\chapter{The critical limit process and the limiting Gibbs law}\label{sec:crit-process}

In this section and the next, we study the dynamics in the critical window $c_N=1+\sigma N^{-1/3}$ on the time scale $N^{2/3}$. The goal of this section is to construct the critical limit process and the limiting Gibbs law, in terms of which \cref{sec:crit-profile} expresses the critical profile $F_\sigma$, and to prove that the rescaled dynamics started from $\nu_N$ converge to its normalization. \Cref{cd-mean} identifies the limit $\psi_R$ of the mean of the rescaled squared norm of the unnormalized process; we call $\psi_R$ the mean function. \Cref{cd-functional} constructs the critical limit process, a countable system of stochastic differential equations whose coordinates share a common random noise amplitude, and shows that finitely many rescaled eigencoordinates of the Langevin diffusion started from $\nu_N$ converge to the corresponding normalized coordinates $\xi_i=U_i/\sqrt Z$. \Cref{cd-pin} identifies the limit $\rho_R$ of the rescaled Gibbs measures, a Gaussian sequence conditioned on a renormalized radial constraint. The uniqueness statement behind \cref{cd-functional} is proved in \cref{cd-clipped-picard}. In \cref{sec:crit-profile}, the proof of the results on the critical profile uses the critical limit process of \cref{cd-functional} and the limiting Gibbs law, and the continuity, positivity, and strict decrease of the profile are proved there with estimates for finite-dimensional semigroups.

We use the unnormalized process of \cref{lem:proc-normalization}, the heat matrix \eqref{cd-heat-matrix} and its estimates in \cref{cd-heat-profiles}, the moment bounds of \cref{cd-lift-moments}, and the countable conditioned Gaussian laws of \cref{sec:conditioning}. All results of this section are deterministic statements about a fixed sequence of half-precisions satisfying the hypotheses of \cref{def:cproc-critical} below; probabilities refer only to the initial point and the driving Brownian motions. For the GOE and every fixed $\eta>0$, \cref{cg-canonical-weyl} gives constants $c_0,C_0$ and events $\mathcal A_N$ with $\liminf_N\PP(\mathcal A_N)\ge1-\eta$ on which the trace condition and the Weyl bounds hold. The inverse scaled half-precisions converge in distribution in $\ell^2$, and their limits have the Weyl asymptotic almost surely. \Cref{sec:crit-profile} applies the deterministic results of this section to the GOE through a localized form of the continuous mapping theorem (\cref{lem:cprof-varying-map}). The results of this section that \cref{sec:crit-profile} uses are \cref{cd-mean,cd-functional,cd-pin}, together with the estimates in their proofs; \cref{cd-mean,cd-pin} are also used in \cref{sec:outer-hot,sec:outer-cold}.

Before entering into details, we describe the critical limit process informally. Its coordinates $U_i$, $i\ge1$, solve
\begin{equation*}
\dd U_i=-2R_iU_i\dd s+\sqrt{2Z}\dd B_i,\qquad U_i(0)=0,
\end{equation*}
where the $B_i$ are independent Brownian motions and the noise amplitude $\sqrt{2Z}$ is common to all coordinates. The ratios $\xi_i=U_i/\sqrt Z$ are the limits of the rescaled eigencoordinates $N^{-1/3}X_i(N^{2/3}s)$. In finite dimension, $Z$ corresponds to the rescaled squared norm $Z_N=N^{-1/3}\sum_i U_{i,N}^2$ of the unnormalized process. We call $Z$ the rescaled squared-norm process. Passing to the limit in this identity coordinate by coordinate is not justified. In the limit, $Z$ is determined by the stochastic Volterra equation \eqref{cd-limiting-system} below. The start is singular: every coordinate starts at zero, while $Z(s)$ is of order $s^{-1/2}$ as $s\downarrow0$ and the weighted norm $\sum_iU_i(s)^2/R_i$ tends to $2$. We explain this in \cref{rem:cproc-entrance}.

\section{Scaled half-precisions and the rescaled unnormalized process}\label{sec:cproc-setting}

\paragraph{Half-precisions.}
Throughout this section, $N$ ranges over an infinite set $\mathcal N$ of integers $N\ge2$, and limits as $N\to\infty$ are taken along $\mathcal N$. For each $N\in\mathcal N$, let $0<r_{1,N}\le\cdots\le r_{N,N}$. The main example is given by the half-precisions $r_{i,N}=p_i/2=(z-\alpha_i)/2$ of a symmetric matrix $A$ with eigenvalues $\alpha_1\ge\cdots\ge\alpha_N$ (\cref{sec:model}). Then $\sum_i(2r_{i,N})^{-1}=\sum_iq_i=N$, and on $S_N$ we have $\sum_ir_{i,N}x_i^2=zN/2-x^\T Ax/2$. The first term is constant on the sphere, so that the generator $L_A$ and the Gibbs measure $\pi_A$ are those of the potential $\sum_ir_{i,N}x_i^2$:
\begin{equation}\label{eq:cproc-generator}
L_A=\Delta_{S_N}-\nabla_{S_N}\Big(\sum_ir_{i,N}x_i^2\Big)\cdot\nabla_{S_N},
\qquad
\pi_A(\dd x)\propto e^{-\sum_ir_{i,N}x_i^2}\,\nu_N(\dd x).
\end{equation}
Conversely, every positive vector $(r_{i,N})$ with $\sum_i(2r_{i,N})^{-1}=N$ is of this form, with $A=-2\diag(r_{i,N})$ and Lagrange multiplier $z=0$. We use \eqref{eq:cproc-generator} to define $L_A$ and $\pi_A$ for every such vector. We write $R_{i,N}=N^{2/3}r_{i,N}$ for the scaled half-precisions, and we set $R_{i,N}^{-1}=0$ for $i>N$.

\begin{definition}[Critical sequences]\label{def:cproc-critical}
Let $0<c_0<C_0<\infty$, and let $R=(R_i)_{i\ge1}$ be a sequence of positive numbers. We say that $(r_{i,N})_{i\le N,\,N\in\mathcal N}$ is a critical sequence with constants $(c_0,C_0)$ and limit $R$ if the following three conditions hold.
\begin{enumerate}
\item For every $N\in\mathcal N$, we have the trace condition
\begin{equation}\label{cd-canonical}
\sum_{i=1}^N\frac{1}{2r_{i,N}}=N.
\end{equation}
\item For every $N\in\mathcal N$, we have the Weyl bounds
\begin{equation}\label{cd-weyl}
c_0\Big(\frac iN\Big)^{2/3}\le r_{i,N}\le C_0\Big(\frac iN\Big)^{2/3},\qquad 1\le i\le N;
\end{equation}
equivalently, $c_0i^{2/3}\le R_{i,N}\le C_0i^{2/3}$ for $1\le i\le N$.
\item The inverse scaled half-precisions converge in $\ell^2$, and the limit has a Weyl asymptotic:
\begin{equation}\label{cd-inverse-convergence}
\lim_{N\to\infty}\sum_{i\ge1}\big|R_{i,N}^{-1}-R_i^{-1}\big|^2=0,
\qquad
\lim_{i\to\infty}\frac{R_i}{i^{2/3}}=\gamma\quad\text{for some }\gamma\in(0,\infty).
\end{equation}
\end{enumerate}
\end{definition}

Let $(r_{i,N})$ be a critical sequence with limit $R$. Since the inverses converge coordinatewise, the Weyl bounds pass to the limit, and $R$ is nondecreasing with
\begin{equation}\label{eq:cproc-weyl-limit}
c_0i^{2/3}\le R_i\le C_0i^{2/3},\qquad i\ge1.
\end{equation}
In the scaled variables, the trace condition \eqref{cd-canonical} reads $\sum_iR_{i,N}^{-1}=2N^{1/3}$. We have $\sum_iR_{i,N}^{-2}\le c_0^{-2}\sum_{i\ge1}i^{-4/3}\le4c_0^{-2}$ by the Weyl bounds, and the same bound holds for $R$. In particular, $\sum_iR_i^{-2}<\infty$, while $\sum_iR_i^{-1}=\infty$ by \eqref{eq:cproc-weyl-limit}.

\paragraph{The critical window.}
For the model with $c_N=1+\sigma N^{-1/3}$, we take $A=c_NW_N$ and $r_{i,N}=p_i/2$, with the Lagrange multiplier $z=z_N(c_N)$. \Cref{cg-canonical-weyl} shows that for every $\eta>0$ there exist constants $c_0,C_0$ and events of probability at least $1-\eta$ for all large $N$ on which \eqref{cd-canonical} and \eqref{cd-weyl} hold, and that the inverse scaled half-precisions converge in distribution in $\ell^2$ to $((R_i^\sigma)^{-1})_{i\ge1}$, where
\begin{equation*}
R_1^\sigma=\frac{\varrho_\sigma}2,\qquad R_i^\sigma=\frac{\Delta_i+\varrho_\sigma}{2}\quad(i\ge2).
\end{equation*}
Here $\Delta_i=\chi_i-\chi_1$, where $(\chi_i)$ is the reflected GOE Airy point process, the limit of $N^{2/3}(2-\lambda_i)$, and $\varrho_\sigma$ is the root of $\mathscr S(\varrho_\sigma)=\sigma$ for the function $\mathscr S$ of \eqref{sp-airy-root-function}.
Almost surely, $\chi_i/(3\pi i/2)^{2/3}\to1$ by \cite[Proposition~1.4(a)]{DuHuang2026}, so that the second assertion of \eqref{cd-inverse-convergence} holds with $\gamma=\gamma_{\mathrm{GOE}}=\frac12(3\pi/2)^{2/3}$.
\Cref{sec:crit-profile} transfers the deterministic results of this section to these random sequences with a localized form of the continuous mapping theorem (\cref{lem:cprof-varying-map}), using the critical spectral conditions with fixed constants at each confidence level. We write $\rho_\sigma=\rho_{R^\sigma}$ for the limiting Gibbs law of \cref{cd-pin} in this case.

\paragraph{The rescaled unnormalized process.}
Let $B_1,\dots,B_N$ be independent standard Brownian motions with respect to a filtration $(\mathcal F_t)_{t\ge0}$, and let $Y$ be the unnormalized process of \cref{lem:proc-normalization} with the quadratic potential coefficients $\rho_i=r_{i,N}$:
\begin{equation}\label{eq:cproc-lift}
\dd Y_i=-2r_{i,N}Y_i\dd t+\sqrt{2/N}\,|Y|\dd B_i,\qquad 1\le i\le N,
\end{equation}
with an $\mathcal F_0$-measurable initial vector $Y(0)\neq0$. Parts (a) and (b) of \cref{lem:proc-normalization}, which allow any $\mathcal F_0$-measurable initial vector that is nonzero almost surely, and \eqref{eq:cproc-generator} imply that \eqref{eq:cproc-lift} has a unique strong solution, that $Y$ never vanishes, and that $X=\sqrt NY/|Y|$ is the Langevin diffusion with generator $L_A$ started from the law of $\sqrt NY(0)/|Y(0)|$. If $Y(0)=g$ is a standard Gaussian vector, then $X$ starts from $\nu_N$. We rescale time by $N^{2/3}$ and define, for $s\ge0$,
\begin{equation}\label{eq:cproc-scaling}
U_{i,N}(s)=N^{-1/6}Y_i(N^{2/3}s),\quad
Z_N(s)=N^{-2/3}|Y(N^{2/3}s)|^2,\quad
B_i^N(s)=N^{-1/3}B_i(N^{2/3}s).
\end{equation}
The $B_i^N$ are independent standard Brownian motions with respect to the filtration $\mathcal F^N_s=\mathcal F_{N^{2/3}s}$. For $i>N$, we set $U_{i,N}=0$ and let $B_i^N$ be further independent Brownian motions, independent of everything else. The rescaled eigencoordinates of the Langevin diffusion are
\begin{equation}\label{eq:cproc-normalized}
N^{-1/3}X_i(N^{2/3}s)=\frac{U_{i,N}(s)}{\sqrt{Z_N(s)}}.
\end{equation}

Let $H=-4\diag(r_{i,N})+\frac2N\mathbf1\mathbf1^\T$ be the heat matrix \eqref{cd-heat-matrix}; this is the matrix $-2\diag(p)+\frac2N\mathbf1\mathbf1^\T$ of \cref{sec:heat} with $p_i=2r_{i,N}$. Let $h(t)=e^{tH}\mathbf1$. We define the scaled kernels and the scaled mean by
\begin{equation}\label{eq:cproc-kernels}
k_{i,N}(s)=N^{-1/3}h_i(N^{2/3}s),
\qquad
\psi_N(s)=N^{-2/3}\sum_{j=1}^Nh_j(N^{2/3}s)=N^{-1/3}\sum_{j=1}^Nk_{j,N}(s).
\end{equation}
In terms of the mean $\bar h(t)=N^{-1}\sum_jh_j(t)$ of \cref{sec:heat}, we have $\psi_N(s)=N^{1/3}\bar h(N^{2/3}s)$. For $i>N$, we set $k_{i,N}=0$; all sums and suprema involving $R_{i,N}k_{i,N}$ are over $1\le i\le N$.
We recall from \cref{sec:heat} that $H$ is symmetric, has nonnegative off-diagonal entries, and is negative semidefinite with kernel spanned by $(r_{i,N}^{-1})_{i\le N}$; that $e^{tH}$ has nonnegative entries; and that $t\mapsto\mathbf1^\T e^{tH}\mathbf1$ is completely monotone, being the Laplace transform of the spectral measure of $-H$ at $\mathbf 1$.

\Cref{tab:cproc-objects} lists the objects of this section, in finite dimension and in the limit.

\begin{table}[ht]
\centering
\small
\begin{tabular}{@{}lll@{}}
\toprule
Object & Finite dimension & Limit \\
\midrule
scaled half-precisions & $R_{i,N}=N^{2/3}r_{i,N}$ & $R_i$, \eqref{cd-inverse-convergence} \\
coordinates & $U_{i,N}(s)=N^{-1/6}Y_i(N^{2/3}s)$ & $U_i(s)$, \cref{cd-functional} \\
rescaled squared norm & $Z_N(s)=N^{-2/3}|Y(N^{2/3}s)|^2$ & $Z(s)=\psi_R(s)+V(s)$ \\
mean function & $\psi_N(s)=\E Z_N(s)$ for $Y(0)=g$ & $\psi_R(s)$, \cref{cd-mean} \\
kernels & $k_{i,N}(s)=N^{-1/3}h_i(N^{2/3}s)$ & $k_i(s)$, \eqref{cd-H-limit} \\
normalized coordinates & $N^{-1/3}X_i(N^{2/3}s)$ & $\xi_i(s)=U_i(s)/\sqrt{Z(s)}$ \\
Gibbs law & law of $(N^{-1/3}x_i)_{i\le N}$ under $\pi_A$ & $\rho_R$, \cref{cd-pin} \\
\bottomrule
\end{tabular}
\caption{The rescaled objects of \cref{sec:crit-process}. Times $s$ are in units of $N^{2/3}$.}
\label{tab:cproc-objects}
\end{table}

The following lemma collects the exact identities satisfied by the rescaled process. It requires only the trace condition.

\begin{lemma}[Exact identities]\label{lem:cproc-exact}
Assume \eqref{cd-canonical}, and let $Y(0)$ be an $\mathcal F_0$-measurable random vector with $Y(0)\ne0$ and $\E|Y(0)|^{2q}<\infty$ for every $q\ge1$. Then the following hold for all $s\ge0$.
\begin{enumerate}
\item We have $\sum_iU_{i,N}(s)^2=N^{1/3}Z_N(s)$, and
\begin{equation}\label{eq:cproc-scaled-sde}
\dd U_{i,N}=-2R_{i,N}U_{i,N}\dd s+\sqrt{2Z_N}\dd B_i^N.
\end{equation}
\item For every $s_0\ge0$,
\begin{multline}\label{eq:cproc-restart-finite}
Z_N(s_0+s)=\sum_{i=1}^Nk_{i,N}(s)U_{i,N}(s_0)^2\\
+\sqrt8\sum_{i=1}^N\int_0^sk_{i,N}(s-v)\sqrt{Z_N(s_0+v)}\,U_{i,N}(s_0+v)\dd B_i^N(s_0+v).
\end{multline}
\item The weighted norm $\|U_N(s)\|_{R_N}^2=\sum_iU_{i,N}(s)^2/R_{i,N}$ is a martingale, and
\begin{equation}\label{eq:cproc-weighted-martingale}
\|U_N(s)\|_{R_N}^2=\|U_N(0)\|_{R_N}^2+\sqrt8\sum_{i=1}^N\int_0^sR_{i,N}^{-1}\sqrt{Z_N(v)}\,U_{i,N}(v)\dd B_i^N(v).
\end{equation}
\item The kernels satisfy
\begin{equation}\label{eq:cproc-finite-duhamel}
k_{i,N}(s)=N^{-1/3}e^{-4R_{i,N}s}+2\int_0^se^{-4R_{i,N}(s-v)}\psi_N(v)\dd v,
\qquad
\sum_{i=1}^N\frac{k_{i,N}(s)}{R_{i,N}}=2.
\end{equation}
\item If $Y(0)=g$ is a standard Gaussian vector, then $\psi_N(s)=\E Z_N(s)$, and
\begin{equation}\label{cd-exact-radial-closure}
Z_N(s)=\psi_N(s)+I_N(s)+\Theta_N(s),
\end{equation}
where
\begin{equation}\label{eq:cproc-IN-ThetaN}
\begin{aligned}
I_N(s)&=N^{-1/3}\sum_{i=1}^Nk_{i,N}(s)(g_i^2-1),\\
\Theta_N(s)&=\sqrt8\sum_{i=1}^N\int_0^sk_{i,N}(s-v)\sqrt{Z_N(v)}\,U_{i,N}(v)\dd B_i^N(v).
\end{aligned}
\end{equation}
\end{enumerate}
\end{lemma}

\begin{proof}
We work in the unscaled time $t$ and write $Q_i=Y_i^2$ and $\mathcal A=|Y|^2=\sum_iQ_i$; we use the letter $\mathcal A$ to distinguish the squared norm from the matrix $A$. Since the coefficients of \eqref{eq:cproc-lift} are globally Lipschitz, we have $\E\sup_{t\le T}|Y(t)|^{2q}<\infty$ for all $q$ and $T$. Every stochastic integral below with a fixed terminal time is square integrable and has mean zero, and the stochastic integrals in \eqref{eq:cproc-weighted-martingale} are true martingales. By It\^o's formula,
\begin{equation}\label{eq:cproc-square-sde}
\dd Q_i=\Big(-4r_{i,N}Q_i+\frac2N\mathcal A\Big)\dd t+2\sqrt{2/N}\,|Y|\,Y_i\dd B_i,
\end{equation}
that is, $\dd Q=HQ\dd t+\dd M$ with $\dd M_i=2\sqrt{2/N}\,|Y|\,Y_i\dd B_i$. It\^o's product rule applied to $e^{-tH}Q(t)$ gives, for $t_0\ge0$ and $t\ge0$,
\begin{equation*}
Q(t_0+t)=e^{tH}Q(t_0)+\int_{t_0}^{t_0+t}e^{(t_0+t-u)H}\dd M(u).
\end{equation*}
We multiply by $\mathbf1^\T$ and use the symmetry of $H$, which gives $\mathbf1^\T e^{tH}=h(t)^\T$. This yields
\begin{equation*}
\mathcal A(t_0+t)=\sum_ih_i(t)Q_i(t_0)+2\sqrt{2/N}\sum_i\int_{t_0}^{t_0+t}h_i(t_0+t-u)\,|Y(u)|\,Y_i(u)\dd B_i(u).
\end{equation*}
We now substitute $t_0=N^{2/3}s_0$, $t=N^{2/3}s$, $u=N^{2/3}(s_0+v)$, and use $|Y|=N^{1/3}\sqrt{Z_N}$, $Y_i=N^{1/6}U_{i,N}$, $h_i=N^{1/3}k_{i,N}$, and $\dd B_i(N^{2/3}\cdot)=N^{1/3}\dd B_i^N$. The powers of $N$ in the stochastic integral are $N^{-2/3}\cdot N^{-1/2}\cdot N^{1/3}\cdot N^{1/3}\cdot N^{1/6}\cdot N^{1/3}=1$, and those in the first term are $N^{-2/3}\cdot N^{1/3}\cdot N^{1/3}=1$. This proves \eqref{eq:cproc-restart-finite}. The identity $\sum_iU_{i,N}^2=N^{1/3}Z_N$ and \eqref{eq:cproc-scaled-sde} follow from the definitions in the same way.

Since $(r_{i,N}^{-1})$ spans the kernel of the symmetric matrix $H$, we have $\sum_ir_{i,N}^{-1}(HQ)_i=0$. Then $\sum_iQ_i/r_{i,N}$ is a martingale with differential $\sum_ir_{i,N}^{-1}\dd M_i$, and rescaling gives \eqref{eq:cproc-weighted-martingale}. For the same reason, $\frac{\dd}{\dd t}\sum_ih_i(t)/r_{i,N}=0$. Hence $\sum_ih_i(t)/r_{i,N}=\sum_ir_{i,N}^{-1}=2N$, where we used \eqref{cd-canonical}. After rescaling, we obtain the second identity in \eqref{eq:cproc-finite-duhamel}. The $i$th coordinate of $h'=Hh$ reads $h_i'=-4r_{i,N}h_i+\frac2N\sum_jh_j$, with $h_i(0)=1$; solving this linear equation and rescaling gives the first identity in \eqref{eq:cproc-finite-duhamel}.

Finally, let $Y(0)=g$. Taking expectations in the identity for $\mathcal A$ with $t_0=0$, and using $\E g_i^2=1$, we find $\E\mathcal A(t)=\sum_ih_i(t)$, that is, $\E Z_N(s)=\psi_N(s)$. With $s_0=0$, the first term of \eqref{eq:cproc-restart-finite} is $N^{-1/3}\sum_ik_{i,N}(s)g_i^2=\psi_N(s)+I_N(s)$, which proves \eqref{cd-exact-radial-closure}.
\end{proof}

We use \eqref{eq:cproc-restart-finite} with $s_0=0$ for the process started from $\nu_N$.

We use the following elementary sums repeatedly. Here and below, $C$ denotes a large constant and $c_*$ a small positive constant; they depend only on the parameters indicated, and their values may change from line to line.

\begin{lemma}[Sums over the scaled half-precisions]\label{lem:cproc-sums}
Let $(R_i)_{i\in I}$ be positive numbers, indexed by $I=\{1,\dots,N\}$ or $I=\{1,2,\dots\}$, with $R_i\ge c_0i^{2/3}$ for all $i\in I$. There exists $C=C(c_0)$ such that for all $t,h>0$,
\begin{gather}
\sum_{i\in I}\frac{e^{-4R_it}}{R_i}\le Ct^{-1/2},\qquad
\sum_{i\in I}e^{-4R_it}\le Ct^{-3/2},\label{eq:cproc-sum-exp}\\
\sum_{i\in I}\frac{1}{(1+R_it)^2}\le Ct^{-3/2},\qquad
\sum_{i\in I}\frac{1}{R_i(1+R_it)^2}\le Ct^{-1/2},\label{eq:cproc-sum-rational}\\
\sum_{i\in I}\frac{\min\{1,R_ih\}^2}{R_i^2}\le C\sqrt h,\qquad
\sum_{i\in I}\frac{\min\{h,R_i^{-1}\}}{R_i}\le C\sqrt h.\label{eq:cproc-sum-increment}
\end{gather}
Moreover, for every $p>2$ there exists $C=C(c_0,p)$ such that for all $J\ge1$,
\begin{equation}\label{eq:cproc-sum-tail}
\sum_{i>J}\frac{\log(2+R_i)}{R_i^3}\le C\frac{\log(2+J)}{J},
\qquad
\sum_{i>J}R_i^{-2+1/p}\le CJ^{-1/3+2/(3p)}.
\end{equation}
\end{lemma}

\begin{proof}
Each summand has the form $f(R_i)$ for a nonnegative nonincreasing function $f$ on $(0,\infty)$. Using $R_i\ge c_0i^{2/3}$, we have
\begin{equation*}
\sum_{i\in I}f(R_i)\le\sum_{i\ge1}f(c_0i^{2/3})\le\int_0^\infty f(c_0x^{2/3})\dd x=\frac32c_0^{-3/2}\int_0^\infty f(y)\,y^{1/2}\dd y,
\end{equation*}
where we substituted $y=c_0x^{2/3}$. For the six functions in \eqref{eq:cproc-sum-exp}--\eqref{eq:cproc-sum-increment}, the last integral equals, respectively, $\sqrt\pi\,(4t)^{-1/2}$, $(\sqrt\pi/2)(4t)^{-3/2}$, $\frac\pi2t^{-3/2}$, $\frac\pi2t^{-1/2}$, $\frac83\sqrt h$, and $4\sqrt h$. For the last two, we split the integral at $y=1/h$. The bounds \eqref{eq:cproc-sum-tail} follow in the same way from $\sum_{i>J}f(c_0i^{2/3})\le\int_J^\infty f(c_0x^{2/3})\dd x$, using $\log(2+c_0x^{2/3})\le C\log(2+x)$ and that $\frac43-\frac2{3p}>1$ for $p>2$.
\end{proof}

\section{Moment and kernel bounds at the critical scale}\label{sec:cproc-bounds}

The next lemma restates the heat estimates of \cref{cd-heat-profiles} and the moment bounds of \cref{cd-lift-moments} in the rescaled variables, and adds bounds on the derivatives of the kernels. Throughout, $N^{-2/3}$ plays the role of a short-time cutoff: in the rescaled time, $s+N^{-2/3}$ corresponds to $(1+t)N^{-2/3}$ in the original time $t=N^{2/3}s$.

\begin{lemma}[Moment and kernel bounds]\label{lem:cproc-bounds}
Let $(r_{i,N})$ satisfy \eqref{cd-canonical} and \eqref{cd-weyl}, and fix $K>0$. There exist $C,c_*>0$, depending only on $c_0,C_0,K$, such that for all $N\in\mathcal N$, $0\le s\le K$, and $1\le i\le N$,
\begin{gather}
\frac{c_*}{\sqrt{s+N^{-2/3}}}\le\psi_N(s)\le\frac{C}{\sqrt{s+N^{-2/3}}},
\qquad
|\psi_N'(s)|\le\frac{C}{(s+N^{-2/3})^{3/2}},\label{eq:cproc-psiN-bounds}\\
\frac{c_*\sqrt{s+N^{-2/3}}}{1+R_{i,N}s}\le k_{i,N}(s)\le\frac{C\sqrt{s+N^{-2/3}}}{1+R_{i,N}s},
\qquad
|k_{i,N}'(s)|\le\frac{C}{\sqrt{s+N^{-2/3}}\,(1+R_{i,N}s)},\label{cd-kernel-bounds}\\
\sum_{j=1}^Nk_{j,N}(s)^2\le\frac{C}{(s+N^{-2/3})^{1/2}},
\qquad
\sum_{j=1}^Nk_{j,N}'(s)^2\le\frac{C}{(s+N^{-2/3})^{5/2}}.\label{eq:cproc-kernel-sums}
\end{gather}
Moreover, for every $p>0$ there exist $C_p$ and $N_0$, depending only on $c_0,C_0,K,p$, such that the following holds. If $Y(0)=g$ is a standard Gaussian vector, then for all $N\ge N_0$, $0\le s\le K$, and $1\le i\le N$,
\begin{equation}\label{cd-scaled-moments}
\begin{gathered}
\|Z_N(s)\|_p\le\frac{C_p}{\sqrt{s+N^{-2/3}}},\qquad
\E Z_N(s)^{-p}\le C_p(s+N^{-2/3})^{p/2},\\
\|U_{i,N}(s)^2\|_p\le\frac{C_p\sqrt{s+N^{-2/3}}}{1+R_{i,N}s},\qquad
\|Z_N(s)U_{i,N}(s)^2\|_p\le\frac{C_p}{1+R_{i,N}s}.
\end{gathered}
\end{equation}
Here $\|F\|_p=(\E|F|^p)^{1/p}$.
\end{lemma}

\begin{proof}
We write $t=N^{2/3}s$, so that $1+t=N^{2/3}(s+N^{-2/3})$ and $r_{i,N}t=R_{i,N}s$.

\paragraph{Step 1: two-sided bounds.}
By \cref{cd-heat-profiles}, uniformly in $0\le t\le KN^{2/3}$, we have $\sum_jh_j(t)\asymp N(1+t)^{-1/2}$ and $h_i(t)\asymp(1+t)^{1/2}(1+r_{i,N}t)^{-1}$, with constants depending only on $c_0,C_0,K$. Multiplying by $N^{-2/3}$ and $N^{-1/3}$, respectively, gives the two-sided bounds in \eqref{eq:cproc-psiN-bounds} and \eqref{cd-kernel-bounds}.

\paragraph{Step 2: the derivative of $\psi_N$.}
The function $\psi_N$ is completely monotone, so $\psi_N(s)=\int e^{-\lambda s}\,\mathfrak m(\dd\lambda)$ for a positive measure $\mathfrak m$ on $[0,\infty)$. Using $\lambda e^{-\lambda s/2}\le 2/(es)$, we find $|\psi_N'(s)|\le 2s^{-1}\psi_N(s/2)$. If $s\ge N^{-2/3}$, then $s+N^{-2/3}\le2s$, and by Step 1 we have $|\psi_N'(s)|\le Cs^{-3/2}\le C(s+N^{-2/3})^{-3/2}$. If $s\le N^{-2/3}$, then $\psi_N'(s)=\mathbf1^\T He^{tH}\mathbf1$ with $t\le1$. Since $e^{tH}$ is a contraction and $\|H\|\le4\max_ir_{i,N}+2\le4C_0+2$, we find $|\psi_N'(s)|\le(4C_0+2)N=(4C_0+2)(N^{-2/3})^{-3/2}$, which is at most $C(s+N^{-2/3})^{-3/2}$.

\paragraph{Step 3: the derivative of $k_{i,N}$.}
Fix $i$ and abbreviate $R=R_{i,N}$ and $k=k_{i,N}$. Differentiating \eqref{eq:cproc-finite-duhamel}, we have $k'=-4Rk+2\psi_N$. Suppose first that $Rs\le1$. Using also $RN^{-2/3}=r_{i,N}\le C_0$, we find $R(s+N^{-2/3})\le1+C_0$, and Step 1 gives
\begin{equation*}
|k'(s)|\le4Rk(s)+2\psi_N(s)\le\frac{4CR(s+N^{-2/3})+2C}{\sqrt{s+N^{-2/3}}}\le\frac{C}{\sqrt{s+N^{-2/3}}},
\end{equation*}
which is the claim, since $1+Rs\le2$. Suppose next that $Rs>1$. Then $s\ge R^{-1}\ge N^{-2/3}/C_0$, which gives $s+N^{-2/3}\le(1+C_0)s$. Since $8R\int_0^se^{-4R(s-v)}\dd v=2(1-e^{-4Rs})$, we may write $2\psi_N(s)=8R\int_0^se^{-4R(s-v)}\psi_N(s)\dd v+2\psi_N(s)e^{-4Rs}$, and \eqref{eq:cproc-finite-duhamel} gives
\begin{equation}\label{eq:cproc-kprime-split}
k'(s)=\big(2\psi_N(s)-4RN^{-1/3}\big)e^{-4Rs}-8R\int_0^se^{-4R(s-v)}\big(\psi_N(v)-\psi_N(s)\big)\dd v.
\end{equation}
This subtraction is the point of the argument: estimating the two terms of $k'=-4Rk+2\psi_N$ separately would lose a factor $Rs$. On $[s/2,s]$, we have $|\psi_N(v)-\psi_N(s)|\le C(s-v)s^{-3/2}$ by Step 2, and $8R\int_0^\infty\ell e^{-4R\ell}\dd\ell=(2R)^{-1}$; this part of the integral is at most $C(Rs^{3/2})^{-1}\le Cs^{-1/2}(1+Rs)^{-1}$. On $[0,s/2]$, we have $0\le\psi_N(v)-\psi_N(s)\le\psi_N(v)$, and $\int_0^{s/2}\psi_N\le C\sqrt s$ by Step 1; this part is at most $CR\sqrt se^{-2Rs}\le Cs^{-1/2}(1+Rs)^{-1}$, because $(Rs)(1+Rs)e^{-2Rs}$ is bounded. For the first term in \eqref{eq:cproc-kprime-split}, we use $\psi_N(s)\le Cs^{-1/2}$ and $N^{-1/3}\le\sqrt{C_0/R}$, which give
\begin{equation*}
\big|2\psi_N(s)-4RN^{-1/3}\big|e^{-4Rs}\le Cs^{-1/2}\big(1+(Rs)^{1/2}\big)e^{-4Rs}\le\frac{C}{\sqrt s\,(1+Rs)}.
\end{equation*}
Combining these bounds with $s\asymp s+N^{-2/3}$ proves the derivative bound in \eqref{cd-kernel-bounds}.

\paragraph{Step 4: the sums.}
Using \eqref{cd-kernel-bounds}, we have $\sum_jk_{j,N}(s)^2\le C(s+N^{-2/3})\sum_j(1+R_{j,N}s)^{-2}$. If $s\ge N^{-2/3}$, then \eqref{eq:cproc-sum-rational} bounds the last sum by $Cs^{-3/2}$, and $s+N^{-2/3}\le2s$. If $s<N^{-2/3}$, then the sum has at most $N=(N^{-2/3})^{-3/2}$ terms. In both cases, the sum is at most $C(s+N^{-2/3})^{-3/2}$, which gives the first bound in \eqref{eq:cproc-kernel-sums}. The second bound follows in the same way from the derivative bound in \eqref{cd-kernel-bounds}.

\paragraph{Step 5: moments.}
Let $Y(0)=g$ and $\mathcal A=|Y|^2$. By \cref{cd-lift-moments}, for every $p>0$ there exist $C_p$ and $N_0$ such that for $N\ge N_0$ and $0\le t\le KN^{2/3}$,
\begin{equation*}
\E\Big[\frac{\mathcal A(t)}{\sum_jh_j(t)}\Big]^p+\E\Big[\frac{\sum_jh_j(t)}{\mathcal A(t)}\Big]^p\le C_p,
\end{equation*}
and, for $p\ge1$, $\|Y_i(t)^2\|_p\le C_ph_i(t)$. Since $Z_N(s)/\psi_N(s)=\mathcal A(t)/\sum_jh_j(t)$, Step 1 implies the first two bounds in \eqref{cd-scaled-moments}. Using $U_{i,N}(s)^2=N^{-1/3}Y_i(t)^2$, we have $\|U_{i,N}(s)^2\|_p\le C_pk_{i,N}(s)$ for $p\ge1$, and the third bound follows from Step 1; for $p<1$ it follows from the case $p=1$. The fourth bound follows from the first and third by the Cauchy--Schwarz inequality, $\|Z_NU_{i,N}^2\|_p\le\|Z_N\|_{2p}\|U_{i,N}^2\|_{2p}$.
\end{proof}

\section{The mean function}\label{sec:cproc-mean}

For a positive sequence $R=(R_i)_{i\ge1}$ with $\sum_iR_i^{-2}<\infty$, we define
\begin{equation}\label{eq:cproc-Lambda}
\Lambda_R(p)=\frac p8\sum_{i\ge1}\frac{1}{R_i(R_i+p/4)}=\frac12\sum_{i\ge1}\Big(\frac1{R_i}-\frac{1}{R_i+p/4}\Big),\qquad p\ge0.
\end{equation}
The function $\Lambda_R$ is finite, continuous, and increasing, with $\Lambda_R(0)=0$. If $\sum_iR_i^{-1}=\infty$, then $\Lambda_R(p)\to\infty$ as $p\to\infty$, by monotone convergence in the second expression. For finite vectors $(R_{i,N})_{i\le N}$, we write $\Lambda_N$ for the same expression with the sum over $i\le N$. We write $\mathcal Lf(p)=\int_0^\infty e^{-ps}f(s)\dd s$ for the Laplace transform.

\begin{lemma}[The mean function]\label{cd-mean}
Let $(r_{i,N})$ be a critical sequence with limit $R$. There exists a unique continuous function $\psi_R\colon(0,\infty)\to(0,\infty)$ such that
\begin{equation}\label{cd-mean-transform}
\int_0^\infty e^{-ps}\psi_R(s)\dd s=\frac{1}{2\Lambda_R(p)},\qquad p>0.
\end{equation}
The function $\psi_R$ is completely monotone. For every $j\ge0$, the derivatives $\psi_N^{(j)}$ converge to $\psi_R^{(j)}$ uniformly on compact subsets of $(0,\infty)$. For every $i\ge1$, $k_{i,N}$ converges uniformly on compact subsets of $[0,\infty)$ to
\begin{equation}\label{cd-H-limit}
k_i(s)=2\int_0^se^{-4R_i(s-v)}\psi_R(v)\dd v.
\end{equation}
Finally, with $\gamma$ as in \eqref{cd-inverse-convergence},
\begin{equation}\label{cd-mean-endpoints}
\lim_{s\downarrow0}\sqrt s\,\psi_R(s)=\psi_*,\qquad\psi_*=\frac{4\gamma^{3/2}}{3\pi^{3/2}},
\qquad\text{and}\qquad
\lim_{s\to\infty}\psi_R(s)=\frac{4}{\sum_iR_i^{-2}}.
\end{equation}
\end{lemma}

The function $\psi_R$ depends only on $R$, and not on the critical sequence converging to $R$. For the GOE value $\gamma_{\mathrm{GOE}}=\frac12(3\pi/2)^{2/3}$, we have $\gamma_{\mathrm{GOE}}^{3/2}=3\pi/2^{5/2}$ and hence $\psi_*=1/\sqrt{2\pi}$.

\begin{proof}
We first derive the Laplace transform of $\psi_N$, then prove compactness of $(\psi_N)$, identify the limit through its transform, and finally compute the two asymptotics in \eqref{cd-mean-endpoints}.

\paragraph{Step 1: the transform of $\psi_N$.}
Let $\bar h(t)=N^{-1}\sum_jh_j(t)$, so that $\psi_N(s)=N^{1/3}\bar h(N^{2/3}s)$, and let
\begin{equation*}
\mathsf G(\lambda)=\frac2N\sum_{i=1}^N\frac{1}{\lambda+4r_{i,N}},\qquad\lambda>0.
\end{equation*}
We recall from \eqref{cd-renewal-resolvent} that the Laplace transform of $\bar h$ is $\mathsf G(\lambda)/(2(1-\mathsf G(\lambda)))$, and
\begin{equation*}
1-\mathsf G(\lambda)=\frac{\lambda}{2N}\sum_{i=1}^N\frac{1}{r_{i,N}(\lambda+4r_{i,N})};
\end{equation*}
the last identity follows from \eqref{cd-canonical} by writing $1=\frac1N\sum_i(2r_{i,N})^{-1}$. Substituting $t=N^{2/3}s$, we find that $\mathcal L\psi_N(p)$ equals $N^{-1/3}$ times the Laplace transform of $\bar h$ at $\lambda=pN^{-2/3}$. At this $\lambda$, we have $1-\mathsf G(\lambda)=N^{-1/3}\Lambda_N(p)$, and therefore
\begin{equation}\label{eq:cproc-LpsiN}
\mathcal L\psi_N(p)=\frac{\mathsf G(pN^{-2/3})}{2\Lambda_N(p)}=\frac{1}{2\Lambda_N(p)}-\frac{1}{2N^{1/3}},\qquad p>0.
\end{equation}

\paragraph{Step 2: convergence of $\Lambda_N$.}
Let $u_{i,N}=R_{i,N}^{-1}$ and $u_i=R_i^{-1}$. Then $\Lambda_N(p)=\frac p8\sum_ig_p(u_{i,N})$ with $g_p(u)=u^2/(1+pu/4)$. A direct computation gives $g_p'(u)=(2u+pu^2/4)(1+pu/4)^{-2}\in[0,2u]$ for $u,p\ge0$. It follows that $|g_p(u)-g_p(w)|\le2\max\{u,w\}|u-w|$, and by the Cauchy--Schwarz inequality we obtain
\begin{equation*}
\Big|\sum_ig_p(u_{i,N})-\sum_ig_p(u_i)\Big|\le2\big(\|u_N\|_{\ell^2}+\|u\|_{\ell^2}\big)\|u_N-u\|_{\ell^2}.
\end{equation*}
Then \eqref{cd-inverse-convergence} implies that $\Lambda_N\to\Lambda_R$ uniformly on bounded subsets of $[0,\infty)$. Then \eqref{eq:cproc-LpsiN} implies that $\mathcal L\psi_N(p)\to(2\Lambda_R(p))^{-1}$ for every $p>0$, uniformly on compact subsets of $(0,\infty)$.

\paragraph{Step 3: compactness.}
Let $f$ be a completely monotone function, $f(s)=\int e^{-\lambda s}\mathfrak m(\dd\lambda)$. Since $f$ is nonincreasing, $f(s)se^{-ps}\le\int_0^se^{-pv}f(v)\dd v\le\mathcal Lf(p)$, so that
\begin{equation}\label{eq:cproc-cm-bounds}
f(s)\le\frac{e^{ps}}{s}\,\mathcal Lf(p),\qquad
|f^{(j)}(s)|=\int\lambda^je^{-\lambda s}\,\mathfrak m(\dd\lambda)\le j!\Big(\frac2s\Big)^jf(s/2),
\end{equation}
where in the second bound we used $\lambda^je^{-\lambda s/2}\le(2j/(es))^j\le j!(2/s)^j$. Using Step 2 and the first bound in \eqref{eq:cproc-cm-bounds} with $p=2/s_0$, we find that $\sup_N\psi_N(s_0/2)<\infty$ for every $s_0>0$; the second bound then shows that every derivative of $\psi_N$ is bounded on $[s_0,\infty)$, uniformly in large $N$. By the Arzel\`a--Ascoli theorem and a diagonal argument, every subsequence of $(\psi_N)$ has a further subsequence along which all derivatives converge uniformly on compact subsets of $(0,\infty)$. Each limit $\psi$ is nonnegative and completely monotone, since $(-1)^j\psi_N^{(j)}\ge0$ for all $j$.

\paragraph{Step 4: identification of the limit.}
Let $\psi$ be such a subsequential limit, and fix $p>0$. For $0<\delta<K$, we split $\mathcal L\psi_N(p)$ into integrals over $[0,\delta]$, $[\delta,K]$, and $[K,\infty)$. Using \eqref{eq:cproc-psiN-bounds}, we have $\int_0^\delta\psi_N\le2C\sqrt{\delta+N^{-2/3}}$. The tail satisfies $\int_K^\infty e^{-ps}\psi_N(s)\dd s\le e^{-pK/2}\mathcal L\psi_N(p/2)$, since $\psi_N$ is nonnegative, and this is at most $Ce^{-pK/2}$ for large $N$ by Step 2. On $[\delta,K]$, $\psi_N\to\psi$ uniformly along the subsequence. Letting $N\to\infty$, then $\delta\downarrow0$ and $K\to\infty$, and using Fatou's lemma for the integrability of $\psi$ near zero, we find that $\mathcal L\psi(p)=(2\Lambda_R(p))^{-1}$ for every $p>0$. A continuous function on $(0,\infty)$ whose Laplace transform is finite on $(0,\infty)$ is determined by it \cite[Section~XIII.1, corollary on p.~433]{Feller1971}.
All subsequential limits therefore coincide, the whole sequence $(\psi_N)$ converges together with all derivatives, and the limit $\psi_R$ is the unique continuous function satisfying \eqref{cd-mean-transform}. It is completely monotone by Step 3, and positive by the second limit in \eqref{cd-mean-endpoints}, which we prove below.

For the kernels, we use \eqref{eq:cproc-finite-duhamel}. For fixed $i$, we have $R_{i,N}\to R_i$, the term $N^{-1/3}e^{-4R_{i,N}s}$ tends to zero uniformly, and the integral $2\int_0^se^{-4R_{i,N}(s-v)}\psi_N(v)\dd v$ converges to $k_i(s)$ uniformly for $s$ in compact subsets of $[0,\infty)$. Indeed, the contribution of $v\in[0,\delta]$ is at most $2\int_0^\delta\psi_N\le C\sqrt{\delta+N^{-2/3}}$ for both $\psi_N$ and $\psi_R$, and on $[\delta,K]$ the integrands converge uniformly.

\paragraph{Step 5: the asymptotic as $s\downarrow0$.}
We first show that
\begin{equation}\label{eq:cproc-Lambda-asymptotic}
\lim_{p\to\infty}\sqrt p\sum_{i\ge1}\frac{1}{R_i(R_i+p/4)}=3\pi\gamma^{-3/2}.
\end{equation}
Fix $\epsilon\in(0,1)$ and choose $i_0$ such that $(1-\epsilon)\gamma i^{2/3}\le R_i\le(1+\epsilon)\gamma i^{2/3}$ for $i\ge i_0$. The terms with $i<i_0$ contribute at most $4p^{-1/2}\sum_{i<i_0}R_i^{-1}\to0$. Since the function $x\mapsto(x(x+p/4))^{-1}$ is decreasing, we may compare the remaining terms, for $a=(1\pm\epsilon)\gamma$, with
\begin{equation*}
\sqrt p\sum_{i\ge i_0}\frac{1}{ai^{2/3}(ai^{2/3}+p/4)}.
\end{equation*}
The summand is decreasing in $i$, so this sum differs from the integral of the same function over $[i_0,\infty)$ by at most its value at $i=i_0$, which tends to zero. The integral over $[0,i_0]$ is at most $\sqrt p\int_0^{i_0}4(apx^{2/3})^{-1}\dd x\to0$. Finally, the substitution $r=ax^{2/3}$ gives
\begin{equation*}
\sqrt p\int_0^\infty\frac{\dd x}{ax^{2/3}(ax^{2/3}+p/4)}=\frac32a^{-3/2}\sqrt p\int_0^\infty\frac{r^{-1/2}\dd r}{r+p/4}=\frac32a^{-3/2}\cdot2\pi=3\pi a^{-3/2},
\end{equation*}
where we used $\int_0^\infty r^{-1/2}(r+b)^{-1}\dd r=\pi b^{-1/2}$ with $b=p/4$. Letting $\epsilon\downarrow0$ proves \eqref{eq:cproc-Lambda-asymptotic}. By \eqref{cd-mean-transform}, we obtain
\begin{equation}\label{eq:cproc-Lpsi-asymptotic}
\mathcal L\psi_R(p)=\frac{1}{2\Lambda_R(p)}\sim C_*p^{-1/2}\quad(p\to\infty),\qquad C_*=\frac{4\gamma^{3/2}}{3\pi}.
\end{equation}
The function $\psi_R$ is nonnegative and nonincreasing. By Karamata's Tauberian theorem for Laplace transforms at infinity \cite[Section~XIII.5, Theorems~2 and~3, p.~445]{Feller1971}, \eqref{eq:cproc-Lpsi-asymptotic} implies that $F(h)=\int_0^h\psi_R(s)\dd s$ satisfies $F(h)\sim C_*h^{1/2}/(\sqrt\pi/2)=2C_*\pi^{-1/2}h^{1/2}$ as $h\downarrow0$.
For completeness, we recall the argument. For $h>0$, let $\nu_h$ be the measure on $[0,\infty)$ with $\nu_h([0,x])=h^{-1/2}F(hx)$. Its Laplace transform at $q>0$ is $h^{-1/2}\mathcal L\psi_R(q/h)$, which tends to $C_*q^{-1/2}$ as $h\downarrow0$, the Laplace transform of the measure $C_*\pi^{-1/2}x^{-1/2}\dd x$. By the extended continuity theorem for Laplace transforms \cite[Section~XIII.1, Theorem~2a, p.~433]{Feller1971}, $\nu_h$ converges vaguely to this measure, which has no atoms; hence $h^{-1/2}F(h)\to2C_*\pi^{-1/2}$. We now pass from $F$ to $\psi_R$ by monotonicity. For $\lambda>1$,
\begin{equation*}
\frac{F(\lambda h)-F(h)}{(\lambda-1)h}\le\psi_R(h)\le\frac{F(h)-F(h/\lambda)}{(1-\lambda^{-1})h}.
\end{equation*}
Multiplying by $\sqrt h$ and letting $h\downarrow0$, the left and right sides tend to $2C_*\pi^{-1/2}(\lambda^{1/2}-1)/(\lambda-1)$ and $2C_*\pi^{-1/2}(1-\lambda^{-1/2})/(1-\lambda^{-1})$. Letting $\lambda\downarrow1$, both tend to $C_*\pi^{-1/2}$. This proves the first limit in \eqref{cd-mean-endpoints}, with $\psi_*=C_*\pi^{-1/2}=4\gamma^{3/2}/(3\pi^{3/2})$.

\paragraph{Step 6: the limit as $s\to\infty$.}
The limit $\ell=\lim_{s\to\infty}\psi_R(s)$ exists because $\psi_R$ is nonincreasing and nonnegative. For $K>0$, we have $p\int_0^Ke^{-ps}\psi_R\le p\int_0^K\psi_R\to0$ as $p\downarrow0$, while $p\int_K^\infty e^{-ps}\psi_R(s)\dd s$ lies between $\ell e^{-pK}$ and $\psi_R(K)e^{-pK}$. Letting $p\downarrow0$ and then $K\to\infty$, we find $\ell=\lim_{p\downarrow0}p\mathcal L\psi_R(p)$. Using monotone convergence, we find that $\Lambda_R(p)/p\to\frac18\sum_iR_i^{-2}$ as $p\downarrow0$. We conclude that $\ell=4/\sum_iR_i^{-2}$, which is positive. This completes the proof.
\end{proof}

The finite-dimensional analog of the second limit in \eqref{cd-mean-endpoints} is the weight of the zero mode of $H$: $\lim_{s\to\infty}\psi_N(s)=4/\sum_iR_{i,N}^{-2}$.

\begin{lemma}[The limiting kernels]\label{lem:cproc-limit-kernels}
Let $(r_{i,N})$ be a critical sequence with limit $R$, and fix $K>0$. There exist $C,c_*>0$, depending only on $c_0,C_0,K$, such that for all $0<t\le K$ and all $i\ge1$,
\begin{gather}
\frac{c_*}{\sqrt t}\le\psi_R(t)\le\frac{C}{\sqrt t},\qquad|\psi_R'(t)|\le\frac{C}{t^{3/2}},\label{eq:cproc-psiR-bounds}\\
\frac{c_*\sqrt t}{1+R_it}\le k_i(t)\le\frac{C\sqrt t}{1+R_it},\qquad|k_i'(t)|\le\frac{C}{\sqrt t\,(1+R_it)},\label{eq:cproc-kernel-limit-bounds}\\
\sum_ik_i(t)^2\le\frac{C}{\sqrt t},\qquad\sum_ik_i'(t)^2\le\frac{C}{t^{5/2}}.\label{eq:cproc-kernel-limit-sums}
\end{gather}
Moreover, for $0<t\le K$ and $u>0$,
\begin{equation}\label{cd-kernel-smoothing}
\sup_{i\ge1}R_ik_i(t)^2\le C,\qquad
\sum_{i\ge1}k_i(t)^2e^{-4R_iu}\le\frac{Ct}{(t+u)^{3/2}},
\end{equation}
and for all $t>0$,
\begin{equation}\label{cd-canonical-weighted-kernel}
\sum_{i\ge1}\frac{k_i(t)}{R_i}=2.
\end{equation}
\end{lemma}

\begin{proof}
The bounds \eqref{eq:cproc-psiR-bounds} and \eqref{eq:cproc-kernel-limit-bounds} follow from \eqref{eq:cproc-psiN-bounds} and \eqref{cd-kernel-bounds} by letting $N\to\infty$, using \cref{cd-mean}, $R_{i,N}\to R_i$, and, for the derivatives, $k_{i,N}'=-4R_{i,N}k_{i,N}+2\psi_N\to-4R_ik_i+2\psi_R=k_i'$. The sums \eqref{eq:cproc-kernel-limit-sums} follow from \eqref{eq:cproc-kernel-limit-bounds}, \eqref{eq:cproc-weyl-limit}, and \eqref{eq:cproc-sum-rational}. The first bound in \eqref{cd-kernel-smoothing} follows from $R_ik_i(t)^2\le C^2R_it(1+R_it)^{-2}\le C^2$. For the second, if $u\le t$, it follows from the first bound in \eqref{eq:cproc-kernel-limit-sums} and $(t+u)^{3/2}\le(2t)^{3/2}$; if $u>t$, it follows from $k_i(t)^2\le C^2t$, \eqref{eq:cproc-sum-exp}, and $u^{-3/2}\le2^{3/2}(t+u)^{-3/2}$. Finally, by \eqref{eq:cproc-finite-duhamel}, $\sum_ik_{i,N}(t)/R_{i,N}=2$ for every $N$. For fixed $t>0$, each term converges to $k_i(t)/R_i$, and by \eqref{cd-kernel-bounds} we have the summable bound $k_{i,N}(t)/R_{i,N}\le C\sqrt{K+1}\,(tR_{i,N}^2)^{-1}\le C\sqrt{K+1}\,t^{-1}c_0^{-2}i^{-4/3}$, uniformly in $N$. We obtain \eqref{cd-canonical-weighted-kernel} by dominated convergence.
\end{proof}

By Tonelli's theorem, $\sum_ik_i(t)/R_i=2\int_0^t\psi_R(v)\Sigma_R(t-v)\dd v$, where
\begin{equation}\label{eq:cproc-Sigma}
\Sigma_R(t)=\sum_{i\ge1}\frac{e^{-4R_it}}{R_i},\qquad t>0.
\end{equation}
In other words, \eqref{cd-canonical-weighted-kernel} states that the mean function solves the Volterra equation $\int_0^t\psi_R(v)\Sigma_R(t-v)\dd v=1$ for $t>0$. In finite dimension, the corresponding object is the renewal equation for $\sum_jh_j$ behind \eqref{cd-renewal-resolvent}. This renewal equation is the finite-dimensional analog of the scalar Volterra equation for the spherical constraint derived by Cugliandolo and Dean \cite[(II.4), (II.6)--(II.7)]{CugliandoloDean1995} for the dynamics with a deterministic Lagrange multiplier in the thermodynamic limit, and the equation above is its rescaled limit at the critical temperature. In our setting, the mean function describes only the mean of the random amplitude $Z$ of the critical limit process.


\section{A clipped system and uniqueness}\label{sec:cproc-clipped}

The critical limit process is obtained in \cref{sec:cproc-construction} as a limit point of the rescaled processes, and its law is identified through a uniqueness statement for the limiting equations. These equations are singular in two ways: the noise amplitude $\sqrt{2Z}$ is not Lipschitz near zero and is unbounded at the entrance, and the equation for $Z$ involves infinitely many stochastic integrals. We first solve a clipped version of the equations, in which the noise amplitude and a weighted norm of the coordinates are truncated, by a contraction argument. We then show that every solution of the untruncated equations agrees with the clipped solutions up to the time at which the truncation becomes active.

\paragraph{The weighted space.}
For a positive sequence $R$, let $E_R$ be the Hilbert space of real sequences $u=(u_i)_{i\ge1}$ with
\begin{equation*}
\|u\|_R^2=\sum_{i\ge1}\frac{u_i^2}{R_i}<\infty.
\end{equation*}
If $R$ satisfies \eqref{eq:cproc-weyl-limit}, then $\|\cdot\|_R$ is equivalent to the norm $(\sum_iu_i^2i^{-2/3})^{1/2}$, with constants depending only on $c_0,C_0$; in particular, the sets $E_R$ coincide for all such $R$. The Borel $\sigma$-field of $E_R$ is the trace of the product $\sigma$-field of $\R^{\N}$, since the coordinates are continuous on $E_R$ and $\|u-w\|_R$ is a supremum of countably many finite sums.

\begin{definition}[Entrance solutions]\label{def:cproc-solution}
Let $R$ be a positive nondecreasing sequence with $R_i/i^{2/3}\to\gamma\in(0,\infty)$, let $\psi=\psi_R$ and $(k_i)$ be the mean and kernels of \cref{cd-mean}, and let $K>0$. Let $(\Omega,\mathcal F,(\mathcal F_t)_{t\in[0,K]},\PP)$ be a filtered probability space satisfying the usual conditions, carrying a sequence $B=(B_i)_{i\ge1}$ of independent $(\mathcal F_t)$-Brownian motions.
An entrance solution on $[0,K]$ is a pair $(U,V)$ of adapted processes, $U=(U_i)_{i\ge1}$, with the following properties. Each $U_i$ is continuous with $U_i(0)=0$. The process $V$ is continuous with $V(0)=0$, and $Z=\psi+V$ is positive on $(0,K]$. For $t\in(0,K]$, $U(t)\in E_R$; the map $t\mapsto U(t)$ is continuous from $(0,K]$ into $E_R$; and $\|U(t)\|_R^2\to2$ as $t\downarrow0$. Finally, for every $t\in[0,K]$, almost surely,
\begin{equation}\label{eq:cproc-entrance-system}
\begin{aligned}
U_i(t)&=\sqrt2\int_0^te^{-2R_i(t-v)}\sqrt{Z(v)}\dd B_i(v),\qquad i\ge1,\\
V(t)&=\sqrt8\sum_{i\ge1}\int_0^tk_i(t-v)\sqrt{Z(v)}\,U_i(v)\dd B_i(v),
\end{aligned}
\end{equation}
where the series converges in probability.
A global entrance solution is a pair defined on $[0,\infty)$ that is an entrance solution on $[0,K]$ for every $K$.
\end{definition}

The stochastic integrals are well defined. We have $\sqrt t\,Z(t)\to\psi_*$ as $t\downarrow0$, because $V$ is continuous with $V(0)=0$; hence $Z$ is integrable on $[0,K]$, and the integrands $k_i(t-v)^2Z(v)U_i(v)^2$ are integrable in $v$ as well.

We introduce the clipping maps. Let $\omega(t)=t^{-1/2}$ for $t>0$, and let $m>1$. We set
\begin{equation}\label{eq:cproc-clip}
a_m(t,z)=\Big(\min\big\{m\,\omega(t),\max\{m^{-1}\omega(t),z\}\big\}\Big)^{1/2},
\qquad
\Pi_mu=\min\Big\{1,\frac{\sqrt m}{\|u\|_R}\Big\}u,
\end{equation}
with $\Pi_m0=0$.
The map $\Pi_m$ is the metric projection of $E_R$ onto the closed ball of radius $\sqrt m$.

\begin{lemma}[A clipped system and uniqueness]\label{cd-clipped-picard}
Let $R$ be positive and nondecreasing, with $R_i/i^{2/3}\to\gamma\in(0,\infty)$. Set $\psi=\psi_R$ and let $k_i$ be the kernels \eqref{cd-H-limit}. These objects are defined by \cref{cd-mean}: the construction of \cref{lem:cprof-approx}, with its first-gap assertion omitted when $R_1=R_2$, supplies a critical sequence with limit $R$. In particular, there exist $c_0,C_0>0$ with $c_0i^{2/3}\le R_i\le C_0i^{2/3}$; by \cref{cd-mean,lem:cproc-limit-kernels}, $\sqrt t\,\psi(t)\to\psi_*\in(0,\infty)$, $\sum_ik_i(t)/R_i=2$, and for every $K>0$ there exists $C_K$ such that
\begin{equation}\label{eq:cproc-picard-hypotheses}
k_i(t)\le\frac{C_K\sqrt t}{1+R_it},\qquad|k_i'(t)|\le\frac{C_K}{\sqrt t\,(1+R_it)},\qquad0<t\le K,\ i\ge1.
\end{equation}
Fix $K>0$ and a filtered probability space as in \cref{def:cproc-solution}. There exists $m_0>\max\{2,\psi_*,\psi_*^{-1}\}$, allowed to depend on $R$, $K$, and the filtered Brownian driver fixed above, such that part (a) holds for every $m\ge m_0$.
\begin{enumerate}
\item The clipped system
\begin{equation}\label{cd-picard-system}
\begin{aligned}
U_i^m(t)&=\sqrt2\int_0^te^{-2R_i(t-v)}a_m\big(v,\psi(v)+V^m(v)\big)\dd B_i(v),\\
V^m(t)&=\sqrt8\sum_{i\ge1}\int_0^tk_i(t-v)\,a_m\big(v,\psi(v)+V^m(v)\big)\big(\Pi_mU^m(v)\big)_i\dd B_i(v)
\end{aligned}
\end{equation}
has a solution on $[0,K]$ such that $V^m$ is continuous with $V^m(0)=0$, each $U_i^m$ is continuous with $U_i^m(0)=0$, and the series converges in probability in $C([0,K])$. The map $t\mapsto U^m(t)$ is continuous from $(0,K]$ into $E_R$, and $\|U^m(t)\|_R^2\to2$ as $t\downarrow0$. Any two such solutions are indistinguishable.
\item There exist $m_{\mathrm{can}}>\max\{2,\psi_*,\psi_*^{-1}\}$ and Borel maps $\mathfrak T_n$, $n\ge0$, from countable Wiener space $C([0,K])^{\N}$ to $C([0,K])^{\N}\times C([0,K])$, such that $\mathfrak T_n$ represents a clipped solution on that space at level $m_{\mathrm{can}}+n$. For every filtered Brownian driver there is a deterministic $n_0$, allowed to depend on that driver, such that for every $n\ge n_0$ and every clipped solution on its space at level $m_{\mathrm{can}}+n$,
\[
(U^{m_{\mathrm{can}}+n},V^{m_{\mathrm{can}}+n})=\mathfrak T_n(B)\quad\text{almost surely on }[0,K].
\]
\item There is a Borel map $\mathfrak S$ on the same input and output spaces as in (b) such that every entrance solution $(U,V)$ on $[0,K]$ satisfies $(U,V)=\mathfrak S(B)$ almost surely. In particular, writing $\mathbb W$ for countable Wiener measure, the joint law of $(U,V,B)$ is the image of $\mathbb W$ under $b\mapsto(\mathfrak S(b),b)$.
\end{enumerate}
\end{lemma}

Part (c) is a uniqueness statement only. The existence of an entrance solution is proved in \cref{cd-functional}. No first-gap assumption on $R$ is needed.

\begin{proof}
On the fixed filtered space, choose $m_0>\max\{\psi_*,\psi_*^{-1},2\}$ large enough for the construction below and fix $m\ge m_0$. We first record deterministic bounds (Step 1). We then construct the clipped solution as the fixed point of a contraction on a weighted $L^2$ space of progressively measurable processes (Step 2), show that the fixed point has continuous paths and that the series converge in path space (Steps 3 and 4), and compute the weighted norm at the entrance (Step 5). Step 6 constructs the Borel maps in (b), and Step 7 proves the uniqueness statement (c) by localization. Throughout, $C$ denotes a constant depending only on $c_0,C_0,K,m,\psi$, and the constants $C_K$ in \eqref{eq:cproc-picard-hypotheses}, unless further dependence is indicated; it never depends on the input process.

\paragraph{Step 1: deterministic bounds.}
By \cref{lem:cproc-sums} and \eqref{eq:cproc-picard-hypotheses}, for $0<t<t+h\le K$,
\begin{equation}\label{cd-picard-heat}
\begin{gathered}
\Sigma_R(t)\le Ct^{-1/2},\qquad
\sup_iR_ik_i(t)^2\le C,\qquad
\sup_iR_ik_i'(t)^2\le Ct^{-2},\\
\sup_iR_i|k_i(t+h)-k_i(t)|^2\le C\min\Big\{1,\frac{h^2}{t^2}\Big\},
\end{gathered}
\end{equation}
with $\Sigma_R$ as in \eqref{eq:cproc-Sigma}. For the second and third bounds, we used $R_it(1+R_it)^{-2}\le1$. For the fourth, we used the second bound and $|k_i(t+h)-k_i(t)|\le h\sup_{[t,t+h]}|k_i'|$ together with the third. The clipping maps satisfy
\begin{equation}\label{eq:cproc-clip-bounds}
\begin{gathered}
a_m(t,z)^2\le m\,\omega(t),\qquad
|a_m(t,z)-a_m(t,z')|^2\le\frac{m}{4\omega(t)}|z-z'|^2,\\
\|\Pi_mu\|_R^2\le m,\qquad\|\Pi_mu-\Pi_mw\|_R\le\|u-w\|_R.
\end{gathered}
\end{equation}
The second bound holds because clipping is $1$-Lipschitz and the square root has Lipschitz constant $(4m^{-1}\omega(t))^{-1/2}$ on $[m^{-1}\omega(t),\infty)$, and the last holds because metric projections onto closed convex subsets of a Hilbert space are $1$-Lipschitz. The function $\omega(t)=t^{-1/2}$ is nonincreasing, $\omega(t)^{-1}\le\sqrt K$ on $(0,K]$, and $\int_0^t\omega(v)\dd v=2\sqrt t$.

\paragraph{Step 2: a contraction.}
For $\lambda>0$, let $\mathcal H_\lambda$ be the space of real progressively measurable processes $V$ on $[0,K]$, modulo $\dd t\otimes\dd\PP$-null sets, with
\begin{equation*}
\|V\|_\lambda^2=\int_0^Ke^{-\lambda t}\,\E|V(t)|^2\dd t<\infty.
\end{equation*}
This is an $L^2$ space, hence complete, and the norms $\|\cdot\|_\lambda$ are equivalent for different $\lambda$. Given $V\in\mathcal H_\lambda$, let $U[V]$ be defined by the first line of \eqref{cd-picard-system} with $V$ in place of $V^m$. Since each coordinate $U_i[V]$ is continuous and adapted, the process $\|U[V](t)\|_R^2$ is progressively measurable, and so is the integrand
\begin{equation*}
\zeta_i(v)=a_m\big(v,\psi(v)+V(v)\big)\big(\Pi_mU[V](v)\big)_i.
\end{equation*}
Note that $\sum_i\zeta_i(v)^2/R_i\le m^2\omega(v)$ pathwise, by \eqref{eq:cproc-clip-bounds}. Using \eqref{cd-picard-heat}, we obtain
\begin{equation*}
\sum_i\E\int_0^tk_i(t-v)^2\zeta_i(v)^2\dd v\le\int_0^t\sup_iR_ik_i(t-v)^2\,\E\sum_i\frac{\zeta_i(v)^2}{R_i}\dd v\le Cm^2\int_0^t\omega(v)\dd v\le C.
\end{equation*}
The stochastic integrals in the second line of \eqref{cd-picard-system} are orthogonal in $L^2$, and the series converges in $L^2$; we define $\mathcal T_mV(t)$ as its sum. We show in Step 3 that $\mathcal T_mV$ has a continuous adapted version; in particular, $\mathcal T_mV\in\mathcal H_\lambda$. Changing $V$ on a $\dd t\otimes\dd\PP$-null set changes neither $U[V]$ nor $\mathcal T_mV$. The coordinates satisfy, by It\^o's isometry and \eqref{eq:cproc-clip-bounds},
\begin{equation}\label{eq:cproc-picard-U-size}
\begin{aligned}
\E\big\|U[V](t)\big\|_R^2&=2\int_0^t\Sigma_R(t-v)\,\E a_m\big(v,\psi(v)+V(v)\big)^2\dd v\\
&\le Cm\int_0^t\frac{\omega(v)}{\sqrt{t-v}}\dd v\le C,
\end{aligned}
\end{equation}
The last integral equals $\pi$ for $t>0$. In particular, $U[V](t)\in E_R$ almost surely.

Let $V,\widetilde V\in\mathcal H_\lambda$ and $\Upsilon(v)=\E|V(v)-\widetilde V(v)|^2$. It\^o's isometry and \eqref{eq:cproc-clip-bounds} give
\begin{equation}\label{eq:cproc-picard-U-diff}
\E\big\|U[V](v)-U[\widetilde V](v)\big\|_R^2\le2\int_0^v\Sigma_R(v-w)\frac{m}{4\omega(w)}\Upsilon(w)\dd w\le C\int_0^v\frac{\Upsilon(w)}{\omega(w)\sqrt{v-w}}\dd w.
\end{equation}
In the radial integrand, we write $a\,\Pi_mU-\widetilde a\,\Pi_m\widetilde U=(a-\widetilde a)\,\Pi_mU+\widetilde a\,(\Pi_mU-\Pi_m\widetilde U)$, with the obvious abbreviations. By It\^o's isometry, \eqref{cd-picard-heat}, and \eqref{eq:cproc-clip-bounds},
\begin{equation*}
\begin{aligned}
\E\big|\mathcal T_mV(s)-\mathcal T_m\widetilde V(s)\big|^2
&\le16\int_0^s\sup_iR_ik_i(s-v)^2\,\frac{m^2\Upsilon(v)}{4\omega(v)}\dd v\\
&\quad+16\int_0^s\sup_iR_ik_i(s-v)^2\,m\,\omega(v)\,\E\big\|U[V](v)-U[\widetilde V](v)\big\|_R^2\dd v\\
&\le C\int_0^s\frac{\Upsilon(v)}{\omega(v)}\dd v\\
&\quad+C\int_0^s\omega(v)\int_0^v\frac{\Upsilon(w)}{\omega(w)\sqrt{v-w}}\dd w\dd v.
\end{aligned}
\end{equation*}
Note that $\omega(v)/\omega(w)\le1$ for $w\le v$, because $\omega$ is nonincreasing, and that $\omega^{-1}$ is bounded on $(0,K]$. Exchanging the order of integration in the second term, we find
\begin{equation*}
\E\big|\mathcal T_mV(s)-\mathcal T_m\widetilde V(s)\big|^2\le C\int_0^s\big(1+(s-w)^{1/2}\big)\Upsilon(w)\dd w.
\end{equation*}
We multiply by $e^{-\lambda s}$, integrate over $s\in[0,K]$, and exchange the order of integration once more. Since $\int_0^\infty e^{-\lambda\ell}(1+\ell^{1/2})\dd\ell=\lambda^{-1}+(\sqrt\pi/2)\lambda^{-3/2}$, we obtain
\begin{equation}\label{cd-picard-contraction}
\big\|\mathcal T_mV-\mathcal T_m\widetilde V\big\|_\lambda^2\le C\big(\lambda^{-1}+(\sqrt\pi/2)\lambda^{-3/2}\big)\|V-\widetilde V\|_\lambda^2.
\end{equation}
We fix $\lambda$ so large that the coefficient in \eqref{cd-picard-contraction} is at most $1/4$. By the Banach fixed point theorem, $\mathcal T_m$ has a unique fixed point $V^m\in\mathcal H_\lambda$, and $(U[V^m],V^m)$ solves \eqref{cd-picard-system}. Conversely, the $V$-component of any solution of \eqref{cd-picard-system} with the properties stated in (a) is a fixed point of $\mathcal T_m$, since its integrands are bounded as above. This proves uniqueness up to modification; by continuity, up to indistinguishability. The fixed point on $[0,K]$ restricts to the fixed point on $[0,K']$ for $K'<K$.

\paragraph{Step 3: continuous paths and convergence of the series.}
Let $V\in\mathcal H_\lambda$ be any input and $\zeta_i$ as in Step 2. For $0\le s<s+h\le K$,
\begin{multline*}
\mathcal T_mV(s+h)-\mathcal T_mV(s)=\sqrt8\sum_i\int_0^s\big(k_i(s+h-v)-k_i(s-v)\big)\zeta_i(v)\dd B_i(v)\\
+\sqrt8\sum_i\int_s^{s+h}k_i(s+h-v)\zeta_i(v)\dd B_i(v).
\end{multline*}
By \eqref{cd-picard-heat} and $\sum_i\zeta_i^2/R_i\le m^2\omega$, the quadratic variations of the two terms are bounded pathwise by
\begin{equation*}
C\int_0^s\omega(v)\min\Big\{1,\frac{h^2}{(s-v)^2}\Big\}\dd v\qquad\text{and}\qquad C\int_s^{s+h}\omega(v)\dd v.
\end{equation*}
The second is at most $C\sqrt h$. For the first, if $s\le2h$ we bound the minimum by $1$ and obtain at most $2\sqrt s\le C\sqrt h$. If $s>2h$, the part $v\le s/2$ is at most $h^2(2/s)^2\cdot2\sqrt s\le C\sqrt h$, and the part $v\ge s/2$ is at most $\sqrt{2/s}\int_0^\infty\min\{1,h^2\ell^{-2}\}\dd\ell=2h\sqrt{2/s}\le C\sqrt h$. By the Burkholder--Davis--Gundy inequality \cite[Theorem~IV.4.1 and Corollary~IV.4.2]{RevuzYor1999},
for every $p\ge2$,
\begin{equation}\label{eq:cproc-picard-holder}
\E\big|\mathcal T_mV(s+h)-\mathcal T_mV(s)\big|^p\le C_ph^{p/4},
\end{equation}
uniformly over inputs $V$. Fix $p>4$. By Kolmogorov's continuity criterion \cite[Theorem~I.2.1]{RevuzYor1999},
$\mathcal T_mV$ has a continuous version, with $\E[\,[\mathcal T_mV]_\alpha^p\,]\le C_{p,\alpha}$ for the H\"older seminorm of order $\alpha<1/4-1/p$, and $\mathcal T_mV(0)=0$. The same bounds hold, uniformly in $J$, for the tails $\mathcal T_m^{>J}V$ obtained by restricting the series to $i>J$. At each fixed $s$, It\^o's isometry gives
\begin{equation*}
\E\big|\mathcal T_m^{>J}V(s)\big|^2\le8m^2\int_0^s\sup_{i>J}R_ik_i(s-v)^2\,\omega(v)\dd v\to0\qquad(J\to\infty),
\end{equation*}
by dominated convergence, since $R_ik_i(\ell)^2\le C(R_i\ell)^{-1}$ tends to zero as $i\to\infty$ for each $\ell>0$. Fix $\eta>0$ and a mesh $\delta>0$. On the event that the H\"older seminorm is at most $M$, the supremum of $|\mathcal T_m^{>J}V|$ over $[0,K]$ exceeds $\eta$ only if its value at some point of the grid $\delta\mathbb Z\cap[0,K]$ exceeds $\eta-M\delta^\alpha$. Choosing $M$, then $\delta$, then $J$, we find that $\sup_{[0,K]}|\mathcal T_m^{>J}V|\to0$ in probability. This proves that the series converges in probability in $C([0,K])$. Since the fixed point $V^m$ equals $\mathcal T_mV^m$, it has a continuous version vanishing at zero; replacing it by this version does not change any stochastic integral. Each coordinate $U_i^m$ is continuous, since $U_i^m(t)=\sqrt2e^{-2R_it}\int_0^te^{2R_iv}a_m\dd B_i$, and the last integral is a continuous martingale because $\int_0^Ka_m^2\le m\int_0^K\omega<\infty$. We have $U_i^m(0)=0$.

\paragraph{Step 4: continuity in $E_R$.}
Fix $\delta\in(0,K)$ and put $t_0=\delta/2$. For $t\ge t_0$, write
\begin{equation*}
U_i^m(t)=e^{-2R_i(t-t_0)}U_i^m(t_0)+\mathcal S_i(t),\qquad
\mathcal S_i(t)=\sqrt2\int_{t_0}^te^{-2R_i(t-v)}a_m\dd B_i.
\end{equation*}
By \eqref{eq:cproc-picard-U-size}, $U^m(t_0)\in E_R$ almost surely. The deterministic semigroup orbit $t\mapsto e^{-2R(t-t_0)}U^m(t_0)$ is continuous into $E_R$ by dominated convergence. On $[t_0,K]$, the clipped amplitude satisfies $a_m^2\le m\,\omega(t_0)$. For $t_0\le t<t+h\le K$,
\begin{equation*}
\mathcal S_i(t+h)-\mathcal S_i(t)=\big(e^{-2R_ih}-1\big)\mathcal S_i(t)+\sqrt2\int_t^{t+h}e^{-2R_i(t+h-v)}a_m\dd B_i.
\end{equation*}
By the Burkholder--Davis--Gundy inequality, $\|\mathcal S_i(t)\|_p^2\le C_{p,\delta}/R_i$ and $\|\int_t^{t+h}\cdots\|_p^2\le C_{p,\delta}\min\{h,R_i^{-1}\}$. Using Minkowski's inequality in $L^{p/2}$ and \eqref{eq:cproc-sum-increment}, we obtain
\begin{equation*}
\Big\|\,\|\mathcal S(t+h)-\mathcal S(t)\|_R^2\Big\|_{p/2}\le C_{p,\delta}\sum_i\frac{\min\{1,R_ih\}^2}{R_i^2}+C_{p,\delta}\sum_i\frac{\min\{h,R_i^{-1}\}}{R_i}\le C_{p,\delta}\sqrt h.
\end{equation*}
For $p>4$, Kolmogorov's criterion gives a continuous $E_R$-valued version of $\mathcal S$, whose coordinates are the continuous processes $\mathcal S_i$. Thus $U^m$ is continuous into $E_R$ on $[\delta,K]$. Taking a countable sequence $\delta\downarrow0$ gives continuity from $(0,K]$ into $E_R$.

\paragraph{Step 5: the weighted norm at the entrance.}
Abbreviate $a(v)=a_m(v,\psi(v)+V^m(v))$. By It\^o's formula and variation of constants, for each $i$ and $t>0$,
\begin{equation*}
U_i^m(t)^2=2\int_0^te^{-4R_i(t-v)}a(v)^2\dd v+2\sqrt2\int_0^te^{-4R_i(t-v)}a(v)\,U_i^m(v)\dd B_i(v).
\end{equation*}
We divide by $R_i$ and sum over $i$. For the first terms we use Tonelli's theorem. For the second terms, by the Burkholder--Davis--Gundy inequality and $a^2\le m\,\omega$, we have $\|U_i^m(v)^2\|_p\le C_p\int_0^ve^{-4R_i(v-w)}w^{-1/2}\dd w\le C_p\sqrt v(1+R_iv)^{-1}$, so that $\|a(v)^2U_i^m(v)^2\|_p\le C_p(1+R_iv)^{-1}$; since $\sum_iR_i^{-2}<\infty$, this shows that the stochastic series converges in every $L^p$. We obtain, for each $t>0$, almost surely,
\begin{equation}\label{cd-picard-norm}
\begin{aligned}
\|U^m(t)\|_R^2&=2\int_0^ta(v)^2\,\Sigma_R(t-v)\dd v+\mathcal J(t),\\
\mathcal J(t)&=2\sqrt2\sum_{i\ge1}\int_0^t\frac{e^{-4R_i(t-v)}}{R_i}\,a(v)\,U_i^m(v)\dd B_i(v).
\end{aligned}
\end{equation}
From the same bounds, we obtain $\|\mathcal J(t)\|_p^2\le C_p\sum_iR_i^{-2}\min\{t,R_i^{-1}\log(1+R_it)\}\le C_pt$, and increments with $\|\mathcal J(t+h)-\mathcal J(t)\|_p^2\le C_p\sqrt h$: the fresh part is bounded using $\sum_iR_i^{-2}\le C$, and the old part using \eqref{eq:cproc-sum-increment}. Then $\mathcal J$ has a continuous version with $\mathcal J(0)=0$ by Kolmogorov's criterion. Since $V^m(0)=0$ and $\sqrt v\,\psi(v)\to\psi_*\in(m^{-1},m)$, there exists a random $\tau_0>0$ such that $a(v)^2=\psi(v)+V^m(v)$ for $v\le\tau_0$. For $t\le\tau_0$, Tonelli's theorem and the hypothesis $\sum_ik_i(t)/R_i=2$ give
\begin{equation*}
2\int_0^ta(v)^2\,\Sigma_R(t-v)\dd v=\sum_i\frac{k_i(t)}{R_i}+2\int_0^tV^m(v)\,\Sigma_R(t-v)\dd v=2+O\Big(\sqrt t\sup_{v\le t}|V^m(v)|\Big),
\end{equation*}
where we used $\int_0^t\Sigma_R\le C\sqrt t$. The right side of \eqref{cd-picard-norm} is therefore continuous on $(0,\tau_0]$, as is the left side by Step 4. Both sides agree almost surely at all rational times, hence at all times in $(0,\tau_0]$, and we conclude that $\|U^m(t)\|_R^2\to2$ as $t\downarrow0$.

\paragraph{Step 6: Borel representations.}
Perform (a) on canonical countable Wiener space $C([0,K])^{\N}$, with its usual augmented filtration, and choose one threshold $m_{\mathrm{can}}>\max\{2,\psi_*,\psi_*^{-1}\}$ there. At the countable levels $m_{\mathrm{can}}+n$, choose Borel versions $\mathfrak T_n$ of these canonical solutions.

To transfer the construction, approximate each stochastic integral by elementary adapted integrands. If $F$ is a progressively measurable process on canonical Wiener space with $\E\int_0^K|F|^2<\infty$, then $F$ is an $L^2$ limit of elementary processes whose coefficients are measurable functions of the Wiener paths up to their left endpoints. Substituting $B$ on another filtered space preserves both these elementary integrals and their $L^2$ limits, since $B$ has law $\mathbb W$. Applying this at every Picard iterate and then at their limit pulls each canonical solution back under the Brownian input map; completing the filtration does not change the stochastic integrals. On the other space, (a) supplies its own threshold $m_0$. For all $n$ large enough that $m_{\mathrm{can}}+n\ge m_0$, uniqueness in (a) identifies every clipped solution with $\mathfrak T_n(B)$. This proves (b).

The same approximation argument also shows that the joint law of stochastic integrals of measurable adapted functionals of $(U,V,B)$ is determined by the joint law of $(U,V,B)$, whenever those integrals are well defined. For locally square-integrable integrands, first stop at increasing bounds for their quadratic integrals and then remove the stopping.

\paragraph{Step 7: uniqueness by localization.}
Let $(U,V)$ be an entrance solution, as in \cref{def:cproc-solution}, on the same filtered space as $(U^m,V^m)$. Let $\tau_m$ be the first time $t\in(0,K]$ at which one of the strict inequalities
\begin{equation*}
m^{-1}<\frac{Z(t)}{\omega(t)}<m,\qquad\|U(t)\|_R^2<m
\end{equation*}
fails, with $\tau_m=K$ if none fails. Then $\tau_m>0$ almost surely, because $Z(t)/\omega(t)=\sqrt t\,Z(t)\to\psi_*\in(m^{-1},m)$ and $\|U(t)\|_R^2\to2<m$. The function $Z/\omega$ extends to a positive continuous function on $[0,K]$, and $\|U(t)\|_R^2$ is bounded on $(0,K]$. Then $\tau_m=K$ for all large $m$, almost surely.

For $v<\tau_m$, the clipping is inactive: $\sqrt{Z(v)}=a_m(v,\psi(v)+V(v))$ and $U(v)=\Pi_mU(v)$. We define the stopped processes $(U^{\tau},V^{\tau})$ by inserting the predictable indicator $\ind_{\{v\le\tau_m\}}$ into all stochastic integrands of \eqref{eq:cproc-entrance-system}, keeping the full kernels $e^{-2R_i(t-v)}$ and $k_i(t-v)$. Then $(U^\tau,V^\tau)$ coincides with $(U,V)$ on $[0,\tau_m]$, and it solves the system obtained from \eqref{cd-picard-system} by inserting the same indicator into the stochastic integrands. The clipped solution $(U^m,V^m)$, stopped in the same way, solves the same modified system. All the estimates of Step 2 remain valid with the indicator, and the modified map is again a contraction on $\mathcal H_\lambda$, with both stopped processes as fixed points. By uniqueness of the fixed point, $V^\tau$ and the stopped $V^m$ agree, and by continuity $(U,V)=(U^m,V^m)$ on $[0,\tau_m]$ almost surely. We stress that stopping the process $V$ itself, as $V(t\wedge\tau_m)$, would not preserve the Volterra kernels; this is why we stop the integrands.

Use the countable maps $\mathfrak T_n$ of (b), and define $\mathfrak S(b)$ to be their eventual constant value whenever it exists, and a fixed path otherwise. The eventual-constancy set and the resulting map are Borel, since the output path space is Polish and the sequence is countable. On any filtered space carrying an existing entrance solution, Step 6 identifies the clipped solutions for all sufficiently large indices, and the preceding localization argument makes them equal to $(U,V)$ once $\tau_m=K$. Countably many almost-sure identities can be imposed on one event. Thus the maps are eventually constant at $(U,V)$ almost surely, proving (c). No existence assertion for the untruncated equations is used here.
\end{proof}

\section{Construction of the critical limit process}\label{sec:cproc-construction}

We now construct the critical limit process as the limit of the rescaled unnormalized processes started from a standard Gaussian vector, whose radial normalizations are the Langevin diffusions started from $\nu_N$.

\begin{theorem}[The critical limit process]\label{cd-functional}
Let $(r_{i,N})$ be a critical sequence with limit $R$, let $\psi_R$ and $k_i$ be as in \cref{cd-mean}, and let $Y(0)=g$ be a standard Gaussian vector. Then the following hold.
\begin{enumerate}
\item There exists, on some filtered probability space carrying independent Brownian motions $B=(B_i)_{i\ge1}$, a global entrance solution $(U,V)$ in the sense of \cref{def:cproc-solution}, with $\psi=\psi_R$. That is, with $Z=\psi_R+V$,
\begin{equation}\label{cd-limiting-system}
\begin{aligned}
U_i(s)&=\sqrt2\int_0^se^{-2R_i(s-v)}\sqrt{Z(v)}\dd B_i(v),\qquad i\ge1,\\
V(s)&=\sqrt8\sum_{i\ge1}\int_0^sk_i(s-v)\sqrt{Z(v)}\,U_i(v)\dd B_i(v),\\
\|U(s)\|_R^2&=2+\sqrt8\sum_{i\ge1}\int_0^s\frac{1}{R_i}\sqrt{Z(v)}\,U_i(v)\dd B_i(v),\qquad s>0.
\end{aligned}
\end{equation}
The joint law of $(U,V,B)$ is unique. We call $(U,Z)$ the critical limit process, and $\xi=(\xi_i)_{i\ge1}$, $\xi_i=U_i/\sqrt Z$, the normalized critical limit process.
\item For every $K$, the series in \eqref{cd-limiting-system} converge in probability in $C([0,K])$. The process $s\mapsto\|U(s)\|_R^2$, extended by the value $2$ at $s=0$, is continuous on $[0,\infty)$. Almost surely, $Z(s)>0$ for all $s>0$, and $\sqrt s\,Z(s)\to\psi_*$ as $s\downarrow0$. For every $p>0$, $K>0$, $0<s\le K$, and $i\ge1$,
\begin{equation}\label{eq:cproc-limit-moments}
\begin{gathered}
\|Z(s)\|_p\le\frac{C_p}{\sqrt s},\qquad\E Z(s)^{-p}\le C_ps^{p/2},\\
\|U_i(s)^2\|_p\le\frac{C_p\sqrt s}{1+R_is},\qquad\|Z(s)U_i(s)^2\|_p\le\frac{C_p}{1+R_is},
\end{gathered}
\end{equation}
with $C_p$ as in \eqref{cd-scaled-moments}.
\item As $N\to\infty$,
\begin{equation*}
\big((U_{i,N})_{i\ge1},\,Z_N-\psi_N,\,\|U_N\|_{R_N}^2,\,(B_i^N)_{i\ge1}\big)\Longrightarrow\big((U_i)_{i\ge1},\,V,\,\|U\|_R^2,\,(B_i)_{i\ge1}\big)
\end{equation*}
in law in $C([0,K])^{\N}\times C([0,K])^2\times C([0,K])^{\N}$, with the product topology, for every $K$. In particular, for every $J\ge1$ and $0<s_0<K$,
\begin{equation}\label{cd-angular-functional}
\big(N^{-1/3}X_i(N^{2/3}s)\big)_{i\le J}\Longrightarrow\big(\xi_i(s)\big)_{i\le J}\qquad\text{in law in }C([s_0,K],\R^J).
\end{equation}
\end{enumerate}
\end{theorem}

The denominator $Z$ in $\xi_i=U_i/\sqrt Z$ is random. The coordinates $U_i$ are driven by independent Brownian motions but are not independent Ornstein--Uhlenbeck processes: they share the noise amplitude $\sqrt{2Z}$, and $Z$ depends on all coordinates through the second line of \eqref{cd-limiting-system}.

\begin{remark}[Behavior as $s\downarrow0$]\label{rem:cproc-entrance}
The start of the critical limit process is singular. Each coordinate starts at zero, because $U_{i,N}(0)=N^{-1/6}g_i$. On the other hand, $Z(s)\sim\psi_*s^{-1/2}$ as $s\downarrow0$, and the weighted norm $\|U(s)\|_R^2$ tends to $2$, not to $0$. In finite dimension, $\|U_N(s)\|_{R_N}^2$ is a martingale by \eqref{eq:cproc-weighted-martingale}, with initial value $N^{-1/3}\sum_ig_i^2/R_{i,N}$, which has mean $2$ and variance $2N^{-2/3}\sum_iR_{i,N}^{-2}\to0$. At time zero, the contribution of any fixed finite set of coordinates to this weighted norm tends to zero in every $L^p$, since $U_{i,N}(0)^2/R_{i,N}=N^{-1/3}g_i^2/R_{i,N}$. At a small positive time $s$, the kernel bounds give $\E\,U_i(s)^2/R_i\asymp\sqrt s\,R_i^{-1}(1+R_is)^{-1}$, and the weighted norm is carried by the coordinates whose index is of order $s^{-3/2}$. As $s\downarrow0$ these indices tend to infinity, and $U(s)$ has no limit in $E_R$. The normalized coordinates $\xi_i(s)$ tend to zero as $s\downarrow0$, in agreement with the fact that the coordinates of a uniform point of $S_N$, multiplied by $N^{-1/3}$, are of order $N^{-1/3}$.
\end{remark}

In Step~4 of the proof below, we identify the stochastic integrals of a limit point through the joint laws of the rescaled processes. A stochastic integral is not a continuous function of its integrand and of the driving path, and we use the following elementary lemma, in which only the function $f$ and the approximating functions $S$ are required to be continuous.

\begin{lemma}[Identification from joint laws]\label{lem:cproc-joint-law-identification}
Let $E$ be a Polish space, and let $\mathfrak Y_n$, $n\ge1$, and $\mathfrak Y$ be $E$-valued random variables, defined on possibly different probability spaces, such that the laws of the $\mathfrak Y_n$ converge weakly to the law of $\mathfrak Y$. Let $f\colon E\to\R$ be continuous, and let $\Xi=G(\mathfrak Y)$ for a Borel function $G\colon E\to\R$. Assume that for every $\eps>0$ there exists a continuous function $S\colon E\to\R$ such that
\begin{equation}\label{eq:cproc-joint-law-hypothesis}
\E\big|f(\mathfrak Y_n)-S(\mathfrak Y_n)\big|^2\le\eps\quad\text{for all sufficiently large }n,
\qquad\text{and}\qquad
\E\big|\Xi-S(\mathfrak Y)\big|^2\le\eps.
\end{equation}
Then $\Xi=f(\mathfrak Y)$ almost surely. More precisely, every such pair $(\eps,S)$ satisfies $\E|f(\mathfrak Y)-\Xi|^2\le4\eps$.
\end{lemma}

The first condition in \eqref{eq:cproc-joint-law-hypothesis} concerns only the laws of the $\mathfrak Y_n$. The functions $f$ and $S$ need not be bounded, and $G$ need not be continuous. In Step~4 of the proof of \cref{cd-functional}, $\Xi$ is a stochastic integral of the limit point, and $S$ is a Riemann sum.

\begin{proof}
Fix $\eps$ and $S$ as in the statement. For $L>0$, the function $\Phi_L=\min\{L,|f-S|^2\}$ is bounded and continuous on $E$, and $\Phi_L\le|f-S|^2$. By weak convergence \cite[Section~2]{Billingsley1999}, we have
\begin{equation*}
\E\Phi_L(\mathfrak Y)=\lim_{n\to\infty}\E\Phi_L(\mathfrak Y_n)\le\eps.
\end{equation*}
Letting $L\to\infty$, we find $\E|f(\mathfrak Y)-S(\mathfrak Y)|^2\le\eps$ by the monotone convergence theorem.
Since $|f(\mathfrak Y)-\Xi|^2\le2|f(\mathfrak Y)-S(\mathfrak Y)|^2+2|S(\mathfrak Y)-\Xi|^2$, we obtain $\E|f(\mathfrak Y)-\Xi|^2\le4\eps$. As $\eps>0$ is arbitrary, $\Xi=f(\mathfrak Y)$ almost surely.
\end{proof}

\begin{proof}[Proof of \cref{cd-functional}]
We first prove increment bounds for the rescaled processes, which give tightness (Step 2), and estimates for the coordinates with large index (Step 3). We then identify the equations satisfied by any limit point (Step 4), show that its rescaled squared norm $Z$ is positive (Step 5), and conclude with the uniqueness statement of \cref{cd-clipped-picard} (Step 6). \Cref{tab:cproc-parameters} lists the parameters and the order in which they are chosen.

\begin{table}[ht]
\centering
\small
\begin{tabular}{@{}lll@{}}
\toprule
Parameter & Role & Order of choice \\
\midrule
$K$ & time horizon & fixed first \\
$p$ & moment order ($p>4$; $p=16$ in Step 5) & fixed with $K$ \\
$\eta$ & accuracy of a probability bound & after $K$ and $p$ \\
$M$ & bound on a H\"older seminorm & after $\eta$ \\
$\delta,h$ & cutoff near the entrance; mesh of a time grid & after $M$ \\
$J$ & number of coordinates treated explicitly & after $\delta$ and $h$ \\
$N$ & dimension & first limit: $N\to\infty$, then $J\to\infty$ \\
\bottomrule
\end{tabular}
\caption{Parameters in the proof of \cref{cd-functional}, in the order used in Step 3. In Step 4, a coordinate or $J$ is fixed first, then $\delta$ and $h$, and $N\to\infty$ is the first limit. In Step 5, $\delta$ is fixed first, and the limits $h\downarrow0$ and then $M\to\infty$ are taken.}
\label{tab:cproc-parameters}
\end{table}

Constants in the estimates for whole sums depend only on $c_0,C_0,K,p$, and on $\delta$ where indicated. Constants in the estimates for a fixed coordinate, and in the arguments of Step~4 for a fixed $J$, may also depend on that coordinate or on $J$. No constant depends on $N$, $h$, or $M$. The moment bounds \eqref{cd-scaled-moments} and the kernel bounds of \cref{lem:cproc-bounds} are used with $K+1$ in place of $K$ and for $N\ge N_0$. We use two standard facts. First, for progressively measurable $f_i$ and $p\ge2$, the Burkholder--Davis--Gundy inequality \cite[Theorem~IV.4.1 and Corollary~IV.4.2]{RevuzYor1999} and Minkowski's inequality in $L^{p/2}$ give
\begin{equation}\label{eq:cproc-bdg}
\Big\|\sum_i\int_a^bf_i(v)\dd B_i^N(v)\Big\|_p^2\le C_p\int_a^b\sum_i\|f_i(v)\|_p^2\dd v.
\end{equation}
Second, by \cref{lem:cond-chaos}, for independent standard Gaussian variables $g_i$ and real numbers $w_i$, of which finitely many are nonzero,
\begin{equation}\label{eq:cproc-chaos}
\Big\|\sum_iw_i(g_i^2-1)\Big\|_p\le C_p\Big(\sum_iw_i^2\Big)^{1/2}.
\end{equation}

\paragraph{Step 1: the exact equations.}
We recall from \cref{lem:cproc-exact} that $U_{i,N}$ satisfies \eqref{eq:cproc-scaled-sde} with $U_{i,N}(0)=N^{-1/6}g_i$, the rescaled squared norm satisfies $Z_N=\psi_N+I_N+\Theta_N$, and the weighted norm satisfies $\|U_N(s)\|_{R_N}^2=\|U_N(0)\|_{R_N}^2+\mathcal M_N(s)$, where
\begin{equation*}
\mathcal M_N(s)=\sqrt8\sum_{i=1}^N\int_0^sR_{i,N}^{-1}\sqrt{Z_N(v)}\,U_{i,N}(v)\dd B_i^N(v).
\end{equation*}
Recall that $\sum_iR_{i,N}^{-1}=2N^{1/3}$. The initial value is therefore $\|U_N(0)\|_{R_N}^2=2+N^{-1/3}\sum_iR_{i,N}^{-1}(g_i^2-1)$. It converges to $2$ in every $L^p$ by \eqref{eq:cproc-chaos} and $\sum_iR_{i,N}^{-2}\le4c_0^{-2}$. For $J\ge1$, we write $\Theta_N^{\le J}$ and $\mathcal M_N^{\le J}$ for the sums over $i\le J$ in $\Theta_N$ and $\mathcal M_N$, and $\Theta_N^{>J}$, $\mathcal M_N^{>J}$ for the remaining sums.

\paragraph{Step 2: increment bounds and tightness.}
We first show that for $0<h\le1$,
\begin{equation}\label{cd-kernel-translation}
\int_0^h\sum_ik_{i,N}(u)^2\dd u+\int_0^K\sum_i\big(k_{i,N}(u+h)-k_{i,N}(u)\big)^2\dd u\le C\sqrt h.
\end{equation}
By \eqref{eq:cproc-kernel-sums}, the first integral is at most $C\int_0^hu^{-1/2}\dd u\le C\sqrt h$, and so is the part $u<h$ of the second. For $u\ge h$, the Cauchy--Schwarz inequality and \eqref{eq:cproc-kernel-sums} give
\begin{equation*}
\sum_i\big(k_{i,N}(u+h)-k_{i,N}(u)\big)^2\le h\int_u^{u+h}\sum_ik_{i,N}'(w)^2\dd w\le Ch^2u^{-5/2},
\end{equation*}
whose integral over $[h,\infty)$ is $\frac23Ch^{1/2}$. This proves \eqref{cd-kernel-translation}.

We next bound the increments of $\Theta_N$. For $0\le s<s+h\le K$, the increment $\Theta_N(s+h)-\Theta_N(s)$ is the sum of a stochastic integral over $[0,s]$ with integrands $\sqrt8(k_{i,N}(s+h-v)-k_{i,N}(s-v))\sqrt{Z_N}U_{i,N}$ and a stochastic integral over $[s,s+h]$ with integrands $\sqrt8k_{i,N}(s+h-v)\sqrt{Z_N}U_{i,N}$. Note that $\|\sqrt{Z_N(v)}U_{i,N}(v)\|_p^2=\|Z_N(v)U_{i,N}(v)^2\|_{p/2}\le C$, by \eqref{cd-scaled-moments}. By \eqref{eq:cproc-bdg} and \eqref{cd-kernel-translation}, for $p\ge2$,
\begin{equation}\label{eq:cproc-Theta-increment}
\big\|\Theta_N(s+h)-\Theta_N(s)\big\|_p\le C_ph^{1/4}.
\end{equation}
The same bound holds for $\Theta_N^{\le J}$ and $\Theta_N^{>J}$, uniformly in $J$. For the initial fluctuation, we find from \eqref{eq:cproc-chaos} and \eqref{eq:cproc-kernel-sums} that $\|I_N(s)\|_p\le C_pN^{-1/3}(\sum_ik_{i,N}(s)^2)^{1/2}\le C_pN^{-1/6}$ and
\begin{equation*}
\big\|I_N(s+h)-I_N(s)\big\|_p\le C_pN^{-1/3}\Big(h^2\sup_{0\le w\le K}\sum_ik_{i,N}'(w)^2\Big)^{1/2}\le C_pN^{1/2}h.
\end{equation*}
Note that $\min\{N^{-1/6},N^{1/2}h\}\le h^{1/4}$ (consider $h\ge N^{-2/3}$ and $h<N^{-2/3}$ separately). We obtain $\|I_N(s+h)-I_N(s)\|_p\le C_ph^{1/4}$. For fixed $i$, \eqref{eq:cproc-scaled-sde}, \eqref{eq:cproc-bdg}, and \eqref{cd-scaled-moments} give
\begin{equation*}
\begin{aligned}
\big\|U_{i,N}(s+h)-U_{i,N}(s)\big\|_p&\le2R_{i,N}\int_s^{s+h}\|U_{i,N}(v)\|_p\dd v+C_p\Big(\int_s^{s+h}\|Z_N(v)\|_{p/2}\dd v\Big)^{1/2}\\
&\le C_{p,i}h^{1/4},
\end{aligned}
\end{equation*}
where we used $R_{i,N}\le C_0i^{2/3}$ and $\int_s^{s+h}(v+N^{-2/3})^{-1/2}\dd v\le2\sqrt h$. Finally, the integrands of $\mathcal M_N$ satisfy $\sum_iR_{i,N}^{-2}\|Z_NU_{i,N}^2\|_{p/2}\le C\sum_iR_{i,N}^{-2}\le C$, and \eqref{eq:cproc-bdg} gives $\|\mathcal M_N(s+h)-\mathcal M_N(s)\|_p\le C_ph^{1/2}$, and the same holds for $\mathcal M_N^{\le J}$.

We fix $p>4$. By Kolmogorov's criterion, each of the processes $U_{i,N}$, $I_N$, $\Theta_N^{\le J}$, $\Theta_N^{>J}$, $\mathcal M_N^{\le J}$, $\mathcal M_N$ has a H\"older seminorm of order $\alpha\in(0,1/4-1/p)$ on $[0,K]$ whose $p$th moment is bounded uniformly in $N$ (and in $J$). Since their initial values are tight, the Arzel\`a--Ascoli theorem implies that each of them is tight in $C([0,K])$. Further, $I_N\to0$ in probability in $C([0,K])$: its values at the points of a finite grid tend to zero in $L^p$, and its H\"older seminorm is bounded in $L^p$.

\paragraph{Step 3: coordinates with large index.}
Fix $\delta\in(0,K)$ and $s\in[\delta,K]$. By It\^o's isometry and \eqref{cd-scaled-moments},
\begin{equation*}
\E\big|\Theta_N^{>J}(s)\big|^2\le C\sum_{i>J}\int_0^s\frac{k_{i,N}(s-v)^2}{1+R_{i,N}v}\dd v.
\end{equation*}
We split the integral at $v=s/2$ and abbreviate $R=R_{i,N}$. For $v\le s/2$, we have $k_{i,N}(s-v)^2\le C_\delta R^{-2}$ by \eqref{cd-kernel-bounds}, and $\int_0^{s/2}(1+Rv)^{-1}\dd v\le R^{-1}\log(1+RK)$. For $v\ge s/2$, we have $(1+Rv)^{-1}\le2(R\delta)^{-1}$ and
\begin{equation*}
\int_0^{s/2}k_{i,N}(u)^2\dd u\le C\int_0^K\frac{u+N^{-2/3}}{(1+Ru)^2}\dd u\le C\Big(\frac{\log(1+RK)}{R^2}+\frac{N^{-2/3}}{R}\Big).
\end{equation*}
Together with \eqref{eq:cproc-sum-tail}, this gives
\begin{equation}\label{cd-radial-tail}
\E\big|\Theta_N^{>J}(s)\big|^2\le C_\delta\sum_{i>J}\Big(\frac{\log(2+R_{i,N})}{R_{i,N}^3}+\frac{N^{-2/3}}{R_{i,N}^2}\Big)\le C_\delta\Big(\frac{\log(2+J)}{J}+N^{-2/3}\Big).
\end{equation}
We now show that
\begin{equation}\label{eq:cproc-Theta-tail-path}
\lim_{J\to\infty}\limsup_{N\to\infty}\PP\Big(\sup_{s\in[0,K]}\big|\Theta_N^{>J}(s)\big|>\eta\Big)=0\qquad\text{for every }\eta>0.
\end{equation}
Let $[\cdot]_\alpha$ denote the H\"older seminorm of Step 2. For $M,\delta,h>0$, we have
\begin{multline*}
\PP\Big(\sup_{[0,K]}|\Theta_N^{>J}|>\eta\Big)\le\PP\big([\Theta_N^{>J}]_\alpha>M\big)+\ind_{\{M\delta^\alpha\ge\eta/2\}}+\ind_{\{Mh^\alpha\ge\eta/2\}}\\
+\sum_{j}\PP\big(|\Theta_N^{>J}(s_j)|>\eta/2\big),
\end{multline*}
where $s_j$ ranges over the grid $(\delta+h\mathbb Z)\cap[\delta,K]$; here we used $\Theta_N^{>J}(0)=0$ for $s\le\delta$. We choose $M$ so that the first term is small uniformly in $N$ and $J$, then $\delta$ and $h$ so that the indicators vanish, and then $J$ so that, by \eqref{cd-radial-tail} and Chebyshev's inequality, the sum is small for all large $N$. This proves \eqref{eq:cproc-Theta-tail-path}. For the weighted norm, Doob's inequality \cite[Chapter~II, (1.7)]{RevuzYor1999}, \eqref{cd-scaled-moments}, and \eqref{eq:cproc-sum-tail} give
\begin{equation}\label{eq:cproc-M-tail}
\E\sup_{s\le K}\big|\mathcal M_N^{>J}(s)\big|^2\le C\sum_{i>J}\frac{1}{R_{i,N}^2}\int_0^K\frac{\dd v}{1+R_{i,N}v}\le C\frac{\log(2+J)}{J}.
\end{equation}

Finally, we bound the weighted norm of the coordinates with large index, uniformly in time. Fix $i>J$ and abbreviate $R=R_{i,N}$. We cover $[\delta,K]$ by at most $KR+1$ intervals $[a,b_a]$, where $b_a=\min\{a+R^{-1},K\}$. On such an interval, we have by \eqref{eq:cproc-scaled-sde} that
\begin{equation*}
\sup_{a\le s\le b_a}|U_{i,N}(s)|\le|U_{i,N}(a)|+\sqrt2\sup_{a\le s\le b_a}\Big|\int_a^se^{2R(v-a)}\sqrt{Z_N(v)}\dd B_i^N(v)\Big|,
\end{equation*}
since $e^{-2R(s-a)}\le1$. By the Burkholder--Davis--Gundy inequality and \eqref{cd-scaled-moments}, the $L^{2p}$ norm of the supremum on the right is at most $C_p(e^4R^{-1}\sup_{\delta\le v\le K}\|Z_N(v)\|_p)^{1/2}\le C_{p,\delta}R^{-1/2}$, and $\|U_{i,N}(a)^2\|_p\le C_p(1+R\delta)^{-1}\le C_{p,\delta}R^{-1}$. Combining these bounds, we find that the $p$th moment of $\sup_{[a,b_a]}U_{i,N}^2$ is at most $(C_{p,\delta}R^{-1})^p$. Bounding the maximum over the intervals by the $\ell^p$ norm, we find $\E\sup_{[\delta,K]}U_{i,N}^2\le C_{p,\delta}(KR+1)^{1/p}R^{-1}$. Summing over $i>J$ and using \eqref{eq:cproc-sum-tail}, we obtain for every fixed $p>2$
\begin{equation}\label{cd-weighted-path-tail}
\E\sup_{\delta\le s\le K}\sum_{i>J}\frac{U_{i,N}(s)^2}{R_{i,N}}\le C_{p,\delta}\sum_{i>J}R_{i,N}^{-2+1/p}\le C_{p,\delta}J^{-1/3+2/(3p)}.
\end{equation}

\paragraph{Step 4: limit points and their equations.}
Let $\mathcal X$ be the countable product of copies of $C([0,\infty))$, with the topology of locally uniform convergence in each factor; it is a Polish space. Consider the $\mathcal X$-valued random variable
\begin{equation*}
\mathfrak X_N=\Big((U_{i,N})_{i\ge1},\,Z_N-\psi_N,\,\|U_N\|_{R_N}^2,\,(\Theta_N^{\le J})_{J\ge1},\,(\mathcal M_N^{\le J})_{J\ge1},\,(B_i^N)_{i\ge1}\Big).
\end{equation*}
Every factor is tight by Step 2, so $(\mathfrak X_N)$ is tight. By Prokhorov's theorem \cite[Theorem~5.1]{Billingsley1999} and Skorokhod's representation theorem \cite[Theorem~6.7]{Billingsley1999}, every subsequence has a further subsequence along which, on some probability space, $\mathfrak X_N\to\mathfrak X$ almost surely, where
\begin{equation*}
\mathfrak X=\Big((U_i)_{i\ge1},\,V,\,\mathcal W,\,(\Theta^{\le J})_{J\ge1},\,(\mathcal M^{\le J})_{J\ge1},\,(B_i)_{i\ge1}\Big).
\end{equation*}
We fix such a subsequence and work on this probability space. Let $(\mathcal G_s)$ be the usual augmentation of the filtration generated by $\mathfrak X$, and let $Z=\psi_R+V$ on $(0,\infty)$. \Cref{cd-mean} implies that $Z_N\to Z$ locally uniformly on $(0,\infty)$, almost surely. Note that $Z\ge0$, and that $V(0)=0$ because $Z_N(0)-\psi_N(0)=I_N(0)\to0$.

\emph{The Brownian motions.} For finite $N$, every component of $\mathfrak X_N(v)$, $v\le s$, is $\mathcal F^N_s$-measurable; this includes the coordinates, the amplitude, the weighted norm, the partial sums $\Theta_N^{\le J}$ and $\mathcal M_N^{\le J}$, and the Brownian motions. The increments of $(B_i^N)_i$ after time $s$ are therefore independent of $(\mathfrak X_N(v))_{v\le s}$. For bounded continuous functions $F$ of finitely many coordinates of $(\mathfrak X_N(v))_{v\le s}$ and $G$ of finitely many increments after $s$, we have $\E[F\,G]=\E[F]\,\E[G]$, and this identity passes to the limit. Let $\mathcal H_s^0$ be the $\sigma$-field generated by $\mathfrak X(v)$, $0\le v\le s$. By a monotone class argument \cite[Chapter~0, Section~2]{RevuzYor1999}, the increments of $B$ after $s$ are independent of $\mathcal H_s^0$. Since the law of $(B_i^N)_{i\ge1}$ does not depend on $N$, the $B_i$ are independent standard Brownian motions.
It remains to replace $\mathcal H_s^0$ by its usual augmentation $\mathcal G_s$ \cite[Chapter~I, Section~4]{RevuzYor1999}. Let $Y$ be a bounded $\mathcal G_s$-measurable random variable, let $t_1,\dots,t_m>s$ and $d\ge1$, let $G$ be a bounded continuous function on $\R^{dm}$, and for $r\in[s,\min_jt_j)$ write $G(r)=G\big((B_i(t_j)-B_i(r))_{i\le d,\,j\le m}\big)$. Let $r_k\downarrow s$ with $r_k<\min_jt_j$. For each $k$, the $\sigma$-field $\mathcal G_s$ is contained in the completion of $\mathcal H^0_{r_k}$, so $Y$ is almost surely equal to an $\mathcal H^0_{r_k}$-measurable random variable, and $\E[Y\,G(r_k)]=\E[Y]\,\E[G(r_k)]$. Letting $k\to\infty$, and using the continuity of $B$ and dominated convergence, we obtain $\E[Y\,G(s)]=\E[Y]\,\E[G(s)]$. By a monotone class argument, this identity extends to bounded measurable functions of the increments.
We conclude that the increments of $B$ after $s$ are independent of $\mathcal G_s$, and that $(B_i)_{i\ge1}$ is a sequence of independent $(\mathcal G_s)$-Brownian motions.

\emph{Moments.} By Fatou's lemma, the bounds \eqref{cd-scaled-moments} pass to the limit and give \eqref{eq:cproc-limit-moments}. The increment bounds of Step 2 also pass to the limit; for instance, $\|V(s+h)-V(s)\|_p\le C_ph^{1/4}$.

\emph{Stochastic integrals.} We identify the stochastic integrals of the limit point through the joint laws of the random variables $\mathfrak X_N$, using \cref{lem:cproc-joint-law-identification}. For $x\in\mathcal X$, we write $u_i$, $\zeta$, $\theta^J$, $m^J$, and $b_i$ for the components of $x$ in the positions of $U_{i,N}$, $Z_N-\psi_N$, $\Theta_N^{\le J}$, $\mathcal M_N^{\le J}$, and $B_i^N$ in $\mathfrak X_N$, and for $v>0$ we let
\begin{equation}\label{eq:cproc-common-amplitude}
q_x(v)=\big(\max\{\psi_R(v)+\zeta(v),0\}\big)^{1/2}.
\end{equation}
For each $v>0$, the map $x\mapsto q_x(v)$ is continuous on $\mathcal X$. Since $Z=\psi_R+V\ge0$, we have $q_{\mathfrak X}(v)=\sqrt{Z(v)}$. For finite $N$, $q_{\mathfrak X_N}(v)$ is the square root of the positive part of $Z_N(v)+\psi_R(v)-\psi_N(v)$. The expression inside the positive part may be negative.

Fix $K>0$ and $s\in(0,K]$. We treat three families of stochastic integrals at once. Each family is given by a constant $c$, a finite set $I$ of indices, deterministic weights $\kappa_{i,N}$ and $\kappa_i$ on $[0,s]$, and a continuous function $f$ on $\mathcal X$:
\begin{enumerate}
\item the coordinates: for fixed $i$, we take $c=\sqrt2$, $I=\{i\}$, $\kappa_{i,N}(v)=e^{-2R_{i,N}(s-v)}$, $\kappa_i(v)=e^{-2R_i(s-v)}$, and $f(x)=u_i(s)$;
\item the rescaled squared norm: for fixed $J$, we take $c=\sqrt8$, $I=\{1,\dots,J\}$, $\kappa_{i,N}(v)=k_{i,N}(s-v)$, $\kappa_i(v)=k_i(s-v)$, and $f(x)=\theta^J(s)$;
\item the weighted norm: for fixed $J$, we take $c=\sqrt8$, $I=\{1,\dots,J\}$, $\kappa_{i,N}=R_{i,N}^{-1}$, $\kappa_i=R_i^{-1}$, and $f(x)=m^J(s)$.
\end{enumerate}
For the coordinates, we set $\beta_{i,N}(v)=\kappa_{i,N}(v)\sqrt{Z_N(v)}$ and $\beta_i(v;x)=\kappa_i(v)\,q_x(v)$. For the two other families, we set $\beta_{i,N}(v)=\kappa_{i,N}(v)\sqrt{Z_N(v)}\,U_{i,N}(v)$ and $\beta_i(v;x)=\kappa_i(v)\,q_x(v)\,u_i(v)$. By \eqref{eq:cproc-scaled-sde} and the definitions of $\Theta_N^{\le J}$ and $\mathcal M_N^{\le J}$ in Step~1, we have
\begin{equation}\label{eq:cproc-observed-finite}
f(\mathfrak X_N)=\iota_N+c\sum_{i\in I}\int_0^s\beta_{i,N}(v)\dd B_i^N(v),
\end{equation}
where $\iota_N=e^{-2R_{i,N}s}N^{-1/6}g_i$ for the coordinates and $\iota_N=0$ for the two other families. Here and below, we evaluate expectations of functions of $\mathfrak X_N$ on the original probability space, with the filtration $(\mathcal F^N_v)$. This is legitimate, since they depend only on the law of $\mathfrak X_N$. We show that $f(\mathfrak X)=\Xi$ almost surely, where
\begin{equation}\label{eq:cproc-proposed-limit}
\Xi=c\sum_{i\in I}\int_0^s\beta_i(v;\mathfrak X)\dd B_i(v).
\end{equation}
We choose a Borel version of $\Xi$ as a function of $\mathfrak X$: elementary adapted integral approximations depend measurably on its paths and Brownian coordinates, and an almost surely convergent subsequence of their probability limits supplies such a version. Set this function to zero on the exceptional set where the chosen approximations do not converge.

For each $i\in I$, the process $\beta_i(\cdot\,;\mathfrak X)$ is continuous and $(\mathcal G_v)$-adapted on $(0,s]$; with the value zero at $v=0$, it is progressively measurable. We have $\E Z(v)\le C_1v^{-1/2}$ and $\E[Z(v)U_i(v)^2]\le C_1$ by \eqref{eq:cproc-limit-moments}, and the weights $\kappa_i$ are bounded on $[0,s]$ by \eqref{eq:cproc-weyl-limit} and \eqref{eq:cproc-kernel-limit-bounds}. By Tonelli's theorem, we find $\E\int_0^s\beta_i(v;\mathfrak X)^2\dd v<\infty$. Then \eqref{eq:cproc-proposed-limit} is a sum of square-integrable It\^o integrals with respect to the $(\mathcal G_v)$-Brownian motions $B_i$ \cite[Chapter~IV, Section~2]{RevuzYor1999}.

The approximating functions are left Riemann sums with the limiting weights. For $0<\delta<s$ and an integer $n\ge K$, let $v_j=\delta+j(s-\delta)/n$ for $0\le j\le n$, let $h=(s-\delta)/n$, so that $h\le1$, and define
\begin{equation*}
S_{\delta,n}(x)=c\sum_{i\in I}\sum_{j=0}^{n-1}\beta_i(v_j;x)\big(b_i(v_{j+1})-b_i(v_j)\big).
\end{equation*}
This is a continuous function on $\mathcal X$, since it is a finite sum of products of continuous functions of the values of $x$ at the times $v_j\ge\delta$. The same function $S_{\delta,n}$ is evaluated at $\mathfrak X_N$ and at $\mathfrak X$. A sum formed with $\psi_N$, $R_{i,N}$, and $k_{i,N}$ would change with $N$, and \cref{lem:cproc-joint-law-identification} would not apply to it. Let $\beta_i^n(v;x)=\beta_i(v_j;x)$ for $v\in(v_j,v_{j+1}]$. The value $\beta_i(v_j;\mathfrak X_N)$ is $\mathcal F^N_{v_j}$-measurable, $\beta_i(v_j;\mathfrak X)$ is $\mathcal G_{v_j}$-measurable, and both are square integrable by the bounds below. Then
\begin{equation*}
S_{\delta,n}(\mathfrak X_N)=c\sum_{i\in I}\int_\delta^s\beta_i^n(v;\mathfrak X_N)\dd B_i^N(v),\qquad
S_{\delta,n}(\mathfrak X)=c\sum_{i\in I}\int_\delta^s\beta_i^n(v;\mathfrak X)\dd B_i(v).
\end{equation*}

We now bound $f(\mathfrak X_N)-S_{\delta,n}(\mathfrak X_N)$ and $\Xi-S_{\delta,n}(\mathfrak X)$ in $L^2$. In the following four estimates, $C$ depends only on $c_0$, $C_0$, $K$, and the coordinate $i$ or the number $J$, and $C_\delta$ depends also on $\delta$. The estimates for finite $N$ hold for all $N$ larger than a threshold depending on the same parameters. We use It\^o's isometry and the orthogonality of stochastic integrals with respect to different Brownian motions throughout.

First, we bound the contribution of the entrance. We have
\begin{equation}\label{eq:cproc-entrance-error}
\begin{aligned}
\E\Big|c\sum_{i\in I}\int_0^\delta\beta_{i,N}(v)\dd B_i^N(v)\Big|^2&=c^2\sum_{i\in I}\int_0^\delta\E\beta_{i,N}(v)^2\dd v\\
&\le\begin{cases}C\delta^{1/2}&\text{for the coordinates},\\ CJ\delta&\text{for the two other families}.\end{cases}
\end{aligned}
\end{equation}
For the coordinates, we used $\kappa_{i,N}\le1$ and $\E Z_N(v)=\psi_N(v)\le Cv^{-1/2}$, from \eqref{eq:cproc-psiN-bounds}. For the two other families, we used $\E[Z_N(v)U_{i,N}(v)^2]\le C$, from \eqref{cd-scaled-moments}, and $\kappa_{i,N}\le C$, from \eqref{cd-kernel-bounds} and $R_{i,N}\ge c_0$. In the same way, using \eqref{eq:cproc-limit-moments} and \eqref{eq:cproc-kernel-limit-bounds}, we find that the integral over $[0,\delta]$ in \eqref{eq:cproc-proposed-limit} satisfies the same bound. These are bounds for second moments, and both are at most $C\delta^{1/2}$.

Second, we show that the coefficients $\beta_i(\cdot\,;\mathfrak X_N)$ and $\beta_i(\cdot\,;\mathfrak X)$ have a common modulus of continuity in $L^2$: for $v,v'\in[\delta,s]$ with $|v-v'|\le h\le1$,
\begin{equation}\label{eq:cproc-coefficient-modulus}
\E\big|\beta_i(v;\mathfrak X_N)-\beta_i(v';\mathfrak X_N)\big|^2\le C_\delta h^{1/4},\qquad
\E\big|\beta_i(v;\mathfrak X)-\beta_i(v';\mathfrak X)\big|^2\le C_\delta h^{1/4}.
\end{equation}
Let $A_N(v)=Z_N(v)+\psi_R(v)-\psi_N(v)$, so that $q_{\mathfrak X_N}(v)=(\max\{A_N(v),0\})^{1/2}$, and let $A(v)=Z(v)$. By \eqref{cd-scaled-moments} and the uniform convergence $\psi_N\to\psi_R$ on $[\delta,K]$ from \cref{cd-mean}, we have $\|A_N(v)\|_2\le C_\delta$ and $\|U_{i,N}(v)\|_4\le C$. The increment $A_N(v)-A_N(v')$ is the sum of $\psi_R(v)-\psi_R(v')$, which is at most $C\delta^{-3/2}h$ in absolute value by \eqref{eq:cproc-psiR-bounds}, and of the increment of $Z_N-\psi_N=I_N+\Theta_N$. Then Step~2 implies that $\|A_N(v)-A_N(v')\|_2\le C_\delta h^{1/4}$ and $\|U_{i,N}(v)-U_{i,N}(v')\|_4\le Ch^{1/4}$. Since $A_N(v)\to A(v)$ and $U_{i,N}(v)\to U_i(v)$ almost surely for each $v$, Fatou's lemma shows that these four bounds also hold for $A$ and $U_i$.
We now argue in the same way in the finite model and in the limit, and drop $N$, $\mathfrak X_N$, and $\mathfrak X$ from the notation. For real $a$ and $b$, we have the elementary inequality
\begin{equation}\label{eq:cproc-sqrt-elementary}
\big|(\max\{a,0\})^{1/2}-(\max\{b,0\})^{1/2}\big|^2\le|a-b|.
\end{equation}
It gives
\begin{gather*}
\E|q(v)-q(v')|^2\le\|A(v)-A(v')\|_2\le C_\delta h^{1/4},\qquad
\|q(v)\|_4^2\le\|A(v)\|_2\le C_\delta,\\
\|q(v)-q(v')\|_4^2\le\|A(v)-A(v')\|_2\le C_\delta h^{1/4}.
\end{gather*}
Writing $q(v)U_i(v)-q(v')U_i(v')=(q(v)-q(v'))U_i(v)+q(v')(U_i(v)-U_i(v'))$ and using Minkowski's and H\"older's inequalities, we find
\begin{equation*}
\begin{aligned}
\|q(v)U_i(v)-q(v')U_i(v')\|_2&\le\|q(v)-q(v')\|_4\|U_i(v)\|_4+\|q(v')\|_4\|U_i(v)-U_i(v')\|_4\\
&\le C_\delta h^{1/8},
\end{aligned}
\end{equation*}
and $\|q(v)U_i(v)\|_2\le\|q(v)\|_4\|U_i(v)\|_4\le C_\delta$. Finally, the weights satisfy $|\kappa_i|\le C$ and $|\kappa_i(v)-\kappa_i(v')|\le Ch^{1/2}$ on $[0,s]$. For the coordinates, this follows from $|\partial_ve^{-2R_i(s-v)}|\le2R_i$ and $h\le h^{1/2}$. For the rescaled squared norm, we have $|k_i'(u)|\le Cu^{-1/2}$ by \eqref{eq:cproc-kernel-limit-bounds}, and hence $|k_i(u+h)-k_i(u)|\le2C\sqrt h$ for all $u\ge0$, including $u=0$. The weights $R_i^{-1}$ are constant. We write $\beta_i(v)-\beta_i(v')$ as $\kappa_i(v)$ times the increment of $q$ or $qU_i$, plus $\kappa_i(v)-\kappa_i(v')$ times the value of $q$ or $qU_i$ at $v'$, and we obtain \eqref{eq:cproc-coefficient-modulus}.

Third, we replace the coefficients of the finite model by the common coefficients. For $z\ge0$ and real $d$, $y$, $\kappa$, and $\kappa'$,
\begin{equation}\label{eq:cproc-coefficient-replacement}
\big|\kappa'\sqrt z\,y-\kappa(\max\{z+d,0\})^{1/2}y\big|^2\le2|\kappa'-\kappa|^2zy^2+2\kappa^2|d|y^2.
\end{equation}
To see this, we write the difference as $(\kappa'-\kappa)\sqrt z\,y+\kappa(\sqrt z-(\max\{z+d,0\})^{1/2})y$ and use \eqref{eq:cproc-sqrt-elementary} with $a=z$ and $b=z+d$. No sign condition on $z+d$ is needed. We apply \eqref{eq:cproc-coefficient-replacement} with $z=Z_N(v)$, $d=\psi_R(v)-\psi_N(v)$, $\kappa'=\kappa_{i,N}(v)$, $\kappa=\kappa_i(v)$, and $y=1$ or $y=U_{i,N}(v)$, so that the left side is $|\beta_{i,N}(v)-\beta_i(v;\mathfrak X_N)|^2$. For $v\ge\delta$, \eqref{cd-scaled-moments} gives $\E[Z_N(v)y^2]\le C_\delta$ and $\E y^2\le C$, and we obtain
\begin{equation}\label{eq:cproc-replacement-error}
\begin{gathered}
\sup_{v\in[\delta,s]}\E\big|\beta_{i,N}(v)-\beta_i(v;\mathfrak X_N)\big|^2\le a_{N,\delta},\\
a_{N,\delta}=C_\delta\Big(\max_{i\in I}\sup_{[\delta,s]}|\kappa_{i,N}-\kappa_i|^2+\sup_{[\delta,s]}|\psi_N-\psi_R|\Big).
\end{gathered}
\end{equation}
We have $a_{N,\delta}\to0$ as $N\to\infty$. Indeed, the mean functions converge uniformly on $[\delta,K]$ by \cref{cd-mean}; $|e^{-2R_{i,N}(s-v)}-e^{-2R_i(s-v)}|\le2K|R_{i,N}-R_i|$, and $R_{i,N}\to R_i$ by \eqref{cd-inverse-convergence}; the kernels $k_{i,N}$ converge to $k_i$ uniformly on $[0,K]$ by \cref{cd-mean}; and $R_{i,N}^{-1}\to R_i^{-1}$.

Fourth, using \eqref{eq:cproc-coefficient-modulus}, we have
\begin{equation}\label{eq:cproc-mesh-error}
\E\Big|c\sum_{i\in I}\int_\delta^s\big(\beta_i(v;\mathfrak X_N)-\beta_i^n(v;\mathfrak X_N)\big)\dd B_i^N(v)\Big|^2\le c^2|I|(s-\delta)C_\delta h^{1/4},
\end{equation}
and the same bound holds with $\mathfrak X$ and $B_i$ in place of $\mathfrak X_N$ and $B_i^N$.

We now combine these estimates. By \eqref{eq:cproc-observed-finite}, the difference $f(\mathfrak X_N)-S_{\delta,n}(\mathfrak X_N)$ is the sum of $\iota_N$, of the integral over $[0,\delta]$ in \eqref{eq:cproc-observed-finite}, of $c\sum_{i\in I}\int_\delta^s(\beta_{i,N}(v)-\beta_i(v;\mathfrak X_N))\dd B_i^N(v)$, and of the integral in \eqref{eq:cproc-mesh-error}. Similarly, $\Xi-S_{\delta,n}(\mathfrak X)$ is the sum of the integral over $[0,\delta]$ in \eqref{eq:cproc-proposed-limit} and of the limiting version of the integral in \eqref{eq:cproc-mesh-error}. Using $(a_1+\dots+a_k)^2\le k(a_1^2+\dots+a_k^2)$, $\E\iota_N^2\le N^{-1/3}$, \eqref{eq:cproc-entrance-error}, \eqref{eq:cproc-replacement-error}, and \eqref{eq:cproc-mesh-error}, we obtain
\begin{equation}\label{eq:cproc-left-sum-errors}
\begin{aligned}
\E\big|f(\mathfrak X_N)-S_{\delta,n}(\mathfrak X_N)\big|^2&\le4N^{-1/3}+C\delta^{1/2}+Ca_{N,\delta}+C_\delta h^{1/4},\\
\E\big|\Xi-S_{\delta,n}(\mathfrak X)\big|^2&\le C\delta^{1/2}+C_\delta h^{1/4}.
\end{aligned}
\end{equation}
Given $\eps>0$, we choose $\delta$ so that $C\delta^{1/2}\le\eps/4$ in \eqref{eq:cproc-left-sum-errors}, then $n$ so large that $C_\delta h^{1/4}\le\eps/4$, and then $N_\eps$ so that $4N^{-1/3}+Ca_{N,\delta}\le\eps/2$ for $N\ge N_\eps$. The cutoff $\delta$ and the mesh $h$ are fixed before $N\to\infty$. Then $S_{\delta,n}$ satisfies \eqref{eq:cproc-joint-law-hypothesis} along our subsequence, with $\mathfrak Y_N=\mathfrak X_N$ and $\mathfrak Y=\mathfrak X$, and \cref{lem:cproc-joint-law-identification} implies that $f(\mathfrak X)=\Xi$ almost surely.

\emph{The coordinates.} Fix $i$ and $s>0$. For the coordinates, we have $f(\mathfrak X)=U_i(s)$, and we have shown that
\begin{equation*}
U_i(s)=\sqrt2\int_0^se^{-2R_i(s-v)}\sqrt{Z(v)}\dd B_i(v)\qquad\text{almost surely}.
\end{equation*}
Moreover, $U_i(0)=\lim_NN^{-1/6}g_i=0$. The right side has the continuous version $\sqrt2e^{-2R_is}\int_0^se^{2R_iv}\sqrt{Z(v)}\dd B_i(v)$, since $\E\int_0^Ke^{4R_iv}Z(v)\dd v<\infty$ by \eqref{eq:cproc-limit-moments}. Both sides are continuous in $s$ and agree almost surely at all rational times, and hence at all times. This proves the first line of \eqref{cd-limiting-system}, almost surely for all $s\ge0$ and $i\ge1$ simultaneously.

\emph{The rescaled squared norm.} Fix $J$ and $s>0$. For the rescaled squared norm, we have $f(\mathfrak X)=\Theta^{\le J}(s)$, and we have shown that
\begin{equation*}
\Theta^{\le J}(s)=\sqrt8\sum_{i\le J}\int_0^sk_i(s-v)\sqrt{Z(v)}\,U_i(v)\dd B_i(v)\qquad\text{almost surely}.
\end{equation*}
As a function of $s$, the right side is a stochastic convolution, not a martingale. The continuous process $\Theta^{\le J}$, which vanishes at $s=0$, is a version of it, and we use this version. Since $V-\Theta^{\le J}=\lim_N(I_N+\Theta_N^{>J})$ almost surely in $C([0,K])$, where on the probability space of the representation $I_N+\Theta_N^{>J}$ stands for the continuous function $Z_N-\psi_N-\Theta_N^{\le J}$ of $\mathfrak X_N$, the bound \eqref{eq:cproc-Theta-tail-path} and $I_N\to0$ give
\begin{equation*}
\PP\Big(\sup_{[0,K]}|V-\Theta^{\le J}|>\eta\Big)\le\liminf_{N\to\infty}\PP\Big(\sup_{[0,K]}|I_N+\Theta_N^{>J}|>\eta\Big)\to0\qquad(J\to\infty).
\end{equation*}
This proves the second line of \eqref{cd-limiting-system}, with convergence of the series in probability in $C([0,K])$.

\emph{The weighted norm.} Fix $J$ and $s>0$. For the weighted norm, we have $f(\mathfrak X)=\mathcal M^{\le J}(s)$, and we have shown that
\begin{equation*}
\mathcal M^{\le J}(s)=\sqrt8\sum_{i\le J}\int_0^sR_i^{-1}\sqrt{Z(v)}\,U_i(v)\dd B_i(v)\qquad\text{almost surely}.
\end{equation*}
As a function of $s$, the right side is a square-integrable martingale, and the continuous process $\mathcal M^{\le J}$, which vanishes at $s=0$, is a version of it. For finite $N$, Step~1 gives $\|U_N(s)\|_{R_N}^2=\|U_N(0)\|_{R_N}^2+\mathcal M_N^{\le J}(s)+\mathcal M_N^{>J}(s)$, and $\|U_N(0)\|_{R_N}^2\to2$ in probability. On the probability space of the representation, the value at zero of the component $\|U_N\|_{R_N}^2$ of $\mathfrak X_N$ converges almost surely to $\mathcal W(0)$, so that $\mathcal W(0)=2$, and $\mathcal M_N^{>J}$ stands for the continuous function $\|U_N\|_{R_N}^2-\|U_N(0)\|_{R_N}^2-\mathcal M_N^{\le J}$ of $\mathfrak X_N$. Then $\mathcal W-2-\mathcal M^{\le J}=\lim_N\mathcal M_N^{>J}$ uniformly on $[0,K]$, almost surely, and Fatou's lemma and \eqref{eq:cproc-M-tail} give
\begin{equation*}
\E\sup_{s\le K}\big|\mathcal W(s)-2-\mathcal M^{\le J}(s)\big|^2\le C\frac{\log(2+J)}{J}.
\end{equation*}
In particular, the series in the third line of \eqref{cd-limiting-system} converges in probability in $C([0,K])$, with sum $\mathcal W-2$. The process $\mathcal W$ is continuous on $[0,\infty)$ with $\mathcal W(0)=2$, and it equals the right side of the third line of \eqref{cd-limiting-system}. It remains to identify $\mathcal W(s)$ with $\|U(s)\|_R^2$ for $s>0$. Fix $0<\delta<K$. For fixed $J$, the process $\sum_{i\le J}U_{i,N}^2/R_{i,N}$ converges uniformly on $[\delta,K]$ to $\sum_{i\le J}U_i^2/R_i$, so
\begin{equation*}
\sup_{[\delta,K]}\Big(\mathcal W-\sum_{i\le J}\frac{U_i^2}{R_i}\Big)=\lim_{N\to\infty}\sup_{[\delta,K]}\sum_{i>J}\frac{U_{i,N}^2}{R_{i,N}}
\end{equation*}
almost surely. Using Fatou's lemma and \eqref{cd-weighted-path-tail}, we see that the expectation of the left side is at most $C_{p,\delta}J^{-1/3+2/(3p)}$. The left side is nonnegative and nonincreasing in $J$, and therefore it tends to zero almost surely as $J\to\infty$. This shows that, almost surely, $U(s)\in E_R$ and $\|U(s)\|_R^2=\mathcal W(s)$ for all $s\in[\delta,K]$, and the series $\sum_iU_i^2/R_i$ converges uniformly on $[\delta,K]$. Since each $U_i$ is continuous, we deduce from the uniform convergence that $s\mapsto U(s)$ is continuous from $[\delta,K]$ into $E_R$. As $\delta$ and $K$ are arbitrary, we find that $U$ is continuous from $(0,\infty)$ into $E_R$, and $\|U(s)\|_R^2=\mathcal W(s)\to2$ as $s\downarrow0$.

\paragraph{Step 5: positivity of $Z$.}
Fix $0<\delta<K$. On $[\delta,K]$, $\psi_R$ has a bounded derivative by \eqref{eq:cproc-psiR-bounds}, and $V$ satisfies $\|V(s+h)-V(s)\|_{p}\le C_ph^{1/4}$ for every $p$. Taking $p=16$ in Kolmogorov's criterion, we find that the H\"older seminorm $[Z]_{1/8}$ of $Z$ on $[\delta,K]$ is almost surely finite. We also have $\E Z(s)^{-16}\le C$ for $s\in[\delta,K]$, by \eqref{eq:cproc-limit-moments}. Let $h\in(0,1)$, and let $s_j$ range over the grid $(\delta+h\mathbb Z)\cap[\delta,K]$, which has at most $K/h+1$ points. If $\min_{[\delta,K]}Z=0$ and $[Z]_{1/8}\le M$, then $Z(s_j)\le Mh^{1/8}$ for some $j$. By Markov's inequality,
\begin{equation*}
\begin{aligned}
\PP\Big(\min_{[\delta,K]}Z=0,\ [Z]_{1/8}\le M\Big)&\le\sum_j\PP\big(Z(s_j)\le Mh^{1/8}\big)\\
&\le\Big(\frac Kh+1\Big)C\,(Mh^{1/8})^{16}\le C(K+1)M^{16}h.
\end{aligned}
\end{equation*}
Letting $h\downarrow0$ and then $M\to\infty$, we find that $\min_{[\delta,K]}Z>0$ almost surely. Since the parameters $\delta$ and $K$ are arbitrary, $Z(s)>0$ for all $s>0$, almost surely. Finally, $\sqrt s\,Z(s)=\sqrt s\,\psi_R(s)+\sqrt s\,V(s)\to\psi_*$ by \eqref{cd-mean-endpoints} and $V(0)=0$.

\paragraph{Step 6: uniqueness and convergence.}
By Steps 4 and 5, $(U,V)$ is a global entrance solution in the sense of \cref{def:cproc-solution}, with $\psi=\psi_R$, on the filtered space $(\mathcal G_s)$. The hypotheses of \cref{cd-clipped-picard} hold for $(R,\psi_R,(k_i))$ by \eqref{eq:cproc-weyl-limit}, \cref{cd-mean}, and \cref{lem:cproc-limit-kernels}. By \cref{cd-clipped-picard}(c), on every interval $[0,K]$, the joint law of $(U,V,B)$ is the image of $\mathbb W$ under $b\mapsto(\mathfrak S(b),b)$, and so it does not depend on the subsequence. The other components of $\mathfrak X$ are determined by $(U,V,B)$: $\mathcal W=\|U\|_R^2$, and $\Theta^{\le J}$ and $\mathcal M^{\le J}$ are stochastic integrals of functionals of $(U,V,B)$, whose laws are determined by the law of $(U,V,B)$ by the argument of Step 6 of the proof of \cref{cd-clipped-picard}. It follows that all subsequential limits of $(\mathfrak X_N)$ have the same law, and the whole sequence converges in law. This proves the existence and uniqueness in (a), the properties in (b), and the first claim in (c). Finally, on $[s_0,K]$ the limit amplitude $Z$ is bounded below by a positive random constant, and $\psi_N\to\psi_R$ uniformly on $[s_0,K]$. The map $(u_1,\dots,u_J,v)\mapsto(u_i/\sqrt{\psi_R+v})_{i\le J}$ is continuous on the set of paths with $\min_{[s_0,K]}(\psi_R+v)>0$. Using \eqref{eq:cproc-normalized} and the continuous mapping theorem (applied on the Skorokhod space, where the convergence is almost sure), we obtain \eqref{cd-angular-functional}.
\end{proof}

\section{The limiting Gibbs law}\label{sec:cproc-stationary-law}

We recall the countable conditioned Gaussian laws of \cref{sec:cond-countable}; at the critical temperature, this construction is due to Du and Huang \cite[Section~2.2]{DuHuang2026}.
Let $v=(v_i)_{i\ge1}$ be a variance sequence, that is, a sequence of positive numbers with $\sum_iv_i^2<\infty$, and suppose that $\sum_iv_i=\infty$. Let $g=(g_i)$ be independent standard Gaussian variables. The series $T_v=\sum_iv_i(g_i^2-1)$ converges in $L^2$ and almost surely, and for $J\ge0$ the tail $\sum_{i>J}v_i(g_i^2-1)$ has a bounded continuous density $p_{v,>J}$, which is strictly positive on $\R$ (\cref{lem:cond-three}); we write $p_v=p_{v,>0}$. The conditioned law $\mu_v$ (\cref{def:cond-countable}) is the unique probability measure on $\R^{\N}$ under which, for every $J\ge1$, the standardized coordinates $z_i=y_i/\sqrt{v_i}$, $i\le J$, have the density
\begin{equation}\label{cd-pin-head-density}
\frac{p_{v,>J}\big(-\sum_{i\le J}v_i(z_i^2-1)\big)}{p_v(0)}\,\prod_{i\le J}\frac{e^{-z_i^2/2}}{\sqrt{2\pi}}
\end{equation}
on $\R^J$ (\cref{lem:cond-consistent,gp-head-density}). Under $\mu_v$, the centered radial series $\sum_i(y_i^2-v_i)$ converges and vanishes almost surely (\cref{prop:cond-constraint}). In other words, $\mu_v$ is the law of a centered Gaussian sequence with variances $v_i$ conditioned on the renormalized radial constraint $\sum_i(y_i^2-v_i)=0$. For a finite variance sequence with $N$ coordinates, $\mu_v$ is the law of a centered Gaussian vector with variances $v_i$ conditioned on $|y|^2=\sum_iv_i$, that is, the Bingham law \eqref{gp-pin-definition}; the analogous marginal-density formula holds for $J\le N-3$, with the density of the finite sum over the remaining coordinates. We write $z_i$ for the standardized coordinates, which are denoted $Z_i$ in \cref{sec:conditioning}, to avoid a clash with the amplitude $Z$.

For a critical sequence with limit $R$, we set
\begin{equation}\label{eq:cproc-pin-variances}
v_i^R=\frac1{2R_i},\qquad v_{i,N}=\frac{1}{2R_{i,N}}\quad(i\le N),\qquad\rho_R=\mu_{v^R}.
\end{equation}
We call $\rho_R$ the limiting Gibbs law. In the critical window, $\rho_\sigma=\rho_{R^\sigma}$ is the law of a Gaussian sequence with variances $\varrho_\sigma^{-1}$ (for $i=1$) and $(\Delta_i+\varrho_\sigma)^{-1}$ (for $i\ge2$), conditioned on the renormalized radial constraint.

\begin{corollary}[The limiting Gibbs law]\label{cd-pin}
Let $(r_{i,N})$ be a critical sequence with limit $R$, and use the notation \eqref{eq:cproc-pin-variances}. Then the following hold.
\begin{enumerate}
\item The law $\rho_R$ is well defined. The density $p_{v^R}$ is continuous and strictly positive on $\R$; in particular, $p_{v^R}(0)>0$. The marginals of $\rho_R$ are given by \eqref{cd-pin-head-density}, and $\sum_i(y_i^2-v_i^R)=0$ $\rho_R$-almost surely.
\item Let $x$ have law $\pi_A$, as in \eqref{eq:cproc-generator}, and let $y^N=(N^{-1/3}x_i)_{i\le N}$. Then $y^N$ has law $\mu_{v_N}$, with $v_N=(v_{i,N})_{i\le N}$. For every $J\ge1$, the law of $(y_i^N)_{i\le J}$ converges in total variation to the law of $(y_i)_{i\le J}$ under $\rho_R$.
\item There exists $N_0$ such that the following holds for every $\mu\in\{\rho_R\}\cup\{\mu_{v_N}:N\ge N_0\}$, with standardized coordinates $z_i=y_i/\sqrt{v_i}$ (where $v=v^R$ or $v=v_N$, respectively). For every $q>0$, $\sup_i\E_\mu|z_i|^q\le C_{q,R}$. For every $p\ge1$ and every real sequence $(w_i)\in\ell^2$, the series $\sum_iw_i(z_i^2-1)$ converges in $L^p(\mu)$, and
\begin{equation}\label{cd-pin-chaos}
\Big\|\sum_iw_i(z_i^2-1)\Big\|_{L^p(\mu)}\le C_{p,R}\Big(\sum_iw_i^2\Big)^{1/2}.
\end{equation}
The constants depend on $q$, $p$, and the sequence $(r_{i,N})$, but not on $N$, $i$, or $w$. More generally, for every $m\ge3$ there exists $N_0(m)$ such that, for $\mu$ as above with $N\ge N_0(m)$, the law of $(z_i)_{i\le m}$ under $\mu$ has density at most $C_{m,R}$ with respect to the standard Gaussian law on $\R^m$.
\item We have $\rho_R(E_R)=1$, and $\E_{\rho_R}\|y\|_R^2\le C_{2,R}\sum_i(2R_i^2)^{-1}<\infty$.
\end{enumerate}
\end{corollary}

\begin{proof}
Note that $v^R$ is a variance sequence, by \eqref{eq:cproc-weyl-limit}, with $\sum_i(v_i^R)^2=\frac14\sum_iR_i^{-2}<\infty$ and $\sum_iv_i^R=\infty$. Part (a) follows from \cref{lem:cond-three,lem:cond-consistent,gp-head-density,prop:cond-constraint}; for the positivity of $p_{v^R}$ on all of $\R$, we use that $\sum_iv^R_i=\infty$.

For (b), \cref{prop:cond-bingham} shows that $\pi_A=\mu_q$ in the eigencoordinates, with $q_i=(z-\alpha_i)^{-1}=(2r_{i,N})^{-1}$, and that $N^{-1/3}x$ has law $\mu_{N^{-2/3}q}=\mu_{v_N}$. Using \eqref{cd-inverse-convergence}, we have $\sum_i(v_{i,N}-v_i^R)^2=\frac14\sum_i|R_{i,N}^{-1}-R_i^{-1}|^2\to0$. The second part of \cref{prop:cond-continuity}, with $v^{(N)}=v_N$ and $A=\{1,\dots,J\}$, implies the convergence of $(y_i^N)_{i\le J}$ in total variation.

For (c), let $B=\{1,\dots,m\}$ with $m\ge3$. The first part of \cref{prop:cond-continuity} gives the bound $M(v^R,B)$ on the density of $(z_i)_{i\le m}$ under $\rho_R$ with respect to the standard Gaussian law, the bound $\sup_i\E_{\rho_R}|z_i|^q\le M(v^R,B)\E|G|^q$ for $q\ge1$, and \eqref{cd-pin-chaos} with $C_{p,R}=2C_pM(v^R,B)^{1/p}$, where $G$ is a standard Gaussian variable and $C_p$ is the constant of \cref{lem:cond-chaos}. By the second part of \cref{prop:cond-continuity}, the same bounds hold for $\mu_{v_N}$, $N\ge N_0(m)$, with $M(v^R,B)$ replaced by $2M(v^R,B)$. We take $N_0=N_0(3)$ for the moment bounds and the moment bounds for quadratic statistics; for $q<1$, the moment bound follows from the case $q=1$.

Finally, the first part of \cref{prop:cond-continuity} with $u_i=v_i^R/R_i$ gives $\E_{\rho_R}\|y\|_R^2=\E_{\rho_R}\sum_iu_iz_i^2\le M(v^R,B)\sum_i(2R_i^2)^{-1}$, which proves (d).
\end{proof}