\chapter{The critical profile}\label{sec:crit-profile}

The goal of this section is to prove \cref{thm:intro-critical}, which describes the mixing profile in the critical window $c_N=1+\sigma N^{-1/3}$. The proof has a deterministic part and a probabilistic part. The deterministic part is \cref{cd-profile}. Consider a sequence of half-precisions that satisfies the critical Weyl bounds and a lower bound on the first gap, and whose scaled inverses converge. We show that the total variation distance $D_N(N^{2/3}s)$ then converges, locally uniformly in $s>0$, to the total variation distance $F_R(s)$ between the time-$s$ law of the normalized critical limit process of \cref{sec:crit-process} and the limiting Gibbs law. The limit is continuous, positive, and strictly decreasing, and \cref{cd-quantiles} transfers the convergence to the mixing times at every fixed accuracy. The probabilistic part is \cref{cd-goe-profile}. We transfer the deterministic theorem to the GOE model with a localized form of the continuous mapping theorem (\cref{lem:cprof-varying-map}), applied to Borel maps that send the spectral data to the finite profiles and mixing times. The limiting curves are measurable functions of a single Airy point process, the profile is random, and from the limit of the ratio of two mixing times we deduce that there is no cutoff. Finally, in \cref{cl3-density-clock} we bound the mixing times above and below by multiples of $N^{2/3}\log(1/\eps)$, for every accuracy $\eps\in(0,1/2)$, and in \cref{critical-finite-time-positivity} we deduce the lower bound $F_R(s)\ge c_1e^{-C_1s}$.

The proofs rely on several earlier results. \Cref{sec:process} provides the representation of the dynamics through the unnormalized process (\cref{cd-lift}) and its moments (\cref{cd-lift-moments}), and \cref{sec:heat} provides the heat estimates of \cref{cd-heat-profiles}. \Cref{sec:inequalities} contains the functional inequalities of \cref{cg-many-soft-pi,cg-three-soft-lsi,cg-critical-lsi} and the density bounds of \cref{cg-mean-level-density}. \Cref{sec:density} bounds moments of the density of the time-$t$ law and gives exponential relaxation from positive times (\cref{cl3-actual-source-density,cg-source-l2-decay}). \Cref{sec:crit-process} constructs the critical limit process; finitely many rescaled coordinates converge in law to its normalization (\cref{cd-functional}), and the rescaled Gibbs measures converge to the conditioned Gaussian law of \cref{cd-pin}. \Cref{sec:spectral} establishes the spectral geometry of the GOE in the critical window (\cref{cg-canonical-weyl}). \Cref{tab:cprof-inputs} records the precise statements we use. Conversely, \cref{cd-profile} is used again in \cref{sec:outer-hot,sec:outer-cold}, where the critical profile is studied far from the critical temperature.

\section{Conditions, notation, and main results}\label{sec:cprof-setup}

\subsection{The critical spectral conditions}

Throughout this section $b=N^{2/3}$. We work with the Gibbs measure written through half-precisions: for a vector $r=(r_1,\dots,r_N)$ we set
\begin{equation}\label{eq:cprof-gibbs}
\pi_r(dx)=\frac1{\mathcal Z_r}\exp\Big(-\sum_{i=1}^Nr_ix_i^2\Big)\,\nu_N(dx),
\end{equation}
and we let $(P^r_t)_{t\ge0}$ be the semigroup of the Langevin diffusion with generator $L_r=\Delta_{S_N}-\nabla_{S_N}(\sum_ir_ix_i^2)\cdot\nabla_{S_N}$. For the model with matrix $A=cW_N$, written in an eigenbasis, the half-precisions are $r_i=(z-c\lambda_i)/2$, where $z$ is the Lagrange multiplier (see \eqref{eq:model-precision-form}); since adding a multiple of $|x|^2$ to the potential changes neither the generator nor the Gibbs measure, $\pi_r$ and $P^r_t$ are then the Gibbs measure $\pi_{N,c}$ and the semigroup $P_t$ of \cref{sec:model}. When $r$ is fixed we drop it from the notation and write $\pi_N=\pi_r$, $P_t=P^r_t$, $D_N(t)=\tvnorm{\nu_NP_t-\pi_N}$, and $t_N(\eps)=\inf\{t\ge0:D_N(t)\le\eps\}$.

Recall from \cref{def:spec-critical-conditions} that, for constants $0<c_0\le C_0$ and $\delta>0$, a vector $r=(r_1,\dots,r_N)$ with $0<r_1\le r_2\le\cdots\le r_N$ satisfies the \emph{critical spectral conditions with constants $(c_0,C_0,\delta)$} if
\begin{gather}
\sum_{i=1}^N\frac1{2r_i}=N,\label{eq:cprof-trace}\\
c_0\Big(\frac iN\Big)^{2/3}\le r_i\le C_0\Big(\frac iN\Big)^{2/3}\qquad(1\le i\le N),\label{eq:cprof-weyl}\\
r_2-r_1\ge\delta/b.\label{eq:cprof-gap}
\end{gather}
These are the three conditions in \eqref{cg-weyl-hypotheses}. As in \cref{def:cproc-critical}, \eqref{eq:cprof-trace} is the trace condition and \eqref{eq:cprof-weyl} are the Weyl bounds; we call \eqref{eq:cprof-gap} the first-gap bound. The trace condition and the Weyl bounds suffice for the moment estimates of \cref{sec:process} and for the lower bound of \cref{cg-entrance-lower}; the first-gap bound enters through the functional inequalities of \cref{cg-critical-lsi} and the density estimates of \cref{sec:density}.

As in \cref{def:cproc-critical}, sequences $r_{\cdot,N}=(r_{i,N})_{i\le N}$ of such vectors are indexed by an infinite set $\mathcal N$ of integers, and limits as $N\to\infty$ are taken along $\mathcal N$. We write $R_{i,N}=br_{i,N}$ and set $R_{i,N}^{-1}=0$ for $i>N$. A sequence $(r_{\cdot,N})_{N\in\mathcal N}$ \emph{converges to} a sequence $R=(R_i)_{i\ge1}$ of positive numbers if
\begin{equation}\label{eq:cprof-conv}
\lim_{N\to\infty}\sum_{i\ge1}\big(R_{i,N}^{-1}-R_i^{-1}\big)^2=0
\qquad\text{and}\qquad
\lim_{i\to\infty}\frac{R_i}{i^{2/3}}=\gamma\ \text{ for some }\gamma\in(0,\infty).
\end{equation}
A sequence that satisfies the critical spectral conditions with fixed constants for every $N\in\mathcal N$ and converges to $R$ is a critical sequence with limit $R$ in the sense of \cref{def:cproc-critical}, which in addition satisfies the first-gap bound. For such a sequence, $R$ is nondecreasing and $R_2-R_1\ge\delta$, since $R_{i,N}\to R_i$ for each $i$. We denote by $\mathcal P$ the set of sequences $R=(R_i)_{i\ge1}$ with
\begin{equation}\label{eq:cprof-class}
0<R_1<R_2\le R_3\le\cdots\qquad\text{and}\qquad\lim_{i\to\infty}R_i/i^{2/3}\in(0,\infty).
\end{equation}
Every $R\in\mathcal P$ satisfies $\inf_iR_i/i^{2/3}>0$ and $\sup_iR_i/i^{2/3}<\infty$. The next lemma shows that every $R\in\mathcal P$ is the limit of a sequence satisfying the critical spectral conditions; we use it to define the limiting objects for every $R\in\mathcal P$, and in \cref{sec:cprof-goe} to prove that they depend measurably on $R$.

\begin{lemma}[Finite approximations of a limiting sequence]\label{lem:cprof-approx}
Identify $R\in\mathcal P$ with the sequence $u=(R_i^{-1})_{i\ge1}\in\ell^2$. Then $\mathcal P$ is a Borel subset of $\ell^2$, and there exist Borel maps $R\mapsto r^{(N)}(R)\in\R^N$, $N\ge2$, with the following property. For every $R\in\mathcal P$ there exists $N_1(R)$ such that, for $N\ge N_1(R)$, the vector $r^{(N)}(R)$ satisfies the critical spectral conditions with constants
\begin{equation}\label{eq:cprof-approx-constants}
c_0=\min\Big(1,\inf_i\frac{R_i}{i^{2/3}}\Big),\qquad C_0=\max\Big(2,\sup_i\frac{R_i}{i^{2/3}}\Big),\qquad \delta=R_2-R_1,
\end{equation}
and the sequence $(r^{(N)}(R))_{N}$ converges to $R$.
\end{lemma}

\begin{proof}
The set $\mathcal P$ is Borel, because it is defined by countably many conditions on the coordinates $u_i$ (positivity, monotonicity, and existence of $\lim_ii^{2/3}u_i$ in $(0,\infty)$). Fix $R\in\mathcal P$, let $c_R=\inf_iR_i/i^{2/3}$, and let $M=\lfloor N^{1/2}\rfloor$. Define
\begin{equation*}
\Theta_N=\Big(2N^{1/3}-\sum_{i\le M}R_i^{-1}\Big)^{-1}\sum_{M<i\le N}i^{-2/3}
\end{equation*}
whenever the bracket is positive, let $\hat R_{i,N}=R_i$ for $i\le M$ and $\hat R_{i,N}=\Theta_Ni^{2/3}$ for $M<i\le N$, let $(R_{i,N})_{i\le N}$ be the nondecreasing rearrangement of $(\hat R_{i,N})_{i\le N}$, and set $r^{(N)}_i(R)=R_{i,N}/b$. When the bracket is not positive, set $r^{(N)}(R)=(1/2,\dots,1/2)$. These maps are Borel, since $\Theta_N$ is a continuous function of finitely many coordinates of $u$ on the open set where the bracket is positive, and rearrangement is continuous.

We now verify the critical spectral conditions for large $N$. Recall that $\sum_{i\le n}i^{-2/3}=3n^{1/3}+O(1)$, which gives $\sum_{M<i\le N}i^{-2/3}=3N^{1/3}+O(N^{1/6})$. Further, $\sum_{i\le M}R_i^{-1}\le c_R^{-1}\sum_{i\le M}i^{-2/3}=O(N^{1/6})$. Then $\Theta_N\to3/2$, and $\Theta_N\in[1,2]$ for $N\ge N_1(R)$. By the definition of $\Theta_N$,
\begin{equation*}
\sum_{i\le N}\frac1{2r^{(N)}_i(R)}=\frac b2\Big(\sum_{i\le M}R_i^{-1}+\Theta_N^{-1}\sum_{M<i\le N}i^{-2/3}\Big)=\frac b2\cdot2N^{1/3}=N,
\end{equation*}
which is the trace condition. Before rearrangement we have $c_0i^{2/3}\le\hat R_{i,N}\le C_0i^{2/3}$ for every $i$, with the constants \eqref{eq:cprof-approx-constants}. These bounds survive the rearrangement: at most $i-1$ indices $j$ satisfy $\hat R_{j,N}<c_0i^{2/3}$ (all of them have $j<i$), so the $i$-th smallest value is at least $c_0i^{2/3}$, and at least $i$ indices satisfy $\hat R_{j,N}\le C_0i^{2/3}$ (namely $j\le i$), which bounds the $i$-th smallest value by $C_0i^{2/3}$. Dividing by $b=N^{2/3}$ gives the Weyl bounds. For large $N$ we have $\Theta_N(M+1)^{2/3}\ge(M+1)^{2/3}>R_2$. Then the two smallest entries are $R_1$ and $R_2$, and the first-gap bound holds with $\delta=R_2-R_1$.

Finally, let $\hat u_N$ be the vector $(\hat R_{i,N}^{-1})_{i\le N}$, padded with zeros. Then
\begin{equation*}
\|\hat u_N-u\|_2^2=\sum_{M<i\le N}\big(\Theta_N^{-1}i^{-2/3}-R_i^{-1}\big)^2+\sum_{i>N}R_i^{-2}\le2(1+c_R^{-2})\sum_{i>M}i^{-4/3},
\end{equation*}
which tends to zero. The vector $(R_{i,N}^{-1})_{i\ge1}$ is the nonincreasing rearrangement of $\hat u_N$, and $u$ is nonincreasing. Rearrangement does not increase the $\ell^2$ distance to $u$: for $a\ge b$ and $c\ge d$,
\[
(a-d)^2+(b-c)^2-(a-c)^2-(b-d)^2=2(a-b)(c-d)\ge0.
\]
Successively exchanging inverted pairs therefore shows that sorting the first $N$ entries of $\hat u_N$ can only decrease their squared distance to $(u_i)_{i\le N}$; the entries after $N$ remain zero. Consequently the inverse vectors converge to $u$ in $\ell^2$. The second condition in \eqref{eq:cprof-conv} is part of the definition of $\mathcal P$.
\end{proof}

\subsection{Notation}

We fix the following notation for a vector $r$ satisfying the critical spectral conditions.
\begin{itemize}
\item We write $q_i=(2r_i)^{-1}$ for the variances of the associated conditioned Gaussian law. We have $\sum_iq_i=N$ by \eqref{eq:cprof-trace}, and $\pi_N$ is the law of independent centered Gaussians with variances $q_i$ conditioned on $|x|^2=N$ (\cref{sec:conditioning}).
\item For $t>0$ the law $\nu_NP_t$ has a positive density of class $C^2$ in space and $C^1$ in time with respect to $\pi_N$, by the regularity argument of \cref{lem:dens-derivative}. We write $f_{N,s}=d(\nu_NP_{bs})/d\pi_N$ for $s>0$. By reversibility, $f_{N,s}=P_{bs}f_{N,0}$, where $f_{N,0}=d\nu_N/d\pi_N$. Poincar\'e inequalities stated on smooth functions extend to $H^1$ by smooth approximation on the compact weighted spheres; the integration by parts arguments below require only this $C^2$ spatial regularity.
\item For a probability measure $\mu$ on $S_N$ and $J\le N$, we write $\mu^{\le J}$ for the law of the rescaled first $J$ coordinates $(N^{-1/3}x_i)_{i\le J}$ under $\mu$. For a probability measure $\mu$ on $\R^{\N}$ (with the product Borel $\sigma$-field), $\mu^{\le J}$ is the law of the first $J$ coordinates. We set $D_N^{\le J}(t)=\tvnorm{(\nu_NP_t)^{\le J}-\pi_N^{\le J}}$. Total variation distances do not change under the rescaling.
\item We let $Y$ be the unnormalized process of \cref{cd-lift} with half-precisions $r_i$ and $Y(0)=g$, a standard Gaussian vector, so that $X=\sqrt NY/|Y|$ is the Langevin diffusion started from $\nu_N$. We write $\mathcal A_t=|Y(t)|^2$, as in \cref{sec:process}, and consider the scaled processes
\begin{equation}\label{eq:cprof-scaled}
U_{i,N}(s)=N^{-1/6}Y_i(bs),\qquad Z_N(s)=b^{-1}\mathcal A_{bs},\qquad N^{-1/3}X(bs)=\frac{U_N(s)}{\sqrt{Z_N(s)}}.
\end{equation}
The processes $B_i^{(N)}(s)=b^{-1/2}B_i(bs)$ are independent Brownian motions, and the equation $dY_i=-2r_iY_i\,dt+\sqrt{2/N}\,|Y|\,dB_i$ becomes
\begin{equation}\label{eq:cprof-scaled-sde}
dU_{i,N}=-2R_{i,N}U_{i,N}\,ds+\sqrt{2Z_N}\,dB_i^{(N)}.
\end{equation}
\end{itemize}
Constants denoted $C,C'$ may change from line to line, and small positive constants always carry a subscript, since the letter $c$ alone denotes the rescaled inverse temperature. Unless stated otherwise, constants depend only on $(c_0,C_0,\delta)$ and on the parameters indicated in their subscripts, and never on $N$, $s$, or the conditioning values. A supremum of nonnegative errors over the empty set is defined to be zero.

\subsection{The critical limit process}\label{sec:cprof-limit}

Let $R\in\mathcal P$, and let $(r_{\cdot,N})$ be a sequence that satisfies the critical spectral conditions with fixed constants for every $N\in\mathcal N$ and converges to $R$ (such sequences exist by \cref{lem:cprof-approx}). We use the following objects from \cref{sec:crit-process}. They depend only on $R$, and not on the approximating sequence.

\begin{itemize}
\item \emph{The critical limit process} (\cref{cd-functional}). This is a pair $(U,Z)$ of processes, where $U=(U_i)_{i\ge1}$ is a sequence of continuous processes with $U_i(0)=0$, and $Z$ is continuous on $(0,\infty)$ with $Z(s)>0$ for every $s>0$. Almost surely, $s^{1/2}Z(s)$ converges to a positive constant as $s\downarrow0$. For every $J$ and $0<s_0<s_1$, the processes $(N^{-1/3}X_i(bs))_{i\le J}$, $s\in[s_0,s_1]$, converge in distribution in $C([s_0,s_1],\R^J)$ to $(U_i(s)/\sqrt{Z(s)})_{i\le J}$, by \eqref{cd-angular-functional}. Recall from \cref{cd-functional} that
\begin{equation}\label{eq:cprof-normalized}
\xi(s)=\big(U_i(s)/\sqrt{Z(s)}\big)_{i\ge1},\qquad s>0,
\end{equation}
is the normalized critical limit process. We write $\mu^\infty_s$ for its law on $\R^\N$.
\item \emph{The limiting Gibbs law} (\cref{cd-pin}). The probability measure $\rho_R$ on $\R^\N$ is the Gaussian sequence with variances $(2R_i)^{-1}$ conditioned on the vanishing of its centered radial series; at the critical temperature this law is due to Du and Huang \cite[Section~2.2]{DuHuang2026}. For each $J$, the marginal $\rho_R^{\le J}$ has a density with respect to Lebesgue measure on $\R^J$, and $\pi_N^{\le J}\to\rho_R^{\le J}$ in total variation.
\end{itemize}
We define the \emph{critical mixing profile} by
\begin{equation}\label{eq:cprof-profile}
F_R(s)=\tvnorm{\mu^\infty_s-\rho_R},\qquad s>0,
\end{equation}
the total variation distance on $\R^\N$ between the law of the normalized critical limit process at time $s$ and the limiting Gibbs law.

\Cref{tab:cprof-inputs} lists the results from other sections that the proofs use, with the form in which we use them. The constants in the middle column depend only on the constants of the critical spectral conditions and on the indicated parameters.

\begin{table}[!htbp]
\centering
\small
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.21\textwidth}>{\raggedright\arraybackslash}p{0.5\textwidth}>{\raggedright\arraybackslash}p{0.22\textwidth}@{}}
\toprule
Result & What is used & Hypotheses\\
\midrule
\cref{cd-lift} & $X=\sqrt NY/|Y|$ is the Langevin diffusion started from the law of $\sqrt NY(0)/|Y(0)|$; with $Y(0)=g$ this law is $\nu_N$ & any real $r$; $Y(0)\ne0$ almost surely, independent of the Brownian motions\\
\cref{cd-heat-profiles} & $\bar h(t)\asymp(1+t)^{-1/2}$ and $h_i(t)\le C(1+t)^{1/2}(1+r_it)^{-1}$ for $0\le t\le Kb$ & \eqref{eq:cprof-trace}, \eqref{eq:cprof-weyl}\\
\cref{cd-lift-moments} & $\E[(N\bar h(t)/\mathcal A_t)^p]\le C_{K,p}$ and $\|Y_i(t)^2\|_p\le C_{K,p}h_i(t)$ for $t\le Kb$, $N\ge N_0$ & \eqref{eq:cprof-trace}, \eqref{eq:cprof-weyl}\\
\cref{cg-mean-level-density} & density of $\sum_iq_iG_i^2$ at its mean is at least $c_{\delta'}m_2^{-1/2}$; with one term removed, the density is at most $C_{\delta'}m_2^{-1/2}$, where $m_2=\sum_iq_i^2$ & three weights $\ge\delta'\max_iq_i$\\
\cref{cg-many-soft-pi} & Poincar\'e constant at most $1/t$ & see \cref{lem:cprof-radius}\\
\cref{cg-three-soft-lsi} & $C_{\mathrm{LS}}\le C_{\delta'}/t+C_{\delta'}C_{\mathrm P}$ & see \cref{lem:cprof-radius}\\
\cref{cg-critical-lsi} & $C_{\mathrm P}(\pi_N)\le K_{\mathrm P}b$ & critical spectral conditions\\
\cref{cl3-actual-source-density} & $\sup_{t\ge s_0b}\|d(\nu_NP_t)/d\pi_N\|_{L^p(\pi_N)}\le C_{p,s_0}$, $N\ge N_0$ & critical spectral conditions\\
\cref{cd-functional,cd-pin} & the critical limit process, convergence of its normalized coordinates, the limiting Gibbs law, and the convergence of $\pi_N^{\le J}$, as listed in \cref{sec:cprof-limit} & critical spectral conditions, convergence to $R$\\
\cref{prop:cond-continuity}; \cref{cd-pin}(c) & for $m\ge3$ and $N\ge N_0(m)$, the law of $(\sqrt{2R_{i,N}}\,N^{-1/3}x_i)_{i\le m}$ under $\pi_N$ has a density with respect to the standard Gaussian law on $\R^m$ that is bounded uniformly in $N$, by domination and continuity in the variance sequence & critical spectral conditions, convergence to $R$\\
\cref{gp-head-density} & marginal densities of a Gaussian law conditioned on a sphere & three remaining coordinates\\
\cref{cg-canonical-weyl} & GOE: critical spectral conditions on events of probability close to one; joint convergence in distribution of the spectral data & $|\sigma|\le M$\\
\bottomrule
\end{tabular}
\caption{Results from other sections used in \cref{sec:crit-profile}. Here $h_i(t)$ and $\bar h(t)=N^{-1}\sum_ih_i(t)$ are the heat quantities of \cref{cd-heat-profiles}, $\mathcal A_t=|Y(t)|^2$, and $G_i$ are independent standard Gaussians.}
\label{tab:cprof-inputs}
\end{table}

\subsection{Main results}

\begin{theorem}[The critical profile along a deterministic sequence]\label{cd-profile}
Let $R\in\mathcal P$, and let $(r_{\cdot,N})_{N\in\mathcal N}$ satisfy the critical spectral conditions with constants $(c_0,C_0,\delta)$ for every $N\in\mathcal N$ and converge to $R$. Then
\begin{equation}\label{cd-full-tv-limit}
\lim_{N\to\infty}\sup_{s\in[s_0,s_1]}\big|D_N(bs)-F_R(s)\big|=0\qquad\text{for every }0<s_0<s_1<\infty.
\end{equation}
The function $F_R$ is continuous and strictly decreasing on $(0,\infty)$, it satisfies $F_R(s)>0$ for every $s>0$, and
\begin{equation*}
\lim_{s\downarrow0}F_R(s)=1,\qquad\lim_{s\to\infty}F_R(s)=0.
\end{equation*}
\end{theorem}

Thus $F_R$ is a continuous strictly decreasing bijection from $(0,\infty)$ onto $(0,1)$, and it has a continuous inverse $F_R^{-1}\colon(0,1)\to(0,\infty)$.

\begin{corollary}[Mixing times in the critical window]\label{cd-quantiles}
Under the hypotheses of \cref{cd-profile}, for every compact interval $I\subset(0,1)$,
\begin{equation}\label{cd-quantile-limit}
\lim_{N\to\infty}\sup_{\eps\in I}\big|N^{-2/3}t_N(\eps)-F_R^{-1}(\eps)\big|=0.
\end{equation}
In particular, $F_R^{-1}(\eps)/F_R^{-1}(1-\eps)>1$ for every $\eps\in(0,1/2)$.
\end{corollary}

We now turn to the GOE model, and we recall the limiting spectral objects of \cref{sec:spectral}. Let $\chi_1<\chi_2<\cdots$ be the reflected GOE Airy point process, the limit of $(N^{2/3}(2-\lambda_i))_{i\le k}$ for each fixed $k$ \cite[Theorem~1.1]{RamirezRiderVirag2011}, and let $\Delta_i=\chi_i-\chi_1$. Let $\Xi$ be the regularized inverse-gap statistic of \cref{sp-raw-gap-resolvent}, constructed in \cite[Theorem~6.1]{LandonSosoe2022}, and let
\begin{equation}\label{cd-airy-root-display}
\mathscr S(x)=\Xi+\frac1x+\sum_{i\ge2}\Big(\frac1{x+\Delta_i}-\frac1{\Delta_i}\Big),\qquad x>0,
\end{equation}
as in \eqref{sp-airy-root-function}. By \cref{sp-airy-counts} (see also \cite[Proposition~1.4]{DuHuang2026}), almost surely $\mathscr S$ is continuous and strictly decreasing, with limits $+\infty$ at $0$ and $-\infty$ at $\infty$. For each $\sigma\in\R$, there exists a unique $\varrho_\sigma>0$ with $\mathscr S(\varrho_\sigma)=\sigma$. The limiting scaled half-precisions are those of \eqref{eq:spec-limiting-half-precisions}:
\begin{equation}\label{eq:cprof-airy-rates}
R_1^\sigma=\frac{\varrho_\sigma}2,\qquad R_i^\sigma=\frac{\Delta_i+\varrho_\sigma}2\quad(i\ge2).
\end{equation}
Almost surely the $\chi_i$ are distinct \cite[Proposition~3.5]{RamirezRiderVirag2011} and $\chi_i/(3\pi i/2)^{2/3}\to1$ \cite[Proposition~1.4]{DuHuang2026}. These facts show that $R^\sigma=(R^\sigma_i)_{i\ge1}$ belongs to $\mathcal P$, with $R^\sigma_i/i^{2/3}\to(3\pi/2)^{2/3}/2$. We write $\rho_\sigma=\rho_{R^\sigma}$ and $F_\sigma=F_{R^\sigma}$ on this event, and $F_\sigma(s)=e^{-s}$ on its complement, which has probability zero. By \cref{lem:cprof-measurable} below, $F_\sigma$ and $F_\sigma^{-1}$ are random elements of $C((0,\infty))$ and $C((0,1))$, where these spaces carry the topology of uniform convergence on compact subsets, and they are measurable functions of $(\Xi,(\Delta_i^{-1})_{i\ge2})$. Finally, we use the augmented spectral data $(\Xi_N,\mathbf b_N)$ of \cref{sp-raw-gap-resolvent}: with $\Delta_{i,N}=N^{2/3}(\lambda_1-\lambda_i)$,
\begin{equation}\label{eq:cprof-augmented}
\Xi_N=\sum_{i=2}^N\Delta_{i,N}^{-1}-N^{1/3},\qquad \mathbf b_N=\big(\Delta_{2,N}^{-1},\dots,\Delta_{N,N}^{-1},0,0,\dots\big)\in\ell^2.
\end{equation}

\begin{theorem}[Critical profiles of the GOE model]\label{cd-goe-profile}
Fix $m\ge1$ and $\sigma_1,\dots,\sigma_m\in\R$, and let $c_{N,j}=1+\sigma_jN^{-1/3}$. For each $j$, let $D_{N,j}$ and $t_{N,j}$ be the total variation distance and the mixing time of the Langevin dynamics with $c=c_{N,j}$ and the same matrix $W_N$, started from $\nu_N$ independently of $W_N$. Then, as $N\to\infty$,
\begin{equation}\label{eq:cprof-goe-limit}
\Big(\Xi_N,\mathbf b_N,\big(D_{N,j}(N^{2/3}\,\cdot\,),\,N^{-2/3}t_{N,j}(\cdot)\big)_{j\le m}\Big)
\Longrightarrow
\Big(\Xi,(\Delta_i^{-1})_{i\ge2},\big(F_{\sigma_j},\,F_{\sigma_j}^{-1}\big)_{j\le m}\Big)
\end{equation}
in distribution in $\R\times\ell^2\times\big(C((0,\infty))\times C((0,1))\big)^m$. Moreover, for every $j$ and every $\eps\in(0,1/2)$, with the finite-$N$ quotient defined to be $1$ on the event $\{t_{N,j}(1-\eps)=0\}$,
\begin{equation}\label{eq:cprof-ratio}
\frac{t_{N,j}(\eps)}{t_{N,j}(1-\eps)}\Longrightarrow\frac{F_{\sigma_j}^{-1}(\eps)}{F_{\sigma_j}^{-1}(1-\eps)},
\end{equation}
and the limit is strictly greater than $1$ almost surely. In particular, for each fixed $\sigma$ there is no cutoff.
\end{theorem}

We note that the limit in \eqref{eq:cprof-goe-limit} is a function of a single Airy point process for all $j$. The convergence is stated in distribution: the theorem does not specify a joint coupling across matrix sizes with the limiting Airy point process. Its proof applies \cref{lem:cprof-varying-map} to the joint law of the spectral data, and it uses, at each confidence level, the critical spectral conditions with fixed constants.

\begin{corollary}[Exponential lower bound for the critical profile]\label{critical-finite-time-positivity}
\begin{enumerate}
\item Under the hypotheses of \cref{cd-profile}, there exist $c_1,C_1>0$, depending only on $(c_0,C_0,\delta)$, such that $F_R(s)\ge c_1e^{-C_1s}$ for every $s>0$.
\item For every $\sigma\in\R$, almost surely there exist $c_1,C_1>0$ such that $F_\sigma(s)\ge c_1e^{-C_1s}$ for every $s>0$.
\end{enumerate}
\end{corollary}

\subsection{Outline of the proofs}

To prove \cref{cd-profile}, we first establish pointwise convergence at every positive time (\cref{cd-pointwise-profile}). We combine two bounds.
\begin{enumerate}
\item The coordinates after the first $J$ contribute little to the total variation distance: for $s_0>0$, we have $0\le D_N(bs)-D_N^{\le J}(bs)\le CJ^{-1/3}$ for every $s\ge s_0$ (\cref{cd-averaged-tail}), where $C$ depends on $s_0$ but not on $s$ or $J$. The density $f_{N,s_0/2}$ is bounded in $L^2(\pi_N)$ by \cref{cl3-actual-source-density}, and finite semigroup smoothing bounds the energy of $f_{N,s}$ for $s\ge s_0$ (\cref{cd-finite-smoothing}). We combine this bound with a Poincar\'e inequality for the conditional law of the remaining coordinates, deduced from the logarithmic Sobolev inequality of \cref{cd-tail-lsi}. Its constant is of order $bJ^{-2/3}$ uniformly over all conditioning values. Here $J$ is fixed before $N\to\infty$.
\item For fixed $J$ and $s>0$, the first $J$ rescaled coordinates converge in total variation to their limiting law (\cref{cd-head-current}). Starting from the weak convergence of \cref{cd-functional}, we integrate by parts under $\pi_N$ along fields that move the first $J$ coordinates and compensate in the remaining ones. Gaussian domination of the Gibbs measure bounds the fields and their divergences (\cref{cd-stationary-fields}). Together with the energy estimate, this gives a translation bound uniform in $N$. Mollification and Scheff\'e's lemma then give total variation convergence at the chosen time.
\end{enumerate}
The Gibbs marginals converge in total variation by \cref{cd-pin}, and total variation on $\R^\N$ is the limit of the distances between finite heads (\cref{lem:cprof-product-tv}). Taking $N\to\infty$ and then $J\to\infty$ therefore proves \cref{cd-pointwise-profile}.

Finite semigroup smoothing also gives $\tvnorm{\nu_NP_{bt}-\nu_NP_{bs}}\le L_{s_0}|t-s|$ for $s,t\ge s_0$, uniformly in $N$. Weak convergence of finite heads and \cref{lem:cprof-product-tv} pass this estimate to the limiting laws, so $F_R$ is locally Lipschitz (\cref{lem:cprof-limit-continuity}). Monotonicity and a finite grid then upgrade pointwise convergence to locally uniform convergence. The finite exponential lower bound \eqref{eq:cprof-exp-lower} gives positivity of $F_R$. If $F_R$ were constant on an interval, correlations of the positive and negative parts of the finite density minus one would tend to zero there, while their constant spectral coefficients would tend to a positive number. The uniform finite spectral gap and a Chebyshev-polynomial argument exclude this (\cref{cd-finite-correlation,cd-vanishing-correlations}). The degree is fixed before $N\to\infty$ and tends to infinity afterwards. \Cref{tab:cprof-order} records the other parameter choices.

\begin{table}[!htbp]
\centering
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{@{}lll@{}}
\toprule
Order & Parameter & Depends on\\
\midrule
1 & times $s\ge s_0$, for an anchor $s_0>0$ & --\\
2 & number $J\ge J_0$ of retained coordinates & $R$ (through $J_0$)\\
3 & mollifier scale $\alpha>0$ & chosen after $J$\\
4 & dimension $N\ge N_0(J,s_0)$ & $J$, $s_0$, the sequence $(r_{\cdot,N})$\\
5 & limit $N\to\infty$, then $\alpha\downarrow0$, then $J\to\infty$ & --\\
\bottomrule
\end{tabular}
\caption{Order of the choices in the proofs of \cref{cd-tail-lsi,cd-averaged-tail,cd-head-current,cd-pointwise-profile}: the conditional logarithmic Sobolev inequality, the bound on the remaining coordinates, total variation convergence of the first coordinates, and pointwise convergence of the full profile. The density is bounded in $L^2(\pi_N)$ at the time $s_0/2$. For convergence of the first coordinates alone, any fixed $J\ge1$ is allowed.}
\label{tab:cprof-order}
\end{table}

The lower bounds of \cref{sec:cprof-lower} do not use the profile. We derive them from a finite-dimensional estimate at a short time (\cref{cg-entrance-lower}) and a lower bound for reversible semigroups (\cref{cl3-density-two-time-lower}). In \cref{critical-finite-time-positivity} we pass the finite lower bound to $F_R$ through the pointwise convergence of \cref{cd-pointwise-profile}. Besides \cref{cd-profile}, the proof of \cref{cd-goe-profile} needs measurable dependence of $F_R$ on $R$, which follows from \cref{lem:cprof-approx} and \cref{cd-profile}. We then transfer the joint law of the spectral data to profiles and mixing times with the localized continuous mapping theorem \cref{lem:cprof-varying-map}, using the critical spectral conditions with fixed constants on events of probability close to one (\cref{cg-canonical-weyl}).

\section{General facts about Markov semigroups and densities}\label{sec:cprof-general}

In this section we collect a reversible lower bound, facts about total variation, and a criterion for vanishing spectral correlations. Throughout, $\tvnorm{\mu-\rho}=\sup_B|\mu(B)-\rho(B)|$, which equals $(1/2)\int|f-g|\,d\theta$ if $\mu=f\theta$ and $\rho=g\theta$.

\begin{lemma}[Reversible lower bound from two times]\label{cl3-density-two-time-lower}
Let $(P_t)_{t\ge0}$ be a Markov semigroup with reversible probability measure $\pi$, and let $\mu$ be a probability measure. Let $\ell>0$ and suppose that $\mu P_\ell$ has a density $h_\ell$ with respect to $\pi$ such that $\|h_\ell-1\|_{L^4(\pi)}\le M$, and that $\tvnorm{\mu P_{2\ell}-\pi}\ge d>0$. Then $4d^2\le M^2$, and
\begin{equation}\label{eq:cprof-two-time}
\tvnorm{\mu P_t-\pi}\ge\frac{4d^3}{M^2}\exp\Big(-\frac32\log(1/\vartheta)\,\frac t\ell\Big)\qquad(t\ge0),\qquad \vartheta=\min\Big(\frac12,\frac{4d^2}{M^2}\Big).
\end{equation}
\end{lemma}

\begin{proof}
All norms are in $L^p(\pi)$. Let $f=h_\ell-1$. By reversibility, $\mu P_{\ell+u}=(P_uh_\ell)\pi$ for $u\ge0$, so that
\begin{equation*}
\tvnorm{\mu P_{\ell+u}-\pi}=\frac12\|P_uf\|_1.
\end{equation*}
We set $a_j=\|P_{j\ell}f\|_2^2$ for $j\ge0$. Using that $P_\ell$ is selfadjoint and the Cauchy--Schwarz inequality, we find
\begin{equation*}
a_j=\ip{P_{(j-1)\ell}f}{P_{(j+1)\ell}f}\le\sqrt{a_{j-1}a_{j+1}}\qquad(j\ge1).
\end{equation*}
Further, $a_0\le\|f\|_4^2\le M^2$ and $a_1\ge\|P_\ell f\|_1^2=4\tvnorm{\mu P_{2\ell}-\pi}^2\ge4d^2$. From the last display, we see that $a_{j+1}=0$ would force $a_j=0$. Then all $a_j$ are positive, and the ratios $a_{j+1}/a_j$ are nondecreasing in $j$. As $P_\ell$ is a contraction on $L^2(\pi)$, we have $4d^2\le a_1\le a_0\le M^2$, and $a_{j+1}/a_j\ge a_1/a_0\ge\vartheta$. We conclude that $a_j\ge4d^2\vartheta^{j-1}$ for $j\ge1$.

Recall that $\|g\|_2\le\|g\|_1^{1/3}\|g\|_4^{2/3}$ by H\"older's inequality, and note that $\|P_uf\|_4\le\|f\|_4\le M$ by Jensen's inequality. We obtain $\|P_uf\|_1\ge\|P_uf\|_2^3/M^2$. For $u\ge\ell$ let $j=\lceil u/\ell\rceil$, so that $j\ell\ge u$ and $j-1<u/\ell$. The contraction property of $P_{j\ell-u}$ gives $\|P_uf\|_2^2\ge a_j$, and we find that
\begin{equation*}
\tvnorm{\mu P_{\ell+u}-\pi}\ge\frac{a_j^{3/2}}{2M^2}\ge\frac{4d^3}{M^2}\vartheta^{3(j-1)/2}\ge\frac{4d^3}{M^2}\exp\Big(-\frac32\log(1/\vartheta)\,\frac u\ell\Big).
\end{equation*}
For $t\ge2\ell$ we apply this with $u=t-\ell\le t$. For $t\le2\ell$, the distance is at least its value at time $2\ell$, because it is nonincreasing in time, and this value is at least $d\ge4d^3/M^2$. This proves \eqref{eq:cprof-two-time}.
\end{proof}

The argument uses only the logarithmic convexity of $t\mapsto\|P_tf\|_2^2$ for a reversible semigroup, a consequence of the spectral theorem.

\begin{lemma}[Total variation on a countable product]\label{lem:cprof-product-tv}
Let $\mu$ and $\rho$ be probability measures on $\R^\N$ with its product Borel $\sigma$-field. Then $J\mapsto\tvnorm{\mu^{\le J}-\rho^{\le J}}$ is nondecreasing and converges to $\tvnorm{\mu-\rho}$ as $J\to\infty$.
\end{lemma}

\begin{proof}
Let $\theta=(\mu+\rho)/2$ and $g=d\mu/d\theta-d\rho/d\theta$. Then $|g|\le2$ and $\tvnorm{\mu-\rho}=(1/2)\int|g|\,d\theta$. Let $\mathcal F_J$ be the $\sigma$-field generated by the first $J$ coordinates. The conditional expectation $g_J=\E_\theta[g\mid\mathcal F_J]$ is the difference of the densities of $\mu^{\le J}$ and $\rho^{\le J}$ with respect to $\theta^{\le J}$, evaluated at the first $J$ coordinates. Hence $\tvnorm{\mu^{\le J}-\rho^{\le J}}=(1/2)\int|g_J|\,d\theta$. Conditional Jensen gives $|g_J|\le\E_\theta[|g_{J+1}|\mid\mathcal F_J]$, and monotonicity follows. The $\sigma$-fields $\mathcal F_J$ generate the product $\sigma$-field, and the martingale convergence theorem \cite[Section~14.2]{Williams1991} implies that $g_J\to g$ in $L^1(\theta)$. This proves the convergence.
\end{proof}

\begin{lemma}[Total variation convergence from a translation bound]\label{cd-mollifier-criterion}
Let $J\ge1$, and let $\mu_n$, $n\ge1$, and $\mu$ be Borel probability measures on $\R^J$ such that $\mu_n$ converges weakly to $\mu$. For $w\in\R^J$, let $\tau_w(x)=x+w$, and let $\tau_w\mu$ be the image of $\mu$ under $\tau_w$.
\begin{enumerate}
\item If $\rho_n$, $n\ge1$, and $\rho$ are Borel probability measures on $\R^J$ such that $\rho_n$ converges weakly to $\rho$, then $\tvnorm{\mu-\rho}\le\liminf_{n\to\infty}\tvnorm{\mu_n-\rho_n}$.
\item Suppose that there exist $B<\infty$ and $n_0$ such that
\begin{equation}\label{cd-mollifier-criterion-bound}
\tvnorm{\tau_w\mu_n-\mu_n}\le B|w|\qquad\text{for all }n\ge n_0\text{ and }w\in\R^J.
\end{equation}
Then $\mu$ is absolutely continuous with respect to Lebesgue measure, and $\tvnorm{\mu_n-\mu}\to0$. In particular, if $\mu_n$ has the density $p_n$ and $\mu$ has the density $p$, then $p_n\to p$ in $L^1(\R^J)$.
\end{enumerate}
\end{lemma}

The second part involves no integration in time: it concerns one sequence of measures on $\R^J$ with $J$ fixed. The limit $\mu$ is the given weak limit, and its absolute continuity is part of the conclusion.

\begin{proof}
For the first part, we recall that, for probability measures on $\R^J$, the total variation distance is the supremum of $\mu(\phi)-\rho(\phi)$ over continuous functions $\phi$ with values in $[0,1]$. Indeed, let $E$ be a Borel set. Since the finite measure $\mu+\rho$ is regular \cite[Theorem~1.1]{Billingsley1999}, there exist a closed set $F\subset E$ and an open set $O\supset E$ such that $(\mu+\rho)(O\setminus F)$ is arbitrarily small. A continuous function with values in $[0,1]$ that equals $1$ on $F$ and $0$ off $O$ then approximates $\ind_E$ in $L^1(\mu+\rho)$. For each such $\phi$, the difference $\mu_n(\phi)-\rho_n(\phi)$ converges to $\mu(\phi)-\rho(\phi)$, and $\mu_n(\phi)-\rho_n(\phi)\le\tvnorm{\mu_n-\rho_n}$. Taking the supremum over $\phi$ proves the first part.

For the second part, the maps $\tau_w$ are continuous, and $\tau_w\mu_n$ converges weakly to $\tau_w\mu$. Then the first part and \eqref{cd-mollifier-criterion-bound} imply that
\begin{equation}\label{eq:cprof-limit-translation}
\tvnorm{\tau_w\mu-\mu}\le B|w|\qquad(w\in\R^J).
\end{equation}
Let $\eta$ be a smooth probability density supported in the unit ball, and let $\eta_\alpha(x)=\alpha^{-J}\eta(x/\alpha)$ for $\alpha>0$. For a Borel probability measure $\sigma$ on $\R^J$, the convolution $\sigma*\eta_\alpha$ has the density
\begin{equation*}
q_{\sigma,\alpha}(x)=\int\eta_\alpha(x-y)\,\sigma(dy),
\end{equation*}
which has integral one by Tonelli's theorem. For a Borel function $\phi$ with values in $[0,1]$, we have $(\sigma*\eta_\alpha)(\phi)=\int\eta_\alpha(w)\,(\tau_w\sigma)(\phi)\,dw$, and then
\begin{equation*}
\big|\sigma(\phi)-(\sigma*\eta_\alpha)(\phi)\big|\le\int\eta_\alpha(w)\,\tvnorm{\sigma-\tau_w\sigma}\,dw.
\end{equation*}
Since $|w|\le\alpha$ on the support of $\eta_\alpha$, the bounds \eqref{cd-mollifier-criterion-bound} and \eqref{eq:cprof-limit-translation} imply that
\begin{equation}\label{eq:cprof-mollifier-error}
\tvnorm{\mu_n-\mu_n*\eta_\alpha}\le B\alpha\quad(n\ge n_0),\qquad\tvnorm{\mu-\mu*\eta_\alpha}\le B\alpha.
\end{equation}
Now fix $\alpha>0$. For every $x$, the function $\eta_\alpha(x-\cdot)$ is bounded and continuous, and by weak convergence we have $q_{\mu_n,\alpha}(x)\to q_{\mu,\alpha}(x)$. These functions are probability densities. Writing $|q-q'|=q+q'-2\min(q,q')$ and applying dominated convergence to $\min(q_{\mu_n,\alpha},q_{\mu,\alpha})\le q_{\mu,\alpha}$, we find that $q_{\mu_n,\alpha}\to q_{\mu,\alpha}$ in $L^1(\R^J)$; this is Scheff\'e's lemma \cite[pp.~435--437]{Scheffe1947}, \cite[Section~5.10]{Williams1991}.
Using the triangle inequality and \eqref{eq:cprof-mollifier-error}, we obtain
\begin{equation*}
\limsup_{n\to\infty}\tvnorm{\mu_n-\mu}\le2B\alpha+\lim_{n\to\infty}\frac12\|q_{\mu_n,\alpha}-q_{\mu,\alpha}\|_{L^1(\R^J)}=2B\alpha.
\end{equation*}
As $\alpha$ is arbitrary, $\tvnorm{\mu_n-\mu}\to0$. Finally, let $E$ be a Borel set of Lebesgue measure zero. Then $(\mu*\eta_\alpha)(E)=0$, and \eqref{eq:cprof-mollifier-error} gives $\mu(E)\le B\alpha$ for every $\alpha>0$. Thus $\mu(E)=0$, and $\mu$ is absolutely continuous. The statement about densities follows from \cref{lem:model-tv}.
\end{proof}

\begin{lemma}[Vanishing spectral correlations]\label{cd-vanishing-correlations}
Let $\lambda_0>0$, $M<\infty$, and $0\le t_1<t_2$. For each $n\ge1$, let $c_n$ and $w_{n,j}$, $j\ge1$, be real numbers with $\sum_j|w_{n,j}|\le M$, let $\lambda_{n,j}\ge\lambda_0$ for $j\ge1$, and let
\begin{equation*}
h_n(t)=c_n+\sum_{j\ge1}w_{n,j}e^{-\lambda_{n,j}t}\qquad(t\ge0).
\end{equation*}
If $c_n\to c$ and $h_n(t)\to0$ for every $t\in[t_1,t_2]$, then $c=0$.
\end{lemma}

The coefficients $w_{n,j}$ may have either sign, and we assume no convergence of the $\lambda_{n,j}$ or of the $w_{n,j}$ as $n\to\infty$. The proof uses both the lower bound $\lambda_0$, which is uniform in $n$, and the bound on $\sum_j|w_{n,j}|$.

\begin{proof}
We use the Chebyshev polynomials $T_m$, defined by $T_0=1$, $T_1(x)=x$, and $T_{m+2}(x)=2xT_{m+1}(x)-T_m(x)$ \cite[(18.9.1) and Table~18.9.1]{OlverLozierBoisvertClark2010}. The identity $T_m(\cos\vartheta)=\cos(m\vartheta)$ \cite[(18.5.1)]{OlverLozierBoisvertClark2010} shows that $|T_m(x)|\le1$ for $x\in[-1,1]$.
We also need the lower bound
\begin{equation}\label{eq:cprof-chebyshev-growth}
T_m(x)\ge1+m^2(x-1)\qquad(x\ge1,\ m\ge0).
\end{equation}
To prove it, fix $x\ge1$, and let $d=x-1$ and $\Delta_m=T_{m+1}(x)-T_m(x)$. We show by induction that $T_m(x)\ge1+m^2d$ and $\Delta_m\ge(2m+1)d$. For $m=0$, we have $T_0=1$ and $\Delta_0=d$. If both bounds hold for $m$, then $T_{m+1}(x)=T_m(x)+\Delta_m\ge1+(m+1)^2d$, and the recursion gives $\Delta_{m+1}=\Delta_m+2dT_{m+1}(x)\ge(2m+1)d+2d=(2m+3)d$, since $T_{m+1}(x)\ge1$. This proves \eqref{eq:cprof-chebyshev-growth}.

Fix an integer $m\ge1$, and let $\tau=(t_2-t_1)/m$ and $\kappa=e^{-\lambda_0\tau}\in(0,1)$. We define the polynomial
\begin{equation*}
p_m(z)=\frac{T_m(2z/\kappa-1)}{T_m(2/\kappa-1)}=\sum_{k=0}^ma_kz^k.
\end{equation*}
The denominator is at least $1$ by \eqref{eq:cprof-chebyshev-growth}, and $p_m(1)=1$. For $z\in[0,\kappa]$ we have $2z/\kappa-1\in[-1,1]$, and \eqref{eq:cprof-chebyshev-growth} gives
\begin{equation}\label{eq:cprof-chebyshev-small}
|p_m(z)|\le\frac1{1+m^2(2/\kappa-2)}\le\frac1{1+2m\lambda_0(t_2-t_1)}\qquad(0\le z\le\kappa).
\end{equation}
In the last inequality, we used $2/\kappa-2=2(e^{\lambda_0\tau}-1)\ge2\lambda_0\tau$. Since $\sum_j|w_{n,j}|<\infty$, we may exchange the finite sum over $k$ with the sum over $j$, and we find
\begin{equation*}
\sum_{k=0}^ma_kh_n(t_1+k\tau)=c_np_m(1)+\sum_{j\ge1}w_{n,j}e^{-\lambda_{n,j}t_1}p_m\big(e^{-\lambda_{n,j}\tau}\big).
\end{equation*}
Here $p_m(1)=1$, $e^{-\lambda_{n,j}t_1}\le1$, and $e^{-\lambda_{n,j}\tau}\in(0,\kappa]$. Using \eqref{eq:cprof-chebyshev-small} and $\sum_j|w_{n,j}|\le M$, we obtain
\begin{equation*}
|c_n|\le\sum_{k=0}^m|a_k|\,|h_n(t_1+k\tau)|+\frac M{1+2m\lambda_0(t_2-t_1)}.
\end{equation*}
The times $t_1+k\tau$, $0\le k\le m$, lie in $[t_1,t_2]$, and the coefficients $a_k$ do not depend on $n$. Letting $n\to\infty$, we find that $|c|\le M/(1+2m\lambda_0(t_2-t_1))$. Letting $m\to\infty$, we conclude that $c=0$.
\end{proof}

We do not claim that the coefficients $a_k$ are bounded uniformly in $m$. This is why the degree $m$ is fixed before the limit $n\to\infty$.

\section{Lower bounds on the critical time scale}\label{sec:cprof-lower}

In this section we prove that the total variation distance stays above $3/4$ up to a time of order $b=N^{2/3}$ (\cref{cg-entrance-lower}), and then that it decays no faster than exponentially on the scale $b$ (\cref{cl3-density-clock}). The first estimate needs only the trace condition and the Weyl bounds.

\begin{lemma}[Lower bound at short critical times]\label{cg-entrance-lower}
There exist $s_*\in(0,1]$ and $N_0$, depending only on $c_0$ and $C_0$, such that the following holds. If $N\ge N_0$ and $r$ satisfies \eqref{eq:cprof-trace} and \eqref{eq:cprof-weyl}, then
\begin{equation*}
D_N(s_*b)\ge3/4.
\end{equation*}
\end{lemma}

\begin{proof}
We find an event that has probability at most $1/8$ under $\pi_N$ and probability at least $7/8$ under $\nu_NP_{s_*b}$.

\paragraph{Step 1: the stationary law.} By the Weyl bounds, the variances $q_i=(2r_i)^{-1}$ satisfy
\begin{equation}\label{eq:cprof-q-bounds}
\frac b{2C_0i^{2/3}}\le q_i\le\frac b{2c_0i^{2/3}}.
\end{equation}
In particular, $q_1,q_2,q_3\ge\delta'\max_iq_i$ with $\delta'=c_0/(3^{2/3}C_0)$, and $m_2=\sum_iq_i^2$ satisfies $b^2/(4C_0^2)\le m_2\le b^2/c_0^2$, since $\sum_ii^{-4/3}\le4$. Let $G_1,\dots,G_N$ be independent standard Gaussians, let $\varphi$ be the density of $\sum_iq_iG_i^2$, and let $\varphi_{-1}$ be the density of $\sum_{i\ge2}q_iG_i^2$. Recall that $\pi_N$ is the Gaussian law with variances $q_i$ conditioned on $|x|^2=N=\sum_iq_i$. By \cref{gp-head-density}, which applies since $N\ge4$ (we take $N_0\ge4$), the first coordinate has the density
\begin{equation*}
\pi_N(x_1\in dx)=\frac{\varphi_{-1}(N-x^2)}{\varphi(N)}\,\frac{e^{-x^2/(2q_1)}}{\sqrt{2\pi q_1}}\,dx,\qquad |x|<\sqrt N.
\end{equation*}
Indeed, in the notation of \cref{gp-head-density} with $A=\{1\}$ and $x=\sqrt{q_1}\,z$, the density of the centered sum $\sum_iq_i(G_i^2-1)$ at $0$ is $\varphi(N)$, and the density of $\sum_{i\ge2}q_i(G_i^2-1)$ at $-q_1(z^2-1)$ is $\varphi_{-1}(N-q_1z^2)$.
By \cref{cg-mean-level-density} and the bounds on $m_2$, there exist $c_3,C_3>0$, depending only on $c_0$ and $C_0$, such that $\varphi(N)\ge c_3/b$ and $\sup\varphi_{-1}\le C_3/b$. Inserting these bounds into the last display, we obtain, for every $a\in(0,1)$,
\begin{equation*}
\pi_N\big(x_1^2\le ab\big)\le\frac{C_3}{c_3}\int_{-\sqrt{ab}}^{\sqrt{ab}}\frac{dx}{\sqrt{2\pi q_1}}\le C_1\sqrt a,
\end{equation*}
where we used $q_1\ge b/(2C_0)$. We fix $a\in(0,1)$ with $C_1\sqrt a\le1/8$.

\paragraph{Step 2: the time-$t$ law.} By \cref{cd-lift}, $X(t)=\sqrt NY(t)/|Y(t)|$ has law $\nu_NP_t$. Using the Cauchy--Schwarz inequality and \cref{cd-lift-moments} with $K=1$ and $p=2$, we have, for $N\ge N_0$ and $0\le t\le b$,
\begin{equation*}
\E\big[X_1(t)^2\big]=N\,\E\Big[\frac{Y_1(t)^2}{\mathcal A_t}\Big]\le N\,\|Y_1(t)^2\|_2\,\big\|\mathcal A_t^{-1}\big\|_2\le C\,\frac{h_1(t)}{\bar h(t)}\le C_2(1+t).
\end{equation*}
In the last step we used \cref{cd-heat-profiles}: $h_1(t)\le C(1+t)^{1/2}$ and $\bar h(t)\ge c_4(1+t)^{-1/2}$, where $\bar h(t)=N^{-1}\sum_ih_i(t)$ and $c_4$ depends only on $c_0$ and $C_0$. We now fix $s_*\in(0,1]$ with $2C_2s_*\le a/8$, and we increase $N_0$ so that $s_*b\ge1$ for $N\ge N_0$. By Markov's inequality, we have
\begin{equation*}
\nu_NP_{s_*b}\big(x_1^2>ab\big)\le\frac{C_2(1+s_*b)}{ab}\le\frac{2C_2s_*}a\le\frac18.
\end{equation*}

\paragraph{Step 3: conclusion.} The event $\{x_1^2\le ab\}$ has probability at least $7/8$ under $\nu_NP_{s_*b}$ and at most $1/8$ under $\pi_N$. Hence $D_N(s_*b)\ge3/4$. The choices of $a$, $s_*$, and $N_0$ depend only on $c_0$ and $C_0$.
\end{proof}

\begin{corollary}[Mixing times at every accuracy]\label{cl3-density-clock}
\begin{enumerate}
\item There exist $c_1,C_1,c_2,C_2>0$ and $N_0$, depending only on $(c_0,C_0,\delta)$, such that, if $N\ge N_0$ and $r$ satisfies the critical spectral conditions with constants $(c_0,C_0,\delta)$, then
\begin{equation}\label{eq:cprof-exp-lower}
D_N(t)\ge c_1e^{-C_1t/b}\qquad(t\ge0),
\end{equation}
and, for every $\eps\in(0,1/2)$,
\begin{equation}\label{eq:cprof-clock}
c_2\,b\log(1/\eps)\le t_N(\eps)\le C_2\,b\log(1/\eps).
\end{equation}
\item For every $\eta>0$ and $M<\infty$, there exist $C_{\eta,M}$, $N_0$, and events $\mathcal A_N$ with $\liminf_{N\to\infty}\PP(\mathcal A_N)\ge1-\eta$ such that, on $\mathcal A_N$ and for $N\ge N_0$, the mixing times of the GOE model with $c=1+\sigma N^{-1/3}$ satisfy
\begin{equation*}
C_{\eta,M}^{-1}N^{2/3}\log(1/\eps)\le t_N(\eps)\le C_{\eta,M}N^{2/3}\log(1/\eps)
\end{equation*}
simultaneously for all $\sigma\in[-M,M]$ and all $\eps\in(0,1/2)$.
\end{enumerate}
\end{corollary}

The bounds hold for all $\eps\in(0,1/2)$ at once, so they apply to accuracies that depend on $N$.

\begin{proof}
We begin with the lower bound \eqref{eq:cprof-exp-lower}. Let $s_*$ be as in \cref{cg-entrance-lower} and $\ell=s_*b/2$. By \cref{cl3-actual-source-density} with $p=4$ and $s_0=s_*/2$, the density $h_\ell$ of $\nu_NP_\ell$ with respect to $\pi_N$ satisfies $\|h_\ell-1\|_{L^4(\pi_N)}\le\|h_\ell\|_{L^4(\pi_N)}+1\le M_4$, where $M_4$ depends only on $(c_0,C_0,\delta)$. Further, $D_N(2\ell)\ge3/4$ by \cref{cg-entrance-lower}. Then \cref{cl3-density-two-time-lower} with $d=3/4$ implies \eqref{eq:cprof-exp-lower}, with $M=M_4$, $c_1=27/(16M_4^2)$, and $C_1=3\log(1/\vartheta)/s_*$, where $\vartheta=\min(1/2,9/(4M_4^2))$.

For the lower bound in \eqref{eq:cprof-clock}, let $t\ge0$ with $D_N(t)\le\eps$. Then $c_1e^{-C_1t/b}\le\eps$ by \eqref{eq:cprof-exp-lower}, that is, $t\ge(b/C_1)\log(c_1/\eps)$. Taking the infimum over such $t$, we have $t_N(\eps)\ge(b/C_1)\log(c_1/\eps)$. For $\eps\le c_1^2$ we have $\log(c_1/\eps)\ge\log(1/\eps)/2$, and then $t_N(\eps)\ge(b/2C_1)\log(1/\eps)$. For $c_1^2<\eps<1/2$, we note that $D_N(t)\ge D_N(s_*b)\ge3/4$ for $t\le s_*b$. Then $t_N(\eps)\ge s_*b\ge s_*b\log(1/\eps)/\log(c_1^{-2})$; here $c_1<1$, because $M_4^2\ge9/4$ by \cref{cl3-density-two-time-lower}. The two cases together prove the lower bound, with $c_2=\min(1/(2C_1),s_*/\log(c_1^{-2}))$.

For the upper bound, we use the Poincar\'e inequality and the $L^2$ bound on the density at time $b$. Let $K_{\mathrm P}$ be as in \cref{tab:cprof-inputs}, so that $C_{\mathrm P}(\pi_N)\le K_{\mathrm P}b$ by the first part of \cref{cg-critical-lsi}, and let $C_{2,1}$ be the constant of \cref{cl3-actual-source-density} with $p=2$ and $s_0=1$, so that $\|f_{N,1}\|_{L^2(\pi_N)}\le C_{2,1}$. We increase $N_0$ so that both bounds hold for $N\ge N_0$; the constants $K_{\mathrm P}$, $C_{2,1}$, and $N_0$ depend only on $(c_0,C_0,\delta)$. The density $f_{N,1}=d(\nu_NP_b)/d\pi_N$ is of class $C^2$, and $\pi_N(f_{N,1})=1$. By reversibility and the semigroup property, $\nu_NP_{b+u}$ has the density $P_uf_{N,1}$ for $u\ge0$. Let $g=f_{N,1}-1$ and $V(u)=\|P_ug\|_{L^2(\pi_N)}^2$. Since $g$ is of class $C^2$ and $\pi_N(P_ug)=\pi_N(g)=0$, the integration by parts formula \eqref{eq:model-dirichlet} and the Poincar\'e inequality give
\begin{equation*}
V'(u)=-2\int|\nabla_{S_N}P_ug|^2\,d\pi_N\le-\frac2{K_{\mathrm P}b}\,V(u).
\end{equation*}
Then $e^{2u/(K_{\mathrm P}b)}V(u)$ is nonincreasing, and $\|P_ug\|_{L^2(\pi_N)}\le e^{-u/(K_{\mathrm P}b)}\|g\|_{L^2(\pi_N)}$; this is the usual exponential decay in $L^2$ implied by a Poincar\'e inequality \cite[Section~4.2]{BakryGentilLedoux2014}. Then \cref{lem:model-tv} and the Cauchy--Schwarz inequality imply that
\begin{equation}\label{eq:cprof-l2-upper}
D_N(b+u)=\frac12\|P_ug\|_{L^1(\pi_N)}\le\frac12\|f_{N,1}-1\|_{L^2(\pi_N)}\,e^{-u/(K_{\mathrm P}b)}\le C_5e^{-u/(K_{\mathrm P}b)}\qquad(u\ge0),
\end{equation}
where $C_5=(C_{2,1}+1)/2$. In the last step we used $\|1\|_{L^2(\pi_N)}=1$ and the triangle inequality. Note that $C_{2,1}\ge\|f_{N,1}\|_{L^1(\pi_N)}=1$, so that $C_5\ge1$. For $\eps\in(0,1/2)$, let $u_\eps=K_{\mathrm P}b\log(C_5/\eps)\ge0$. Then $D_N(b+u_\eps)\le\eps$ by \eqref{eq:cprof-l2-upper}, and $t_N(\eps)\le b+K_{\mathrm P}b\log(C_5/\eps)$. For $\eps<1/2$ we have $1\le\log(1/\eps)/\log2$ and $\log(C_5/\eps)\le(1+\log C_5/\log 2)\log(1/\eps)$, and the upper bound in \eqref{eq:cprof-clock} follows, with $C_2=1/\log2+K_{\mathrm P}(1+\log C_5/\log2)$.

For the second part, \cref{cg-canonical-weyl} provides, for given $\eta$ and $M$, constants $(c_0,C_0,\delta)$ and events $\mathcal A_N$ with $\liminf_N\PP(\mathcal A_N)\ge1-\eta$ on which, for every $\sigma\in[-M,M]$, the half-precisions of the model in increasing order satisfy the critical spectral conditions with these constants. The constants in the first part depend only on $(c_0,C_0,\delta)$, so the first part, applied for each $\sigma$ on $\mathcal A_N$, proves the claim.
\end{proof}

\section{The remaining coordinates}\label{sec:cprof-tail}

In this section we show that, at every time $s\ge s_0>0$, the total variation distance is determined by the first $J$ coordinates up to an error $O(J^{-1/3})$, with a constant that depends on $s_0$ but not on $s$. There are two inputs. The first is a bound on the energy of the density $f_{N,s}$, which follows from the $L^2$ bound on the density at time $s_0/2$ and the smoothing of the semigroup. The second is a Poincar\'e inequality for the conditional law of the remaining coordinates, which holds for every conditioning value. For $t\ge0$ let $Q_{N,t}=P_{bt}$, whose generator is $bL_r$, and let
\begin{equation*}
\mathcal E_N(u)=b\int_{S_N}|\nabla_{S_N}u|^2\,d\pi_N
\end{equation*}
be its Dirichlet form, defined on the closure of the smooth functions in the norm $(\|u\|_{L^2(\pi_N)}^2+\mathcal E_N(u))^{1/2}$. There is no factor $1/2$ in $\mathcal E_N$, in accordance with \eqref{eq:model-dirichlet}. By reversibility and the semigroup property, $f_{N,s}=Q_{N,s-t}f_{N,t}$ for $s\ge t>0$; indeed, both sides have the same integral against every bounded measurable function.

\begin{lemma}[Finite semigroup smoothing]\label{cd-finite-smoothing}
Let $N\ge2$ and $r\in\R^N$. For every $u\in L^2(\pi_r)$, $t>0$, and $h\ge0$, the function $Q_{N,t}u$ belongs to the domain of $\mathcal E_N$, and
\begin{equation}\label{eq:cprof-smoothing}
\|(Q_{N,h}-I)Q_{N,t}u\|_{L^2(\pi_r)}\le\frac h{et}\|u\|_{L^2(\pi_r)},\qquad\mathcal E_N(Q_{N,t}u)\le\frac1{2t}\|u\|_{L^2(\pi_r)}^2.
\end{equation}
\end{lemma}

These bounds do not depend on $N$ or on $r$.

\begin{proof}
By the elliptic spectral theory recalled after \eqref{sgl-euclidean-ibp} \cite[Section~2, p.~30]{Setti1998}, there exists an orthonormal basis $(e_j)_{j\ge0}$ of $L^2(\pi_r)$ consisting of smooth eigenfunctions of $-bL_r$, with eigenvalues $\lambda_j\ge0$ and $e_0=1$. We write $u=\sum_ju_je_j$. Then $Q_{N,t}u=\sum_je^{-t\lambda_j}u_je_j$, and Parseval's identity \cite[Corollary~5.10]{Brezis2011} gives
\begin{equation*}
\|(Q_{N,h}-I)Q_{N,t}u\|_{L^2(\pi_r)}^2=\sum_j\big(1-e^{-h\lambda_j}\big)^2e^{-2t\lambda_j}u_j^2\le\Big(\sup_{\lambda\ge0}h\lambda e^{-t\lambda}\Big)^2\sum_ju_j^2.
\end{equation*}
Here we used $1-e^{-h\lambda}\le h\lambda$. Since $\sup_{\lambda\ge0}\lambda e^{-t\lambda}=1/(et)$, this proves the first bound. For the second bound, let $S_n=\sum_{j\le n}e^{-t\lambda_j}u_je_j$, which is smooth. By \eqref{sgl-euclidean-ibp} and orthonormality, we have $\mathcal E_N(S_n-S_m)=\sum_{m<j\le n}\lambda_je^{-2t\lambda_j}u_j^2$ for $m<n$. Since $\lambda e^{-2t\lambda}\le1/(2et)$ for $\lambda\ge0$, the partial sums $S_n$ converge to $Q_{N,t}u$ in the norm of the domain of $\mathcal E_N$, and
\begin{equation*}
\mathcal E_N(Q_{N,t}u)=\sum_j\lambda_je^{-2t\lambda_j}u_j^2\le\frac1{2et}\sum_ju_j^2\le\frac1{2t}\|u\|_{L^2(\pi_r)}^2.
\end{equation*}
\end{proof}

We apply \cref{cd-finite-smoothing} to the densities $f_{N,s}$. Fix $s_0>0$. By \cref{cl3-actual-source-density} with $p=2$, there exist $M_{s_0}\ge1$ and $N_{s_0}$, depending only on $s_0$ and $(c_0,C_0,\delta)$, such that $\|f_{N,s}\|_{L^2(\pi_N)}\le M_{s_0}$ for every $s\ge s_0/2$, every $N\ge N_{s_0}$, and every $r$ that satisfies the critical spectral conditions with constants $(c_0,C_0,\delta)$. Let $s\ge s_0$ and $N\ge N_{s_0}$. Applying \cref{cd-finite-smoothing} with $u=f_{N,s_0/2}$ and $t=s-s_0/2\ge s_0/2$, we have
\begin{equation}\label{eq:cprof-positive-energy}
\mathcal E_N(f_{N,s})\le\frac{M_{s_0}^2}{2(s-s_0/2)}\le\frac{M_{s_0}^2}{s_0}\qquad(s\ge s_0,\ N\ge N_{s_0}).
\end{equation}
We use the density at time $s_0/2$, and not at time $s_0$, so that the semigroup acts for a time at least $s_0/2$; no upper bound on $s$ is needed. Similarly, let $s_0\le s\le s'$. The first bound in \eqref{eq:cprof-smoothing}, applied with $u=f_{N,s_0/2}$, the time $s-s_0/2\ge s_0/2$, and $h=s'-s$, gives $\|f_{N,s'}-f_{N,s}\|_{L^2(\pi_N)}\le2M_{s_0}(s'-s)/(es_0)$. Since $\pi_N$ is a probability measure, the $L^1(\pi_N)$ norm is at most the $L^2(\pi_N)$ norm, and \cref{lem:model-tv} gives
\begin{equation}\label{eq:cprof-time-modulus}
\tvnorm{\nu_NP_{bt}-\nu_NP_{bs}}\le L_{s_0}|t-s|,\qquad L_{s_0}=\frac{M_{s_0}}{es_0}\qquad(s,t\ge s_0,\ N\ge N_{s_0}).
\end{equation}

The functional inequalities of \cref{sec:inequalities} are stated on the sphere $S_N$ of squared radius $N$. The conditional laws below live on spheres of arbitrary radius, so we first record the rescaled form. Logarithmic Sobolev and Poincar\'e constants are taken with respect to the tangent gradient of the sphere on which the measure lives.

\begin{lemma}[Spheres of arbitrary radius]\label{lem:cprof-radius}
Fix $\delta'\in(0,1]$. Let $n\ge3$, $S>0$, and let $\pi$ be the probability measure on $\{w\in\R^n:|w|^2=S\}$ with density proportional to $\exp(-\sum_i\omega_iw_i^2)$ with respect to the uniform measure, where $\omega_i>0$ and $\sum_i(2\omega_i)^{-1}=S$. Let $t=\min_i\omega_i$. Suppose that $St\ge4$, that at least $260$ indices satisfy $\omega_i\le2t$, and that three distinct weights $(2\omega_i)^{-1}$ are at least $\delta'\max_j(2\omega_j)^{-1}$. Then
\[
C_{\mathrm P}(\pi)\le1/t,\qquad C_{\mathrm{LS}}(\pi)\le C_{\delta'}/t,
\]
where $C_{\delta'}$ depends only on $\delta'$.
\end{lemma}

\begin{proof}
This is the scaling recorded in \cref{rem:ineq-scaling}; we include the short argument. Let $w=\sqrt{S/n}\,w'$, so that $w'$ lies on the sphere of squared radius $n$. The density becomes proportional to $\exp(-\sum_i\omega'_iw_i'^2)$ with $\omega'_i=(S/n)\omega_i$, and $\sum_i(2\omega'_i)^{-1}=(n/S)\sum_i(2\omega_i)^{-1}=n$. The minimum is $t'=(S/n)t$, and $nt'=St\ge4$. The hypotheses on the ratios $\omega_i/t$ and on the weights are unchanged. For a function $g$ on the sphere of squared radius $S$, let $g'(w')=g(\sqrt{S/n}\,w')$. Then $|\nabla g'|^2=(S/n)|\nabla g|^2$ at corresponding points, while variances and entropies are unchanged. We obtain $C_{\mathrm P}(\pi)=(S/n)C_{\mathrm P}(\pi')$ and $C_{\mathrm{LS}}(\pi)=(S/n)C_{\mathrm{LS}}(\pi')$, where $\pi'$ is the image measure. \Cref{cg-many-soft-pi} with $L=2$ and $k=260$ shows that $C_{\mathrm P}(\pi')\le1/t'$, and then $C_{\mathrm P}(\pi)\le1/t$. Similarly, \cref{cg-three-soft-lsi} implies that $C_{\mathrm{LS}}(\pi')\le C'_{\delta'}/t'+C'_{\delta'}C_{\mathrm P}(\pi')\le2C'_{\delta'}/t'$, where $C'_{\delta'}$ is the constant of that corollary. We conclude that $C_{\mathrm{LS}}(\pi)\le C_{\delta'}/t$ with $C_{\delta'}=2C'_{\delta'}$.
\end{proof}

For $h\in\R^J$ with $|h|^2<N$, we write $\pi_N(\cdot\mid x_{\le J}=h)$ for the conditional law of $(x_i)_{i>J}$ under $\pi_N$ given $x_{\le J}=h$. Conditioned on its first $J$ coordinates, the uniform measure $\nu_N$ is the uniform measure on the sphere $\{z\in\R^{N-J}:|z|^2=N-|h|^2\}$, and the factor $\exp(-\sum_{i\le J}r_ih_i^2)$ is constant on it. Thus $\pi_N(\cdot\mid x_{\le J}=h)$ is the measure on this sphere with density proportional to $\exp(-\sum_{i>J}r_iz_i^2)$.

\begin{lemma}[Logarithmic Sobolev inequality for the remaining coordinates]\label{cd-tail-lsi}
Let $(r_{\cdot,N})$ satisfy the critical spectral conditions with fixed constants for every $N\in\mathcal N$ and converge to $R$, and let $\gamma$ be as in \eqref{eq:cprof-conv}. There exist an absolute constant $C$ and an integer $J_0$, depending on $R$, such that for every $J\ge J_0$ there exists $N_0(J)$ with the following property: for all $N\ge N_0(J)$ and all $h\in\R^J$ with $|h|^2<N$,
\begin{equation}\label{cd-tail-lsi-bound}
C_{\mathrm{LS}}\big(\pi_N(\cdot\mid x_{\le J}=h)\big)\le\frac{C\,b}{\gamma J^{2/3}}.
\end{equation}
\end{lemma}

The order of quantifiers is $\exists J_0\ \forall J\ge J_0\ \exists N_0(J)\ \forall N\ge N_0(J)\ \forall h$; no uniformity in $J$ is claimed for $J$ growing with $N$. The bound holds for every conditioning value, including those for which the remaining sphere is very small.

\begin{proof}
The proof has four steps. We first describe the conditional law through a common shift of the half-precisions, which reduces the claim to a lower bound on its smallest coefficient. In Steps 2 and 3 we identify this lower bound as $N\to\infty$ and then as $J\to\infty$. In Step 4 we apply \cref{lem:cprof-radius}. Throughout the proof, $k=260$.

\paragraph{Step 1: a common shift.} Fix $J$ and $N>J+k$, and let $S=N-|h|^2\in(0,N]$. Adding a constant to all coefficients $r_i$, $i>J$, does not change the conditional law on the sphere $|z|^2=S$. The function $\alpha\mapsto\sum_{i>J}[2(r_i+\alpha)]^{-1}$ decreases continuously from $+\infty$ to $0$ on $(-r_{J+1},\infty)$, because the $r_i$ are nondecreasing. Then there exists a unique $\alpha(S)>-r_{J+1}$ with
\begin{equation*}
\sum_{i>J}\frac1{2(r_i+\alpha(S))}=S,
\end{equation*}
and $\alpha(S)$ is nonincreasing in $S$. With $\omega_i=r_i+\alpha(S)$, the conditional law is the measure of \cref{lem:cprof-radius} with $n=N-J$, and its smallest coefficient is $t(S)=r_{J+1}+\alpha(S)$. Since $S\le N$, we have $\alpha(S)\ge\alpha(N)$. At $S=N$, the trace condition implies that $\sum_{i>J}(2r_i)^{-1}<N$, and then $\alpha(N)<0$. We write $\alpha(N)=-d_{N,J}/b$ with $0<d_{N,J}<R_{J+1,N}$. Multiplying the defining equation by $2/b$ and subtracting the trace condition $\sum_iR_{i,N}^{-1}=2N^{1/3}$, we find that $d_{N,J}$ is the unique root in $(0,R_{J+1,N})$ of
\begin{equation}\label{eq:cprof-root}
G_{N,J}(d)=A_{N,J},\qquad G_{N,J}(d)=\sum_{i>J}\frac d{R_{i,N}(R_{i,N}-d)},\qquad A_{N,J}=\sum_{i\le J}\frac1{R_{i,N}}.
\end{equation}
We have established that, for every $S\in(0,N]$,
\begin{equation}\label{eq:cprof-min-coef}
t(S)\ge t(N)=\frac{R_{J+1,N}-d_{N,J}}b.
\end{equation}

\paragraph{Step 2: the limit $N\to\infty$ at fixed $J$.} Let $G_J(d)=\sum_{i>J}d/(R_i(R_i-d))$ for $0\le d<R_{J+1}$ and $A_J=\sum_{i\le J}R_i^{-1}$. The function $G_J$ is continuous and strictly increasing, with $G_J(0)=0$ and $G_J(d)\to\infty$ as $d\uparrow R_{J+1}$, and there exists a unique $d_J\in(0,R_{J+1})$ with $G_J(d_J)=A_J$. We claim that $d_{N,J}\to d_J$. Note that $R_{i,N}\to R_i$ for each $i$ by \eqref{eq:cprof-conv}, and $A_{N,J}\to A_J$. Fix $0<D<R_{J+1}$ and $u_0$ with $R_{J+1}^{-1}<u_0<D^{-1}$. For large $N$ we have $R_{J+1,N}^{-1}<u_0$, and then all $u=R_{i,N}^{-1}$ and $u=R_i^{-1}$ with $i>J$ lie in $[0,u_0]$. The summand is $g_d(u)=du^2/(1-du)$, and for $d\in[0,D]$ and $u\in[0,u_0]$,
\begin{equation*}
|g_d'(u)|=\frac{du(2-du)}{(1-du)^2}\le\frac{2Du}{(1-Du_0)^2}.
\end{equation*}
Integrating this bound, we have $|g_d(u)-g_d(u')|\le C_D(u+u')|u-u'|$, and the Cauchy--Schwarz inequality together with \eqref{eq:cprof-conv} shows that $G_{N,J}\to G_J$ uniformly on $[0,D]$. Given $0<\zeta<\min\{d_J,R_{J+1}-d_J\}$, we have $G_J(d_J-\zeta)<A_J<G_J(d_J+\zeta)$. For large $N$, uniform convergence gives $G_{N,J}(d_J-\zeta)<A_{N,J}<G_{N,J}(d_J+\zeta)$. Since $G_{N,J}$ is increasing on $[0,R_{J+1,N})$, we obtain $|d_{N,J}-d_J|<\zeta$.

\paragraph{Step 3: the limit $J\to\infty$.} We show that
\begin{equation}\label{eq:cprof-xstar}
\lim_{J\to\infty}\frac{d_J}{\gamma J^{2/3}}=x_*,\qquad\text{where }x_*\in(0,1)\text{ solves }\sqrt{x_*}\operatorname{artanh}\sqrt{x_*}=1.
\end{equation}
Numerically $x_*=0.69481\ldots$; we only use $x_*<1$. Let $\mathcal G(x)=3\sqrt x\operatorname{artanh}\sqrt x$ for $x\in(0,1)$, a continuous increasing function from $0$ to $\infty$. We first show that, uniformly for $x$ in compact subsets of $(0,1)$,
\begin{equation}\label{eq:cprof-riemann}
\frac\gamma{J^{1/3}}\,G_J(\gamma J^{2/3}x)\longrightarrow\int_1^\infty\frac{x\,du}{u^{2/3}(u^{2/3}-x)}=\mathcal G(x).
\end{equation}
To evaluate the integral, we substitute $u=v^3$ and obtain
\begin{equation*}
\int_1^\infty\frac{x\,du}{u^{2/3}(u^{2/3}-x)}=3x\int_1^\infty\frac{dv}{v^2-x}=\frac{3\sqrt x}2\log\frac{1+\sqrt x}{1-\sqrt x}=\mathcal G(x).
\end{equation*}
Fix $0<x_1<x_2<1$ and let $\tau_i=R_i/(\gamma J^{2/3})$. The left side of \eqref{eq:cprof-riemann} equals $J^{-1}\sum_{i>J}x/(\tau_i(\tau_i-x))$. Let $\zeta_J=\sup_{i>J}|R_i/(\gamma i^{2/3})-1|$, which tends to zero. Then $\tau_i=(i/J)^{2/3}(1+O(\zeta_J))$ uniformly in $i>J$. For large $J$, both $\tau_i$ and $(i/J)^{2/3}$ are at least $\tau_0=(1+x_2)/2$ for all $i>J$. For $\tau\ge\tau_0$ and $x\le x_2$ we have $\tau-x\ge c_2\tau$ with $c_2=(1-x_2)/(1+x_2)$, and the derivative of $\tau\mapsto x/(\tau(\tau-x))$ is then at most $2/(c_2^2\tau^3)$ in absolute value. As $|\tau_i-(i/J)^{2/3}|\le C\zeta_J(i/J)^{2/3}$, replacing $\tau_i$ by $(i/J)^{2/3}$ changes the sum by at most
\begin{equation*}
\frac CJ\sum_{i>J}\zeta_J\Big(\frac iJ\Big)^{-4/3}\le C'\zeta_J.
\end{equation*}
The function $\phi_x(u)=x/(u^{2/3}(u^{2/3}-x))$ is decreasing on $[1,\infty)$, so that
\begin{equation*}
\int_{1+1/J}^\infty\phi_x(u)\,du\le\frac1J\sum_{i>J}\phi_x(i/J)\le\int_1^\infty\phi_x(u)\,du,
\end{equation*}
and $\int_1^{1+1/J}\phi_x\le\phi_x(1)/J\le1/(J(1-x_2))$. This proves \eqref{eq:cprof-riemann}. Next, since $R_i\sim\gamma i^{2/3}$ and $\sum_{i\le J}i^{-2/3}\sim3J^{1/3}$, we have $\gamma A_J/J^{1/3}\to3$. Let $0<\zeta<\min\{x_*,1-x_*\}$. The function $\mathcal G$ is increasing with $\mathcal G(x_*)=3$, so $\mathcal G(x_*-\zeta)<3<\mathcal G(x_*+\zeta)$. For large $J$, $\gamma J^{2/3}(x_*+\zeta)<R_{J+1}$ because $R_{J+1}/(\gamma J^{2/3})\to1$, and we deduce from \eqref{eq:cprof-riemann} and $\gamma A_J/J^{1/3}\to3$ that $G_J(\gamma J^{2/3}(x_*-\zeta))<A_J<G_J(\gamma J^{2/3}(x_*+\zeta))$. As $G_J$ is increasing, it follows that $d_J/(\gamma J^{2/3})\in(x_*-\zeta,x_*+\zeta)$, which proves \eqref{eq:cprof-xstar}.

From \eqref{eq:cprof-xstar} and $R_{J+1}/(\gamma J^{2/3})\to1$, we obtain $(R_{J+1}-d_J)/(\gamma J^{2/3})\to1-x_*>0$. Further, $R_{J+k}-R_{J+1}=o(J^{2/3})$ as $J\to\infty$, because $R_{J+k}$ and $R_{J+1}$ are both $\gamma J^{2/3}(1+o(1))$. We choose $J_0$ such that, for all $J\ge J_0$,
\begin{equation}\label{eq:cprof-J0}
R_{J+1}-d_J\ge\frac{1-x_*}2\gamma J^{2/3},\qquad R_{J+k}-R_{J+1}\le\frac{1-x_*}8\gamma J^{2/3}.
\end{equation}
Then, for $J\ge J_0$, Step 2 and the convergence $R_{i,N}\to R_i$ provide $N_0(J)$ such that, for $N\ge N_0(J)$,
\begin{equation}\label{eq:cprof-N0}
R_{J+1,N}-d_{N,J}\ge\frac{1-x_*}4\gamma J^{2/3},\qquad R_{J+k,N}-R_{J+1,N}\le\frac{1-x_*}4\gamma J^{2/3}.
\end{equation}

\paragraph{Step 4: conclusion.} Let $J\ge J_0$, $N\ge N_0(J)$, and $S\in(0,N]$. Combining \eqref{eq:cprof-min-coef} and \eqref{eq:cprof-N0}, we find that $bt(S)\ge(1-x_*)\gamma J^{2/3}/4\ge R_{J+k,N}-R_{J+1,N}$. Then, for $J<i\le J+k$,
\begin{equation*}
\omega_i=t(S)+r_i-r_{J+1}\le t(S)+\frac{R_{J+k,N}-R_{J+1,N}}b\le2t(S).
\end{equation*}
Thus at least $k=260$ coefficients are at most $Lt(S)$ with $L=2$, and $k-2=258\ge64L^2$. Using the defining equation of $\alpha(S)$, we have
\begin{equation*}
St(S)=\frac12\sum_{i>J}\frac{t(S)}{\omega_i}\ge\frac12\sum_{i=J+1}^{J+k}\frac{t(S)}{2t(S)}=\frac k4\ge4.
\end{equation*}
The three weights $(2\omega_i)^{-1}$, $i=J+1,J+2,J+3$, are at least $(4t(S))^{-1}=\max_j(2\omega_j)^{-1}/2$. Now \cref{lem:cprof-radius} with $\delta'=1/2$ implies that
\begin{equation*}
C_{\mathrm{LS}}\big(\pi_N(\cdot\mid x_{\le J}=h)\big)\le\frac{C_{1/2}}{t(S)}\le\frac{4C_{1/2}}{1-x_*}\cdot\frac b{\gamma J^{2/3}},
\end{equation*}
which is \eqref{cd-tail-lsi-bound}.
\end{proof}

\begin{lemma}[The remaining coordinates at positive times]\label{cd-averaged-tail}
Under the hypotheses of \cref{cd-tail-lsi}, let $s_0>0$. There exists $C$, depending only on $s_0$, $\gamma$, and $(c_0,C_0,\delta)$, such that for every $J\ge J_0$ there exists $N_0(J,s_0)$ with
\begin{equation}\label{cd-averaged-tail-bound}
0\le D_N(bs)-D_N^{\le J}(bs)\le CJ^{-1/3}\qquad\text{for all }N\in\mathcal N\text{ with }N\ge N_0(J,s_0)\text{ and all }s\ge s_0.
\end{equation}
\end{lemma}

As in \cref{cd-tail-lsi}, $J$ is fixed before $N\to\infty$, and no uniformity is claimed for $J$ growing with $N$. The bound holds for all $s\ge s_0$ at once.

\begin{proof}
Let $J\ge J_0$, and let $C'$ be the absolute constant in \eqref{cd-tail-lsi-bound}. We choose $N_0(J,s_0)\ge\max\{N_0(J),N_{s_0},J+3\}$, with $N_0(J)$ as in \cref{cd-tail-lsi} and $N_{s_0}$ as in \eqref{eq:cprof-positive-energy}. Fix $N\ge N_0(J,s_0)$ and $s\ge s_0$, and let $\mu=\nu_NP_{bs}$ and $f=f_{N,s}$, which is of class $C^2$.

\paragraph{Step 1: a conditional Poincar\'e inequality.} A logarithmic Sobolev inequality implies a Poincar\'e inequality with half its constant \cite[Section~5.1]{BakryGentilLedoux2014}. In the convention \eqref{eq:model-fi}, this follows by applying the logarithmic Sobolev inequality to $1+\zeta g$ for a smooth function $g$ and letting $\zeta\to0$: we have $\Ent((1+\zeta g)^2)=2\zeta^2\Var(g)+o(\zeta^2)$, while the energy of $1+\zeta g$ is $\zeta^2$ times the energy of $g$. Let $h\in\R^J$ with $|h|^2<N$. By \eqref{cd-tail-lsi-bound}, for every smooth function $g$ on the sphere $\{z\in\R^{N-J}:|z|^2=N-|h|^2\}$,
\begin{equation*}
\Var_{\pi_N(\cdot\mid x_{\le J}=h)}(g)\le\frac{C'b}{2\gamma J^{2/3}}\int|\nabla g|^2\,d\pi_N(\cdot\mid x_{\le J}=h),
\end{equation*}
where $\nabla$ is the tangent gradient of this sphere. This holds for every conditioning value $h$, including those for which the sphere is very small.

\paragraph{Step 2: the conditional variance.} We write $x=(h,z)$ with $h\in\R^J$ and $z\in\R^{N-J}$, we let $\pi_H$ be the law of $h$ under $\pi_N$, and we let $f_H(h)=\E_{\pi_N}[f\mid x_{\le J}=h]$, viewed also as a function on $S_N$. The set $\{|h|^2=N\}$ has $\pi_H$-measure zero, since $N\ge J+3$ (\cref{gp-head-density}). The tangent space of the fiber $\{(h,z):|z|^2=N-|h|^2\}$ at $(h,z)$ consists of the vectors $(0,w)$ with $w\cdot z=0$, which are tangent to $S_N$ at $(h,z)$. The fiber gradient $\nabla_{\mathrm{fib}}f$ is the orthogonal projection of $\nabla_{S_N}f$ onto this subspace, and then $|\nabla_{\mathrm{fib}}f|\le|\nabla_{S_N}f|$. Applying Step 1 to the restriction of $f$ to each fiber, whose tangent gradient is $\nabla_{\mathrm{fib}}f$, and integrating over $\pi_H$, we find
\begin{equation}\label{eq:cprof-conditional-variance}
\|f-f_H\|_{L^2(\pi_N)}^2=\int\Var_{\pi_N(\cdot\mid x_{\le J}=h)}(f)\,\pi_H(dh)\le\frac{C'b}{2\gamma J^{2/3}}\int|\nabla_{S_N}f|^2\,d\pi_N=\frac{C'}{2\gamma J^{2/3}}\,\mathcal E_N(f).
\end{equation}

\paragraph{Step 3: comparison of distances.} Let $\tilde\mu=f_H\,\pi_N$. Since $f_H$ is nonnegative and $\int f_H\,d\pi_N=1$, this is a probability measure. It has the same law of the first $J$ coordinates as $\mu$, and the conditional law of $\pi_N$ given the first $J$ coordinates. Then $\tvnorm{\tilde\mu-\pi_N}=D_N^{\le J}(bs)$, since total variation distances do not change under the rescaling of the coordinates. Using \cref{lem:model-tv}, the triangle inequality, and the fact that projection does not increase total variation, we have
\begin{equation*}
0\le D_N(bs)-D_N^{\le J}(bs)\le\tvnorm{\mu-\tilde\mu}=\frac12\|f-f_H\|_{L^1(\pi_N)}\le\frac12\|f-f_H\|_{L^2(\pi_N)}.
\end{equation*}
Inserting \eqref{eq:cprof-conditional-variance} and \eqref{eq:cprof-positive-energy}, we obtain \eqref{cd-averaged-tail-bound} with $C=(M_{s_0}/2)\big(C'/(2\gamma s_0)\big)^{1/2}$.
\end{proof}

\section{Total variation of finitely many coordinates}\label{sec:cprof-head}

In this section we prove convergence of the first $J$ rescaled coordinates in total variation at every positive time (\cref{cd-head-current}). The proof uses bounds under $\pi_N$ for vector fields that translate the first $J$ coordinates, the energy bound \eqref{eq:cprof-positive-energy}, and mollification at the chosen time.

Let $\nabla_{S_N}$ and $\operatorname{div}_{S_N}$ be the tangent gradient and divergence on $S_N$. For a smooth tangent vector field $V$, we write $\operatorname{div}_{\pi_N}V=\operatorname{div}_{S_N}V-V\cdot\nabla(\sum_ir_ix_i^2)$, so that $\int(V\phi)\,d\pi_N=-\int\phi\operatorname{div}_{\pi_N}V\,d\pi_N$ for smooth $\phi$.

\begin{lemma}[Stationary bounds for the translation fields]\label{cd-stationary-fields}
Let $(r_{\cdot,N})$ satisfy the critical spectral conditions with constants $(c_0,C_0,\delta)$ for every $N\in\mathcal N$ and converge to $R$, and fix $J\ge1$. For $N>J$ and $a\le J$, let $\Sigma_1(x)=\sum_{i>J}q_ix_i^2$, let $V_a$ be the vector field on $\{x\in S_N:x_{>J}\ne0\}$ with components
\begin{equation*}
(V_a)_a=1,\qquad(V_a)_i=0\quad(i\le J,\ i\ne a),\qquad(V_a)_i=-\frac{x_aq_ix_i}{\Sigma_1}\quad(i>J),
\end{equation*}
and let $\tilde V_a=N^{1/3}V_a$. There exist $C_J$ and $N_J$ such that, for all $N\in\mathcal N$ with $N\ge N_J$ and all $a\le J$,
\begin{equation}\label{eq:cprof-field-moments}
\big\|\operatorname{div}_{\pi_N}\tilde V_a\big\|_{L^2(\pi_N)}\le C_J,\qquad\int|\tilde V_a|^2\,d\pi_N\le C_Jb.
\end{equation}
The constants $C_J$ and $N_J$ depend on $J$ and on the sequence $(r_{\cdot,N})$, but not on $N$.
\end{lemma}

The bounds \eqref{eq:cprof-field-moments} are integrals under the Gibbs measure $\pi_N$; they do not involve the law of the diffusion at positive times. The set $\{x_{>J}=0\}$ has $\pi_N$-measure zero, and the fields are defined $\pi_N$-almost everywhere.

\begin{proof}
We first compute the divergences of the fields, and we then bound them using moments of the coordinates under $\pi_N$.

\paragraph{Step 1: the divergence.} Note that $x\cdot V_a=x_a-x_a\Sigma_1/\Sigma_1=0$. Then $V_a$ is tangent to every sphere centered at the origin, and $V_a$ is homogeneous of degree zero. For such a field the spherical divergence equals the Euclidean divergence, since the Euclidean divergence equals the spherical divergence plus $\langle\partial_nV_a,n\rangle$ for the unit normal $n=x/|x|$, and $\partial_nV_a=0$ by homogeneity. Let $\Sigma_2(x)=\sum_{i>J}q_i^2x_i^2$. Differentiating, we find
\begin{equation*}
\operatorname{div}V_a=\sum_{i>J}\partial_i(V_a)_i=-x_a\frac{\sum_{i>J}q_i}{\Sigma_1}+2x_a\frac{\Sigma_2}{\Sigma_1^2}.
\end{equation*}
Using $2r_iq_i=1$, we find that on $S_N$ $-V_a\cdot\nabla(\sum_ir_ix_i^2)=-2r_ax_a+x_a\sum_{i>J}x_i^2/\Sigma_1=-2r_ax_a+x_a(N-|x_{\le J}|^2)/\Sigma_1$. Using $\sum_iq_i=N$, we obtain
\begin{equation}\label{cd-horizontal-divergence}
\operatorname{div}_{\pi_N}V_a=-2r_ax_a+x_a\Big\{\frac{\sum_{i\le J}q_i-|x_{\le J}|^2}{\Sigma_1}+\frac{2\Sigma_2}{\Sigma_1^2}\Big\},\qquad|V_a|^2=1+\frac{x_a^2\Sigma_2}{\Sigma_1^2}.
\end{equation}

\paragraph{Step 2: the rescaled fields.} Let $y=N^{-1/3}x$. Then $\tilde V_a$ acts on functions of $y_{\le J}$ as $\partial/\partial y_a$: if $\Phi(x)=\phi(N^{-1/3}x_{\le J})$, then $\tilde V_a\Phi=(\partial_a\phi)(y_{\le J})$. We write
\begin{equation*}
\sigma_1(y)=\sum_{i>J}\frac{y_i^2}{R_{i,N}},\qquad\sigma_2(y)=\sum_{i>J}\frac{y_i^2}{R_{i,N}^2},\qquad H(y)=\sum_{i\le J}y_i^2,
\end{equation*}
and we recall the sum $A_{N,J}=\sum_{i\le J}R_{i,N}^{-1}$ of \eqref{eq:cprof-root}. Recall that $q_i=b/(2R_{i,N})$ and $N^{2/3}=b$. Then $\Sigma_1=b^2\sigma_1/2$, $\Sigma_2=b^3\sigma_2/4$, $\sum_{i\le J}q_i=bA_{N,J}/2$, and $|x_{\le J}|^2=bH$. Substituting into \eqref{cd-horizontal-divergence}, we obtain
\begin{equation}\label{cd-scaled-horizontal}
\operatorname{div}_{\pi_N}\tilde V_a=-2R_{a,N}y_a+2y_a\Big\{\frac{A_{N,J}/2-H}{\sigma_1}+\frac{\sigma_2}{\sigma_1^2}\Big\},\qquad\frac{|\tilde V_a|^2}b=1+\frac{y_a^2\sigma_2}{\sigma_1^2}.
\end{equation}
For instance, $N^{1/3}r_ax_a=br_ay_a=R_{a,N}y_a$, and the term with $\Sigma_2$ becomes $N^{1/3}\cdot2x_a\Sigma_2/\Sigma_1^2=2by_a\cdot(b^3\sigma_2/4)(b^2\sigma_1/2)^{-2}=2y_a\sigma_2/\sigma_1^2$. The other terms are converted in the same way, and the identity for $|\tilde V_a|^2$ follows from $x_a^2=by_a^2$.

\paragraph{Step 3: moments under $\pi_N$.} By \cref{cd-pin}(b), $y$ has law $\mu_{v_N}$ under $\pi_N$, the conditioned Gaussian law with variances $v_{i,N}=(2R_{i,N})^{-1}$. Let $z_i=\sqrt{2R_{i,N}}\,y_i$ be the standardized coordinates. By \cref{cd-pin}(c), whose proof rests on the domination in \cref{prop:cond-continuity}, there exist $N_0$ and $B_1$ such that $\sup_i\int|z_i|^{16}\,d\pi_N\le B_1$ for $N\ge N_0$. By the same result, for every $m\ge3$ there exist $N_0(m)$ and $\mathfrak M_m$ such that, for $N\ge N_0(m)$, the law of $(z_i)_{i\le m}$ under $\pi_N$ has density at most $\mathfrak M_m$ with respect to the standard Gaussian law on $\R^m$. Since $R_{i,N}\ge c_0$ by the Weyl bounds, we have
\begin{equation*}
\sup_i\int|y_i|^{16}\,d\pi_N\le B_0,\qquad\int H^8\,d\pi_N\le J^7\sum_{i\le J}\int|y_i|^{16}\,d\pi_N\le J^8B_0,\qquad B_0=\frac{B_1}{(2c_0)^8}.
\end{equation*}
In the second bound we used $(\sum_{i\le J}y_i^2)^8\le J^7\sum_{i\le J}y_i^{16}$, which follows from H\"older's inequality. For the inverse moments of $\sigma_1$, let $I=\{J+1,\dots,J+19\}$ and $N\ge J+19$. By the Weyl bounds, $R_{i,N}\le D_J=C_0(J+19)^{2/3}$ for $i\in I$, and then
\begin{equation*}
\sigma_1\ge\sum_{i\in I}\frac{y_i^2}{R_{i,N}}=\sum_{i\in I}\frac{z_i^2}{2R_{i,N}^2}\ge\frac1{2D_J^2}\sum_{i\in I}z_i^2.
\end{equation*}
For $N\ge N_0(J+19)$, the law of $(z_i)_{i\in I}$ has density at most $\mathfrak M_{J+19}$ with respect to the standard Gaussian law on $\R^I$, since it is a marginal of the law of $(z_i)_{i\le J+19}$. In particular, $\sigma_1>0$ $\pi_N$-almost everywhere, and
\begin{equation}\label{eq:cprof-inverse-moment}
\int\sigma_1^{-8}\,d\pi_N\le B_J,\qquad B_J=(2D_J^2)^8\,\mathfrak M_{J+19}\,\E\Big(\sum_{i=1}^{19}G_i^2\Big)^{-8},
\end{equation}
where $G_1,\dots,G_{19}$ are independent standard Gaussian variables. The last expectation is finite: by integration in polar coordinates in $\R^{19}$ \cite[Chapter~2, Section~6, (6.3), p.~66]{Simon2014}, it equals
\begin{equation*}
\int_0^\infty\varrho^2e^{-\varrho^2/2}\,d\varrho\,\Big/\int_0^\infty\varrho^{18}e^{-\varrho^2/2}\,d\varrho,
\end{equation*}
since $\varrho^{-16}\varrho^{18}=\varrho^2$. All integrals in this step are under the Gibbs measure $\pi_N$, and no independence of the coordinates under $\pi_N$ is used. Bounds on positive moments alone would not give \eqref{eq:cprof-inverse-moment}; the inverse moment is finite because the law of a block of $19>16$ coordinates is dominated by a Gaussian law.

\paragraph{Step 4: bounds on the fields.} Let $N_J=\max\{N_0,N_0(J+19),J+19\}$, let $N\ge N_J$ and $a\le J$, and let
\begin{equation*}
E_a=1+y_a^8+H^8+\sigma_1^{-8}.
\end{equation*}
The Weyl bounds \eqref{eq:cprof-weyl} imply that $\sigma_2\le c_0^{-1}\sigma_1$, $R_{a,N}\le C_0J^{2/3}$, and $A_{N,J}\le J/c_0$. By the inequality between arithmetic and geometric means and the bound $u^k\le1+u^8$ for $u\ge0$ and $0\le k\le8$, each of the quantities $y_a^2$, $y_a^2\sigma_1^{-1}$, $y_a^2\sigma_1^{-2}$, and $y_a^2H^2\sigma_1^{-2}$ is at most $E_a$. Using \eqref{cd-scaled-horizontal} and $(u_1+u_2+u_3+u_4)^2\le4(u_1^2+u_2^2+u_3^2+u_4^2)$, we find
\begin{equation*}
\big|\operatorname{div}_{\pi_N}\tilde V_a\big|^2\le16R_{a,N}^2y_a^2+\frac{4A_{N,J}^2y_a^2}{\sigma_1^2}+\frac{16y_a^2H^2}{\sigma_1^2}+\frac{16y_a^2\sigma_2^2}{\sigma_1^4}\le L_JE_a,
\end{equation*}
where $L_J=16C_0^2J^{4/3}+4J^2/c_0^2+16+16/c_0^2$. In the last inequality, we used $\sigma_2\le c_0^{-1}\sigma_1$ for the last term. Similarly, $|\tilde V_a|^2/b=1+y_a^2\sigma_2/\sigma_1^2\le1+c_0^{-1}y_a^2\sigma_1^{-1}\le(1+c_0^{-1})E_a$. By Step 3 and $y_a^8\le1+y_a^{16}$, we have
\begin{equation*}
\int E_a\,d\pi_N\le E_J,\qquad E_J=2+B_0+J^8B_0+B_J.
\end{equation*}
This proves \eqref{eq:cprof-field-moments} with $C_J=\max\{(L_JE_J)^{1/2},(1+c_0^{-1})E_J\}$. The fields and their divergences are smooth off the set $\{x_{>J}=0\}$, which has $\pi_N$-measure zero.
\end{proof}

\begin{lemma}[Total variation convergence of the first coordinates]\label{cd-head-current}
Let $(r_{\cdot,N})$ satisfy the critical spectral conditions with constants $(c_0,C_0,\delta)$ for every $N\in\mathcal N$ and converge to $R$. Then, for every $J\ge1$ and every $s>0$,
\begin{equation}\label{eq:cprof-head-tv}
\lim_{N\to\infty}\tvnorm{(\nu_NP_{bs})^{\le J}-(\mu^\infty_s)^{\le J}}=0.
\end{equation}
\end{lemma}

\begin{proof}
Fix $J\ge1$ and $s>0$, and choose $s_0\in(0,s]$. Let $N\in\mathcal N$ with $N\ge\max\{N_J,N_{s_0},J+3\}$, with $N_J$ as in \cref{cd-stationary-fields} and $N_{s_0}$ as in \eqref{eq:cprof-positive-energy}. Write $f=f_{N,s}$, which is of class $C^2$, and $\mu^J_{N,s}=(\nu_NP_{bs})^{\le J}$. We establish a translation bound whose constant is independent of $N$.

\paragraph{Step 1: integration by parts.} Let $\phi\in C_c^\infty(\R^J)$ and $\Phi(x)=\phi(N^{-1/3}x_{\le J})$, and let $a\le J$. We show that
\begin{equation}\label{eq:cprof-ibp}
\int f\,(\partial_a\phi)(y_{\le J})\,d\pi_N=-\int\Phi\,\big(f\operatorname{div}_{\pi_N}\tilde V_a+\tilde V_a\cdot\nabla_{S_N}f\big)\,d\pi_N,\qquad y=N^{-1/3}x.
\end{equation}
Both terms on the right are integrable, because $\Phi$ is bounded and each term is a product of two functions in $L^2(\pi_N)$, by \eqref{eq:cprof-field-moments}. No division by $f$ occurs. For $\eta\in(0,1)$, let $\Omega_\eta=\{x\in S_N:|x_{>J}|>\eta\}$. The field $V_a$ is smooth on $\Omega_\eta$, and by the divergence theorem \cite[Chapter~2, Section~4, (4.21)]{Simon2014},
\begin{equation*}
\int_{\Omega_\eta}(V_a\Phi)f\,d\pi_N=-\int_{\Omega_\eta}\Phi\big(f\operatorname{div}_{\pi_N}V_a+V_af\big)\,d\pi_N+\mathcal B_\eta,
\end{equation*}
where the boundary term $\mathcal B_\eta$ is the integral of $\Phi f\,(V_a\cdot n_\eta)$ over $\partial\Omega_\eta$ against the density of $\pi_N$ times the surface measure of $\partial\Omega_\eta$, and $n_\eta$ is the outward unit normal. Since $f$ is of class $C^2$ on the compact manifold $S_N$, $f$ and $\nabla_{S_N}f$ are bounded. Near $\{x_{>J}=0\}$ we have $\Sigma_1\asymp|x_{>J}|^2$. Then $|V_a|=O(|x_{>J}|^{-1})$ and $|\operatorname{div}_{\pi_N}V_a|=O(|x_{>J}|^{-2})$, while $\partial\Omega_\eta$ is the product of spheres $\{|x_{>J}|=\eta\}\times\{|x_{\le J}|^2=N-\eta^2\}$, of area $O(\eta^{N-J-1})$. The term $\mathcal B_\eta$ is therefore $O(\eta^{N-J-2})$, which tends to zero for $N-J>2$, and $|x_{>J}|^{-2}$ is integrable near $\{x_{>J}=0\}$ for the same reason. Letting $\eta\downarrow0$, multiplying by $N^{1/3}$, and using $\tilde V_a\Phi=(\partial_a\phi)(y_{\le J})$, we obtain \eqref{eq:cprof-ibp}.

\paragraph{Step 2: a translation bound.} By \eqref{eq:cprof-ibp}, the Cauchy--Schwarz inequality, and \eqref{eq:cprof-field-moments}, we have
\begin{equation}\label{cd-head-score-bound}
\Big|\int\partial_a\phi\,d\mu^J_{N,s}\Big|\le\|\phi\|_\infty\Big(C_J\|f\|_{L^2(\pi_N)}+(C_Jb)^{1/2}\Big(\int|\nabla_{S_N}f|^2\,d\pi_N\Big)^{1/2}\Big)\le C_{J,s_0}\|\phi\|_\infty,
\end{equation}
where $C_{J,s_0}=M_{s_0}\big(C_J+(C_J/s_0)^{1/2}\big)$. In the last inequality, we used $\|f\|_{L^2(\pi_N)}\le M_{s_0}$ and $b\int|\nabla_{S_N}f|^2\,d\pi_N=\mathcal E_N(f)\le M_{s_0}^2/s_0$, by \eqref{eq:cprof-positive-energy}; the factors $b^{1/2}$ and $b^{-1/2}$ cancel. Let $w\in\R^J$. Since $\phi(y+w)-\phi(y)=\int_0^1\sum_{a\le J}w_a(\partial_a\phi)(y+\theta w)\,d\theta$, applying \eqref{cd-head-score-bound} to the functions $\phi(\cdot+\theta w)$, $\theta\in[0,1]$, we obtain
\begin{equation*}
\Big|\int\phi(y+w)\,\mu^J_{N,s}(dy)-\int\phi(y)\,\mu^J_{N,s}(dy)\Big|\le C_{J,s_0}\|\phi\|_\infty\sum_{a\le J}|w_a|\le C_{J,s_0}J^{1/2}|w|\,\|\phi\|_\infty.
\end{equation*}
For two probability measures on $\R^J$, twice their total variation distance is the supremum of the differences of their integrals of functions $\phi\in C_c^\infty(\R^J)$ with $|\phi|\le1$. Indeed, let $\mu$ and $\nu$ be the two measures. Since finite Borel measures on $\R^J$ are regular \cite[Theorem~1.1]{Billingsley1999} and closed bounded sets are compact, every Borel function with values in $[-1,1]$ is a limit in $L^1(\mu+\nu)$ of continuous compactly supported functions with values in $[-1,1]$. Each of these is a uniform limit of its mollifications, which are smooth, compactly supported, and bounded by one in absolute value \cite[Proposition~4.21]{Brezis2011}. Taking the supremum over such $\phi$, we find
\begin{equation}\label{eq:cprof-head-translation}
\tvnorm{\tau_w\mu^J_{N,s}-\mu^J_{N,s}}\le\frac12C_{J,s_0}J^{1/2}|w|\qquad(w\in\R^J,\ s\ge s_0),
\end{equation}
with $\tau_w$ as in \cref{cd-mollifier-criterion}.

\paragraph{Step 3: passage to total variation.} By \eqref{cd-angular-functional}, $\mu^J_{N,s}$ converges weakly to $(\mu^\infty_s)^{\le J}$. The translation estimate \eqref{eq:cprof-head-translation} has a constant independent of $N$, so \cref{cd-mollifier-criterion}(b) applies to this sequence at the fixed time $s$. It gives \eqref{eq:cprof-head-tv}. Equivalently, at a fixed mollifier scale the weak convergence gives convergence of the mollified densities in $L^1$ by Scheff\'e's lemma; the translation estimate bounds both mollification errors by a constant times the scale, which is then sent to zero.
\end{proof}

\section{Continuity and pointwise convergence}\label{sec:cprof-average}

We now combine the estimates of the last two sections. For $s>0$ and $J\ge1$ let
\begin{equation*}
F_R^{(J)}(s)=\tvnorm{(\mu^\infty_s)^{\le J}-\rho_R^{\le J}}.
\end{equation*}
By \cref{lem:cprof-product-tv},
\begin{equation}\label{cd-product-profile-formula}
F_R(s)=\sup_{J\ge1}F_R^{(J)}(s)=\lim_{J\to\infty}F_R^{(J)}(s).
\end{equation}
Each $F_R^{(J)}$ is lower semicontinuous on $(0,\infty)$, since $s\mapsto(\mu^\infty_s)^{\le J}$ is weakly continuous and total variation is a supremum of differences of integrals of continuous functions with values in $[0,1]$. As a supremum of such functions, $F_R$ is lower semicontinuous, and in particular measurable.

\begin{lemma}[Continuity of the limiting laws]\label{lem:cprof-limit-continuity}
Under the hypotheses of \cref{cd-profile}, for every $s_0>0$ and all $s,t\ge s_0$,
\begin{equation}\label{eq:cprof-profile-lipschitz}
\tvnorm{\mu^\infty_s-\mu^\infty_t}\le L_{s_0}|s-t|,\qquad
|F_R(s)-F_R(t)|\le L_{s_0}|s-t|,
\end{equation}
where $L_{s_0}$ is the constant of \eqref{eq:cprof-time-modulus}. In particular, $F_R$ is continuous on $(0,\infty)$.
\end{lemma}

\begin{proof}
Fix $J\ge1$. Projecting \eqref{eq:cprof-time-modulus} gives
\[
\tvnorm{(\nu_NP_{bs})^{\le J}-(\nu_NP_{bt})^{\le J}}\le L_{s_0}|s-t|.
\]
By \eqref{cd-angular-functional}, both heads converge weakly to the corresponding heads of $\mu^\infty_s$ and $\mu^\infty_t$. Lower semicontinuity of total variation, \cref{cd-mollifier-criterion}(a), passes the same bound to those heads. Its right side does not depend on $J$, so \cref{lem:cprof-product-tv} gives the first inequality in \eqref{eq:cprof-profile-lipschitz}. The second follows from the triangle inequality. This argument uses only weak convergence of the heads and finite semigroup smoothing.
\end{proof}

\begin{proposition}[Pointwise convergence of the profile]\label{cd-pointwise-profile}
Let $R\in\mathcal P$, and let $(r_{\cdot,N})$ satisfy the critical spectral conditions with constants $(c_0,C_0,\delta)$ for every $N\in\mathcal N$ and converge to $R$. Then
\begin{equation}\label{eq:cprof-pointwise}
\lim_{N\to\infty}D_N(bs)=F_R(s)\qquad(s>0).
\end{equation}
\end{proposition}

\begin{proof}
Fix $s>0$ and $J\ge J_0$, with $J_0$ as in \cref{cd-tail-lsi}. For all sufficiently large $N$, \cref{cd-averaged-tail} gives
\[
0\le D_N(bs)-D_N^{\le J}(bs)\le C_sJ^{-1/3}.
\]
By the triangle inequality,
\[
|D_N^{\le J}(bs)-F_R^{(J)}(s)|
\le\tvnorm{(\nu_NP_{bs})^{\le J}-(\mu^\infty_s)^{\le J}}
+\tvnorm{\pi_N^{\le J}-\rho_R^{\le J}},
\]
which tends to zero at fixed $J$ by \cref{cd-head-current,cd-pin}. Since $F_R^{(J)}(s)\le F_R(s)$, we obtain
\[
\limsup_{N\to\infty}|D_N(bs)-F_R(s)|
\le C_sJ^{-1/3}+F_R(s)-F_R^{(J)}(s).
\]
Letting $J\to\infty$ and using \eqref{cd-product-profile-formula} proves the result. The dimension limit is taken before the number of retained coordinates tends to infinity.
\end{proof}

\section{Proofs of the deterministic results}\label{sec:cprof-proof}

The proof of strict decrease in \cref{cd-profile} uses the following estimate for the finite-dimensional semigroups $Q_{N,t}=P_{bt}$, with $K_{\mathrm P}$ as in \cref{tab:cprof-inputs}.

\begin{lemma}[Finite semigroup correlations]\label{cd-finite-correlation}
Fix $s>0$. There exist $M\ge1$ and $N_0$, depending only on $s$ and $(c_0,C_0,\delta)$, with the following property. Let $N\ge N_0$, let $r$ satisfy the critical spectral conditions with constants $(c_0,C_0,\delta)$, write $f_{N,s}-1=g_+-g_-$ with $g_+=\max(f_{N,s}-1,0)$ and $g_-=\max(1-f_{N,s},0)$, and let
\begin{equation*}
h_N(t)=\ip{g_+}{Q_{N,t}g_-}_{L^2(\pi_N)}\qquad(t\ge0).
\end{equation*}
Then there exist real numbers $w_{N,j}$ and $\lambda_{N,j}\ge1/K_{\mathrm P}$, $j\ge1$, with $\sum_j|w_{N,j}|\le M$, such that
\begin{equation}\label{eq:cprof-correlation-expansion}
h_N(t)=D_N(bs)^2+\sum_{j\ge1}w_{N,j}e^{-\lambda_{N,j}t}\qquad(t\ge0).
\end{equation}
Moreover, for every $u\ge0$,
\begin{equation}\label{eq:cprof-correlation-overlap}
0\le h_N(2u)\le(M^4+1)^{1/3}\big[D_N(bs)-D_N(b(s+u))\big]^{2/3}.
\end{equation}
\end{lemma}

The coefficients $w_{N,j}$ may have either sign. The bounds are uniform over all $r$ that satisfy the critical spectral conditions with constants $(c_0,C_0,\delta)$; the lower bound on $\lambda_{N,j}$ comes from the Poincar\'e inequality on the full sphere.

\begin{proof}
By \cref{cl3-actual-source-density} with $p=4$ and $s_0=s$, there exist $M\ge1$ and $N_0$, depending only on $s$ and $(c_0,C_0,\delta)$, such that $\|f_{N,s}\|_{L^4(\pi_N)}\le M$ for $N\ge N_0$. We increase $N_0$ so that also $C_{\mathrm P}(\pi_N)\le K_{\mathrm P}b$ for $N\ge N_0$. Let $f=f_{N,s}$ and $d_N=D_N(bs)$.

\paragraph{Step 1: the decrease of the distance.} Since $f\ge0$ and $\int f\,d\pi_N=1$, we have $\int g_+\,d\pi_N=\int g_-\,d\pi_N=d_N$ by \cref{lem:model-tv}. Further, $0\le g_+\le f$ and $0\le g_-\le1$. For $u\ge0$, the functions $Q_{N,u}g_\pm$ are nonnegative and have integral $d_N$, and $Q_{N,u}g_+-Q_{N,u}g_-=Q_{N,u}(f-1)=f_{N,s+u}-1$. Using $|a-a'|=a+a'-2\min(a,a')$ and \cref{lem:model-tv}, we find
\begin{equation}\label{eq:cprof-tv-drop}
d_N-D_N(b(s+u))=\int\min(Q_{N,u}g_+,Q_{N,u}g_-)\,d\pi_N.
\end{equation}

\paragraph{Step 2: the correlation and the decrease of the distance.} For $a,a'\ge0$, we have
\begin{equation*}
aa'\le\min(a,a')^{2/3}\big(a^4+a'^4\big)^{1/3}.
\end{equation*}
Indeed, if $a\le a'$, then $(aa')^3=a^3a'^3\le a^2a'^4\le a^2(a^4+a'^4)$, and the other case is symmetric. Since $Q_{N,u}$ is selfadjoint on $L^2(\pi_N)$ and $Q_{N,2u}=Q_{N,u}Q_{N,u}$, we have $h_N(2u)=\int Q_{N,u}g_+\,Q_{N,u}g_-\,d\pi_N\ge0$. H\"older's inequality with exponents $3/2$ and $3$ gives
\begin{equation*}
h_N(2u)\le\Big(\int\min(Q_{N,u}g_+,Q_{N,u}g_-)\,d\pi_N\Big)^{2/3}\Big(\int\big((Q_{N,u}g_+)^4+(Q_{N,u}g_-)^4\big)\,d\pi_N\Big)^{1/3}.
\end{equation*}
By Jensen's inequality for the Markov operator $Q_{N,u}$ and the invariance of $\pi_N$, we have $\int(Q_{N,u}g_+)^4\,d\pi_N\le\int g_+^4\,d\pi_N\le M^4$ and $\int(Q_{N,u}g_-)^4\,d\pi_N\le\int g_-^4\,d\pi_N\le1$. Inserting \eqref{eq:cprof-tv-drop}, we obtain \eqref{eq:cprof-correlation-overlap}.

\paragraph{Step 3: the spectral expansion.} Let $(e_j)_{j\ge0}$ be an orthonormal basis of $L^2(\pi_N)$ of smooth eigenfunctions of $-bL_r$, with eigenvalues $\lambda_{N,j}\ge0$ and $e_0=1$, as in the proof of \cref{cd-finite-smoothing}. For $j\ge1$, $e_j$ is orthogonal to $e_0$, and then $\Var_{\pi_N}(e_j)=1$. The Poincar\'e inequality gives $1\le K_{\mathrm P}b\int|\nabla_{S_N}e_j|^2\,d\pi_N=K_{\mathrm P}\lambda_{N,j}$, that is, $\lambda_{N,j}\ge1/K_{\mathrm P}$. Let $\alpha_j=\ip{g_+}{e_j}$ and $\beta_j=\ip{g_-}{e_j}$, so that $\alpha_0=\beta_0=d_N$. Expanding $Q_{N,t}g_-=\sum_j\beta_je^{-\lambda_{N,j}t}e_j$ in $L^2(\pi_N)$, we have
\begin{equation*}
h_N(t)=\sum_{j\ge0}\alpha_j\beta_je^{-\lambda_{N,j}t}=d_N^2+\sum_{j\ge1}w_{N,j}e^{-\lambda_{N,j}t},\qquad w_{N,j}=\alpha_j\beta_j.
\end{equation*}
By the Cauchy--Schwarz inequality and Parseval's identity, we have
\begin{equation*}
\sum_{j\ge1}|w_{N,j}|\le\|g_+\|_{L^2(\pi_N)}\|g_-\|_{L^2(\pi_N)}\le\|f\|_{L^2(\pi_N)}\le M.
\end{equation*}
In the second inequality, we used $0\le g_+\le f$ and $0\le g_-\le1$. In particular, the series converges absolutely for every $t\ge0$, and this proves \eqref{eq:cprof-correlation-expansion}.
\end{proof}

\begin{proof}[Proof of \cref{cd-profile}]
The proof uses pointwise convergence, finite spectral correlations, and the continuity estimate already established.

\paragraph{Step 1: continuity and monotonicity.} By \cref{lem:cprof-limit-continuity}, $F_R$ is continuous. The convergence \eqref{eq:cprof-pointwise} holds by \cref{cd-pointwise-profile}. Since every function $s\mapsto D_N(bs)$ is nonincreasing, its pointwise limit $F_R$ is nonincreasing.

\paragraph{Step 2: positivity.} By \eqref{eq:cprof-exp-lower}, there exist $c_1,C_1>0$ and $N_0$, depending only on $(c_0,C_0,\delta)$, such that $D_N(bs)\ge c_1e^{-C_1s}$ for all $s\ge0$ and $N\in\mathcal N$ with $N\ge N_0$. This bound was proved in \cref{sec:cprof-lower} without \cref{cd-profile}. Letting $N\to\infty$ and using \eqref{eq:cprof-pointwise}, we obtain
\begin{equation}\label{eq:cprof-profile-lower}
F_R(s)\ge c_1e^{-C_1s}>0\qquad(s>0).
\end{equation}

\paragraph{Step 3: strict decrease.} Suppose that $0<s_1<s_2$ and $F_R(s_1)=F_R(s_2)$. Since $F_R$ is nonincreasing, it is constant on $[s_1,s_2]$. Let $\tau=s_2-s_1$, and let $h_N$, $M$, and $N_0$ be as in \cref{cd-finite-correlation} with $s=s_1$. For $t\in[\tau,2\tau]$, we have $s_1+t/2\le s_2$. Since $D_N$ is nonincreasing, \eqref{eq:cprof-correlation-overlap} with $u=t/2$ implies that
\begin{equation*}
0\le h_N(t)\le(M^4+1)^{1/3}\big[D_N(bs_1)-D_N(bs_2)\big]^{2/3}\qquad(N\ge N_0).
\end{equation*}
By \eqref{eq:cprof-pointwise}, the right side tends to $(M^4+1)^{1/3}[F_R(s_1)-F_R(s_2)]^{2/3}=0$. Thus $h_N(t)\to0$ for every $t\in[\tau,2\tau]$, while the constant term $D_N(bs_1)^2$ in \eqref{eq:cprof-correlation-expansion} tends to $F_R(s_1)^2$. We apply \cref{cd-vanishing-correlations} along the increasing enumeration of $\{N\in\mathcal N:N\ge N_0\}$, with $t_1=\tau$, $t_2=2\tau$, $\lambda_0=1/K_{\mathrm P}$, the coefficients $w_{N,j}$ and rates $\lambda_{N,j}$ of \eqref{eq:cprof-correlation-expansion}, and the bound $M$ on $\sum_j|w_{N,j}|$. It gives $F_R(s_1)^2=0$, which contradicts \eqref{eq:cprof-profile-lower}. We conclude that $F_R(s_2)<F_R(s_1)$ whenever $0<s_1<s_2$. Here $M$ bounds the norm of $f_{N,s_1}$ in $L^4(\pi_N)$. It should not be confused with the bound $M_4$ in the proof of \eqref{eq:cprof-exp-lower}, which concerns the centered density at the time $s_*b/2$.

\paragraph{Step 4: uniform convergence on compact sets and endpoints.} Let $0<s_0<s_1$ and $\zeta>0$. Since $F_R$ is nonincreasing and uniformly continuous on $[s_0,s_1]$, there exist $s_0=u_0<u_1<\cdots<u_k=s_1$ with $F_R(u_{l-1})-F_R(u_l)<\zeta$ for all $l$. For $s\in[u_{l-1},u_l]$, the monotonicity of $D_N$ and of $F_R$ implies that
\begin{equation*}
D_N(bu_l)-F_R(u_l)-\zeta\le D_N(bs)-F_R(s)\le D_N(bu_{l-1})-F_R(u_{l-1})+\zeta.
\end{equation*}
By \eqref{eq:cprof-pointwise} at the finitely many points $u_l$, we find that $\limsup_{N\to\infty}\sup_{s\in[s_0,s_1]}|D_N(bs)-F_R(s)|\le\zeta$. As $\zeta$ is arbitrary, this proves \eqref{cd-full-tv-limit}.

For the limit at zero, \cref{cd-functional} gives $U_1(s)\to0$ and $Z(s)\to\infty$ almost surely, and hence $\xi_1(s)\to0$ almost surely. For $a>0$, let $B_a=\{y:|y_1|<a\}$. Then $F_R(s)\ge\mu^\infty_s(B_a)-\rho_R(B_a)$, so $\liminf_{s\downarrow0}F_R(s)\ge1-\rho_R(B_a)$. Since $\rho_R^{\le1}$ has a density, letting $a\downarrow0$ gives $F_R(0+)=1$.

For the limit at infinity, the finite $L^2$ decay estimate \eqref{eq:cprof-l2-upper} gives $D_N(bs)\le C_5e^{-(s-1)/K_{\mathrm P}}$ for all large $N$ and all $s\ge1$. Passing to the limit in \eqref{eq:cprof-pointwise} yields the same bound for $F_R(s)$, so $F_R(\infty)=0$. This completes the proof.
\end{proof}

\begin{proof}[Proof of \cref{cd-quantiles}]
The assertion for an empty interval is immediate. Let $I=[\eps_1,\eps_2]\subset(0,1)$ be nonempty, and let $[u_1,u_2]=F_R^{-1}(I)$, a compact subinterval of $(0,\infty)$ because $F_R^{-1}$ is continuous. Fix $0<h<u_1/2$. The function
\begin{equation*}
\eps\mapsto\min\Big\{F_R\big(F_R^{-1}(\eps)-h\big)-\eps,\ \eps-F_R\big(F_R^{-1}(\eps)+h\big)\Big\}
\end{equation*}
is continuous and positive on $I$, since $F_R$ is continuous and strictly decreasing; let $\zeta>0$ be its minimum. By \eqref{cd-full-tv-limit} on $[u_1-h,u_2+h]$, for large $N$ we have $|D_N(bs)-F_R(s)|<\zeta$ for all $s$ in this interval. Then, for every $\eps\in I$, $D_N(b(F_R^{-1}(\eps)-h))>\eps$ and $D_N(b(F_R^{-1}(\eps)+h))<\eps$. Since $D_N$ is nonincreasing, the first inequality implies that $t_N(\eps)\ge b(F_R^{-1}(\eps)-h)$, and the second implies that $t_N(\eps)\le b(F_R^{-1}(\eps)+h)$. Thus $\sup_{\eps\in I}|N^{-2/3}t_N(\eps)-F_R^{-1}(\eps)|\le h$ for large $N$. As $h$ is arbitrary, this proves \eqref{cd-quantile-limit}. For $\eps\in(0,1/2)$ we have $\eps<1-\eps$, and since $F_R^{-1}$ is strictly decreasing, $F_R^{-1}(\eps)>F_R^{-1}(1-\eps)>0$.
\end{proof}

\begin{proof}[Proof of the first part of \cref{critical-finite-time-positivity}]
By \eqref{eq:cprof-exp-lower}, with $c_1,C_1$ and $N_0$ depending only on $(c_0,C_0,\delta)$, we have $D_N(bs)\ge c_1e^{-C_1s}$ for all $s\ge0$ and $N\in\mathcal N$ with $N\ge N_0$. Letting $N\to\infty$ and using \eqref{eq:cprof-pointwise} of \cref{cd-pointwise-profile}, we obtain $F_R(s)\ge c_1e^{-C_1s}$ for every $s>0$. The finite bound is proved using only \cref{cg-entrance-lower,cl3-actual-source-density,cl3-density-two-time-lower}, and its passage to the limit uses only \cref{cd-pointwise-profile}. Neither argument uses positivity or strict decrease of the limiting profile.
\end{proof}

\section{The GOE critical window}\label{sec:cprof-goe}

To apply \cref{cd-profile} to the GOE model, we need to know that the finite profiles and the limiting profile are measurable functions of the spectral data. We first treat a fixed dimension.

\begin{lemma}[Profiles in fixed dimension]\label{lem:cprof-finite}
Let $N\ge2$ and $r\in\R^N$, and let $D^r(t)=\tvnorm{\nu_NP^r_t-\pi_r}$ and $t^r(\eps)=\inf\{t\ge0:D^r(t)\le\eps\}$.
\begin{enumerate}
\item The function $t\mapsto D^r(t)$ is continuous and nonincreasing on $[0,\infty)$ and tends to zero at infinity. If the $r_i$ are not all equal, it is positive and strictly decreasing; otherwise it vanishes identically.
\item The function $\eps\mapsto t^r(\eps)$ is continuous on $(0,1)$.
\item The maps $r\mapsto D^r\in C([0,\infty))$ and $r\mapsto t^r\in C((0,1))$ are Borel.
\end{enumerate}
\end{lemma}

\begin{proof}
The density $\varpi=d\pi_r/d\nu_N$ is bounded above and below, and $\nu_N$ satisfies a Poincar\'e inequality with some constant $C_{\nu}$, since the Laplacian on $S_N$ has a spectral gap. For smooth $g$, we have
\begin{equation*}
\Var_{\pi_r}(g)\le\int(g-\nu_N(g))^2\,d\pi_r\le\sup\varpi\,C_\nu\int|\nabla_{S_N}g|^2\,d\nu_N\le\frac{\sup\varpi}{\inf\varpi}\,C_\nu\int|\nabla_{S_N}g|^2\,d\pi_r,
\end{equation*}
so $\pi_r$ satisfies a Poincar\'e inequality with some constant $C_{\mathrm P}$; this is the perturbation argument of Holley and Stroock \cite[Lemma~5.1, pp.~1183--1184]{HolleyStroock1987}. The semigroup $(P^r_t)$ is a strongly continuous semigroup of selfadjoint Markov operators on $L^2(\pi_r)$, and $\|P^r_tg-\pi_r(g)\|_2\le e^{-t/C_{\mathrm P}}\|g-\pi_r(g)\|_2$.

For the first part, let $f_0=d\nu_N/d\pi_r$, which is bounded. Then $D^r(t)=\|P^r_tf_0-1\|_{L^1(\pi_r)}/2$ by reversibility. The continuity of $D^r$ follows from the strong continuity in $L^2(\pi_r)$, its monotonicity from the contraction property, and the bound $D^r(t)\le e^{-t/C_{\mathrm P}}\|f_0-1\|_2/2$ from the spectral gap. If the $r_i$ are all equal, then $\pi_r=\nu_N$ and $D^r\equiv0$. Otherwise $f_0\ne1$ and $D^r(0)>0$. If $D^r(t_0)=0$ for some $t_0>0$, then $P^r_{t_0}(f_0-1)=0$, and the injectivity of $P^r_{t_0}=e^{-t_0\mathcal A}$ forces $f_0=1$, a contradiction. Thus $D^r>0$. For $0\le s<s'$, write $P^r_sf_0-1=g_+-g_-$, where $g_\pm$ are continuous and nonnegative. Their integrals both equal $D^r(s)>0$. The finite diffusion sends every nonzero nonnegative continuous function to a strictly positive function at every positive time. Indeed, its heat equation is uniformly parabolic in each relatively compact coordinate chart of the sphere. The strong maximum principle implies that a zero at a positive time forces zeros at earlier positive times throughout that chart; connectedness propagates this through a chain of overlapping charts. This would contradict preservation of the positive integral. Applying this to $g_+$ and $g_-$ at time $u=s'-s>0$, we obtain
\[
D^r(s')=D^r(s)-\int\min\{P^r_ug_+,P^r_ug_-\}\,d\pi_r<D^r(s).
\]


For the second part, if $D^r\equiv0$ then $t^r\equiv0$. Otherwise $D^r$ is a continuous strictly decreasing bijection from $[0,\infty)$ onto $(0,D^r(0)]$. Then $t^r(\eps)=(D^r)^{-1}(\eps)$ for $\eps\in(0,D^r(0)]$, which is continuous and tends to zero as $\eps\uparrow D^r(0)$, and $t^r(\eps)=0$ for $\eps\ge D^r(0)$.

For the third part, fix $t\ge0$. By \cref{cd-lift}, $\nu_NP^r_t$ is the law of $\sqrt NY^r(t)/|Y^r(t)|$, where $Y^r$ solves $dY_i=-2r_iY_i\,dt+\sqrt{2/N}|Y|\,dB_i$ with $Y(0)=g$. We couple all these solutions by using the same random Gaussian vector $g$, independent of the same Brownian motions $(B_i)$. If $r^{(n)}\to r$, It\^o's isometry and Gronwall's lemma, applied to the Lipschitz equation, give $\E|Y^{r^{(n)}}(t)-Y^r(t)|^2\to0$, where the expectation is over $g$ and the Brownian motions. As $|Y^r(t)|>0$ almost surely, it follows that $\nu_NP^{r^{(n)}}_t\to\nu_NP^r_t$ weakly. The measures $\pi_r$ are also weakly continuous in $r$. Total variation on $S_N$ is the supremum of $\mu(\phi)-\pi(\phi)$ over a countable dense set of continuous $\phi\colon S_N\to[0,1]$ (\cref{lem:model-tv}), and therefore $r\mapsto D^r(t)$ is lower semicontinuous, hence Borel. As $D^r$ is continuous in $t$ and the Borel $\sigma$-field of $C([0,\infty))$ is generated by evaluations at rational times, the map $r\mapsto D^r$ is Borel. Finally, for $\eps\in(0,1)$ and $s\ge0$, we have $t^r(\eps)>s$ if and only if $D^r(s)>\eps$, because $D^r$ is continuous and nonincreasing. We conclude that $r\mapsto t^r(\eps)$ is Borel for every $\eps$, and by continuity in $\eps$, $r\mapsto t^r$ is Borel.
\end{proof}

\begin{lemma}[Measurable dependence of the critical profile]\label{lem:cprof-measurable}
Let the spaces $C((0,\infty))$ and $C((0,1))$ carry the topology of uniform convergence on compact subsets.
\begin{enumerate}
\item The maps $R\mapsto F_R\in C((0,\infty))$ and $R\mapsto F_R^{-1}\in C((0,1))$ are Borel on $\mathcal P$, where $\mathcal P$ is viewed as a Borel subset of $\ell^2$ as in \cref{lem:cprof-approx}.
\item For every $\sigma\in\R$, there exists a Borel map $\Phi^\sigma\colon\R\times\ell^2\to C((0,\infty))\times C((0,1))$ such that, almost surely, $(F_\sigma,F_\sigma^{-1})=\Phi^\sigma\big(\Xi,(\Delta_i^{-1})_{i\ge2}\big)$. In particular, $F_\sigma$ and $F_\sigma^{-1}$ are random elements of these spaces, and they are measurable functions of $(\Xi,(\Delta_i^{-1})_{i\ge2})$.
\end{enumerate}
\end{lemma}

\begin{proof}
For $R\in\mathcal P$ let $r^{(N)}(R)$ be the vectors of \cref{lem:cprof-approx}, and define
\begin{equation*}
\Phi_N(R)=\Big(D^{r^{(N)}(R)}\big(N^{2/3}\,\cdot\,\big),\ N^{-2/3}t^{r^{(N)}(R)}(\cdot)\Big)\in C((0,\infty))\times C((0,1)).
\end{equation*}
\Cref{lem:cprof-approx,lem:cprof-finite} show that $\Phi_N$ is Borel on $\mathcal P$; restriction to $(0,\infty)$ and rescaling of time are continuous operations. Fix $R\in\mathcal P$. By \cref{lem:cprof-approx}, the vectors $r^{(N)}(R)$ with $N\ge N_1(R)$ satisfy the hypotheses of \cref{cd-profile} with $\mathcal N=\{N\ge N_1(R)\}$. Then \cref{cd-profile,cd-quantiles} imply that $\Phi_N(R)\to(F_R,F_R^{-1})$, since discarding the finitely many dimensions $N<N_1(R)$ does not affect the limit. A pointwise limit of Borel maps into a metric space is Borel, and the first part follows.

For the second part, let $(\xi,\mathbf b)\in\R\times\ell^2$ with $b_i>0$ for all $i$, where $\mathbf b=(b_i)_{i\ge2}$. For $x>0$, let $\mathscr S_{\xi,\mathbf b}(x)=\xi+x^{-1}-\sum_{i\ge2}xb_i^2/(1+xb_i)$; this is \eqref{cd-airy-root-display} with $b_i=\Delta_i^{-1}$, and for fixed $x$ it is a continuous function of $(\xi,\mathbf b)$, since the series is bounded by $x\|\mathbf b\|_2^2$ and is Lipschitz in $\mathbf b$ on bounded sets. Let
\begin{equation*}
\varrho(\xi,\mathbf b)=\sup\big\{x\in\mathbb Q\cap(0,\infty):\mathscr S_{\xi,\mathbf b}(x)>\sigma\big\},
\end{equation*}
a Borel function with values in $[0,\infty]$, and on the Borel set where $0<\varrho<\infty$ let $u(\xi,\mathbf b)\in\ell^2$ have coordinates $u_1=2/\varrho$ and $u_i=2b_i/(1+\varrho b_i)$ for $i\ge2$. The map $u$ is Borel, because the Borel $\sigma$-field of $\ell^2$ is generated by the coordinates. Let $G$ be the Borel set of $(\xi,\mathbf b)$ for which $u(\xi,\mathbf b)$ is defined and is the inverse sequence of an element $R$ of $\mathcal P$. We set $\Phi^\sigma(\xi,\mathbf b)=(F_R,F_R^{-1})$ on $G$, which is Borel by the first part, and $\Phi^\sigma(\xi,\mathbf b)=(s\mapsto e^{-s},\eps\mapsto\log(1/\eps))$ off $G$. On the event of probability one where $\mathscr S$ is continuous and strictly decreasing from $+\infty$ to $-\infty$, the supremum over rationals equals the root, so that $\varrho(\Xi,(\Delta_i^{-1}))=\varrho_\sigma$ and $u(\Xi,(\Delta_i^{-1}))=((R^\sigma_i)^{-1})_{i\ge1}$ by \eqref{eq:cprof-airy-rates}. Since also $R^\sigma\in\mathcal P$ almost surely, the convention for $F_\sigma$ in \cref{sec:cprof-setup} gives $(F_\sigma,F_\sigma^{-1})=\Phi^\sigma(\Xi,(\Delta_i^{-1}))$ almost surely.
\end{proof}

The proof of \cref{cd-goe-profile} transfers \cref{cd-profile,cd-quantiles} from deterministic sequences to the joint law of the spectral data. The maps that send the spectral data to the finite profiles depend on $N$, and they are continuous only along sequences that satisfy the critical spectral conditions with fixed constants. We use the following localized form of the continuous mapping theorem; compare \cite[Theorem~2.7]{Billingsley1999}.

\begin{lemma}[Convergence for varying measurable maps]\label{lem:cprof-varying-map}
Let $E$ and $G$ be Polish spaces, let $\mu_n$, $n\ge1$, and $\mu$ be Borel probability measures on $E$ such that $\mu_n$ converges weakly to $\mu$, and let $\mathcal D\subset E$ be a Borel set with $\mu(\mathcal D)=1$. Let $f_n,f\colon E\to G$ be Borel maps. Suppose that for every $\eta>0$ there exist Borel sets $A_n^\eta\subset E$, $n\ge1$, with the following two properties.
\begin{enumerate}
\item $\limsup_{n\to\infty}\mu_n(E\setminus A_n^\eta)\le\eta$.
\item For every strictly increasing sequence $(n_k)_{k\ge1}$ of integers, every sequence of points $x_k\in A_{n_k}^\eta$, and every $x\in\mathcal D$ with $x_k\to x$, we have $f_{n_k}(x_k)\to f(x)$.
\end{enumerate}
Then the image of $\mu_n$ under the map $x\mapsto(x,f_n(x))$ converges weakly to the image of $\mu$ under $x\mapsto(x,f(x))$, as probability measures on $E\times G$.
\end{lemma}

Equivalently, if $X_n$ and $X$ are random elements of $E$ with laws $\mu_n$ and $\mu$, then $(X_n,f_n(X_n))$ converges in distribution to $(X,f(X))$; the random elements need not be defined on a common probability space. The sets $A_n^\eta$ may depend on $\eta$, but $\mathcal D$ and $f$ do not, and the points $x_k$ in the second hypothesis need not lie in $\mathcal D$. No continuity of $f_n$ or of $f$ is assumed. If $f_n=f$ for all $n$ and $f$ is continuous at every point of $\mathcal D$, then the hypotheses hold with $A_n^\eta=E$, and the lemma reduces to the continuous mapping theorem.

\begin{proof}
Let $\phi$ be a bounded continuous function on $E\times G$, let $B=\sup|\phi|$, and let $h(x)=\phi(x,f(x))$, a Borel function on $E$ with $|h|\le B$. Fix $\zeta>0$, and let $\theta=\min\{1/2,\zeta/(8+20B)\}$.

\paragraph{Step 1: a continuous approximation.} We claim that there exist a compact set $K\subset\mathcal D$ with $\mu(K)>1-\theta$ and a continuous function $H\colon E\to[-B,B]$ with $H=h$ on $K$. Indeed, $\mu$ is a Radon measure, since $E$ is Polish \cite[Theorem~7.1.7]{Bogachev2007}. By the continuous approximation in Lusin's theorem \cite[Theorem~7.1.13]{Bogachev2007}, which applies to real functions on a metric space, there exists a continuous function $H_0\colon E\to\R$ with $\mu(H_0\ne h)<\theta/2$. Let $H=\max(-B,\min(H_0,B))$. Then $H$ is continuous, $H=h$ wherever $H_0=h$, and the Borel set $\mathcal D\cap\{H=h\}$ has measure greater than $1-\theta/2$. By inner regularity, it contains a compact set $K$ with $\mu(K)>1-\theta$.

\paragraph{Step 2: a uniform estimate near $K$.} Let $A_n=A_n^\theta$. We claim that there exist an integer $k_0\ge1$ and $n_0$ such that
\begin{equation}\label{eq:cprof-varying-near}
\big|\phi(y,f_n(y))-H(y)\big|\le2\theta\qquad\text{whenever }n\ge n_0,\ y\in A_n,\text{ and }d_E(y,K)<1/k_0.
\end{equation}
Otherwise, we can choose recursively $n_k>n_{k-1}$, $y_k\in A_{n_k}$, and $z_k\in K$ such that $d_E(y_k,z_k)<1/k$ and $|\phi(y_k,f_{n_k}(y_k))-H(y_k)|>2\theta$. Since $K$ is compact, a subsequence $z_{k_l}$ converges to some $z\in K\subset\mathcal D$, and then $y_{k_l}\to z$. The integers $n_{k_l}$ are strictly increasing, and by the second hypothesis, $f_{n_{k_l}}(y_{k_l})\to f(z)$. By the continuity of $\phi$ and $H$, the differences $|\phi(y_{k_l},f_{n_{k_l}}(y_{k_l}))-H(y_{k_l})|$ converge to $|h(z)-H(z)|=0$, which contradicts the lower bound $2\theta$. This proves \eqref{eq:cprof-varying-near}. We have not used any continuity of $f$ or of the $f_n$.

\paragraph{Step 3: integration.} Let $U=\{y\in E:d_E(y,K)<1/k_0\}$, an open set containing $K$. Since $\mu(E\setminus U)\le\mu(E\setminus K)<\theta$ and $E\setminus U$ is closed, the portmanteau theorem \cite[Theorem~2.1]{Billingsley1999} implies that $\mu_n(E\setminus U)<2\theta$ for large $n$. The first hypothesis with $\eta=\theta$ implies that also $\mu_n(E\setminus A_n)<2\theta$ for large $n$, and then $\mu_n(E\setminus(A_n\cap U))<4\theta$. For a probability measure $P$, Borel functions $u,v$ with $|u|,|v|\le B$, and a Borel set $S$ on which $|u-v|\le c$, splitting the integral over $S$ and its complement gives
\begin{equation*}
\Big|\int u\,dP-\int v\,dP\Big|\le c+2B\,P(E\setminus S).
\end{equation*}
We apply this with $P=\mu_n$, $u(x)=\phi(x,f_n(x))$, $v=H$, $S=A_n\cap U$, and $c=2\theta$, using \eqref{eq:cprof-varying-near}, and with $P=\mu$, $u=H$, $v=h$, $S=K$, and $c=0$. We obtain, for large $n$,
\begin{equation*}
\Big|\int\phi(x,f_n(x))\,\mu_n(dx)-\int H\,d\mu_n\Big|\le2\theta+8B\theta,\qquad\Big|\int H\,d\mu-\int h\,d\mu\Big|\le2B\theta.
\end{equation*}
Since $H$ is bounded and continuous, $|\int H\,d\mu_n-\int H\,d\mu|<\theta$ for large $n$. Combining the three bounds, we find that
\begin{equation*}
\Big|\int\phi(x,f_n(x))\,\mu_n(dx)-\int\phi(x,f(x))\,\mu(dx)\Big|<(3+10B)\theta<\zeta
\end{equation*}
for large $n$. The order of the choices is $\phi$, $\zeta$, $\theta$, then $K$ and $H$, then the sets $A_n^\theta$, then $k_0$ and $n_0$. As $\zeta$ and $\phi$ are arbitrary, this proves the lemma.
\end{proof}

\begin{proof}[Proof of \cref{cd-goe-profile}]
We apply \cref{lem:cprof-varying-map} to the spectral data and to the maps that send them to the finite profiles and mixing times.

\paragraph{Step 1: spaces and maps.} Let $\mathcal Y=C((0,\infty))\times C((0,1))$, where both spaces carry the topology of uniform convergence on compact subsets. This space is Polish. For instance, $d_+(g,g')=\sum_{k\ge1}2^{-k}\min\{1,\sup_{s\in[1/k,k]}|g(s)-g'(s)|\}$ is a complete metric for this topology on $C((0,\infty))$. The piecewise linear functions with rational breakpoints and rational values on some $[1/k,k]$, extended to be constant outside this interval, form a countable dense set. For $C((0,1))$ we use the intervals $[1/k,1-1/k]$, $k\ge3$, in the same way. Let
\begin{equation*}
E=\R\times\ell^2\times(\ell^2)^m,\qquad G=\mathcal Y^m,
\end{equation*}
which are Polish spaces. Let $M=1+\max_{j\le m}|\sigma_j|$, and let $N>(2M)^3$; then $c_{N,j}>0$ for all $j$, and discarding finitely many $N$ does not affect convergence in distribution. Let $r^{(j)}_N$ be the half-precisions of the model with $c=c_{N,j}$, in increasing order, $R^{(j)}_{i,N}=br^{(j)}_{i,N}$, and $\mathbf v_{N,\sigma_j}=((R^{(j)}_{i,N})^{-1})_{i\le N}$, padded with zeros, as in \cref{cg-canonical-weyl}. We set
\begin{equation*}
X_N=\big(\Xi_N,\mathbf b_N,(\mathbf v_{N,\sigma_j})_{j\le m}\big)\in E.
\end{equation*}
For $v\in\ell^2$ with $v_i>0$ for $i\le N$, let $r(v)\in\R^N$ have the coordinates $r_i(v)=1/(bv_i)$, and let
\begin{equation*}
\Psi_N(v)=\big(s\mapsto D^{r(v)}(bs),\ \eps\mapsto b^{-1}t^{r(v)}(\eps)\big)\in\mathcal Y,
\end{equation*}
in the notation of \cref{lem:cprof-finite}. For the other $v\in\ell^2$, let $\Psi_N(v)=y_*$, where $y_*=(s\mapsto e^{-s},\eps\mapsto\log(1/\eps))$. The map $\Psi_N$ is Borel: the set where $v_i>0$ for $i\le N$ is open, $r(v)$ is continuous on it, the maps $r\mapsto D^r$ and $r\mapsto t^r$ are Borel by \cref{lem:cprof-finite}, and rescaling of time and restriction to $(0,\infty)$ are continuous. We define the Borel map $f_N\colon E\to G$ by $f_N(\xi,\mathbf b,(v^{(j)})_{j\le m})=(\Psi_N(v^{(j)}))_{j\le m}$. By the reduction to eigencoordinates in \cref{sec:model-model} and \eqref{eq:model-precision-form}, we have $D_{N,j}=D^{r^{(j)}_N}$ and $t_{N,j}=t^{r^{(j)}_N}$, and $r(\mathbf v_{N,\sigma_j})=r^{(j)}_N$. Thus
\begin{equation}\label{eq:cprof-goe-finite-curves}
f_N(X_N)=\big(D_{N,j}(N^{2/3}\,\cdot\,),\,N^{-2/3}t_{N,j}(\cdot)\big)_{j\le m}.
\end{equation}
All $m$ curves are computed from the same matrix $W_N$.

\paragraph{Step 2: the limit.} Let $\mathcal U=\{(R_i^{-1})_{i\ge1}:R\in\mathcal P\}$, a Borel subset of $\ell^2$ by \cref{lem:cprof-approx}. For $v=(R_i^{-1})_{i\ge1}\in\mathcal U$, let $\Theta(v)=(F_R,F_R^{-1})$, and for $v\in\ell^2\setminus\mathcal U$, let $\Theta(v)=y_*$. This map is Borel by the first part of \cref{lem:cprof-measurable}. Let
\begin{equation*}
\mathcal D=\R\times\ell^2\times\mathcal U^m,\qquad f\big(\xi,\mathbf b,(v^{(j)})_{j\le m}\big)=\big(\Theta(v^{(j)})\big)_{j\le m}.
\end{equation*}
By the third part of \cref{cg-canonical-weyl}, from which we omit the roots $\varrho_{N,\sigma_j}$, the data $X_N$ converge in distribution to
\begin{equation*}
X=\big(\Xi,\mathbf b,(\mathbf v_{\sigma_j})_{j\le m}\big),\qquad\mathbf b=(\Delta_i^{-1})_{i\ge2},\quad\mathbf v_{\sigma_j}=((R^{\sigma_j}_i)^{-1})_{i\ge1},
\end{equation*}
and all these limits are functions of the same Airy point process. Since $R^{\sigma_j}\in\mathcal P$ almost surely for each $j$, the law $\mu$ of $X$ satisfies $\mu(\mathcal D)=1$. By the convention for $F_\sigma$ in \cref{sec:cprof-setup} and the second part of \cref{lem:cprof-measurable}, we have
\begin{equation}\label{eq:cprof-goe-limit-curves}
f(X)=\big(F_{\sigma_j},F_{\sigma_j}^{-1}\big)_{j\le m}=\big(\Phi^{\sigma_j}(\Xi,\mathbf b)\big)_{j\le m}\qquad\text{almost surely}.
\end{equation}
The inverse vectors $\mathbf v_{\sigma_j}$ in $X$ are functions of $(\Xi,\mathbf b)$, and they add no randomness.

\paragraph{Step 3: sets of good data.} Fix $\eta>0$, and let $\eta_0=\min\{\eta/2,1/2\}$. By the second part of \cref{cg-canonical-weyl}, with confidence $\eta_0$ and with $M$ as above, there exist constants $0<c_\eta\le C_\eta$ and $\delta_\eta>0$, depending only on $\eta$ and $M$, and events $\mathcal B_N^\eta$ with $\liminf_N\PP(\mathcal B_N^\eta)\ge1-\eta_0$, with the following property. On $\mathcal B_N^\eta$, the half-precisions satisfy the critical spectral conditions with constants $(c_\eta,C_\eta,\delta_\eta)$ simultaneously for every $\sigma\in[-M,M]$. Let $A_N^\eta$ be the set of $x=(\xi,\mathbf b,(v^{(j)})_{j\le m})\in E$ such that, for every $j\le m$, we have $v^{(j)}_i>0$ for $i\le N$ and $v^{(j)}_i=0$ for $i>N$, and the vector $r(v^{(j)})$ is nondecreasing and satisfies the critical spectral conditions with constants $(c_\eta,C_\eta,\delta_\eta)$. This is a Borel set: the conditions on the coordinates $i>N$ are countably many closed conditions, and the others are finitely many continuous equalities and inequalities on the open set where $v^{(j)}_i>0$ for $i\le N$. On $\mathcal B_N^\eta$, we have $X_N\in A_N^\eta$. With $\mu_N$ the law of $X_N$, it follows that
\begin{equation*}
\limsup_{N\to\infty}\mu_N(E\setminus A_N^\eta)\le\limsup_{N\to\infty}\PP\big((\mathcal B_N^\eta)^c\big)\le\eta_0\le\eta.
\end{equation*}
The sets $A_N^\eta$ are subsets of the space $E$ of spectral data, and they place no condition on $\xi$ or $\mathbf b$.

\paragraph{Step 4: the deterministic theorem.} Fix $\eta>0$, a strictly increasing sequence $(N_k)_{k\ge1}$, points $x_k\in A_{N_k}^\eta$, and $x\in\mathcal D$ with $x_k\to x$ in $E$. Let $v^{(j)}_k$ and $v^{(j)}$ be the $j$-th inverse vectors of $x_k$ and $x$, and let $R^{(j)}\in\mathcal P$ be the sequence with inverse sequence $v^{(j)}$. For each $j$, the vectors $r(v^{(j)}_k)\in\R^{N_k}$ satisfy the critical spectral conditions with the constants $(c_\eta,C_\eta,\delta_\eta)$ for every $k$, and their scaled inverse vectors, padded with zeros, are the vectors $v^{(j)}_k$. Thus
\begin{equation*}
\sum_{i\ge1}\Big(\big(N_k^{2/3}r_i(v^{(j)}_k)\big)^{-1}-\big(R^{(j)}_i\big)^{-1}\Big)^2=\big\|v^{(j)}_k-v^{(j)}\big\|_{\ell^2}^2\longrightarrow0,
\end{equation*}
with the convention that the finite inverse vector vanishes for $i>N_k$, and the second condition in \eqref{eq:cprof-conv} holds because $R^{(j)}\in\mathcal P$. We may therefore apply \cref{cd-profile,cd-quantiles} with $\mathcal N=\{N_k:k\ge1\}$, and we obtain $\Psi_{N_k}(v^{(j)}_k)\to(F_{R^{(j)}},F_{R^{(j)}}^{-1})$ in $\mathcal Y$ for every $j\le m$. That is, $f_{N_k}(x_k)\to f(x)$ in $G$. This step uses \cref{cd-profile,cd-quantiles} for general critical sequences, and not only for the sequences of \cref{lem:cprof-approx}: the vectors $r(v^{(j)}_k)$ are arbitrary vectors that satisfy the critical spectral conditions with fixed constants.

\paragraph{Step 5: conclusion.} By Steps 2--4, \cref{lem:cprof-varying-map} applies to the laws $\mu_N$ and $\mu$, the set $\mathcal D$, the maps $f_N$ and $f$, and the sets $A_N^\eta$. We obtain that $(X_N,f_N(X_N))$ converges in distribution to $(X,f(X))$ in $E\times G$. We apply the continuous projection $((\xi,\mathbf b,(v^{(j)})_{j\le m}),y)\mapsto(\xi,\mathbf b,y)$ and the continuous mapping theorem \cite[Theorem~2.7]{Billingsley1999}, and we use \eqref{eq:cprof-goe-finite-curves} and \eqref{eq:cprof-goe-limit-curves}. This proves \eqref{eq:cprof-goe-limit}, jointly for all $j\le m$ and jointly with the spectral data.

\paragraph{The ratio.} Fix $j$ and $\eps\in(0,1/2)$. The map $g\mapsto g(\eps)/g(1-\eps)$ on $C((0,1))$, defined to be $1$ when $g(1-\eps)=0$, is continuous at every $g$ with $g(1-\eps)>0$. The finite-$N$ quotient in \eqref{eq:cprof-ratio} is this map applied to $N^{-2/3}t_{N,j}$, with the convention of \cref{cd-goe-profile} on $\{t_{N,j}(1-\eps)=0\}$. Since $N^{-2/3}t_{N,j}(1-\eps)$ converges in distribution to $F_{\sigma_j}^{-1}(1-\eps)$, which is positive almost surely, we have $\PP(t_{N,j}(1-\eps)=0)\to0$, and the convention does not affect the limit. We deduce \eqref{eq:cprof-ratio} from \eqref{eq:cprof-goe-limit} and the continuous mapping theorem \cite[Theorem~2.7]{Billingsley1999}; note that $t_{N,j}(\eps)/t_{N,j}(1-\eps)$ does not change when both times are multiplied by $N^{-2/3}$. The limit is strictly greater than $1$ almost surely by \cref{cd-quantiles}. If there were cutoff in the sense of \cref{def:model-cutoff}, the ratio would converge in probability to $1$, hence in distribution to the point mass at $1$, which is not the law of the limit in \eqref{eq:cprof-ratio}. We conclude that there is no cutoff.
\end{proof}

\begin{proof}[Proof of the second part of \cref{critical-finite-time-positivity}]
Almost surely $R^\sigma\in\mathcal P$. For such an outcome, \cref{lem:cprof-approx} provides a sequence satisfying the critical spectral conditions with constants \eqref{eq:cprof-approx-constants} computed from $R^\sigma$ and converging to $R^\sigma$. The first part of the corollary then shows that $F_\sigma(s)\ge c_1e^{-C_1s}$, with $c_1,C_1$ depending on these constants.
\end{proof}

\begin{proof}[Proof of \cref{thm:intro-critical}]
We use the notation $F_{\sigma}=F_{R^\sigma}$ of \eqref{eq:cprof-airy-rates}. The joint convergence in distribution of the profiles and the mixing times, on a common disorder, was proved in \cref{cd-goe-profile}. Almost surely $R^\sigma\in\mathcal P$. Using \cref{cd-profile} for the sequence of \cref{lem:cprof-approx} that converges to $R^\sigma$, we find that $F_\sigma$ is continuous and strictly decreasing, with $F_\sigma(0+)=1$ and $F_\sigma(\infty)=0$. We obtain the lower bound $F_\sigma(s)\ge c_1e^{-C_1s}$, with random constants, from \cref{critical-finite-time-positivity}. The convergence of the ratio of mixing times and the absence of cutoff were also shown in \cref{cd-goe-profile}. Finally, we obtain the bounds $C_\eta^{-1}N^{2/3}\log(1/\eps)\le t_N(\eps)\le C_\eta N^{2/3}\log(1/\eps)$ for all $\eps\in(0,1/2)$, on events $\mathcal A_N$ with $\liminf_N\PP(\mathcal A_N)\ge1-\eta$ and uniformly for $\sigma$ in a compact set, from the second part of \cref{cl3-density-clock}.
\end{proof}