate $2d$ instead of converging.
\end{remark}

The next lemma expresses the information through the function $\bar h$ alone. This is how the information is computed in \cref{sec:hot-meso}.

\begin{lemma}[An exact formula for the information]\label{mc-exact-info}
Let $p$ satisfy \eqref{eq:heat-trace}. For every $t\ge0$,
\begin{equation}\label{mc-info-formula}
\cI_N(t)=\frac12\sum_{i=1}^Ne_i(t)^2=\frac{N\,\bar h''(2t)}{8\,\bar h(t)^2} .
\end{equation}
Moreover, there exists a finite positive measure $\rho$ supported on $[2d,\infty)$ such that
\begin{equation}\label{mc-positive-measure}
\bar h(t)=\vartheta+\int e^{-\lambda t}\,\rho(d\lambda),\qquad \rho([2d,\infty))=1-\vartheta .
\end{equation}
\end{lemma}

\begin{proof}
The measure $\rho$ of \cref{lem:heat-algebra} has the required properties. Since $H$ is symmetric and commutes with $e^{tH}$,
\[
\bar h''(2t)=\frac1N\mathbf 1^\T H^2e^{2tH}\mathbf 1=\frac1N\big(He^{tH}\mathbf 1\big)^\T\big(He^{tH}\mathbf 1\big)=\frac1N\big|Hh(t)\big|^2 .
\]
By \eqref{eq:heat-derivative-identity}, $(Hh(t))_i=h_i'(t)=-2\bar h(t)e_i(t)$. Inserting this into the last display gives $\bar h''(2t)=4\bar h(t)^2N^{-1}\sum_ie_i(t)^2=8\bar h(t)^2\cI_N(t)/N$, which is \eqref{mc-info-formula}.
\end{proof}

At high temperature the relevant times are of order $d^{-1}$, and the transient part of $\bar h$ is best viewed at that scale. The total mass of the measure $\mathfrak n_N$ below is $(1-\vartheta)/\sqrt d$, which need not stay bounded as $N\to\infty$.

\begin{lemma}[The transient spectral measure at the scale of the smallest precision]\label{lem:heat-rescaled}
Let $p$ satisfy \eqref{eq:heat-trace}, and let $\rho$ be the measure of \eqref{mc-positive-measure}. Let $\mathfrak n_N$ be $d^{-1/2}$ times the image of $\rho$ under the map $\lambda\mapsto\lambda/(2d)-1$; it is a finite positive measure on $[0,s/d-1]$. For $w\ge0$, let
\begin{equation}\label{mc-j-definitions}
f_N(w)=\frac{\bar h(w/d)}{\sqrt d},\qquad
j_N(w)=f_N(w)-\frac\vartheta{\sqrt d}=e^{-2w}\int e^{-2wx}\,\mathfrak n_N(dx).
\end{equation}
Then $f_N(w)$ decreases to $\vartheta/\sqrt d$ as $w\to\infty$. For every complex $\zeta\notin\{0\}\cup[-2s/d,-2]$, and in particular for every real $\zeta\in(-2,\infty)$ with $\zeta\ne0$,
\begin{equation}\label{mc-transform}
\int\frac{\mathfrak n_N(dx)}{\zeta+2+2x}=\frac{\sqrt d}{2\big(1-\mathsf G(d\zeta)\big)}-\frac{\sqrt d}2-\frac\vartheta{\sqrt d\,\zeta} .
\end{equation}
For real $\zeta>0$, the left side of \eqref{mc-transform} is $\widehat{j_N}(\zeta)$, and $\widehat{f_N}(\zeta)=\sqrt d/(2(1-\mathsf G(d\zeta)))-\sqrt d/2$.
\end{lemma}

\begin{proof}
The measure $\mathfrak n_N$ is supported in $[0,s/d-1]$ because $\rho$ is supported in $[2d,2s]$. By definition of $\mathfrak n_N$, we have $\int g(\lambda/(2d)-1)\,\rho(d\lambda)=\sqrt d\int g(x)\,\mathfrak n_N(dx)$ for every bounded measurable $g$. With $\lambda=2d(1+x)$, we find from \eqref{mc-positive-measure} that
\[
\bar h(w/d)-\vartheta=\int e^{-\lambda w/d}\rho(d\lambda)=\sqrt d\int e^{-2w(1+x)}\,\mathfrak n_N(dx),
\]
which is \eqref{mc-j-definitions}. By \cref{lem:heat-algebra}, $f_N$ decreases to $\vartheta/\sqrt d$. Let $\zeta\notin\{0\}\cup[-2s/d,-2]$. Then $d\zeta\notin\{0\}\cup[-2s,-2d]$, and
\[
\int\frac{\rho(d\lambda)}{d\zeta+\lambda}=\frac1d\int\frac{\rho(d\lambda)}{\zeta+\lambda/d}=\frac1{\sqrt d}\int\frac{\mathfrak n_N(dx)}{\zeta+2+2x} .
\]
Inserting this into \eqref{hi3-hot-renewal-transform} at $d\zeta$, multiplying by $\sqrt d$, and writing $\mathsf G/(2(1-\mathsf G))=1/(2(1-\mathsf G))-1/2$, we obtain \eqref{mc-transform}. For real $\zeta>0$, the substitution $t=w/d$ yields $\widehat{f_N}(\zeta)=\sqrt d\,\widehat{\bar h}(d\zeta)$, and the formula for $\widehat{f_N}$ follows from the sixth part of \cref{lem:heat-algebra}. Finally, $\widehat{j_N}(\zeta)=\int\mathfrak n_N(dx)/(\zeta+2+2x)$ by Fubini's theorem.
\end{proof}

\section{The covariance evolution and the Bingham comparison law}\label{sec:heat-covariance}

Let $A$ be a real symmetric matrix with eigenvalues $\alpha_1\ge\dots\ge\alpha_N$. We work in its eigencoordinates, in which $A=\diag(\alpha_1,\dots,\alpha_N)$, and we let $z$ be its Lagrange multiplier and $p_i=z-\alpha_i$ its precisions (\cref{def:model-multiplier}). Then $p$ satisfies \eqref{eq:heat-trace}, and $d=p_1\le p_2\le\dots\le p_N=s$. All notation of this section refers to this sequence $p$.

\begin{definition}[Covariance evolution and Bingham comparison law]\label{def:heat-covariance}
Let $m(t)\in\R^N$ be the solution of the linear system
\begin{equation}\label{ht-covariance-ode}
m_i'(t)=2\alpha_im_i(t)+\frac2N\sum_{j=1}^Nm_j(t),\qquad m_i(0)=1,\qquad\text{and let}\quad v_i(t)=\frac{Nm_i(t)}{\sum_jm_j(t)} .
\end{equation}
We define the moving multiplier $z(t)$, the coefficients $b_i(t)$, the matrix $B_t$, and the Bingham comparison law $\mu_t$ by
\begin{equation}\label{ht-bingham-proxy-definition}
\begin{gathered}
z(t)=1+\frac1N\sum_{i=1}^N\alpha_iv_i(t),\qquad b_i(t)=z(t)-\frac1{v_i(t)},\qquad B_t=\diag\big(b_i(t)\big),\\
\mu_t(dx)=\frac{e^{x^\T B_tx/2}\,\nu_N(dx)}{\int_{S_N}e^{y^\T B_ty/2}\,\nu_N(dy)} .
\end{gathered}
\end{equation}
We call $v(t)$ the covariance evolution of $A$.
\end{definition}

The matrix of \eqref{ht-covariance-ode} is $2\diag(\alpha)+(2/N)\mathbf 1\mathbf 1^\T=H+2zI$, so that $m(t)=e^{2zt}h(t)$. By \cref{lem:heat-algebra}, $m(t)$ has positive coordinates, and $v_i(t)=h_i(t)/\bar h(t)$ agrees with \eqref{eq:heat-errors}. In particular, $e_i(t)=v_i(t)/q_i-1$ and $\cI_N(t)$ are the relative errors and the information of the covariance evolution. On $S_N$ we have $x^\T B_tx/2=Nz(t)/2-\sum_ix_i^2/(2v_i(t))$. Since $\sum_iv_i(t)=N$, the second part of \cref{prop:cond-bingham} implies that $\mu_t=\mu_{v(t)}$ is the Gaussian law with variances $v_i(t)$ conditioned on $S_N$.

The covariance evolution has the following interpretation. Consider the linear system $dx_i=(\alpha_i-z(t))x_i\,dt+\sqrt2\,d\mathcal B_i(t)$, driven by independent standard Brownian motions $\mathcal B_i$ and started from independent standard Gaussian values, where $z(t)$ is chosen so that $\E|x(t)|^2=N$ for all $t$. The variances $\E x_i(t)^2$ then solve $v_i'=2+2(\alpha_i-z(t))v_i$, and the constraint forces $z(t)=1+N^{-1}\sum_i\alpha_iv_i(t)$; this is the system \eqref{ht-covariance-identities} below. In this system the spherical constraint holds only in expectation. For the spherical Sherrington--Kirkpatrick model, Cugliandolo and Dean solved this system in the limit $N\to\infty$ by diagonalizing the interaction matrix \cite[(II.2)--(II.4)]{CugliandoloDean1995}, and they reduced the multiplier to a scalar Volterra equation \cite[(II.6)--(II.7)]{CugliandoloDean1995}; see also \cite[Section~II, (2)--(6)]{ZippoldKuhnHorner2000}.
In our notation, the function $e^{2zt}\bar h(t)=N^{-1}\sum_jm_j(t)$ is equal to $\exp(2\int_0^tz(u)\,du)$, by \eqref{eq:heat-derivative-identity} and the identity $N^{-1}\sum_je_j(t)=z-z(t)$, and averaging the variation of constants formula for \eqref{ht-covariance-ode} shows that it solves
\[
e^{2zt}\bar h(t)=\frac1N\sum_ie^{2\alpha_it}+2\int_0^t\Big(\frac1N\sum_ie^{2\alpha_i(t-u)}\Big)e^{2zu}\bar h(u)\,du .
\]
This is the finite-dimensional form of the Volterra equation of Cugliandolo and Dean, at temperature one, with unit initial variances, and with the average over the spectrum taken with respect to the empirical spectral measure of $A$. What we add is the comparison of the Bingham comparison law $\mu_t$ with the time-$t$ law of the diffusion on $S_N$, which is carried out in \cref{sec:comparison}.

\begin{lemma}[Positive covariance evolution and uniform variance bounds]\label{ht-covariance-system}
All $m_i(t)$ are positive, $\sum_iv_i(t)=N$, and, for all $t\ge0$,
\begin{equation}\label{ht-covariance-identities}
v_i'=2+2\big(\alpha_i-z(t)\big)v_i,\qquad
b_i'=z'(t)+\frac{2(\alpha_i-b_i)}{v_i},\qquad
\sum_{i=1}^N\big(\alpha_i-b_i(t)\big)v_i(t)=0 .
\end{equation}
With $q_i=(z-\alpha_i)^{-1}$, $d=z-\alpha_1$, and $s=z-\alpha_N$, we have, for all $t\ge0$ and all $i$,
\begin{equation}\label{ht-covariance-variance-range}
\frac1s\le v_i(t)\le\frac1d,\qquad
|v_i(t)-q_i|\le\frac{s-d}{d^2}\,e^{-2dt} .
\end{equation}
If $\|A\|\le M$, then, uniformly in $t\ge0$,
\begin{equation}\label{ht-covariance-field-relaxation}
\|B_t\|\le2+3M,\qquad
\|A-B_t\|\le C_Md^{-1}e^{-2dt},\qquad
|z(t)-z|\le(1+2M)\,d^{-1}e^{-2dt},
\end{equation}
where $C_M=4M(1+M)$. Moreover, $\mu_0=\nu_N$, and $\mu_t\to\pi_A$ in total variation as $t\to\infty$, for fixed $N$ and $A$.
\end{lemma}

\begin{proof}
The proof has five steps: the identities \eqref{ht-covariance-identities}, a transformed linear system and the decay of its oscillation, the range of $v_i$, the bounds on $z(t)$ and $B_t$, and the values at $t=0$ and $t=\infty$.

\paragraph{Step 1: identities.}
We showed above that $m_i(t)>0$. Let $\Lambda(t)=N^{-1}\sum_jm_j(t)$. By \eqref{ht-covariance-ode}, $\Lambda'=N^{-1}\sum_j(2\alpha_jm_j+2\Lambda)=2\Lambda\big(1+N^{-1}\sum_j\alpha_jv_j\big)=2z(t)\Lambda$. Using $v_i=m_i/\Lambda$, we find
\[
v_i'=\frac{m_i'}\Lambda-\frac{m_i\Lambda'}{\Lambda^2}=2\alpha_iv_i+2-2z(t)v_i,
\]
which is the first identity in \eqref{ht-covariance-identities}. Also, $\sum_iv_i=N$ by definition. Differentiating $b_i=z(t)-v_i^{-1}$ and using the first identity, we obtain
\[
b_i'=z'(t)+\frac{v_i'}{v_i^2}=z'(t)+\frac2{v_i}\Big(\frac1{v_i}+\alpha_i-z(t)\Big)=z'(t)+\frac{2(\alpha_i-b_i)}{v_i} .
\]
Finally, $\sum_i(\alpha_i-b_i)v_i=\sum_i\alpha_iv_i-z(t)\sum_iv_i+N=N(z(t)-1)-Nz(t)+N=0$.

\paragraph{Step 2: a transformed system.}
Let $w_i(t)=p_ih_i(t)=e^{-2zt}m_i(t)/q_i$. Recall that $h_i'=-2p_ih_i+2\bar h$ and $\bar h=N^{-1}\sum_ih_i$. Multiplying by $p_i$, we find
\begin{equation}\label{ht-transformed-markov-system}
w_i'=2p_i\big(\bar h-w_i\big),\qquad w_i(0)=p_i,\qquad \bar h=\frac1N\sum_{i=1}^Nq_iw_i,\qquad v_i=\frac{q_iw_i}{\bar h} .
\end{equation}
Note that $\bar h(t)$ is an average of the numbers $w_i(t)$ with positive weights $q_i/N$ that sum to one. In particular, $\bar h(t)$ lies between $\underline w(t)=\min_iw_i(t)$ and $\overline w(t)=\max_iw_i(t)$. We claim that
\begin{equation}\label{ht-markov-oscillation}
\osc w(t)=\overline w(t)-\underline w(t)\le(s-d)\,e^{-2dt},\qquad t\ge0 .
\end{equation}
Fix $t$. By continuity, for small $\eps>0$ the maximum of $w_j(t+\eps)$ over $j$ is attained at an index $j$ that maximizes $w_j(t)$. It follows that the right derivative of $\overline w$ at $t$ exists and equals $\max\{w_i'(t):w_i(t)=\overline w(t)\}$. For such $i$, we have $w_i'(t)=2p_i(\bar h-\overline w)\le2d(\bar h-\overline w)$, since $\bar h-\overline w\le0$ and $p_i\ge d$. In the same way, the right derivative of $\underline w$ is at least $2d(\bar h-\underline w)$. Subtracting, we find that the right derivative of $\osc w$ is at most $-2d\osc w$. A continuous function whose right derivative exists everywhere and is nonpositive is nonincreasing. We conclude that $e^{2dt}\osc w(t)$ is nonincreasing, and \eqref{ht-markov-oscillation} follows from $\osc w(0)=s-d$. The same argument shows that $\overline w$ is nonincreasing and $\underline w$ is nondecreasing; in particular, $w_i(t)\in[d,s]$ for all $i$ and $t$.

By \cref{lem:heat-algebra}, we have $\bar h(t)\ge\vartheta\ge d$. Using also $q_i\le1/d$, we obtain
\[
|v_i-q_i|=\frac{q_i|w_i-\bar h|}{\bar h}\le\frac{\osc w}{d^2}\le\frac{s-d}{d^2}\,e^{-2dt},
\]
which is the second bound in \eqref{ht-covariance-variance-range}.

\paragraph{Step 3: the range of $v_i$.}
The right side of \eqref{eq:heat-duhamel} is a nonincreasing function of $p_i$, because $\bar h\ge0$. Then $v_1(t)\ge v_2(t)\ge\dots\ge v_N(t)$, and it suffices to show $v_1\le1/d$ and $v_N\ge1/s$.

For the upper bound, let $k_i=w_i-w_1$ for $2\le i\le N$. We have $\bar h-w_1=N^{-1}\sum_jq_j(w_j-w_1)=N^{-1}\sum_{j\ge2}q_jk_j$, and \eqref{ht-transformed-markov-system} yields
\[
k_i'=2p_i(\bar h-w_i)-2d(\bar h-w_1)=\frac{2(p_i-d)}N\sum_{j\ge2}q_jk_j-2p_ik_i,\qquad k_i(0)=p_i-d\ge0 .
\]
This is a linear system whose matrix has the nonnegative off-diagonal entries $2(p_i-d)q_j/N$. As noted after \cref{lem:heat-algebra}, its solution stays nonnegative, since it starts from nonnegative values. Thus $w_1=\underline w\le \bar h$, and $v_1=q_1w_1/\bar h\le q_1=1/d$.

For the lower bound, \eqref{eq:heat-duhamel} with $p_N=s$ implies that
\[
w_N(t)=se^{-2st}+2s\int_0^te^{-2s(t-u)}\bar h(u)\,du\ge se^{-2st}+\bar h(t)\big(1-e^{-2st}\big)\ge \bar h(t),
\]
where we used that $\bar h$ is nonincreasing and $s\ge1\ge \bar h(t)$ (\cref{lem:heat-algebra}). We conclude that $v_N=q_Nw_N/\bar h\ge q_N=1/s$. This proves the first bound in \eqref{ht-covariance-variance-range}.

\paragraph{Step 4: the multiplier and the quadratic coefficient matrix.}
Let $w_{\mathrm{av}}=N^{-1}\sum_iw_i$. Since $N^{-1}\sum_ip_iv_i=w_{\mathrm{av}}/\bar h$ and $N^{-1}\sum_ip_iv_i=z-N^{-1}\sum_i\alpha_iv_i=z-z(t)+1$, we have
\begin{equation}\label{eq:heat-multiplier-identity}
z(t)-z=\frac{\bar h-w_{\mathrm{av}}}{\bar h},\qquad v_i^{-1}-p_i=\frac{\bar h-w_i}{\bar h\,v_i},
\end{equation}
where the second identity follows from $v_i=q_iw_i/\bar h$. Both $\bar h$ and $w_{\mathrm{av}}$ lie between $\underline w$ and $\overline w$. Since also $v_i\ge1/s$, we conclude that
\begin{equation}\label{eq:heat-field-bounds}
|z(t)-z|\le\frac{\osc w(t)}{\bar h(t)},\qquad |v_i^{-1}-p_i|\le\frac{s\osc w(t)}{\bar h(t)},\qquad
\|A-B_t\|\le\frac{(1+s)\osc w(t)}{\bar h(t)},
\end{equation}
where for the last bound we used $\alpha_i-b_i=(v_i^{-1}-p_i)-(z(t)-z)$. Suppose now that $\|A\|\le M$. Then $s-d=\alpha_1-\alpha_N\le2M$ and, since $d\le1$, also $s\le1+2M$. By \eqref{ht-markov-oscillation} and $\bar h\ge d$, we have $|z(t)-z|\le2Md^{-1}e^{-2dt}$ and $\|A-B_t\|\le(2+2M)2Md^{-1}e^{-2dt}$, which are the last two bounds in \eqref{ht-covariance-field-relaxation}. Also $|z(t)|\le1+MN^{-1}\sum_iv_i=1+M$ and $v_i^{-1}\le s\le1+2M$, so that $\|B_t\|=\max_i|b_i(t)|\le2+3M$.

\paragraph{Step 5: the endpoints.}
At $t=0$, $v_i(0)=1$ and $b_i(0)=z(0)-1=N^{-1}\Tr A$ for all $i$. The density of $\mu_0$ with respect to $\nu_N$ is then constant on $S_N$, and $\mu_0=\nu_N$. As $t\to\infty$, we have $b_i(t)\to\alpha_i$ by \eqref{ht-covariance-variance-range} and \eqref{eq:heat-field-bounds}. Then the densities of $\mu_t$ with respect to $\nu_N$ converge uniformly on $S_N$ to that of $\pi_A$, and the total variation distance, which is half the $L^1(\nu_N)$ distance of the densities, tends to zero.
\end{proof}

For the model $A=cW_N$, where $\alpha_i=c\lambda_i$, the first identity in \eqref{ht-covariance-identities} shows that $v$ solves the system \eqref{eq:intro-covariance} of the introduction. Its right side is a polynomial in $v$, and in particular locally Lipschitz. Therefore $v$ is the unique solution of that system, and the two definitions agree.

The proof of \cref{ht-covariance-system} contains a conserved quantity and a more precise form of the bounds on $z(t)-z$ and $A-B_t$, which we record separately.

\begin{lemma}[A conserved quantity and the weight of the zero eigenvalue]\label{ht-effective-normalization}
For the covariance evolution of \cref{def:heat-covariance}, let $w_i(t)=p_ih_i(t)=e^{-2zt}m_i(t)/q_i$; then $\bar h(t)=N^{-1}\sum_iq_iw_i(t)$. For all $t\ge0$,
\begin{equation}\label{ht-effective-normalization-bound}
\sum_{i=1}^Nq_i^2w_i(t)=N,\qquad 1\ge \bar h(t)\ge\vartheta,\qquad \bar h(t)\longrightarrow\vartheta\quad(t\to\infty).
\end{equation}
Moreover, with $\Omega_t=(s-d)e^{-2dt}$,
\begin{equation}\label{ht-effective-field-relaxation}
\osc w(t)\le\Omega_t,\qquad |z(t)-z|\le\frac{\Omega_t}{\bar h(t)},\qquad \|A-B_t\|\le\frac{(1+s)\,\Omega_t}{\bar h(t)} .
\end{equation}
\end{lemma}

\begin{proof}
By the fifth part of \cref{lem:heat-algebra}, we have $\sum_iq_i^2w_i=\sum_iq_ih_i=N$, and the remaining assertions of \eqref{ht-effective-normalization-bound} are contained in the third part of that lemma. The bounds \eqref{ht-effective-field-relaxation} are \eqref{ht-markov-oscillation} and \eqref{eq:heat-field-bounds}.
\end{proof}

Note that $\vartheta$ is the long-time limit of $\bar h$, not a rate of decay. The decay of $\bar h$ towards $\vartheta$ is governed by the spectral measure $\rho$ on $[2d,2s]$. Under the hypothesis \eqref{hi3-indexed-geometry} of the next subsection, \cref{hi3-finite-heat} shows that $\bar h(t)$ is comparable to $(1+t)^{-1/2}$ for $0\le t\le d^{-1}$ and to $\sqrt d$ for $t\ge d^{-1}$, with constants depending only on the constant $K$ in \eqref{hi3-indexed-geometry}.

\begin{remark}[The variances are not second moments]\label{rem:heat-not-moments}
The Bingham comparison law $\mu_t$ is an explicit comparison law. In general, it is not equal to the time-$t$ law $\nu_NP^A_t$ of the diffusion, since the dynamics does not preserve the family of Bingham laws; the two laws agree at $t=0$, and for all $t$ when $A$ is a multiple of the identity. The numbers $v_i(t)$ are the variances of the Gaussian law that is conditioned on $S_N$ to obtain $\mu_t$. In general, they are equal neither to the second moments $\mu_t(x_i^2)$ nor to the second moments of the time-$t$ law, and in general $q_i$ is not equal to $\pi_A(x_i^2)$. The difference is already visible in the first derivative at $t=0$. On $S_N$ we have
\[
L_Ax_i^2=2+2(\alpha_i-1)x_i^2-\frac2N\,x_i^2\,x^\T Ax,
\]
and under $\nu_N$ we have $\int x_i^2x_j^2\,d\nu_N=N(1+2\delta_{ij})/(N+2)$. With $\bar\alpha=N^{-1}\Tr A$, we find
\[
\frac d{dt}\Big|_{t=0}\int x_i^2\,d(\nu_NP^A_t)=\int L_Ax_i^2\,d\nu_N=\frac{2N}{N+2}\,(\alpha_i-\bar\alpha),
\qquad
v_i'(0)=2(\alpha_i-\bar\alpha),
\]
so the two vectors of initial derivatives differ unless $A$ is a multiple of the identity. For a general nonscalar $A$, the equations for the second moments of the coordinates under the time-$t$ law involve mixed fourth moments, since $L_Ax_i^2$ contains the term $x_i^2x^\T Ax$. Our use of $\mu_t$ in \cref{sec:comparison} does not require any identification of moments: it compares the two laws through the forward equation.
\end{remark}

\section{Estimates under two-sided precision-spacing bounds}\label{sec:heat-hot}

In this subsection we fix $K\ge1$ and assume the following two-sided precision-spacing bounds with constant $K$:
\begin{equation}\label{hi3-indexed-geometry}
\begin{gathered}
0<d\le1,\qquad p_1=d,\qquad \frac1K\Big(\frac{i-1}N\Big)^{2/3}\le p_i-d\le K\Big(\frac{i-1}N\Big)^{2/3}\quad(2\le i\le N),\\
\sum_{i=1}^N\frac1{p_i}=N,\qquad Nd^{3/2}\ge1 .
\end{gathered}
\end{equation}
In particular, $d=\min_ip_i$ and $s\le d+K\le1+K$. Constants denoted $C_K$ or $c_K$ are positive and depend only on $K$; their values may change from line to line. For positive $X,Y$, we write $X\asymp_KY$ if $c_KY\le X\le C_KY$. We abbreviate
\begin{equation}\label{eq:heat-envelope}
\mathsf a(t)=\sqrt d+(1+t)^{-1/2},\qquad t\ge0 .
\end{equation}
We use the following elementary properties of $\mathsf a$: it is decreasing, $\mathsf a(t/2)\le\sqrt2\,\mathsf a(t)$, $\int_0^t\mathsf a(u)\,du\le\sqrt d\,t+2\sqrt t$, $\mathsf a(t)\le3(1+t)^{-1/2}$ for $t\le d^{-1}$, and $\mathsf a(t)\le2\sqrt d$ for $t\ge d^{-1}$.

\begin{remark}[Precision-spacing bounds and the order of the coordinates]\label{rem:heat-hot-hypothesis}
The hypothesis does not depend on the order of the indices $i\ge2$, in the following sense. Let $g_1=0$ and $g_i=p_i-d$, and let $g_{(1)}\le\dots\le g_{(N)}$ be the increasing rearrangement. The bounds in \eqref{hi3-indexed-geometry} increase with $i$. Thus the $i$ numbers $g_1,\dots,g_i$ are at most $K((i-1)/N)^{2/3}$, which implies that $g_{(i)}\le K((i-1)/N)^{2/3}$. The $N-i+1$ numbers $g_j$ with $j\ge i$ are at least $K^{-1}((i-1)/N)^{2/3}$; at most $i-1$ of the $g_j$ are smaller, and $g_{(i)}\ge K^{-1}((i-1)/N)^{2/3}$. Hence the increasing rearrangement of $p$ satisfies \eqref{hi3-indexed-geometry} with the same $K$. For the model at fixed high temperature and in the approach to the critical temperature from above, $p_i-d=c(\lambda_1-\lambda_i)$, and \eqref{hi3-indexed-geometry} holds at every fixed confidence in the sense of \cref{sec:model-quantifiers}, with $K$ depending on the confidence (and on $c$ at fixed temperature), by \cref{sp-fixed-confidence-geometry,sp-canonical-saddle}. This is checked in \cref{sec:hot-fixed,sec:hot-meso}.
\end{remark}

\begin{lemma}[Bounds on $\bar h$ under precision-spacing bounds]\label{hi3-finite-heat}
Assume \eqref{hi3-indexed-geometry}. Then, for all $t\ge0$,
\begin{equation}\label{hi3-all-time-heat}
\vartheta\asymp_K\sqrt d,\qquad \bar h(t)\asymp_K\sqrt d+(1+t)^{-1/2}=\mathsf a(t).
\end{equation}
Moreover, $\bar h$ is completely monotone and has the representation
\begin{equation}\label{hi3-positive-transient}
\bar h(t)=\vartheta+\int_{[2d,\infty)}e^{-\lambda t}\,\rho(d\lambda),\qquad \rho\ge0,\qquad \rho([2d,\infty))=1-\vartheta .
\end{equation}
\end{lemma}

\begin{proof}
The representation \eqref{hi3-positive-transient} and complete monotonicity are the third part of \cref{lem:heat-algebra}. It remains to prove \eqref{hi3-all-time-heat}. The proof has four steps: a bound on the sum $\mathsf S$ of \eqref{eq:heat-GS}, from which $\vartheta\asymp_K\sqrt d$ follows; bounds on the Laplace transform of $\bar h$; the upper bound on $\bar h$; and the lower bound on $\bar h$.

\paragraph{Step 1: a rational sum.}
We show that, for $0\le\zeta\le1$,
\begin{equation}\label{hi3-rational-sum}
\mathsf S(\zeta)=\frac1N\sum_{i=1}^N\frac1{p_i(p_i+\zeta/2)}\asymp_K\frac1{\sqrt d+\sqrt{d+\zeta}} .
\end{equation}
Let $d_\zeta=d+\zeta/2$, and let $\phi(x)=[(d+x)(d_\zeta+x)]^{-1}$ for $x\ge0$, a decreasing function. For $i\ge2$ and $y_i=(i-1)/N$, we have $\phi(Ky_i^{2/3})\le[p_i(p_i+\zeta/2)]^{-1}\le\phi(K^{-1}y_i^{2/3})$ by \eqref{hi3-indexed-geometry}. For a decreasing function $g$ on $[0,1]$, we have
\[
\int_{1/N}^1g(y)\,dy\le\frac1N\sum_{j=1}^{N-1}g(j/N)\le\int_0^1g(y)\,dy,\qquad \int_0^{1/N}g(y)\,dy\le\frac{g(0)}N .
\]
The summand $i=1$ in \eqref{hi3-rational-sum} equals $\phi(0)/N=(Ndd_\zeta)^{-1}$. Applying these inequalities to $g(y)=\phi(Ky^{2/3})$ and to $g(y)=\phi(K^{-1}y^{2/3})$, we obtain
\begin{equation}\label{eq:heat-rational-sandwich}
I_K\le\mathsf S(\zeta)\le I_{K^{-1}}+\frac1{Ndd_\zeta},\qquad
I_L=\int_0^1\phi(Ly^{2/3})\,dy=\frac3{2L^{3/2}}\int_0^L\frac{\sqrt x\,dx}{(d+x)(d_\zeta+x)},
\end{equation}
where we substituted $x=Ly^{2/3}$. The substitution $x=y^2$ and partial fractions give, for $d\ne d_\zeta$, and by continuity also for $d=d_\zeta$,
\begin{equation}\label{eq:heat-sqrt-integral}
\int_0^\infty\frac{\sqrt x\,dx}{(x+d)(x+d_\zeta)}=\frac\pi{\sqrt d+\sqrt{d_\zeta}} .
\end{equation}
Since $d_\zeta\le d+\zeta\le2d_\zeta$, we have $\sqrt d+\sqrt{d_\zeta}\le\sqrt d+\sqrt{d+\zeta}\le\sqrt2(\sqrt d+\sqrt{d_\zeta})$. It therefore suffices to prove \eqref{hi3-rational-sum} with $d_\zeta$ in place of $d+\zeta$.

For the upper bound, we have $I_{K^{-1}}\le\frac32K^{3/2}\pi(\sqrt d+\sqrt{d_\zeta})^{-1}$ by \eqref{eq:heat-sqrt-integral}. For the summand $i=1$, we use $Nd^{3/2}\ge1$ and $d\le d_\zeta$ to find
\[
\frac1{Ndd_\zeta}=\frac1{Nd^{3/2}}\cdot\frac{\sqrt d}{d_\zeta}\le\frac{\sqrt d}{d_\zeta}=\frac1{\sqrt d+\sqrt{d_\zeta}}\Big(\frac d{d_\zeta}+\sqrt{\frac d{d_\zeta}}\Big)\le\frac2{\sqrt d+\sqrt{d_\zeta}} .
\]
For the lower bound, we bound $I_K$ from below. If $2d_\zeta\le K$, we restrict the integral in \eqref{eq:heat-rational-sandwich} to $[d_\zeta,2d_\zeta]$, where $\sqrt x\ge\sqrt{d_\zeta}$, $d+x\le3d_\zeta$, and $d_\zeta+x\le3d_\zeta$. This gives
\[
\int_0^K\frac{\sqrt x\,dx}{(d+x)(d_\zeta+x)}\ge d_\zeta\cdot\frac{\sqrt{d_\zeta}}{9d_\zeta^2}=\frac1{9\sqrt{d_\zeta}}\ge\frac1{9(\sqrt d+\sqrt{d_\zeta})} .
\]
If $2d_\zeta>K$, then $d_\zeta\ge1/2$ (since $K\ge1$) and $d_\zeta\le3/2$. Restricting the integral to $[K/4,K/2]$, we find that it is at least a positive constant depending only on $K$, while $(\sqrt d+\sqrt{d_\zeta})^{-1}\le\sqrt2$. In both cases, $I_K\ge c_K(\sqrt d+\sqrt{d_\zeta})^{-1}$, which completes the proof of \eqref{hi3-rational-sum}. At $\zeta=0$, we have $\mathsf S(0)=N^{-1}\sum_iq_i^2=1/\vartheta$, and \eqref{hi3-rational-sum} implies that $\vartheta\asymp_K\sqrt d$.

\paragraph{Step 2: the Laplace transform.}
Let $0<\zeta\le1$. By \eqref{hi3-hot-renewal-transform}, $\widehat{\bar h}(\zeta)=\mathsf G(\zeta)/(\zeta\,\mathsf S(\zeta))$. We have $\mathsf G(\zeta)\le\mathsf G(0)=1$ and $\mathsf G(\zeta)\ge(1+K+1/2)^{-1}$, since $p_i\le1+K$. Using \eqref{hi3-rational-sum} and $\sqrt d+\sqrt{d+\zeta}\asymp\sqrt d+\sqrt\zeta$, we obtain
\begin{equation}\label{hi3-transform-bounds}
\widehat{\bar h}(\zeta)\asymp_K\frac{\sqrt d+\sqrt{d+\zeta}}\zeta\asymp_K\frac{\sqrt d+\sqrt\zeta}\zeta,\qquad0<\zeta\le1 .
\end{equation}

\paragraph{Step 3: the upper bound.}
Since $\bar h$ is nonincreasing and nonnegative, for every $t>0$,
\[
\widehat{\bar h}(1/t)\ge\int_0^te^{-u/t}\bar h(u)\,du\ge e^{-1}t\,\bar h(t) .
\]
For $t\ge1$, \eqref{hi3-transform-bounds} with $\zeta=1/t$ gives $\bar h(t)\le e\,t^{-1}\widehat{\bar h}(1/t)\le C_K(\sqrt d+t^{-1/2})\le C_K\mathsf a(t)$. For $t\le1$, we have $\bar h(t)\le1\le\sqrt2\,\mathsf a(t)$. Thus $\bar h(t)\le C_1\mathsf a(t)$ for all $t\ge0$, for a constant $C_1=C_1(K)$.

\paragraph{Step 4: the lower bound.}
By \eqref{hi3-transform-bounds}, there exists $c_1=c_1(K)>0$ such that $\widehat{\bar h}(1/t)\ge c_1\sqrt t$ for $t\ge1$. Fix $\eps=\eps(K)\in(0,1)$ such that $C_1(\eps+2\sqrt\eps)\le c_1/2$, and let $1\le t\le d^{-1}$. Using Step~3 and $\sqrt d\,t\le\sqrt t$, we have
\[
\int_0^{\eps t}e^{-u/t}\bar h(u)\,du\le C_1\int_0^{\eps t}\mathsf a(u)\,du\le C_1\big(\sqrt d\,\eps t+2\sqrt{\eps t}\big)\le C_1(\eps+2\sqrt\eps)\sqrt t\le\frac{c_1}2\sqrt t .
\]
Since $\bar h$ is nonincreasing, the remaining part of the integral defining $\widehat{\bar h}(1/t)$ is at most $\bar h(\eps t)\int_0^\infty e^{-u/t}\,du=t\bar h(\eps t)$. Then $t\bar h(\eps t)\ge(c_1/2)\sqrt t$, that is, $\bar h(v)\ge(c_1\sqrt\eps/2)v^{-1/2}$ for $v=\eps t\in[\eps,\eps/d]$. On this interval, we have $\sqrt d\le\sqrt{\eps/v}$ and hence $\mathsf a(v)\le(1+\eps^{-1/2})v^{-1/2}$. Then $\bar h(v)\ge c_K\mathsf a(v)$ for $\eps\le v\le\eps/d$. For $0\le v\le\eps$, we have $\bar h(v)\ge e^{-2sv}\ge e^{-2(1+K)}$ by the first part of \cref{lem:heat-algebra}, while $\mathsf a(v)\le2$. For $v\ge\eps/d$, we have $\bar h(v)\ge\vartheta\ge c_K\sqrt d$ by Step~1, while $\mathsf a(v)\le(1+\eps^{-1/2})\sqrt d$. The three intervals cover $[0,\infty)$, since $d\le1$. This proves the lower bound in \eqref{hi3-all-time-heat}.
\end{proof}

\begin{lemma}[The coordinates of the heat vector]\label{iw-all-time-heat}
Assume \eqref{hi3-indexed-geometry}. Uniformly for every $t\ge0$ and $1\le i\le N$,
\[
\bar h(t)\asymp_K\mathsf a(t),\qquad h_i(t)\asymp_K\mathsf a(t)\,\frac{1+t}{1+p_it},\qquad \max_jh_j(t)\le\frac{C_K}{\sqrt d} .
\]
Moreover, for every nonzero $y\in[0,\infty)^N$,
\begin{equation}\label{iw-backward-ratio}
\max_jh_j(L-u)\,\frac{h(u)^\T y}{h(L)^\T y}\le C_K\,\frac{\mathsf a(u)}d,\qquad0\le u\le L<\infty .
\end{equation}
\end{lemma}

\begin{proof}
The bound on $\bar h$ is \cref{hi3-finite-heat}. We use \eqref{eq:heat-duhamel}, $h_i(t)=e^{-2p_it}+2\int_0^te^{-2p_i(t-u)}\bar h(u)\,du$, and recall that $p_i\le1+K$.

For $t\le1$, we have $e^{-2(1+K)}\le h_i(t)\le1+2t\le3$, since $\bar h\le1$, while $1/(2(2+K))\le\mathsf a(t)(1+t)/(1+p_it)\le4$. This proves the bounds on $h_i$ for $t\le1$. Let $t\ge1$. For the lower bound, we keep only the integral over $[t/2,t]$, where $\bar h(u)\ge \bar h(t)$:
\[
h_i(t)\ge2\bar h(t)\int_0^{t/2}e^{-2p_iv}\,dv=\bar h(t)\,\frac{1-e^{-p_it}}{p_i}\ge \bar h(t)\,\frac t{1+p_it}\ge c_K\mathsf a(t)\,\frac{1+t}{1+p_it},
\]
where we used $1-e^{-x}\ge x/(1+x)$ and $t\ge(1+t)/2$. For the upper bound, on $[t/2,t]$ we use $\bar h(u)\le C_K\mathsf a(t/2)\le C_K\mathsf a(t)$ and $2\int_0^{t/2}e^{-2p_iv}\,dv\le\min\{t,p_i^{-1}\}\le2t/(1+p_it)$. On $[0,t/2]$ we use $e^{-2p_i(t-u)}\le e^{-p_it}\le(1+p_it)^{-1}$ and $\int_0^{t/2}\bar h\le C_K(\sqrt d\,t+\sqrt t)\le C_K\mathsf a(t)(1+t)$. Finally, $e^{-2p_it}\le(1+p_it)^{-1}\le\mathsf a(t)(1+t)/(1+p_it)$, since $\mathsf a(t)(1+t)\ge\sqrt{1+t}\ge1$. Adding the three contributions gives the upper bound on $h_i(t)$.

For the maximum, note that $\mathsf a(t)(1+t)/(1+p_it)\le\mathsf a(t)(1+t)/(1+dt)$. Since $d\le1$, we have $\sqrt d(1+t)/(1+dt)\le\sqrt d\max\{1,d^{-1}\}=d^{-1/2}$. Further, $\sqrt{1+t}/(1+dt)$ is at most $\sqrt{1+d^{-1}}\le\sqrt{2/d}$ if $dt\le1$, and at most $\sqrt{2t}/(dt)\le\sqrt2\,d^{-1/2}$ if $dt\ge1$. With the upper bound on $h_j$, this gives $\max_jh_j(t)\le C_Kd^{-1/2}$.

For \eqref{iw-backward-ratio}, let $0\le u\le L$. Since $y\ne0$ has nonnegative coordinates and $h_i(L)>0$, we have $h(u)^\T y/h(L)^\T y\le\max_ih_i(u)/h_i(L)$. By the bounds on $h_i$,
\[
\frac{h_i(u)}{h_i(L)}\le C_K\,\frac{\mathsf a(u)}{\mathsf a(L)}\cdot\frac{1+p_iL}{1+L}\cdot\frac{1+u}{1+p_iu}\le C_K\,\frac{\mathsf a(u)}{\sqrt d}\,\max\{1,p_i\}\le C_K\,\frac{\mathsf a(u)}{\sqrt d} .
\]
In the second step, we used $\mathsf a(L)\ge\sqrt d$ and that $x\mapsto(1+p_ix)/(1+x)$ is monotone with values between $1$ and $p_i$; the ratio of its values at $L$ and at $u\le L$ is therefore at most $\max\{1,p_i\}$. Multiplying by $\max_jh_j(L-u)\le C_Kd^{-1/2}$ yields \eqref{iw-backward-ratio}.
\end{proof}

The next lemma bounds the relative errors $e_i(t)=p_iv_i(t)-1$ of \eqref{eq:heat-errors}, and their mean square $N^{-1}\sum_ie_i(t)^2=2\cI_N(t)/N$, before and after the time $d^{-1}$.

\begin{lemma}[Relative errors before and after the time $d^{-1}$]\label{hi3-information-envelopes}
Assume \eqref{hi3-indexed-geometry}, and let $T=d^{-1}$. There exists $C_K<\infty$ such that, for all $1\le i\le N$,
\begin{align}
|e_i(t)|&\le\frac{C_K}{1+p_it}&&(0\le t\le T),\label{hi3-early-coordinate}\\
\frac1N\sum_{j=1}^Ne_j(t)^2&\le C_K(1+t)^{-3/2}&&(0\le t\le T),\label{hi3-early-information}\\
|e_i(T+\tau)|&\le C_K\,\frac d{p_i}\,(1+d\tau)\,e^{-2d\tau}&&(\tau\ge0),\label{hi3-late-coordinate}\\
\frac1N\sum_{j=1}^Ne_j(T+\tau)^2&\le C_Kd^{3/2}(1+d\tau)^2e^{-4d\tau}&&(\tau\ge0).\label{hi3-late-information}
\end{align}
\end{lemma}

The factor $1+d\tau$ in \eqref{hi3-late-coordinate} comes from the coordinates with $p_i-d\le d$, whose own relaxation rate $2p_i$ is comparable to the rate $2d$ at which $\bar h$ approaches $\vartheta$; it is integrated in later estimates and cannot be dropped in this argument.

\begin{proof}
We first prove the coordinate bounds before $T$, then the bound on the mean square before $T$, and finally the bounds after $T$. By the third part of \cref{lem:heat-algebra} with $j=1$, for every $v>0$,
\begin{equation}\label{hi3-derivative-heat}
-\bar h'(v)\le\frac2{ev}\,\bar h(v/2) .
\end{equation}

\paragraph{Coordinates before $T$.}
For $t\le1$, we have $h_i(t)\le1+2t\le3$ by \eqref{eq:heat-duhamel} and $\bar h\le1$, and $\bar h(t)\ge c_K$ by \cref{hi3-finite-heat}. Since $p_i\le1+K$, this gives $|e_i(t)|\le p_ih_i(t)/\bar h(t)+1\le C_K\le C_K(2+K)/(1+p_it)$. Let now $1\le t\le T$. Then $\bar h(t)\asymp_Kt^{-1/2}$ by \cref{hi3-finite-heat}, since $\mathsf a(t)\le3(1+t)^{-1/2}$ for $t\le d^{-1}$.

If $p_it\le1$, then \eqref{eq:heat-duhamel} and $\sqrt d\,t\le\sqrt t$ give $h_i(t)\le1+2\int_0^t\bar h\le1+C_K(\sqrt d\,t+\sqrt t)\le C_K\sqrt t$. Then $p_ih_i(t)/\bar h(t)\le C_Kp_it\le C_K$, and $|e_i(t)|\le C_K+1\le2(C_K+1)/(1+p_it)$.

If $p_it\ge1$, we split the integral in \eqref{eq:heat-duhamel} at $t/2$ and integrate the part over $[t/2,t]$ by parts, using $\frac d{du}e^{-2p_i(t-u)}=2p_ie^{-2p_i(t-u)}$. The result is the exact identity
\begin{equation}\label{eq:heat-split}
\begin{split}
p_ih_i(t)-\bar h(t)&=p_ie^{-2p_it}+2p_i\int_0^{t/2}e^{-2p_i(t-u)}\bar h(u)\,du\\
&\quad-e^{-p_it}\bar h(t/2)-\int_{t/2}^te^{-2p_i(t-u)}\bar h'(u)\,du .
\end{split}
\end{equation}
We divide \eqref{eq:heat-split} by $\bar h(t)\ge c_Kt^{-1/2}$ and bound the four terms on the right. The first contributes at most $C_Kp_i\sqrt te^{-2p_it}\le C_Kp_ite^{-p_it}$. Using $\int_0^{t/2}\bar h\le C_K\sqrt t$, we see that the second contributes at most $2p_ie^{-p_it}\int_0^{t/2}\bar h/\bar h(t)\le C_Kp_ite^{-p_it}$. The third contributes at most $e^{-p_it}\bar h(t/2)/\bar h(t)\le C_Ke^{-p_it}$. For $u\in[t/2,t]$, \eqref{hi3-derivative-heat} implies that $-\bar h'(u)\le(4/(et))\bar h(t/4)\le C_K\bar h(t)/t$, and the last term contributes at most $C_Kt^{-1}\int_0^\infty e^{-2p_iv}\,dv\le C_K/(p_it)$. Since $(1+x)e^{-x}\le C/x$ for $x\ge1$, we obtain $|e_i(t)|=|p_ih_i(t)-\bar h(t)|/\bar h(t)\le C_K/(p_it)\le2C_K/(1+p_it)$. This proves \eqref{hi3-early-coordinate}.

\paragraph{Mean square before $T$.}
By \eqref{hi3-early-coordinate}, it suffices to show that $N^{-1}\sum_j(1+p_jt)^{-2}\le C_K(1+t)^{-3/2}$ for $t\le T$. At $t=0$ the sum equals one. For $0<t\le T$, the summand $j=1$ is at most $1/N\le d^{3/2}$, and $d^{3/2}\le\min\{1,t^{-3/2}\}\le2^{3/2}(1+t)^{-3/2}$. For $j\ge2$, we have $p_j\ge K^{-1}((j-1)/N)^{2/3}$, and $y\mapsto(1+K^{-1}ty^{2/3})^{-2}$ is decreasing. For $t\ge1$, we find
\[
\frac1N\sum_{j\ge2}\frac1{(1+p_jt)^2}\le\int_0^1\frac{dy}{(1+K^{-1}ty^{2/3})^2}
=\frac32\Big(\frac Kt\Big)^{3/2}\int_0^{t/K}\frac{\sqrt x\,dx}{(1+x)^2}\le\frac{3\pi}4\Big(\frac Kt\Big)^{3/2},
\]
where we substituted $x=K^{-1}ty^{2/3}$ and used $\int_0^\infty\sqrt x(1+x)^{-2}dx=\pi/2$. For $t\le1$, the sum is at most one. Squaring \eqref{hi3-early-coordinate} proves \eqref{hi3-early-information}.

\paragraph{After $T$.}
Using \eqref{hi3-derivative-heat} and \cref{hi3-finite-heat}, we find that $-\bar h'(T)\le2d\,\bar h(T/2)/e\le C_Kd\,\mathsf a(T/2)\le C_Kd^{3/2}$. The last bound in \eqref{eq:heat-derivatives} with $j=1$ implies that
\begin{equation}\label{eq:heat-late-derivative}
-\bar h'(T+u)\le C_Kd^{3/2}e^{-2du},\qquad u\ge0 .
\end{equation}
Let $\kappa_i=p_ih_i-\bar h=\bar h\,e_i$. From $h_i'=-2p_ih_i+2\bar h$, we find $\kappa_i'=-2p_i\kappa_i-\bar h'$. Solving this equation from time $T$ by variation of constants and dividing by $\bar h(T+\tau)$, we obtain
\begin{equation}\label{hi3-deficit-restart}
e_i(T+\tau)=e^{-2p_i\tau}\,\frac{\bar h(T)}{\bar h(T+\tau)}\,e_i(T)-\frac1{\bar h(T+\tau)}\int_0^\tau e^{-2p_i(\tau-u)}\bar h'(T+u)\,du .
\end{equation}
By \cref{hi3-finite-heat}, we have $\bar h(T)\le C_K\mathsf a(T)\le C_K\sqrt d$ and $\bar h(T+\tau)\ge\vartheta\ge c_K\sqrt d$, so that $\bar h(T)/\bar h(T+\tau)\le C_K$. Further, \eqref{hi3-early-coordinate} at $t=T$ gives $|e_i(T)|\le C_K/(1+p_i/d)\le C_Kd/p_i$. The first term in \eqref{hi3-deficit-restart} is at most $C_K(d/p_i)e^{-2d\tau}$. By \eqref{eq:heat-late-derivative}, the second term is at most
\[
\begin{split}
C_K\,\frac{d^{3/2}}{\sqrt d}\int_0^\tau e^{-2p_i(\tau-u)}e^{-2du}\,du&=C_Kd\,e^{-2d\tau}\int_0^\tau e^{-2(p_i-d)(\tau-u)}\,du\\
&\le C_Kd\,e^{-2d\tau}\min\Big\{\tau,\frac1{2(p_i-d)}\Big\},
\end{split}
\]
where the minimum is $\tau$ when $p_i=d$. We claim that $d\min\{\tau,(2(p_i-d))^{-1}\}\le2(1+d\tau)d/p_i$. If $p_i-d\le d$, then $d/p_i\ge1/2$ and $d\tau\le2(1+d\tau)d/p_i$. If $p_i-d>d$, then $2(p_i-d)\ge p_i$ and $d/(2(p_i-d))\le d/p_i$. This proves \eqref{hi3-late-coordinate}. Finally, $N^{-1}\sum_j(d/p_j)^2=d^2/\vartheta\le C_Kd^{3/2}$ by \cref{hi3-finite-heat}, and squaring \eqref{hi3-late-coordinate} proves \eqref{hi3-late-information}.
\end{proof}

\begin{lemma}[Derivatives of the heat vector]\label{hmr-heat-derivatives}
Assume \eqref{hi3-indexed-geometry}. There exists $C_K<\infty$ such that, for every $t\ge0$ and $1\le i\le N$,
\begin{equation}\label{hmr-heat-derivative-bounds}
\max_jp_jh_j(t)^2\le C_K(1+dt),\qquad
\max_j\sqrt{p_j}\,|h_j'(t)|\le\frac{C_K}{1+t},\qquad
|h_i'(t)|\le\frac{C_Kh_i(t)}{1+t} .
\end{equation}
In particular, for all $t,u\ge0$,
\begin{equation}\label{hmr-difference-kernel}
\max_jp_j\big[h_j(t+u)-h_j(t)\big]^2\le C_K\log^2\Big(1+\frac u{1+t}\Big),
\end{equation}
and, for every nonzero $y\in[0,\infty)^N$, $\big|\frac d{dt}h(t)^\T y\big|\le C_K\,h(t)^\T y/(1+t)$.
\end{lemma}

\begin{proof}
Let $T=d^{-1}$. We use \cref{iw-all-time-heat}, the identity $h_i'=-2\bar h\,e_i$ of \eqref{eq:heat-derivative-identity}, and \cref{hi3-information-envelopes}.

For the first bound, let $t\ge1$. Since $(1+x)^2\ge4x$, we have $p_i/(1+p_it)^2\le1/(4t)$. Using \cref{iw-all-time-heat}, we find
\[
\begin{split}
p_ih_i(t)^2&\le C_K\mathsf a(t)^2\,\frac{p_i(1+t)^2}{(1+p_it)^2}\le C_K\mathsf a(t)^2\,\frac{(1+t)^2}{4t}\\
&\le C_K\big(d+(1+t)^{-1}\big)(1+t)\le C_K(1+dt),
\end{split}
\]
where we used $\mathsf a(t)^2\le2(d+(1+t)^{-1})$, $(1+t)/(4t)\le1/2$, and $d\le1$. For $t\le1$, we use $p_i\le1+K$ and $h_i\le3$.

For the third bound, note that \cref{iw-all-time-heat} implies $\bar h(t)/h_i(t)\le C_K(1+p_it)/(1+t)$. If $t\le T$, then \eqref{hi3-early-coordinate} yields
\[
\frac{|h_i'(t)|}{h_i(t)}=\frac{2\bar h(t)|e_i(t)|}{h_i(t)}\le C_K\,\frac{1+p_it}{1+t}\cdot\frac1{1+p_it}=\frac{C_K}{1+t} .
\]
If $t=T+\tau$ with $\tau\ge0$, we use \eqref{hi3-late-coordinate} and $d(1+p_it)/p_i\le1+dt=2+d\tau$ to find
\[
\frac{|h_i'(t)|}{h_i(t)}\le C_K\,\frac{1+p_it}{1+t}\cdot\frac d{p_i}(1+d\tau)e^{-2d\tau}\le\frac{C_K}{1+t}\,(2+d\tau)(1+d\tau)e^{-2d\tau}\le\frac{C_K}{1+t} .
\]

For the second bound, $\sqrt{p_i}|h_i'|=2\sqrt{p_i}\,\bar h|e_i|$. For $t\le1$, this is at most $2\sqrt{1+K}\,C_K$. For $1\le t\le T$, we have $\sqrt{p_i}/(1+p_it)\le1/(2\sqrt t)$, since $1+x\ge2\sqrt x$, and $\bar h(t)\le C_K(1+t)^{-1/2}$. Then \eqref{hi3-early-coordinate} implies that $\sqrt{p_i}|h_i'(t)|\le C_K/(\sqrt t\sqrt{1+t})\le C_K/(1+t)$. For $t=T+\tau$, we have $\bar h(t)\le C_K\mathsf a(t)\le C_K\sqrt d$. Using \eqref{hi3-late-coordinate} and $p_i\ge d$, we obtain
\[
\begin{split}
\sqrt{p_i}\,|h_i'(t)|&\le C_K\,\frac{d^{3/2}}{\sqrt{p_i}}\,(1+d\tau)e^{-2d\tau}\le C_Kd(1+d\tau)e^{-2d\tau}\\
&\le\frac{C_K}{1+t}\,(2+d\tau)(1+d\tau)e^{-2d\tau}\le\frac{C_K}{1+t},
\end{split}
\]
where we used $d(1+t)=d+1+d\tau\le2+d\tau$.

Integrating the second bound in \eqref{hmr-heat-derivative-bounds}, we find $\sqrt{p_j}|h_j(t+u)-h_j(t)|\le C_K\int_t^{t+u}(1+v)^{-1}\,dv=C_K\log(1+u/(1+t))$, and squaring yields \eqref{hmr-difference-kernel}. Finally, for $y\ge0$, we have $|\frac d{dt}h(t)^\T y|\le\sum_j|h_j'(t)|y_j\le C_K(1+t)^{-1}\sum_jh_j(t)y_j$ by the third bound.
\end{proof}

\section{Estimates in the critical window}\label{sec:heat-critical}

In the critical window, the smallest half-precision is of order $N^{-2/3}$ and no gap between the first two half-precisions is assumed. The following lemma uses only the trace condition and two-sided bounds of order $(i/N)^{2/3}$ on all half-precisions. These hypotheses are part of the critical spectral conditions \eqref{cg-weyl-hypotheses}, which hold for the model at every fixed confidence by \cref{cg-canonical-weyl}. We write $X\asymp Y$ if $c_*Y\le X\le C\,Y$ for positive constants $c_*,C$ that depend only on the quantities named in the statement.

\begin{lemma}[Heat estimates in the critical window]\label{cd-heat-profiles}
Let $N\ge2$ and $0<c_0\le C_0$, and let $r_1,\dots,r_N>0$, not necessarily ordered, satisfy
\begin{equation}\label{eq:heat-weyl}
\sum_{i=1}^N\frac1{2r_i}=N,\qquad c_0\Big(\frac iN\Big)^{2/3}\le r_i\le C_0\Big(\frac iN\Big)^{2/3}\quad(1\le i\le N).
\end{equation}
Let $H$, $h(t)$, and $\bar h(t)$ be as in \eqref{eq:heat-objects} with $p_i=2r_i$; then $H$ is the matrix \eqref{cd-heat-matrix}. For every fixed $K<\infty$, uniformly in $0\le t\le KN^{2/3}$ and $1\le i\le N$,
\begin{equation}\label{cd-heat-bounds}
\bar h(t)=\frac1N\sum_{j=1}^Nh_j(t)\asymp\frac1{\sqrt{1+t}},\qquad h_i(t)\asymp\frac{\sqrt{1+t}}{1+r_it},
\end{equation}
with constants depending only on $c_0$, $C_0$, and $K$. There exists $C<\infty$, depending only on $c_0$ and $C_0$, such that for all $u,w\ge0$,
\begin{equation}\label{cd-heat-sum}
\sum_{i=1}^N\frac1{(1+r_iw)^2(1+r_iu)}\le\frac{CN}{(1+w)^{3/2}}\min\Big\{1,\frac{1+w}{1+u}\Big\} .
\end{equation}
The entries of $e^{tH}$ are given by
\begin{equation}\label{cd-rank-one-heat}
(e^{tH})_{ij}=e^{-4r_it}\delta_{ij}+\frac2N\int_0^te^{-4r_i(t-u)}h_j(u)\,du .
\end{equation}
\end{lemma}

\begin{proof}
The identity \eqref{cd-rank-one-heat} is \eqref{eq:heat-entries} with $p_i=2r_i$. In this proof, $c_*$ and $C$ denote positive constants depending only on $c_0$ and $C_0$, and $C_K$ also depends on $K$. Note that $p_i=2r_i\le2C_0$. The proof has five steps: bounds on the Laplace transform of $\bar h$, the upper bound on $\bar h$, the lower bound on $\bar h$, the bounds on $h_i$, and the bound \eqref{cd-heat-sum}.

\paragraph{Step 1: the Laplace transform.}
By \eqref{hi3-hot-renewal-transform} and \eqref{cd-renewal-resolvent}, for $\zeta>0$,
\[
\widehat{\bar h}(\zeta)=\frac{\mathsf G(\zeta)}{\zeta\,\mathsf S(\zeta)},\qquad \mathsf G(\zeta)=\frac1N\sum_i\frac1{2r_i+\zeta/2},\qquad \mathsf S(\zeta)=\frac1N\sum_i\frac1{r_i(\zeta+4r_i)} .
\]
For $0<\zeta\le1$, we have $(2C_0+1/2)^{-1}\le\mathsf G(\zeta)\le\mathsf G(0)=1$. We show that
\begin{equation}\label{eq:heat-crit-S}
\mathsf S(\zeta)\le C\zeta^{-1/2}\quad(\zeta>0),\qquad \mathsf S(\zeta)\ge c_*\,\zeta^{-1/2}\quad(N^{-2/3}\le\zeta\le1).
\end{equation}
For the upper bound, the lower bound in \eqref{eq:heat-weyl} and the monotonicity of $y\mapsto[c_0y^{2/3}(\zeta+4c_0y^{2/3})]^{-1}$ give
\[
\mathsf S(\zeta)\le\int_0^\infty\frac{dy}{c_0y^{2/3}(\zeta+4c_0y^{2/3})}=\frac6{(4c_0)^{3/2}\sqrt\zeta}\int_0^\infty\frac{dx}{\sqrt x\,(1+x)}=\frac{3\pi}{4c_0^{3/2}\sqrt\zeta},
\]
where we substituted $4c_0y^{2/3}=\zeta x$. For the lower bound, suppose first that $N^{-2/3}\le\zeta\le2^{-2/3}$. The interval $[N\zeta^{3/2},2N\zeta^{3/2}]$ has length at least one and lies in $[1,N]$. It contains at least $N\zeta^{3/2}/2$ integers $i$, and for each of them we have $r_i\le C_0(2\zeta^{3/2})^{2/3}\le2C_0\zeta$ and $[r_i(\zeta+4r_i)]^{-1}\ge[2C_0(1+8C_0)\zeta^2]^{-1}$. Summing over these $i$ yields $\mathsf S(\zeta)\ge c_*\,\zeta^{-1/2}$. If $2^{-2/3}\le\zeta\le1$, the bound $r_i\le C_0$ yields $\mathsf S(\zeta)\ge[C_0(1+4C_0)]^{-1}\ge c_*\,\zeta^{-1/2}$. Combining \eqref{eq:heat-crit-S} with the bounds on $\mathsf G$, we obtain
\begin{equation}\label{eq:heat-crit-transform}
c_*\,\zeta^{-1/2}\le\widehat{\bar h}(\zeta)\le C\zeta^{-1/2},\qquad N^{-2/3}\le\zeta\le1 .
\end{equation}

\paragraph{Step 2: the upper bound on $\bar h$.}
As in the proof of \cref{hi3-finite-heat}, $\bar h(t)\le e\,t^{-1}\widehat{\bar h}(1/t)$ for $t>0$. For $1\le t\le N^{2/3}$, this bound and \eqref{eq:heat-crit-transform} imply that $\bar h(t)\le Ct^{-1/2}$. For $t\le1$, we have $\bar h(t)\le1$. For $N^{2/3}\le t\le KN^{2/3}$, monotonicity gives $\bar h(t)\le \bar h(N^{2/3})\le CN^{-1/3}\le C\sqrt K\,t^{-1/2}$. Altogether, $\bar h(t)\le C_K(1+t)^{-1/2}$ for $0\le t\le KN^{2/3}$.

\paragraph{Step 3: the lower bound on $\bar h$.}
Let $\bar\rho=\vartheta\delta_0+\rho$. By \cref{lem:heat-algebra}, we have $\bar h(t)=\int e^{-\lambda t}\bar\rho(d\lambda)$ and $\widehat{\bar h}(\zeta)=\int\bar\rho(d\lambda)/(\zeta+\lambda)$ for $\zeta>0$. Let $c_2,C_2$ be the constants in \eqref{eq:heat-crit-transform}, and fix $L\ge1$ such that $2C_2L^{-1/2}\le c_2/2$. Let $N^{-2/3}\le\zeta\le1/L$. For $\lambda>L\zeta$, we have $(\zeta+\lambda)^{-1}\le\lambda^{-1}\le2(L\zeta+\lambda)^{-1}$, so
\[
\int_{(L\zeta,\infty)}\frac{\bar\rho(d\lambda)}{\zeta+\lambda}\le2\widehat{\bar h}(L\zeta)\le2C_2(L\zeta)^{-1/2}\le\frac{c_2}2\,\zeta^{-1/2},
\]
where we used \eqref{eq:heat-crit-transform} at $L\zeta\in[N^{-2/3},1]$. It follows that $\int_{[0,L\zeta]}\bar\rho(d\lambda)/(\zeta+\lambda)\ge(c_2/2)\zeta^{-1/2}$. The integrand is at most $\zeta^{-1}$, which implies that $\bar\rho([0,L\zeta])\ge(c_2/2)\zeta^{1/2}$. Then
\[
\bar h(1/\zeta)\ge\int_{[0,L\zeta]}e^{-\lambda/\zeta}\,\bar\rho(d\lambda)\ge e^{-L}\bar\rho([0,L\zeta])\ge e^{-L}\frac{c_2}2\,\zeta^{1/2} .
\]
Setting $t=1/\zeta$, we obtain $\bar h(t)\ge c_*t^{-1/2}$ for $L\le t\le N^{2/3}$. For $t\le L$, we have $\bar h(t)\ge e^{-4C_0t}\ge e^{-4C_0L}$ by the first part of \cref{lem:heat-algebra}. For $t\ge N^{2/3}$, we use $\bar h(t)\ge\vartheta$. By \eqref{ht-effective-normalization-definition}, $\vartheta=N/\sum_i(2r_i)^{-2}=4N/\sum_ir_i^{-2}$, and \eqref{eq:heat-weyl} gives
\[
C_0^{-2}N^{4/3}\le r_1^{-2}\le\sum_ir_i^{-2}\le c_0^{-2}N^{4/3}\sum_{i\ge1}i^{-4/3}\le4c_0^{-2}N^{4/3}.
\]
This gives $c_0^2N^{-1/3}\le\vartheta\le4C_0^2N^{-1/3}$; in particular, $\sum_jh_j(t)$ decreases to $N\vartheta\asymp N^{2/3}$. For $t\ge N^{2/3}$, we have $(1+t)^{-1/2}\le N^{-1/3}$ and therefore $\bar h(t)\ge c_0^2(1+t)^{-1/2}$. We conclude that $\bar h(t)\ge c_*(1+t)^{-1/2}$ for all $t\ge0$, and together with Step~2 this proves the first bound in \eqref{cd-heat-bounds}.

\paragraph{Step 4: the coordinates.}
By \eqref{eq:heat-duhamel}, $h_i(t)=e^{-4r_it}+2\int_0^te^{-4r_i(t-u)}\bar h(u)\,du$. For $t\le1$, we have $e^{-4C_0}\le h_i(t)\le3$, while $(1+C_0)^{-1}\le\sqrt{1+t}/(1+r_it)\le\sqrt2$. Let $1\le t\le KN^{2/3}$. For the lower bound, we keep the integral over $[t/2,t]$ and use Step~3 and $1-e^{-x}\ge x/(1+x)$:
\[
h_i(t)\ge2\bar h(t)\int_0^{t/2}e^{-4r_iv}\,dv=\bar h(t)\,\frac{1-e^{-2r_it}}{2r_i}\ge \bar h(t)\,\frac t{1+2r_it}\ge c_*\,\frac{\sqrt{1+t}}{1+r_it} .
\]
For the upper bound, we have $\bar h(u)\le C_K(1+t)^{-1/2}$ for $u\in[t/2,t]$ by Step~2, and $2\int_0^{t/2}e^{-4r_iv}\,dv\le\min\{t,(2r_i)^{-1}\}\le2t/(1+r_it)$. Thus the integral over $[t/2,t]$ is at most $C_K\sqrt{1+t}/(1+r_it)$. The integral over $[0,t/2]$ is at most $2e^{-2r_it}\int_0^{t/2}\bar h\le C_Ke^{-2r_it}\sqrt t\le C_K\sqrt{1+t}/(1+r_it)$, and $e^{-4r_it}\le(1+r_it)^{-1}$. This proves the second bound in \eqref{cd-heat-bounds}.

\paragraph{Step 5: the sum \eqref{cd-heat-sum}.}
Let $\bar w=1+w$ and $\bar u=1+u$. Since $r_i\le C_0$, we have $1+r_i\bar w\le(1+C_0)(1+r_iw)$, and similarly for $u$. The summand in \eqref{cd-heat-sum} is a decreasing function of $r_i$, and the lower bound in \eqref{eq:heat-weyl} gives
\[
\sum_{i=1}^N\frac1{(1+r_iw)^2(1+r_iu)}\le(1+C_0)^3\sum_{i=1}^N\phi\big(c_0(i/N)^{2/3}\big),\qquad \phi(x)=\frac1{(1+\bar wx)^2(1+\bar ux)} .
\]
Since $\phi$ is decreasing, the right side is at most $(1+C_0)^3N\int_0^1\phi(c_0y^{2/3})\,dy$, and the substitution $x=c_0y^{2/3}$ bounds this by $CN\int_0^\infty\sqrt x\,\phi(x)\,dx$. If $\bar u\le\bar w$, we bound the last factor of $\phi$ by one and substitute $x=x'/\bar w$:
\[
\int_0^\infty\sqrt x\,\phi(x)\,dx\le\int_0^\infty\frac{\sqrt x\,dx}{(1+\bar wx)^2}=\bar w^{-3/2}\int_0^\infty\frac{\sqrt{x'}\,dx'}{(1+x')^2}=\frac\pi2\,\bar w^{-3/2} .
\]
If $\bar u\ge\bar w$, we use $(1+\bar ux)^{-1}\le(\bar ux)^{-1}$ and the same substitution:
\[
\int_0^\infty\sqrt x\,\phi(x)\,dx\le\frac1{\bar u}\int_0^\infty\frac{dx}{\sqrt x\,(1+\bar wx)^2}=\frac{\bar w^{-1/2}}{\bar u}\int_0^\infty\frac{dx'}{\sqrt{x'}\,(1+x')^2}=\frac\pi2\,\bar w^{-3/2}\,\frac{\bar w}{\bar u} .
\]
In both cases, the integral is at most $(\pi/2)\bar w^{-3/2}\min\{1,\bar w/\bar u\}$, and \eqref{cd-heat-sum} follows.
\end{proof}

\section{Positive measures from real squared resolvents}\label{pr-sec:real-measures}

The following lemma identifies limits of positive measures on $[0,\infty)$ from the convergence of their squared resolvents at real arguments. It is used in \cref{sec:hot-meso,sec:outer-hot}, where these squared resolvents are computed from exact resolvent identities such as \eqref{hi3-hot-renewal-transform}. The compact-interval moment method goes back to Hausdorff \cite{Hausdorff1921a,Hausdorff1921b}; see \cite[Chapter~III, pp.~100--143]{Widder1941}. After compactifying $[0,\infty)$ to $[0,1]$, we recover the moments from the squared resolvents by the finite-difference calculation below. A related use of compactification and Hausdorff moments for generalized Stieltjes transforms appears in \cite[proof of Theorem~2, (13)]{Sokal2010}. We then approximate continuous functions by Bernstein polynomials \cite[pp.~1--2]{Bernstein1912}.
We include the short proof because we need versions with varying test functions, with uniformity over families of test functions, and with random measures.

\begin{lemma}[Positive measures from real squared resolvents]\label{pr-real-measure-transfer}
Let $\mu_n$, $n\ge1$, and $\mu$ be positive Borel measures on $[0,\infty)$, and let
\[
T_n(\zeta)=\int_{[0,\infty)}\frac{\mu_n(dy)}{(\zeta+y)^2},\qquad T(\zeta)=\int_{[0,\infty)}\frac{\mu(dy)}{(\zeta+y)^2},\qquad\zeta\ge1 .
\]
Suppose that $T_n(1)$ and $T(1)$ are finite and that $T_n(\zeta)\to T(\zeta)$ for every rational $\zeta\ge1$.
\begin{enumerate}
\item For every continuous function $f:[0,1]\to\R$,
\begin{equation}\label{pr-eq:weighted-transform-convergence}
\int_{[0,\infty)}\frac{f\big((1+y)^{-1}\big)}{(1+y)^2}\,\mu_n(dy)\longrightarrow\int_{[0,\infty)}\frac{f\big((1+y)^{-1}\big)}{(1+y)^2}\,\mu(dy).
\end{equation}
In particular, $\mu_n\to\mu$ vaguely.
\item Let $\varphi_n,\varphi:[0,\infty)\to\R$ be continuous functions such that $\sup_{0\le y\le R}|\varphi_n(y)-\varphi(y)|\to0$ for every $R<\infty$, and
\begin{equation}\label{pr-eq:weighted-test-tails}
\lim_{R\to\infty}\sup_n\sup_{y\ge R}(1+y)^2|\varphi_n(y)|=0,\qquad
\lim_{R\to\infty}\sup_{y\ge R}(1+y)^2|\varphi(y)|=0 .
\end{equation}
Then $\int\varphi_n\,d\mu_n\to\int\varphi\,d\mu$.
\item Let $\Theta$ be a set, and let $\varphi_{n,\theta},\varphi_\theta$, $\theta\in\Theta$, be continuous functions on $[0,\infty)$. Suppose that $\sup_\theta\sup_{0\le y\le R}|\varphi_{n,\theta}(y)-\varphi_\theta(y)|\to0$ for every $R$, and that \eqref{pr-eq:weighted-test-tails} holds with $\sup_\theta$ inserted before $\sup_{y\ge R}$ in both conditions. Suppose also that the family $(\varphi_\theta)_{\theta\in\Theta}$ is uniformly bounded and uniformly equicontinuous on $[0,R]$ for every $R$. Then $\sup_\theta|\int\varphi_{n,\theta}\,d\mu_n-\int\varphi_\theta\,d\mu|\to0$.
\item Suppose instead that $\mu_n$ and $\mu$ are random measures, that $T_n(1)$ and $T(1)$ are almost surely finite, and that $T_n(\zeta)-T(\zeta)\to0$ in probability for every rational $\zeta\ge1$. Then the conclusion of the first part holds in probability for every continuous $f$, and also for every random $f$ that is a measurable random element of $C([0,1])$. The conclusions of the second and third parts hold in probability for deterministic tests that satisfy their hypotheses. They also hold in probability for random tests under the following conditions.
\begin{itemize}
\item The tests are measurable random elements of $C([0,\infty))$, with the topology of uniform convergence on compact sets. In the third part, all suprema over $\theta$ that appear in the hypotheses and in the conclusion are measurable.
\item For each $n$, $\int|\varphi_n|\,d\mu_n+\int|\varphi|\,d\mu<\infty$ almost surely, and in the third part the same holds for each $\theta$.
\item For every $R<\infty$, $\sup_\theta\sup_{0\le y\le R}|\varphi_{n,\theta}(y)-\varphi_\theta(y)|\to0$ in probability.
\item For every $\eps>0$, there exist events $\mathcal E_n^\eps$ with $\liminf_n\PP(\mathcal E_n^\eps)\ge1-\eps$ and a deterministic function $\tau_\eps$ with $\tau_\eps(R)\to0$ as $R\to\infty$ such that, for each $n$, on $\mathcal E_n^\eps$,
\[
\sup_{\theta\in\Theta}\sup_{y\ge R}(1+y)^2\big(|\varphi_{n,\theta}(y)|+|\varphi_\theta(y)|\big)\le\tau_\eps(R)\qquad\text{for every }R\ge1 .
\]
In the third part, we also assume that on $\mathcal E_n^\eps$ the functions $\varphi_\theta$ are bounded by a deterministic constant $B_\eps$ and have, on each interval $[0,R]$, a modulus of continuity bounded by a deterministic function $\varpi_{\eps,R}$ with $\varpi_{\eps,R}(r)\to0$ as $r\downarrow0$.
\end{itemize}
In the second part, the suprema over $\theta$ are omitted.
\end{enumerate}
\end{lemma}

\begin{proof}
We first compactify, then recover the moments by finite differences, approximate continuous functions by Bernstein polynomials, and finally treat varying tests, families of tests, and random measures.

\paragraph{Step 1: compactification.}
Let $\lambda_n$ and $\lambda$ be the images of the finite measures $(1+y)^{-2}\mu_n(dy)$ and $(1+y)^{-2}\mu(dy)$ under the map $y\mapsto x=(1+y)^{-1}$ from $[0,\infty)$ to $(0,1]$. They are finite measures on $[0,1]$ with no atom at $0$ and with total masses $T_n(1)$ and $T(1)$. The left side of \eqref{pr-eq:weighted-transform-convergence} is $\int f\,d\lambda_n$. For $t\ge0$, let $F_t(x)=(1+tx)^{-2}$. Since $(1+t/(1+y))^{-2}(1+y)^{-2}=(1+t+y)^{-2}$, we have
\begin{equation}\label{pr-eq:compact-resolvent-test}
\int F_t\,d\lambda_n=T_n(1+t),\qquad\int F_t\,d\lambda=T(1+t).
\end{equation}
In particular, $\lambda_n([0,1])=T_n(1)\to T(1)=\lambda([0,1])$.

\paragraph{Step 2: moments.}
Fix an integer $k\ge1$ and a rational $\delta>0$, and let
\[
D_{k,\delta}(x)=\frac1{(-1)^k(k+1)!\,\delta^k}\sum_{j=0}^k(-1)^{k-j}\binom kjF_{j\delta}(x) .
\]
We have $\partial_t^kF_t(x)=(-1)^k(k+1)!\,x^k(1+tx)^{-k-2}$. The $k$th forward difference of $t\mapsto F_t(x)$ with step $\delta$ is the integral of $\partial_t^kF_t(x)$ at $t=t_1+\dots+t_k$ over $(t_1,\dots,t_k)\in[0,\delta]^k$, by the fundamental theorem of calculus applied $k$ times. Hence
\[
D_{k,\delta}(x)=\delta^{-k}\int_{[0,\delta]^k}\frac{x^k}{(1+(t_1+\dots+t_k)x)^{k+2}}\,dt_1\cdots dt_k .
\]
For $x\in[0,1]$ and $0\le\tau\le k\delta$, Bernoulli's inequality $(1+\tau x)^{-k-2}\ge1-(k+2)\tau x$ implies that $0\le x^k-x^k(1+\tau x)^{-k-2}\le(k+2)k\delta$. Taking $\tau=t_1+\dots+t_k$ and averaging over the cube, we obtain
\begin{equation}\label{pr-eq:finite-difference-error}
\sup_{0\le x\le1}\big|D_{k,\delta}(x)-x^k\big|\le k(k+2)\,\delta .
\end{equation}
By \eqref{pr-eq:compact-resolvent-test}, $\int D_{k,\delta}\,d\lambda_n$ is a finite linear combination of the numbers $T_n(1+j\delta)$, at rational arguments, and it converges to $\int D_{k,\delta}\,d\lambda$. Further, \eqref{pr-eq:finite-difference-error} gives
\[
\Big|\int x^k\,\lambda_n(dx)-\int x^k\,\lambda(dx)\Big|\le\Big|\int D_{k,\delta}\,d\lambda_n-\int D_{k,\delta}\,d\lambda\Big|+k(k+2)\delta\big(T_n(1)+T(1)\big) .
\]
Letting $n\to\infty$ and then $\delta\downarrow0$ through the rationals, we find that every moment of $\lambda_n$ converges to the corresponding moment of $\lambda$. For $k=0$ this is $T_n(1)\to T(1)$.

\paragraph{Step 3: continuous functions.}
For a continuous $f$ on $[0,1]$ and $m\ge1$, let $B_mf(x)=\sum_{j=0}^mf(j/m)\binom mjx^j(1-x)^{m-j}$ be its Bernstein polynomial, and let $\omega_f$ be its modulus of continuity. Let $J$ be a binomial random variable with parameters $m$ and $x$. Then $B_mf(x)-f(x)=\E[f(J/m)-f(x)]$, and $J/m$ has mean $x$ and variance $x(1-x)/m\le1/(4m)$. Splitting according to whether $|J/m-x|\le r$ and using Chebyshev's inequality, we find
\[
\|B_mf-f\|_\infty\le\omega_f(r)+2\|f\|_\infty\,\frac1{4mr^2}=\omega_f(r)+\frac{\|f\|_\infty}{2mr^2} .
\]
With $r=m^{-1/4}$, the right side tends to zero as $m\to\infty$. Then
\[
\Big|\int f\,d\lambda_n-\int f\,d\lambda\Big|\le\Big|\int B_mf\,d\lambda_n-\int B_mf\,d\lambda\Big|+\|B_mf-f\|_\infty\big(T_n(1)+T(1)\big).
\]
The first term tends to zero as $n\to\infty$ by Step~2, since $B_mf$ is a polynomial. Letting $n\to\infty$ and then $m\to\infty$ proves \eqref{pr-eq:weighted-transform-convergence}. If $\varphi$ is continuous with compact support in $[0,\infty)$, then $f(x)=x^{-2}\varphi(x^{-1}-1)$ for $x\in(0,1]$ and $f(0)=0$ defines a continuous function on $[0,1]$ that vanishes near zero, with $\int f\,d\lambda_n=\int\varphi\,d\mu_n$, and vague convergence follows from \eqref{pr-eq:weighted-transform-convergence}.

\paragraph{Step 4: varying tests.}
Define $f_n(x)=x^{-2}\varphi_n(x^{-1}-1)$ and $f(x)=x^{-2}\varphi(x^{-1}-1)$ for $x\in(0,1]$, and $f_n(0)=f(0)=0$. With $y=x^{-1}-1$, we have $|f(x)|=(1+y)^2|\varphi(y)|$. The second condition in \eqref{pr-eq:weighted-test-tails} then shows that $f$ is continuous at zero, and similarly for each $f_n$. Also $\int\varphi_n\,d\mu_n=\int f_n\,d\lambda_n$ and $\int\varphi\,d\mu=\int f\,d\lambda$. Let $R>0$. For $x\ge(1+R)^{-1}$, that is $y\le R$, we have $|f_n(x)-f(x)|\le(1+R)^2\sup_{[0,R]}|\varphi_n-\varphi|$. For $x<(1+R)^{-1}$, we have $|f_n(x)-f(x)|\le\sup_{y\ge R}(1+y)^2(|\varphi_n(y)|+|\varphi(y)|)$. Choosing first $R$ large and then $n$ large, we find $\|f_n-f\|_\infty\to0$. Then
\[
\Big|\int f_n\,d\lambda_n-\int f\,d\lambda\Big|\le\|f_n-f\|_\infty\,T_n(1)+\Big|\int f\,d\lambda_n-\int f\,d\lambda\Big|\longrightarrow0,
\]
by Step~3. This proves the second part.

\paragraph{Step 5: families of tests.}
Define $f_{n,\theta}$ and $f_\theta$ as in Step~4. As in Step~4, we have $\sup_\theta\|f_{n,\theta}-f_\theta\|_\infty\to0$. We claim that $\{f_\theta:\theta\in\Theta\}$ is uniformly bounded and uniformly equicontinuous on $[0,1]$. Let $\tau(R)=\sup_\theta\sup_{y\ge R}(1+y)^2|\varphi_\theta(y)|$, which tends to zero. On $[0,(1+R)^{-1}]$ we have $|f_\theta|\le\tau(R)$. On $[(1+R)^{-1},1]$, the map $x\mapsto x^{-1}-1$ is Lipschitz with constant $(1+R)^2$, and $x\mapsto x^{-2}$ is bounded by $(1+R)^2$ and Lipschitz with constant $2(1+R)^3$. Therefore the uniform bound and the uniform modulus of continuity of $(\varphi_\theta)$ on $[0,R]$ yield a uniform bound and a uniform modulus of continuity of $(f_\theta)$ there. Given $\eps>0$, we choose $R$ with $2\tau(R)\le\eps$; two points of $[0,1]$ at distance at most $r$ that are not both in $[(1+R)^{-1},1]$ can be compared through the point $(1+R)^{-1}$, where $|f_\theta|\le\tau(R)$. This proves the claim. By the Arzel\`a--Ascoli theorem, for every $\eps>0$ there exist finitely many continuous functions $g_1,\dots,g_L$ on $[0,1]$ such that every $f_\theta$ satisfies $\|f_\theta-g_l\|_\infty\le\eps$ for some $l$. Then
\[
\sup_\theta\Big|\int f_\theta\,d\lambda_n-\int f_\theta\,d\lambda\Big|\le\max_{l\le L}\Big|\int g_l\,d\lambda_n-\int g_l\,d\lambda\Big|+\eps\big(T_n(1)+T(1)\big),
\]
and the maximum tends to zero by Step~3. Since $\eps$ is arbitrary and $\sup_\theta\|f_{n,\theta}-f_\theta\|_\infty\,T_n(1)\to0$, the third part follows.

\paragraph{Step 6: random measures.}
Under the hypotheses of the fourth part, $T_n(1)$ converges in probability to the almost surely finite $T(1)$. The total masses are therefore tight, that is, for every $\eps>0$ there exists $M_\eps$ with $\PP(T_n(1)+T(1)>M_\eps)\le\eps$ for all large $n$. In Step~2, at fixed $k$ and $\delta$, $\int D_{k,\delta}\,d\lambda_n-\int D_{k,\delta}\,d\lambda$ tends to zero in probability. Given $\eps>0$, we choose $\delta$ with $k(k+2)\delta M_\eps\le\eps$ and then $n$ large; this shows that every moment difference tends to zero in probability. In Step~3, for deterministic $f$, we choose $m$ with $\|B_mf-f\|_\infty M_\eps\le\eps$. For a random $f$ that is a measurable random element of $C([0,1])$, $\int B_mf\,d\lambda_n-\int B_mf\,d\lambda$ is a sum of $m+1$ products of the almost surely finite random variables $f(j/m)$ with moment differences of $x^j(1-x)^{m-j}$, which tend to zero in probability. Hence it tends to zero in probability, and $\|B_mf-f\|_\infty\to0$ almost surely as $m\to\infty$. This proves the first part in probability.

For random tests, the integrals in the second and third parts are almost surely defined by the integrability assumption, and the measurability assumptions make the events below measurable. Since the limit tests $\varphi_\theta$ do not depend on $n$, the event on which their weighted tails are bounded by $\tau_\eps$, and in the third part also the bound $B_\eps$ and the moduli $\varpi_{\eps,R}$ hold, contains $\mathcal E_n^\eps$ for every $n$, and so has probability at least $1-\eps$. On this event, the transformed limit tests $f_\theta$ are continuous at zero, and in the third part they lie in a fixed deterministic set $\mathcal K_\eps$ of functions on $[0,1]$ with a common bound and a common modulus of continuity, as in Step~5. This set is compact in the uniform topology by the Arzel\`a--Ascoli theorem, so the functions $g_1,\dots,g_L$ of Step~5 can be chosen deterministic. On $\mathcal E_n^\eps$, as in Step~4,
\[
\sup_\theta\|f_{n,\theta}-f_\theta\|_\infty\le(1+R)^2\sup_\theta\sup_{0\le y\le R}|\varphi_{n,\theta}(y)-\varphi_\theta(y)|+\tau_\eps(R)
\]
for every $R\ge1$. We choose $R$ with $\tau_\eps(R)$ small and then let $n\to\infty$; since the complements of the events above have probability at most $\eps$ asymptotically, $\sup_\theta\|f_{n,\theta}-f_\theta\|_\infty\to0$ in probability. The first part, in probability, applied to $g_1,\dots,g_L$ and to $f$, together with the tightness of $T_n(1)+T(1)$, completes the proof as in Steps~4 and~5.
\end{proof}