\chapter{The threshold inputs for general pure models}\label{app:general-selection} In this section, we prove \cref{prop:threshold-inputs} for every $p\ge3$. For $p\in\{3,4\}$, \cref{sec:tree-ceilings} proves both parts of that proposition by a deterministic inequality for tree vectors. The argument given here does not use the tree ceilings and works for every pure model. Part~\ref{item:threshold-ceiling} of \cref{prop:threshold-inputs} is the input to the proof of marginality in \cref{sel:positive-marginality}, and part~\ref{item:threshold-forgetting} is the input to the proof of forgetting in \cref{sel:fixed-wait-forgetting}. The finite forks of \cref{sec:general-forks}, the overlap penalty of \cref{sel:absolute-field-cost}, and the saturation identity \eqref{sel:fork-saturation} are used again in \cref{resp:appendix-p}, where we extend the localization of the response to every $p\ge3$. We use the landscape bounds of \cref{lem:landscape-moments}, the fixed-horizon bridge of \cref{prop:hard-bridge}, and the Brownian form of the averaged response in \cref{lem:brown-force-response}. From \cref{sec:ck-structure}, we use the energy identity of \cref{prop:energy-bridge} and the bound \eqref{eq:finite-lag-flow-moments} on the moments of the response, proved there. From \cref{sec:entropy-clusters}, we use the entropy budget of \cref{lem:entropy-budget}, the integration by parts formula \eqref{eq:gibbs-current-integration-by-parts}, the Fisher majorant of \cref{thm:fisher-clock}, and the stationary clusters of \cref{prop:stationary-clusters}. We also use the formulas \eqref{sel:prefix-norm} and \eqref{sel:prefix-energy} for $d_L$ and $h_L$. The proof of part~\ref{item:threshold-forgetting} uses, in addition, the identities \eqref{sel:limiting-center-energy} and \eqref{sel:center-amgm} and the correlation modulus \eqref{eq:global-modulus}. \paragraph{Ideas of the proof.} We obtain the energy ceiling in part~\ref{item:threshold-ceiling} from the algorithmic threshold of Huang and Sellke. At a late time $T$, we continue the trajectory by $K$ independent Brownian futures of length $L$, which we call a fork, and average their endpoints. The current identity of \cref{sec:entropy-clusters} shows that two branches of the fork have overlap close to $c(a+d)$, where $a$ and $d$ are their lengths. We compute the energy of the average exactly by Gaussian integration by parts. Subtracting the energy of a single branch removes the common past in the dimension limit. Using the hard--soft approximation of Sellke, we approximate the average by a function of the disorder that is Lipschitz with a constant independent of the dimension. The algorithmic threshold then bounds its energy. Letting $K\to\infty$, we obtain part~\ref{item:threshold-ceiling}. For part~\ref{item:threshold-forgetting}, we add a deterministic field to the threshold. By convex duality, the field version of the threshold penalizes the overlap of a near-threshold algorithm with any direction that is independent of the disorder. However, at positive times, an earlier state $X_s$ depends on the disorder. We therefore disclose the disorder in stages, along a finite sequence of gradient queries, and apply the penalty to one new direction at each stage. An Euler--Maruyama approximation places $X_s$ close to the span of finitely many such queries. Under marginality, the normalized average of a fork saturates the threshold, and the penalty forces its overlap with $X_s$ to vanish. Up to normalization, this overlap is $C(t,s)$. \paragraph{Attribution.} The algorithmic threshold is \cite[Theorem~1 and Corollary~1.8]{hs2023}; for even $p$ it was proved earlier in \cite{hs2021}. In \cite[Section~3]{sellke2024}, Sellke combined it with a Lipschitz approximation of Langevin dynamics on a fixed time interval to show that the energy of the dynamics stays below the threshold. We follow that argument in \cref{sec:general-forks}. Trajectories with a common past and independent future noise are a standard construction in the physics of aging. Cugliandolo and Dean considered replicas that share their noise up to a waiting time and receive independent noise afterward in \cite[arXiv version~1, Section~II]{cugliandolodean1995}. Barrat, Burioni, and M\'ezard classified aging through the overlaps of such copies in \cite[Sections~2 and~4]{bbm1996}. Franz and Virasoro proposed averaging cloned futures to estimate a quasi-state magnetization in \cite[arXiv version~3, Section~III]{franzvirasoro2000}, and noise-averaged magnetizations are central to Biroli's dynamical TAP approach in \cite[arXiv version~2, Section~III.B]{biroli1999}. Here we prove the overlap and energy identities of the forks from the current identity of \cref{sec:entropy-clusters} and from Gaussian integration by parts. The disclosure argument of \cref{sec:general-disclosure} is a Gaussian conditioning argument; we credit its antecedents there. \paragraph{Order of limits.} Every horizon, number of branches, field strength, approximation accuracy, and Lipschitz constant below is fixed while the dimension tends to infinity, and we take the long-time and outer limits only afterward. We take dimension limits along the subsequence $(N_j)$ of \cref{thm:fisher-clock}, and we often write $N\to\infty$ for $N=N_j$, $j\to\infty$. The parameters and their order are as follows: \begin{equation}\label{sel:aging-limit-order} N_j\to\infty,\qquad n\to\infty,\qquad \varepsilon\downarrow0,\qquad K\to\infty,\qquad L\to\infty,\qquad M\to\infty. \end{equation} Here $n$ indexes a sequence of late times ($T_n$ in \cref{sec:general-forks} and $t_n$ in \cref{sec:general-transfer}), $\varepsilon$ is the accuracy of the Lipschitz approximation in \cref{sel:finite-forks}, $K$ is the number of branches of a fork, $L$ is their length, and $M$ is the number of steps of the Euler--Maruyama scheme of the proof of \cref{sel:general-fixed-wait}. Each parameter is fixed while the limits to its left are taken. Part~\ref{item:threshold-ceiling} uses only the first four limits, at a fixed $L$; part~\ref{item:threshold-forgetting} uses all six. For the forgetting proof, it suffices to take the approximation accuracies $\varepsilon=1/(m+1)$, $m\in\mathbb N$, so $\varepsilon\downarrow0$ there means $m\to\infty$. We remove the tolerance and the field strengths of the threshold theorem inside the proof of \cref{sel:absolute-field-cost}, after the dimension limit at fixed values of all other parameters. Constants of the Lipschitz maps may depend on every fixed parameter, and they are used only before the limits in which that parameter varies. \section{The algorithmic threshold and its overlap penalty} \label{sec:general-field-threshold} Throughout this section, $\norm{\cdot}$ is the Euclidean norm, \[ B_N=\{x\in\R^N:\norm x\le\sqrt N\}, \] and $\Pi_{B_N}$ is the metric projection onto $B_N$, which is $1$-Lipschitz. The coefficient tensor $J$ of \eqref{eq:hamiltonian} lies in the $N^p$-dimensional space $(\R^N)^{\otimes p}$, with its Euclidean norm, and has the standard Gaussian law there. More explicitly, we keep the ordered coefficients and use the Hilbert norm \[ \norm J_{\mathrm{coef}}^2 =\sum_{i_1,\dots,i_p=1}^N J_{i_1\dots i_p}^2 . \] This norm is not rescaled with the dimension. When the dependence on $J$ matters, we write $H_{N,J}$ for $H_N$. An algorithm is a measurable map $\mathcal A_N=\mathcal A_N(J,\zeta)$ with values in $B_N$, where the seed $\zeta$ is a random variable independent of $J$ (possibly absent). For $0\le L_{\mathrm{Lip}}<\infty$, we say that $\mathcal A_N$ is $L_{\mathrm{Lip}}$-Lipschitz if, for almost every value of $\zeta$, \[ \norm{\mathcal A_N(J,\zeta)-\mathcal A_N(J',\zeta)}\le L_{\mathrm{Lip}}\norm{J-J'}_{\mathrm{coef}} \qquad\text{for all }J,J'\in(\R^N)^{\otimes p}. \] The constant $L_{\mathrm{Lip}}$ is deterministic and does not depend on $N$ or on the seed. Recall the tensor norm $Q_N$ of \eqref{eq:tensor-norm} and the constants $A_p,c_p>0$ of \cref{lem:landscape-moments}, for which $\Prob(Q_N>A_p)\le e^{-c_pN}$. We write \[ Q_*=\sup_N(\E Q_N^2)^{1/2}, \] which is finite by \cref{lem:landscape-moments}. By the definition of $Q_N$, a contraction of $J$ against vectors $u_1,\dots,u_p$ is at most $\sqrt N\,Q_N\norm{u_1}\cdots\norm{u_p}$ in absolute value. A derivative of order $k\le2$ of $H_N$ at $x$, applied to $k$ vectors, is $N^{-(p-1)/2}$ times a sum of $p(p-1)\cdots(p-k+1)$ such contractions in which $p-k$ slots equal $x$. For $R\ge1$ and $\norm x\le R\sqrt N$, this implies that \begin{equation}\label{eq:general-ball-derivatives} \begin{gathered} \frac{|H_N(x)|}N\le R^pQ_N,\qquad \norm{\nabla H_N(x)}\le pR^{p-1}Q_N\sqrt N,\\ \norm{\nabla^2H_N(x)}_\op\le p(p-1)R^{p-2}Q_N. \end{gathered} \end{equation} For $R=1$, the first two bounds read \begin{equation}\label{sel:tensor-ui} \sup_{x\in B_N}\frac{|H_N(x)|}N\le Q_N,\qquad \sup_{x\in B_N}\norm{\nabla H_N(x)}\le pQ_N\sqrt N. \end{equation} In particular, the normalized energy of any algorithm is bounded by $Q_N$, whose moments are bounded uniformly in $N$. Now let $x$ and $y$ be random vectors of norm at most $R\sqrt N$. The segment between them lies in the same ball. Using the mean value theorem, \eqref{eq:general-ball-derivatives}, and the Cauchy--Schwarz inequality, we find \begin{equation}\label{eq:general-energy-transfer} \frac{\E|H_N(x)-H_N(y)|}N \le pR^{p-1}Q_*\left(\E\frac{\norm{x-y}^2}N\right)^{1/2}. \end{equation} For $\kappa\ge0$, let $q_1(\kappa)=(\kappa^2/(p(p-2)))^{1/(p-1)}$, and define \begin{equation}\label{eq:alg-field} \ALG_p(\kappa)= \begin{cases} \Eth+\dfrac{p-2}{p}\sqrt{p(p-1)}\,q_1(\kappa)^{p/2}, &\kappa^2\le p(p-2),\\[3mm] \sqrt{p+\kappa^2},&\kappa^2\ge p(p-2). \end{cases} \end{equation} In terms of $\kappa$, the first line equals $\Eth+\gamma_p\kappa^{p/(p-1)}$ with $\gamma_p=((p-2)/p)\sqrt{p(p-1)}\,[p(p-2)]^{-p/(2(p-1))}$. \begin{theorem}[External-field Lipschitz threshold of Huang and Sellke] \label{sel:published-field-threshold} Fix an integer $p\ge3$, $0\le L_{\mathrm{Lip}}<\infty$, $\kappa\ge0$, and $\eta>0$. There exist $c>0$ and $N_0$, depending only on $p$, $L_{\mathrm{Lip}}$, $\kappa$, and $\eta$, such that the following holds for every $N\ge N_0$. Let $w\in\R^N$ be deterministic with $\norm w=\sqrt N$, and let $\mathcal A_N:(\R^N)^{\otimes p}\to B_N$ be $L_{\mathrm{Lip}}$-Lipschitz. Then \begin{equation}\label{sel:field-threshold} \Prob\left(\frac{H_N(\mathcal A_N(J))+\kappa\ip{w}{\mathcal A_N(J)}}N \ge\ALG_p(\kappa)+\eta\right)\le e^{-cN}. \end{equation} \end{theorem} \Cref{sel:published-field-threshold} is the case of one species and $\xi(q)=q^p$ of \cite[Theorem~1 and Corollary~1.8]{hs2023}. We now check that our setting satisfies the hypotheses there. In \cite{hs2023}, the Hamiltonian is $\ip{h}{\sigma}$ plus a Gaussian part on the ball of radius $\sqrt N$, the field is $h$ times the all-ones vector $\mathbf 1$ with $h\ge0$, and an algorithm is a function of the disorder coefficients, concatenated into one vector with the Euclidean distance. Theorem~1 there states that, for fixed $\tau,\eta>0$ and $N$ large, every $\tau$-Lipschitz algorithm reaches energy at least $\ALG+\eta$ with probability at most $e^{-cN}$, with $c$ and the threshold for $N$ depending on $\tau$, $\eta$, the covariance, and the field only. For one species with $\xi(q)=q^p$, the Gaussian part is exactly $H_N$ of \eqref{eq:hamiltonian}, whose ordered coefficients are independent standard Gaussians; we take $h=\kappa$. Our algorithms are functions of $J$ alone, the field being deterministic. Corollary~1.8 of \cite{hs2023}, which covers odd and even $p$, states that $\ALG=(\xi'(1)+\kappa^2)^{1/2}$ if $\xi'(1)+\kappa^2\ge\xi''(1)$, and otherwise \[ \ALG=q_1\,\xi''(q_1)^{1/2}+\int_{q_1}^1\xi''(q)^{1/2}\,\dd q, \] where $q_1\in[0,1)$ solves $\xi'(q_1)+\kappa^2=q_1\xi''(q_1)$. For $\xi(q)=q^p$, the first condition reads $\kappa^2\ge p(p-2)$, the equation for $q_1$ reads $\kappa^2=p(p-2)q_1^{p-1}$, and \[ \begin{aligned} q_1\xi''(q_1)^{1/2}+\int_{q_1}^1\xi''(q)^{1/2}\,\dd q &=\sqrt{p(p-1)}\left(q_1^{p/2}+\frac2p\big(1-q_1^{p/2}\big)\right)\\ &=\Eth+\frac{p-2}p\sqrt{p(p-1)}\,q_1^{p/2}, \end{aligned} \] where we used $2\sqrt{p(p-1)}/p=\Eth$. This is \eqref{eq:alg-field}. At $\kappa^2=p(p-2)$, we have $q_1=1$ and both lines of \eqref{eq:alg-field} equal $\sqrt{p(p-1)}$. It remains to replace $\mathbf 1$ by an arbitrary direction $w$. Let $O$ be an orthogonal matrix such that $O\mathbf 1=w$. The map $J\mapsto O^{\otimes p}J$ is a linear isometry of $(\R^N)^{\otimes p}$ that preserves the standard Gaussian law, and $H_{N,O^{\otimes p}J}(Ox)=H_{N,J}(x)$ for all $x$. Given $\mathcal A_N$, the map $\mathcal A'_N(J)=O^{\mathsf T}\mathcal A_N(O^{\otimes p}J)$ is $L_{\mathrm{Lip}}$-Lipschitz with values in $B_N$, and, with $J'=O^{\otimes p}J$, \[ H_{N,J}(\mathcal A'_N(J))+\kappa\ip{\mathbf 1}{\mathcal A'_N(J)} =H_{N,J'}(\mathcal A_N(J'))+\kappa\ip{w}{\mathcal A_N(J')}. \] The tensor $J'$ has the law of $J$, and \cite[Theorem~1]{hs2023} applied to $\mathcal A'_N$ gives \eqref{sel:field-threshold}, with constants that do not depend on $w$ or on $\mathcal A_N$. For $\kappa=0$, we have $\ALG_p(0)=\Eth$, and \cref{sel:published-field-threshold} is the pure-model threshold, recorded in this form in \cite[Proposition~1.9]{sellke2024}. In this argument, we may always use the larger positive bound $L_{\mathrm{Lip}}+1$. This includes $L_{\mathrm{Lip}}=0$ and gives the same uniform choice of $c$ and $N_0$ for every algorithm with the prescribed original bound. We need no separate measurability assumption on a deterministic algorithm, since Lipschitz maps are continuous. \paragraph{Constructive finite-tree qualifications.} The branching proof of the threshold uses finite trees whose depth and branching degrees are fixed before the dimension tends to infinity. Two auxiliary statements in \cite[Appendix~A]{hs2023} require qualifications in a constructive version. In Lemma~A.8, if the absolute values of the leaf energies are bounded by $C>0$ and the depth is $D\ge1$, the retained branching degree $\lfloor K\varepsilon/(3CD)\rfloor$ is obtained in the range $0<\varepsilon\le CD$, and $K$ is then taken large enough to make this degree positive. This is the range used when $\varepsilon\downarrow0$; an unrestricted error can make the displayed degree exceed $K$. For the base-preserving fork correction of Lemma~A.9, we impose $k0$, the target increment has squared norm $q'\ge0$, and the approximate increments $y_1,\dots,y_k$ satisfy, for $\varepsilon\ge0$, \[ |\norm{y_i}^2-q'|\le\varepsilon,\qquad |\ip x{y_i}|\le\varepsilon,\qquad |\ip{y_i}{y_j}|\le\varepsilon\quad(i\ne j). \] Under the sufficient small-error conditions $\sqrt\varepsilon\le q_0$ and $4(k+1)\sqrt\varepsilon\le1$, the base can be kept exactly and the increments corrected to mutually orthogonal vectors $z_i\perp x$ with $\norm{z_i}^2=q'$ and $\norm{z_i-y_i}\le e_k(\varepsilon)$, where \[ e_k(\varepsilon) =(4\sqrt{k+1}+1)\sqrt\varepsilon +2\sqrt{\sqrt\varepsilon}. \] This bound permits $q'=0$ and tends to zero for fixed $k$ as $\varepsilon\downarrow0$. It supplies a vanishing modulus, independent of the dimension, in place of the piecewise numerical modulus printed in that lemma. We handle a schedule with zero initial overlap before this correction. For fixed $\delta\in(0,1)$, we scale the normalized vertices, which have norm at most one, by $\sqrt{1-\delta}$ and add a common orthogonal vector of norm $\sqrt\delta$. The new schedule of target overlaps is $q_i^{(\delta)}=(1-\delta)q_i+\delta$, so its root is at least $\delta$; the terminal vectors change by at most $2\sqrt\delta$. We choose the dimension larger than the number of vertices of the retained finite tree, as well as its branching degree, so that the common orthogonal direction and all fork corrections exist. For fixed depth and retained branching, the accumulated correction tends to zero, and $\delta$ can be removed afterward. Repeated overlap levels are allowed: their target increments have norm zero. In the algorithmic application, we can retain a fixed deterministic set of children, because the success event imposes the desired energy bound at every original terminal leaf. Internal vertices are first obtained from averages of the original terminal outputs. Restricting the independent Gaussian increments of the tree to this fixed set preserves their joint law. We therefore need no distributional assertion about an adaptively selected subtree. The depths, branching degrees, error tolerances, and dimension cutoffs are chosen before the algorithm and the deterministic field direction. The calibrated overlap schedule may depend on the algorithm, since the finite-tree ceiling is uniform over all admissible schedules. This is the uniformity asserted in \cref{sel:published-field-threshold}. The field version of the threshold penalizes the overlap of an algorithm with a direction independent of $J$. To state the penalty, recall $\qd=(p-2)/(p-1)$ and $\bd^2=(p-1)^{p-1}/(p(p-2)^{p-2})$. The constant $\bd$ is the dynamical transition point, and it enters here only as the coefficient of the penalty. The inverse temperature $\beta$ of the dynamics plays no role in \cref{sel:published-field-threshold,sel:absolute-field-cost}. \begin{lemma}[Absolute-overlap energy cost]\label{sel:absolute-field-cost} Fix an integer $p\ge3$ and $0\le L_{\mathrm{Lip}}<\infty$. There exist finite real numbers $\phi_N=\phi_N(p,L_{\mathrm{Lip}})\ge0$, for every $N\ge1$, with $\phi_N\to0$ as $N\to\infty$ such that the following holds. Let $\zeta$ be a random variable independent of $J$, let $w=w(\zeta)\in\R^N$ satisfy $\norm w=\sqrt N$, let $\mathcal A_N$ be an $L_{\mathrm{Lip}}$-Lipschitz algorithm with seed $\zeta$, and set $a_N=\ip{w}{\mathcal A_N}/N$. Then, almost surely, \begin{equation}\label{eq:conditional-field-cost} \E\left[\frac{H_N(\mathcal A_N)}N+\bd|a_N|^p\,\middle|\,\zeta\right] \le\Eth+\phi_N. \end{equation} In particular, \begin{equation}\label{sel:absolute-field-cost-equation} \limsup_{N\to\infty} \left\{\E\frac{H_N(\mathcal A_N)}N+\bd\,\E|a_N|^p\right\}\le\Eth. \end{equation} \end{lemma} The lemma makes no assumption on the concentration of $a_N$, and the seed may be absent, in which case $w$ is deterministic. \begin{proof} The proof has three steps. We first compute the best ceiling that the field thresholds give for a given overlap, then we use finitely many fields at once, and finally we refine the set of fields after the dimension limit. \paragraph{Step 1: the conjugate ceiling.} For $0\le a\le1$, we define \[ \mathcal E_p(a)=\inf_{\kappa\ge0}\{\ALG_p(\kappa)-\kappa a\}. \] We claim that \begin{equation}\label{eq:conjugate-ceiling} \mathcal E_p(a)= \begin{cases} \Eth-\bd a^p,&0\le a\le\sqrt{\qd},\\ \sqrt{p(1-a^2)},&\sqrt{\qd}\le a\le1, \end{cases} \qquad\text{and}\qquad \Eth-\mathcal E_p(a)\ge\bd a^p\quad(0\le a\le1). \end{equation} The function $\ALG_p$ is convex on $[0,\infty)$, since $\kappa^{p/(p-1)}$ and $\sqrt{p+\kappa^2}$ are convex. It is also continuously differentiable. Indeed, let $\kappa_*=\sqrt{p(p-2)}$. On $[0,\kappa_*]$, its derivative is $p(p-1)^{-1}\gamma_p\kappa^{1/(p-1)}$, which increases from $0$ to \[ \frac p{p-1}\gamma_p\kappa_*^{1/(p-1)} =\frac{p-2}{p-1}\sqrt{\frac{p(p-1)}{p(p-2)}}=\sqrt{\qd}. \] On $[\kappa_*,\infty)$, its derivative is $\kappa/\sqrt{p+\kappa^2}$, which increases from $\sqrt{\qd}$ to $1$. By convexity, for $0\le a<1$ the infimum defining $\mathcal E_p(a)$ is attained where $\ALG_p'(\kappa)=a$. If $a\le\sqrt{\qd}$, this point is $\kappa=((p-1)a/(p\gamma_p))^{p-1}$. At this point, we have $\gamma_p\kappa^{1/(p-1)}=(p-1)a/p$, and therefore \[ \mathcal E_p(a)=\Eth+\kappa\big(\gamma_p\kappa^{1/(p-1)}-a\big) =\Eth-\frac{\kappa a}p =\Eth-\frac1p\left(\frac{p-1}{p\gamma_p}\right)^{p-1}a^p. \] Inserting the value of $\gamma_p$, we have \[ \begin{aligned} \left(\frac{p-1}{p\gamma_p}\right)^{p-1} &=\left(\frac{p-1}{p-2}\right)^{p-1}\big(p(p-1)\big)^{-(p-1)/2} \big(p(p-2)\big)^{p/2}\\ &=p^{1/2}(p-1)^{(p-1)/2}(p-2)^{-(p-2)/2}, \end{aligned} \] and dividing by $p$, we find that the coefficient of $a^p$ is $((p-1)^{p-1}/(p(p-2)^{p-2}))^{1/2}=\bd$. If $\sqrt{\qd}\le a<1$, the point is $\kappa=\sqrt p\,a/\sqrt{1-a^2}$, and \[ \mathcal E_p(a)=\sqrt{\frac p{1-a^2}}-\frac{\sqrt p\,a^2}{\sqrt{1-a^2}} =\sqrt{p(1-a^2)}. \] At $a=1$, we have $\mathcal E_p(1)=0$, because the function $\sqrt{p+\kappa^2}-\kappa$ decreases to $0$ as $\kappa\to\infty$. The two formulas agree at $a=\sqrt{\qd}$, where both equal $\sqrt{p/(p-1)}$. This proves the first part of \eqref{eq:conjugate-ceiling}, and shows that $\mathcal E_p$ is continuous on $[0,1]$. Let $D_p(a)=\Eth-\mathcal E_p(a)$. For $a\le\sqrt{\qd}$, we have $D_p(a)=\bd a^p$. For $\sqrt{\qd}0$. We define \[ V_{\mathcal G}(a)=\min_{\kappa\in\mathcal G}\{\ALG_p(\kappa)-\kappa a\}, \qquad 0\le a\le1. \] Note that $|V_{\mathcal G}|\le M_{\mathcal G}$, where $M_{\mathcal G}=\Eth+\max\mathcal G$, because $0\in\mathcal G$ and $\ALG_p\ge0$. We condition on $\zeta$. Then $w$ is fixed, $J$ is standard Gaussian, and $J\mapsto\mathcal A_N(J,\zeta)$ is $L_{\mathrm{Lip}}$-Lipschitz. For each $\kappa\in\mathcal G$, we use \cref{sel:published-field-threshold} with the field strength $\kappa$ and the directions $w$ and $-w$. Let $c(\mathcal G,\eta)>0$ and $N_0(\mathcal G,\eta)$ be the smallest constant and the largest threshold for $N$ among these $|\mathcal G|$ applications. For $N\ge N_0(\mathcal G,\eta)$, the event \[ \mathfrak B=\bigcup_{\kappa\in\mathcal G}\left\{ \frac{H_N(\mathcal A_N)}N+\kappa|a_N|\ge\ALG_p(\kappa)+\eta\right\} \] has conditional probability at most $2|\mathcal G|e^{-c(\mathcal G,\eta)N}$, uniformly in $\zeta$. Off $\mathfrak B$, we have $H_N(\mathcal A_N)/N0$ and a stationary cluster $(c,r,\ell)$ of the translated CK kernels, as in \cref{prop:stationary-clusters}, realized along a sequence $T_n\to\infty$. Its plateau is $q$ and its energy is $e_\infty=(\ell-1)/(p\beta)$. Recall from \eqref{eq:cluster-fdt-properties} that $0\le c\le1$, that $r=-c'\ge0$, so that $c$ is nonincreasing, and that $\int_0^\infty r(\tau)\,\dd\tau=1-q$. Recall also that $e_\beta(T_n+a)\to e_\infty$ for every $a\ge0$, by \eqref{eq:energy-identity} and the convergence of the translated multipliers. \paragraph{Entropy notation.} We use the notation of \cref{sec:entropy-clusters}. For fixed disorder, $\rho_t$ is the density of $X_t$ with respect to $\omega_N$ and $g_N$ is the Gibbs density. We write \[ v^{\mathrm{cur}}_t=\nabla_S\log(\rho_t/g_N),\qquad I_H(t)=\int\norm{v^{\mathrm{cur}}_t}^2\rho_t\,\dd\omega_N,\qquad a_N(t)=\frac1N\E I_H(t) \] for the relative score, the Fisher information, and its normalized mean. This is the notation of \cref{sec:entropy-clusters}; the superscript distinguishes the score from the tree vectors $v_t$. Let $\mu_N(\dd t)=a_N(t)\,\dd t$. \Cref{lem:entropy-budget} implies that $\mu_N([0,\infty))\le\beta^2/2$. By \eqref{eq:fisher-density-budget}, $\mu_{N_j}(I)\to\mu(I)$ for every bounded interval $I$, where $\mu(\dd t)=a_F(t)\,\dd t$ is the finite measure of \eqref{eq:entropy-measure}. We write $P^H_t$ for the diffusion semigroup at fixed disorder and $\mathcal L_N$ for its generator. \paragraph{Forks.} For each fixed integer $K\ge1$, we use the product probability space carrying $J$, the uniform initial point $X_0$, a full Brownian path $(B_u)_{u\ge0}$, and $K$ full Brownian paths $B^{(1)},\dots,B^{(K)}$ on $[0,\infty)$. These coordinate blocks are mutually independent. We call \[ \zeta_N=\big(X_0,(B^{(b)})_{1\le b\le K},B\big) \] the fork seed, with its uniform-initial and Wiener product law; the disorder has its independent standard Gaussian law. The seed space and law depend on $K$, which we suppress in the notation. Fix $T\ge0$ and $L\ge0$. The trajectory $X$ solves \eqref{eq:hard-sde} driven by $B$. For $b\le K$, let $Z_b$ be the solution of \eqref{eq:hard-sde} on $[0,T+L]$ with $Z_b(0)=X_0$, driven by the Brownian motion that equals $B_u$ for $u\le T$ and $B_T+B^{(b)}_{u-T}$ for $u\ge T$. Then $Z_b(u)=X_u$ for $u\le T$, and after time $T$ the paths $Z_1,\dots,Z_K$ are continuations of $X$ from $X_T$ by independent noises. We write \[ Y_b=Z_b(T+L),\qquad m_{N,T}=\frac1K\sum_{b=1}^KY_b, \] and call $m_{N,T}$ the center of the fork. When needed, we indicate the dependence on $K$ and $L$ in the notation. The Brownian motion that drives $Z_b$ is a standard Brownian motion independent of $(J,X_0)$. Hence each pair $(J,Z_b)$ has the same law as $(J,(X_u)_{u\le T+L})$. The branches share their past before $T$ and are not independent. Note that $\norm{m_{N,T}}\le\sqrt N$, because $\norm{Y_b}=\sqrt N$. \paragraph{Response moments.} Let $U_b(t,u)$ be the variational response matrix of the path $Z_b$, defined as in \eqref{eq:response-matrix}. Each driving process is Brownian for its own full filtration and is independent of $(J,X_0)$, even though the branches share a common past. Therefore the proof of \eqref{eq:finite-lag-flow-moments} applies to every branch. With the constants $K_S$ and $\Theta_S$ defined there, its coordinate columns and normalized trace have fourth moment at most $K_S$ whenever $0\le t-u\le S$. In particular, \[ \E\left|\frac{\tr U_b(t,u)}N\right|^2\le\Theta_S^2, \qquad \E\norm{U_b(t,u)}_{\op}^2\le N\Theta_S^2. \] We use these estimates on the whole interval $[0,T+L]$, with $S=T+L$, and on the final interval $[T,T+L]$, with $S=L$. The latter bounds are uniform in $T$. \begin{lemma}[Entropy comparison for two branches]\label{sel:fork-moments} Let $T,a,d\ge0$, and let $Y$ and $Z$ be the endpoints of two branches at time $T$ of lengths $a$ and $d$, that is, of two continuations of $X$ from $X_T$ driven by independent Brownian motions. Set $Q_{N,T}(a,d)=\ip{Y}{Z}/N$. Then, for $k\in\{1,2\}$, \begin{equation}\label{sel:fork-moment-comparison} \left|\E Q_{N,T}(a,d)^k-\E C_N(T+2a+d,T+a)^k\right| \le2k\,e^{\beta p^2A_p(a+d)}\sqrt{a\,\mu_N([T,T+a])}+2e^{-c_pN}. \end{equation} Moreover, if $T_n\to\infty$ is a sequence along which the translated kernels converge to a stationary cluster with correlation profile $c$, then, for every $L<\infty$, \begin{equation}\label{sel:fork-overlap-concentration} \lim_{n\to\infty}\limsup_{j\to\infty}\sup_{0\le a,d\le L} \E\left|Q_{N_j,T_n}(a,d)-c(a+d)\right|^2=0. \end{equation} \end{lemma} \begin{proof} For $k\in\{1,2\}$, let $\Phi_k(x)=(x/\sqrt N)^{\otimes k}$, with values in $(\R^N)^{\otimes k}$ and its Euclidean norm, so that $\ip{\Phi_k(x)}{\Phi_k(y)}=(\ip xy/N)^k$. On $S_N$, we have $\norm{\Phi_k}=1$ and $\norm{D_S\Phi_k}_\op\le k/\sqrt N$. Fix the disorder. For $0\le z\le a$, we define \[ f_z=P^H_{a-z}\Phi_k,\qquad g_z=P^H_{d+z}\Phi_k,\qquad F_J(z)=\int_{S_N}\rho_{T+z}\ip{f_z}{g_z}\,\dd\omega_N, \] where the semigroup acts on each coordinate of $\Phi_k$. We first identify the endpoint values. By the Markov property and the conditional independence of the two branches given the past up to time $T$, we have $\E F_J(0)=\E Q_{N,T}(a,d)^k$. By the semigroup property, $F_J(a)$ is the conditional expectation of $\ip{\Phi_k(X_{T+a})}{\Phi_k(X_{T+2a+d})}$ given $J$, so that $\E F_J(a)=\E C_N(T+2a+d,T+a)^k$. Next we compute $F_J'$. We have $\partial_zf_z=-\mathcal L_Nf_z$, $\partial_zg_z=\mathcal L_Ng_z$, and $\partial_z\rho_{T+z}=\mathcal L_N^*\rho_{T+z}$. By the product rule for the generator, applied to each pair of coordinates, $\mathcal L_N\ip{f}{g}=\ip{\mathcal L_Nf}{g}+\ip{f}{\mathcal L_Ng} +2\sum_i\ip{D_Sf[e_i]}{D_Sg[e_i]}$, where $(e_i)$ is an orthonormal frame of the tangent space. Using these identities, we find \[ \begin{aligned} F_J'(z)&=\int\rho_{T+z}\left\{\mathcal L_N\ip{f_z}{g_z} -\ip{\mathcal L_Nf_z}{g_z}+\ip{f_z}{\mathcal L_Ng_z}\right\}\dd\omega_N\\ &=2\int\rho_{T+z}\left\{\ip{f_z}{\mathcal L_Ng_z} +\sum_i\ip{D_Sf_z[e_i]}{D_Sg_z[e_i]}\right\}\dd\omega_N. \end{aligned} \] When we apply \eqref{eq:gibbs-current-integration-by-parts} to each coordinate of $f_z$ and $g_z$, the gradient terms cancel, and we obtain the exact current identity \begin{equation}\label{sel:fork-current-identity} F_J'(z)=-2\int\rho_{T+z} \ip{P^H_{a-z}\Phi_k}{D_S\big(P^H_{d+z}\Phi_k\big)\big[v^{\mathrm{cur}}_{T+z}\big]} \,\dd\omega_N. \end{equation} Note that this identity does not use stationarity of $\rho_{T+z}$: the score $v^{\mathrm{cur}}_{T+z}$ records how far $\rho_{T+z}$ is from the Gibbs density. We now bound the right side of \eqref{sel:fork-current-identity} on the event $G_N=\{Q_N\le A_p\}$. By Jensen's inequality, $\norm{P^H_t\Phi_k}\le1$. In the proof of \cref{lem:waiting-clock-current}, the Bochner formula, the nonnegative Ricci curvature of $S_N$, and the Hessian bound of \eqref{eq:landscape-derivative-bounds} give the gradient bound $\norm{\nabla_SP^H_t\varphi}^2\le e^{2\beta p^2Q_Nt}P^H_t\norm{\nabla_S\varphi}^2$ for smooth scalar $\varphi$; this is the classical gradient bound under a lower bound on the curvature. Applied to $\varphi=\ip{w}{\Phi_k}$ for a unit vector $w$ in $(\R^N)^{\otimes k}$, and followed by the supremum over $w$, it gives \[ \norm{D_SP^H_t\Phi_k}_\op\le\frac k{\sqrt N}e^{\beta p^2A_pt} \qquad\text{on }G_N, \] with no factor depending on the dimension of $(\R^N)^{\otimes k}$. Using the Cauchy--Schwarz inequality, we find that, on $G_N$ and for $z\le a$, \[ |F_J'(z)|\le\frac{2k}{\sqrt N}e^{\beta p^2A_p(a+d)} \int\rho_{T+z}\norm{v^{\mathrm{cur}}_{T+z}}\,\dd\omega_N \le2k\,e^{\beta p^2A_p(a+d)}\left(\frac{I_H(T+z)}N\right)^{1/2}. \] Taking expectations and using Jensen's inequality, then integrating in $z$ and using the Cauchy--Schwarz inequality, we obtain \[ \begin{aligned} \int_0^a\E\big[\1_{G_N}|F_J'(z)|\big]\,\dd z &\le2k\,e^{\beta p^2A_p(a+d)}\int_0^a\sqrt{a_N(T+z)}\,\dd z\\ &\le2k\,e^{\beta p^2A_p(a+d)}\sqrt{a\,\mu_N([T,T+a])}. \end{aligned} \] On $G_N^c$, we have $|F_J(0)|,|F_J(a)|\le1$, and $\Prob(G_N^c)\le e^{-c_pN}$. Writing $\E[F_J(0)-F_J(a)]=-\E[\1_{G_N}\int_0^aF_J'(z)\,\dd z]+\E[\1_{G_N^c}(F_J(0)-F_J(a))]$ and combining the last two bounds with the identification of the endpoint values, we obtain \eqref{sel:fork-moment-comparison}. For \eqref{sel:fork-overlap-concentration}, fix $L$ and let $a,d\le L$. In \eqref{sel:fork-moment-comparison}, we may replace $a\,\mu_N([T,T+a])$ by $L\,\mu_N([T,T+L])$ and the exponential factor by $e^{2\beta p^2A_pL}$. Then \eqref{eq:fisher-density-budget} implies that $\mu_{N_j}([T,T+L])\to\mu([T,T+L])$ for each fixed $T$. By \cref{prop:hard-bridge} on the horizon $T+3L$ and $|C_N|\le1$, the moments $\E C_N(T+2a+d,T+a)^k$ converge to $C(T+2a+d,T+a)^k$ uniformly in $a,d\le L$. Combining these facts with \eqref{sel:fork-moment-comparison}, we find \[ \limsup_{j\to\infty}\sup_{0\le a,d\le L} \left|\E Q_{N_j,T}(a,d)^k-C(T+2a+d,T+a)^k\right| \le4e^{2\beta p^2A_pL}\sqrt{L\,\mu([T,T+L])}. \] Along $T=T_n$, the right side tends to zero because $\mu$ is a finite measure, and $C(T_n+2a+d,T_n+a)\to c(a+d)$ uniformly in $a,d\le L$, by the locally uniform convergence of the translated correlations on $[0,3L]^2$. Finally, writing $Q=Q_{N,T}(a,d)$ and $c_0=c(a+d)\in[0,1]$, we have \[ \E|Q-c_0|^2=(\E Q^2-c_0^2)-2c_0(\E Q-c_0), \] and the convergence of the first two moments, uniformly in $a,d\le L$, proves \eqref{sel:fork-overlap-concentration}. \end{proof} We now state the main result of this section. For $K\ge1$ and $L\ge0$, we define \begin{align} Q_{K,L}&=\frac1K+\left(1-\frac1K\right)c(2L), \label{sel:fork-norm}\\ k_{K,L}(\tau)&=\frac{c(\tau)+(K-1)c(2L-\tau)}K,\qquad 0\le\tau\le L, \label{sel:fork-mixed-overlap}\\ h_{K,L}&=e_\infty-\beta p\int_0^Lr(\tau) \left\{c(\tau)^{p-1}-k_{K,L}(\tau)^{p-1}\right\}\dd\tau. \label{sel:fork-center-energy} \end{align} Note that $Q_{K,L}$ is a convex combination of $1$ and $c(2L)$, so that $q\le c(2L)\le Q_{K,L}\le1$, and that $0\le k_{K,L}\le1$. \begin{proposition}[Finite forks and their center]\label{sel:finite-forks} Fix $\beta>0$ and let $(c,r,\ell)$ be a stationary cluster with plateau $q>0$ and energy $e_\infty$, realized along $T_n\to\infty$. Fix $K\ge1$ and $L\ge0$, and let $m_{N,T}$ be the center of the fork at time $T$ with $K$ branches of length $L$. \begin{enumerate}[label=(\alph*)] \item\label{item:fork-limits} The squared norm and the energy of the center satisfy \begin{align} &\lim_{n\to\infty}\limsup_{j\to\infty} \E\left|\frac{\norm{m_{N_j,T_n}}^2}{N_j}-Q_{K,L}\right|^2=0, \label{sel:fork-center-concentration}\\ &\lim_{n\to\infty}\limsup_{j\to\infty} \left|\frac{\E H_{N_j}(m_{N_j,T_n})}{N_j}-h_{K,L}\right|=0. \label{sel:fork-center-energy-limit} \end{align} \item\label{item:fork-ceiling} The limiting energy is below the threshold: \begin{equation}\label{sel:fork-center-threshold} h_{K,L}\le\Eth\,Q_{K,L}^{p/2}. \end{equation} \item\label{item:fork-lipschitz} For every $T\ge0$ and $\varepsilon>0$, there exist algorithms $\mathcal A^\varepsilon_{N,T}=\mathcal A^\varepsilon_{N,T}(J,\zeta_N)$ with values in $B_N$ and seed $\zeta_N$, which are $L_{\mathrm{Lip}}$-Lipschitz for a constant $L_{\mathrm{Lip}}$ depending only on $p$, $\beta$, $T$, $K$, $L$, and $\varepsilon$, such that \begin{equation}\label{sel:fork-lipschitz} \limsup_{j\to\infty}\frac1{N_j}\E\norm{\mathcal A^\varepsilon_{N_j,T} -\Pi_{B_{N_j}}\big(m_{N_j,T}/\sqrt{Q_{K,L}}\big)}^2\le\varepsilon^2. \end{equation} Moreover, along $T=T_n$, the vector $\Pi_{B_N}(m_{N,T}/\sqrt{Q_{K,L}})$ may be replaced by $m_{N,T}/\sqrt{Q_{K,L}}$, or by $\widehat m_{N,T}=\sqrt N\,m_{N,T}/\norm{m_{N,T}}$ (with $\widehat m_{N,T}=0$ if $m_{N,T}=0$): the scaled mean-square distance between $\Pi_{B_N}(m_{N,T}/\sqrt{Q_{K,L}})$ and either vector tends to zero as $j\to\infty$ and then $n\to\infty$. \end{enumerate} \end{proposition} By \eqref{sel:tensor-ui}, the normalized energies of the algorithms in part~\ref{item:fork-lipschitz} are bounded by $Q_N$. Part~\ref{item:fork-lipschitz} also transfers overlaps. If $V_N$ is a random vector on the same probability space with $\norm{V_N}\le\sqrt N$ (for example $V_N=X_s$ with $s\le T$), then, by the Cauchy--Schwarz inequality, we have \begin{equation}\label{eq:fork-overlap-transfer} \left|\E\frac{\ip{\mathcal A^\varepsilon_{N,T}}{V_N}}N -\E\frac{\ip{m_{N,T}}{V_N}}{N\sqrt{Q_{K,L}}}\right| \le\left(\frac1N\E\norm{\mathcal A^\varepsilon_{N,T} -m_{N,T}/\sqrt{Q_{K,L}}}^2\right)^{1/2}. \end{equation} \begin{proof} The proof has five steps. In the first two, we compute the energy of the center exactly by Gaussian integration by parts and subtract the energy of a single branch. In the third, we show that the common past before $T$ contributes nothing in the dimension limit, and in the fourth we evaluate the contribution of the last $L$ units of time; this proves part~\ref{item:fork-limits}. In the fifth step, we construct the Lipschitz algorithms of part~\ref{item:fork-lipschitz} and deduce part~\ref{item:fork-ceiling} from the threshold. We write $t=T+L$. \paragraph{Step 1: Gaussian integration by parts.} For an ordered index $I=(i_1,\dots,i_p)$, let $G_I(x)=N^{-(p-1)/2}x_{i_1}\cdots x_{i_p}$, so that $H_N=\sum_IJ_IG_I$. Since $\sum_IG_I(x)G_I(z)=N(\ip xz/N)^p$, differentiating first in $x_i$ and then in $z_l$ gives \begin{equation}\label{sel:fork-covariance-derivative} \sum_I\partial_iG_I(x)\,\partial_lG_I(z) =p\left(\frac{\ip xz}N\right)^{p-1}\delta_{il} +\frac{p(p-1)}N\left(\frac{\ip xz}N\right)^{p-2}z_ix_l. \end{equation} Differentiating the actual terminal solution into $L^2$ with respect to the disorder, as in the proof of \cref{prop:energy-bridge}, we have \[ \partial_{J_I}m_{N,T} =\frac\beta K\sum_{b=1}^K\int_0^tU_b(t,u)\nabla G_I(Z_b(u))\,\dd u. \] The seed is independent of $J$, and the response has the column and operator moments above. As in the proof of \cref{prop:energy-bridge}, apply Gaussian integration by parts to the expected coordinate observable in each disorder coefficient, using its genuine $L^2$ derivative: \[ \E\big[J_IG_I(m_{N,T})\big]=\E\big[\ip{\nabla G_I(m_{N,T})}{\partial_{J_I}m_{N,T}}\big], \] and $\E H_N(m_{N,T})$ is the sum of these terms over $I$. For $u\le t$, we write \[ \gamma_b(u)=\frac{\ip{Y_b}{Z_b(u)}}N,\qquad \bar\gamma_b(u)=\frac{\ip{m_{N,T}}{Z_b(u)}}N =\frac1K\sum_{b'=1}^K\frac{\ip{Y_{b'}}{Z_b(u)}}N. \] Contracting \eqref{sel:fork-covariance-derivative} with $U_b(t,u)$ gives the exact identity \begin{align} \frac{\E H_N(m_{N,T})}N ={}&\frac{\beta p}K\sum_{b=1}^K\int_0^t \E\left[\frac{\tr U_b(t,u)}N\,\bar\gamma_b(u)^{p-1}\right]\dd u +\mathcal R_{N,T},\label{sel:fork-exact-ibp}\\ \mathcal R_{N,T}={}&\frac{\beta p(p-1)}K\sum_{b=1}^K\int_0^t \E\left[\bar\gamma_b(u)^{p-2} \frac{\ip{Z_b(u)}{U_b(t,u)m_{N,T}}}{N^2}\right]\dd u.\notag \end{align} Using $\norm{m_{N,T}},\norm{Z_b(u)}\le\sqrt N$ and $|\bar\gamma_b|\le1$, we find that $|\mathcal R_{N,T}|\le\beta p(p-1)t\,\Theta_t/\sqrt N$. \paragraph{Step 2: the exact subtraction.} The same computation with $m_{N,T}$ replaced by $Y_b$ gives $\E H_N(Y_b)/N=\beta p\int_0^t\E[N^{-1}\tr U_b(t,u)\gamma_b(u)^{p-1}]\,\dd u +O(t\,\Theta_t/\sqrt N)$. The pair $(J,Y_b)$ has the law of $(J,X_t)$, and therefore $\E H_N(Y_b)/N=e_N(t)$, where $e_N(t)=\E H_N(X_t)/N$. Averaging over $b$ and subtracting \eqref{sel:fork-exact-ibp}, before taking any limit, we obtain \begin{equation}\label{sel:fork-exact-subtraction} \begin{aligned} e_N(t)-\frac{\E H_N(m_{N,T})}N ={}&\frac{\beta p}K\sum_{b=1}^K\int_0^t \E\left[\frac{\tr U_b(t,u)}N \big\{\gamma_b(u)^{p-1}-\bar\gamma_b(u)^{p-1}\big\}\right]\dd u\\ &+O\big(t\,\Theta_tN^{-1/2}\big), \end{aligned} \end{equation} where the implicit constant depends only on $p$ and $\beta$. Note that this identity retains the whole history of the trajectory. \paragraph{Step 3: the common past.} Let $u\le T$. Then $Z_b(u)=X_u$ for every $b$, and the pair $(Y_b,X_u)$ has the law of $(X_t,X_u)$. By \cref{prop:hard-bridge} and $|C_N|\le1$, we have $\E|\gamma_b(u)-C(t,u)|^2\to0$ as $N\to\infty$, uniformly in $u\le T$. The same holds for $\bar\gamma_b(u)$, which is the average of $\gamma_{b'}(u)$ over $b'$. The function $x\mapsto x^{p-1}$ is $(p-1)$-Lipschitz on $[-1,1]$, and the normalized trace has second moment at most $\Theta_t^2$. Using the Cauchy--Schwarz inequality, we find that the part of the integral in \eqref{sel:fork-exact-subtraction} over $u\le T$ is at most \[ \beta p(p-1)\,T\,\Theta_t \sup_{b\le K,\,u\le T}\big(\E|\gamma_b(u)-\bar\gamma_b(u)|^2\big)^{1/2} \] in absolute value. This tends to zero as $N\to\infty$ with $T$ fixed. We use no estimate on the decay of the memory here. The contribution of the past is small because, at a fixed time horizon, both overlaps concentrate around the same deterministic value $C(t,u)$. \paragraph{Step 4: the last $L$ units of time.} Let $u=T+v$ with $0\le v\le L$, and write $\tau=L-v$. The pair $(Y_b,Z_b(T+v))$ has the law of $(X_t,X_{T+v})$. By \cref{prop:hard-bridge}, the overlap $\gamma_b(T+v)$ converges in $L^2$ to $C(T+L,T+v)$ as $N\to\infty$, uniformly in $v$. Along $T=T_n$, this tends to $c(L-v)=c(\tau)$, uniformly in $v\le L$. For $b'\ne b$, the vectors $Y_{b'}$ and $Z_b(T+v)$ are the endpoints of two branches at $T$ of lengths $L$ and $v$. By \eqref{sel:fork-overlap-concentration}, the overlap $\ip{Y_{b'}}{Z_b(T+v)}/N$ converges in $L^2$ to $c(L+v)=c(2L-\tau)$, in the double limit $j\to\infty$ and then $n\to\infty$, uniformly in $v\le L$. Averaging over $b'$, we find \[ \lim_{n\to\infty}\limsup_{j\to\infty}\sup_{0\le v\le L} \Big(\E|\gamma_b(T_n+v)-c(L-v)|^2 +\E|\bar\gamma_b(T_n+v)-k_{K,L}(L-v)|^2\Big)=0. \] On this interval, we have $t-u\le L$, and the normalized response trace has second moment at most $\Theta_L^2$, uniformly in $T$. As in Step~3, using the Cauchy--Schwarz inequality, we may replace $\gamma_b^{p-1}-\bar\gamma_b^{p-1}$ in \eqref{sel:fork-exact-subtraction} by the deterministic weight \[ \omega(\tau)=c(\tau)^{p-1}-k_{K,L}(\tau)^{p-1},\qquad\tau=t-u, \] with an error whose $\limsup_j$ tends to zero as $n\to\infty$. It remains to pass to the limit in $\beta p\int_T^t\bar r_N(t,u)\,\omega(t-u)\,\dd u$, where $\bar r_N(t,u)=\E\tr U_b(t,u)/N$ is the averaged response of \cref{lem:brown-force-response}, which does not depend on $b$. \Cref{lem:brown-force-response} and \cref{prop:hard-bridge} imply that $\int_{s_1}^{s_2}\bar r_N(t,u)\,\dd u\to\int_{s_1}^{s_2}R(t,u)\,\dd u$ for all $T\le s_1\le s_2\le t$. The signed measure $\bar r_N(t,u)\,\dd u$ on $[T,t]$ has total variation \[ \int_T^t|\bar r_N(t,u)|\,\dd u \le\int_T^t\E\left|\frac{\tr U_b(t,u)}N\right|\,\dd u \le L\,\Theta_L. \] As $R\ge0$, the integral of $R(t,\cdot)$ over $[T,t]$ is the limit of the masses of these signed measures and obeys the same bound. The weight $\omega$ is continuous on $[0,L]$ and is the uniform limit of step functions. For a step function, the integral converges as $N\to\infty$, and the total variation bound controls the error of the approximation, uniformly in $N$. We conclude that, for fixed $T$, \[ \lim_{N\to\infty}\int_T^t\bar r_N(t,u)\,\omega(t-u)\,\dd u =\int_T^tR(t,u)\,\omega(t-u)\,\dd u. \] Along $T=T_n$, we have $R(T_n+L,T_n+v)\to r(L-v)$ uniformly in $v\le L$, by the translated convergence on the closed causal triangle. Moreover, $e_N(t)\to e_\beta(T+L)$ by \cref{prop:energy-bridge}, and $e_\beta(T_n+L)\to e_\infty$. Inserting these limits into \eqref{sel:fork-exact-subtraction}, we obtain \[ \lim_{n\to\infty}\limsup_{j\to\infty} \left|\frac{\E H_{N_j}(m_{N_j,T_n})}{N_j} -e_\infty+\beta p\int_0^Lr(\tau)\,\omega(\tau)\,\dd\tau\right|=0, \] which is \eqref{sel:fork-center-energy-limit} by the definition \eqref{sel:fork-center-energy}. For the norm, we have \[ \frac{\norm{m_{N,T}}^2}N=\frac1K+\frac1{K^2}\sum_{b\ne b'} \frac{\ip{Y_b}{Y_{b'}}}N. \] Each overlap $\ip{Y_b}{Y_{b'}}/N$ with $b\ne b'$ has the law of $Q_{N,T}(L,L)$, which tends to $c(2L)$ in $L^2$ in the double limit, by \eqref{sel:fork-overlap-concentration}. There are $K(K-1)$ such terms, and $K(K-1)/K^2=1-1/K$. This proves \eqref{sel:fork-center-concentration} and completes the proof of part~\ref{item:fork-limits}. \paragraph{Step 5: Lipschitz algorithms and the threshold.} Fix $T$ and $\varepsilon>0$. We may assume that $\varepsilon<1$. Let $\varepsilon_1=\varepsilon/(2\sqrt K)$. We use the hard--soft approximation of \cite[Lemmas~3.1--3.2]{sellke2024}. We first check its hypotheses. In \cite{sellke2024}, the Hamiltonian is our $H_N$, with an ordered tensor of independent standard Gaussian coefficients and the Euclidean distance between coefficient tensors, and the dynamics is $\dd x_v=(\beta_{\mathrm S}\nabla_{\mathrm{sp}}H_N(x_v) -(N-1)(2N)^{-1}x_v)\,\dd v+P_{x_v}\,\dd W_v$, with noise coefficient one. With $\beta_{\mathrm S}=\beta/2$, $X_u=x_{2u}$, and $B_u=W_{2u}/\sqrt2$, this is \eqref{eq:hard-sde}, as in the proof of \cref{prop:hard-bridge}, and our horizon $t$ becomes $2t$. Lemma~3.1 there states that, for a fixed initial point on $S_N$ and a Hamiltonian satisfying their derivative bounds (which hold for $H_N$ with probability at least $1-e^{-c_1N}$, by \cite[Proposition~1.11]{sellke2024}), the hard dynamics stays within $\varepsilon_1\sqrt N$ of a soft spherical dynamics driven by the same Brownian motion, on the whole horizon, with probability at least $1-e^{-c_1N}$. Here the confinement parameter of the soft dynamics is chosen large, depending on $p$, $\beta$, $t$, and $\varepsilon_1$, and $c_1>0$ and the threshold for $N$ depend on the same parameters. Lemma~3.2 there states that, for a fixed initial point and almost every Brownian path, there exists a map $\ell$ from coefficient tensors to $B_N$, Lipschitz with a constant depending only on $p$, $\beta$, $t$, and $\varepsilon_1$, such that $(1+\varepsilon_1)\ell(J)$ equals the soft endpoint with probability at least $1-e^{-c_1N}$, where the probability is over both the disorder and the Brownian motion. The coefficient-Lipschitz extension uses Kirszbraun's theorem in the proof of Lemma~3.2 of \cite{sellke2024}; footnote~2 there gives a constructive choice that depends measurably on the Brownian path. The construction of \cite{sellke2024} is for a fixed initial point. To include the uniform initial point, we choose measurably an orthogonal matrix $O_{x_0}$ such that $O_{x_0}(\sqrt N e_1)=x_0$. Rotating the tensor by $(O_{x_0}^{\mathsf T})^{\otimes p}$ and the Brownian path by $O_{x_0}^{\mathsf T}$, solving from $\sqrt Ne_1$, and rotating the output back by $O_{x_0}$ gives the solution from $x_0$, because $H_N$ and \eqref{eq:hard-sde} are equivariant under rotations. Conditionally on $X_0=x_0$, the rotated tensor and Brownian path have the same joint law as the original ones, and the tensor rotation is an isometry of $(\R^N)^{\otimes p}$, which does not change Lipschitz constants. We apply this to each concatenated path $Z_b$, whose driving Brownian motion is a function of the seed $\zeta_N$. We obtain maps $\ell_b(J)=\ell_b(J,\zeta_N)\in B_N$, $b\le K$, which are Lipschitz in $J$ with a constant $L_{\mathrm{Lip},1}$ depending only on $p$, $\beta$, $t$, and $\varepsilon_1$. Let $\mathcal G_N$ be the event on which $H_N$ satisfies the derivative bounds of \cite{sellke2024} and, for every $b\le K$, the conclusions of both lemmas hold for the branch $b$. On $\mathcal G_N$, we have $\norm{(1+\varepsilon_1)\ell_b(J)-Y_b}\le\varepsilon_1\sqrt N$ for every $b\le K$, and a union bound gives $\Prob(\mathcal G_N^c)\le3Ke^{-c_1N}$. The union bound does not require the branches to be independent. We define \begin{equation}\label{sel:fork-projected-algorithm} \mathcal A^\varepsilon_{N,T}(J) =\Pi_{B_N}\left(\frac{1+\varepsilon_1}{\sqrt{Q_{K,L}}} \frac1K\sum_{b=1}^K\ell_b(J)\right). \end{equation} Since $\Pi_{B_N}$ is $1$-Lipschitz and $Q_{K,L}\ge1/K$, this is an $L_{\mathrm{Lip}}$-Lipschitz algorithm with seed $\zeta_N$ and $L_{\mathrm{Lip}}=2L_{\mathrm{Lip},1}\sqrt K$. On $\mathcal G_N$, the distance between $\mathcal A^\varepsilon_{N,T}$ and $\Pi_{B_N}(m_{N,T}/\sqrt{Q_{K,L}})$ is at most $\varepsilon_1\sqrt N/\sqrt{Q_{K,L}}\le\varepsilon_1\sqrt{KN} =\varepsilon\sqrt N/2$; on $\mathcal G_N^c$, it is at most $2\sqrt N$. Using $\Prob(\mathcal G_N^c)\le3Ke^{-c_1N}$, we find \[ \frac1N\E\norm{\mathcal A^\varepsilon_{N,T} -\Pi_{B_N}(m_{N,T}/\sqrt{Q_{K,L}})}^2 \le\frac{\varepsilon^2}4+12Ke^{-c_1N}, \] which proves \eqref{sel:fork-lipschitz}. Next we compare the three normalizations. Let $\nu_N=\norm{m_{N,T}}^2/N$. For $x\ge0$, we have \begin{equation}\label{eq:sqrt-comparison} \left|\sqrt{x/Q_{K,L}}-1\right| =\frac{|x-Q_{K,L}|}{\sqrt{Q_{K,L}}\,(\sqrt x+\sqrt{Q_{K,L}})} \le\frac{|x-Q_{K,L}|}{Q_{K,L}}. \end{equation} The scaled distance between $m_{N,T}/\sqrt{Q_{K,L}}$ and its projection onto $B_N$ is $(\sqrt{\nu_N/Q_{K,L}}-1)_+$. If $m_{N,T}\ne0$, the scaled distance between the projection and $\widehat m_{N,T}$ is $(1-\sqrt{\nu_N/Q_{K,L}})_+$, and both vectors vanish if $m_{N,T}=0$. By \eqref{eq:sqrt-comparison} and $Q_{K,L}\ge q$, both distances are at most $|\nu_N-Q_{K,L}|/q$. Then \eqref{sel:fork-center-concentration} implies that their mean squares tend to zero as $j\to\infty$ and then $n\to\infty$. This completes the proof of part~\ref{item:fork-lipschitz}. Finally, we prove part~\ref{item:fork-ceiling}. Since $\mathcal A^\varepsilon_{N,T}$ is an $L_{\mathrm{Lip}}$-Lipschitz algorithm with seed $\zeta_N$, \cref{sel:absolute-field-cost}, applied with any $\zeta_N$-measurable direction and with the nonnegative overlap term dropped, gives \[ \limsup_{N\to\infty}\frac{\E H_N(\mathcal A^\varepsilon_{N,T})}N\le\Eth. \] This is the pure-model threshold, the case $\kappa=0$ of \cref{sel:published-field-threshold}, averaged over the seed. Next we compare energies. Note that $m_{N,T}/\sqrt{Q_{K,L}}$ has norm at most $\sqrt{N/q}$. Using \eqref{eq:general-energy-transfer} with $R=q^{-1/2}$ and the homogeneity $H_N(m_{N,T}/\sqrt{Q_{K,L}})=Q_{K,L}^{-p/2}H_N(m_{N,T})$, we find \[ Q_{K,L}^{-p/2}\frac{\E H_N(m_{N,T})}N \le\frac{\E H_N(\mathcal A^\varepsilon_{N,T})}N +p\,q^{-(p-1)/2}Q_*\,\Delta_{N,T}, \] where \[ \Delta_{N,T}=\left(\frac1N\E\norm{\mathcal A^\varepsilon_{N,T} -m_{N,T}/\sqrt{Q_{K,L}}}^2\right)^{1/2}. \] By \eqref{sel:fork-lipschitz} and the comparison of normalizations, we have \[ \limsup_{n\to\infty}\limsup_{j\to\infty}\Delta_{N_j,T_n}\le\varepsilon. \] We take $j\to\infty$ with $T=T_n$ fixed, then $n\to\infty$, and finally $\varepsilon\downarrow0$, in the order \eqref{sel:aging-limit-order}. By \eqref{sel:fork-center-energy-limit}, we obtain $Q_{K,L}^{-p/2}h_{K,L}\le\Eth$, which is \eqref{sel:fork-center-threshold}. The constants of the algorithms depend on $T$, but we use them only at a fixed $T$, before the limit $n\to\infty$. \end{proof} We next record the limits of the fork parameters as $K\to\infty$ and then $L\to\infty$. Recall that $d_L=c(2L)$ and $h_L$ are given by \eqref{sel:prefix-norm} and \eqref{sel:prefix-energy}, with the cluster energy $e_\infty$, and that $h$ is the quantity in \eqref{sel:limiting-center-energy}. \begin{lemma}[The fork parameters as $K\to\infty$ and $L\to\infty$] \label{lem:fork-parameter-limits} Let $(c,r,\ell)$ be a stationary cluster with plateau $q\ge0$ and energy $e_\infty$. \begin{enumerate}[label=(\roman*)] \item\label{item:fork-K-limit} For every $L\ge0$ and $K\ge1$, \begin{equation}\label{eq:fork-K-limit} 0\le Q_{K,L}-d_L\le\frac1K,\qquad |h_{K,L}-h_L|\le\frac{\beta p(p-1)}K. \end{equation} \item\label{item:fork-L-limit} As $L\to\infty$, we have $d_L\to q$ and $h_L\to h=e_\infty-\beta\{1-q^p-pq^{p-1}(1-q)\}$. \item\label{item:fork-saturation} If $q>0$ and $\beta^2p(p-1)q^{p-2}(1-q)^2=1$, then \begin{equation}\label{sel:fork-saturation} \lim_{L\to\infty}\lim_{K\to\infty}\frac{h_{K,L}}{Q_{K,L}^{p/2}}=\Eth. \end{equation} \end{enumerate} \end{lemma} \begin{proof} For part~\ref{item:fork-K-limit}, we have $Q_{K,L}-d_L=(1-c(2L))/K\in[0,1/K]$. By \eqref{sel:prefix-energy} and \eqref{sel:fork-center-energy}, \[ h_{K,L}-h_L=\beta p\int_0^Lr(\tau) \left\{k_{K,L}(\tau)^{p-1}-c(2L-\tau)^{p-1}\right\}\dd\tau. \] The integrand is dominated by $r(\tau)$ times a quantity that tends to zero uniformly. Indeed, $k_{K,L}(\tau)-c(2L-\tau)=(c(\tau)-c(2L-\tau))/K$, and both $k_{K,L}(\tau)$ and $c(2L-\tau)$ lie in $[0,1]$. By the mean value theorem, \[ \left|k_{K,L}(\tau)^{p-1}-c(2L-\tau)^{p-1}\right|\le\frac{p-1}K. \] Together with $\int_0^Lr\le1$, this proves \eqref{eq:fork-K-limit}. Part~\ref{item:fork-L-limit} is part of \cref{lem:center-energy}. For completeness, we recall the computation. Using $r=-c'$, we find that $\int_0^Lr(\tau)c(\tau)^{p-1}\,\dd\tau=(1-c(L)^p)/p$. As $c$ is nonincreasing, we have $c(2L-\tau)\in[q,c(L)]$ for $0\le\tau\le L$, and $\int_0^Lr=1-c(L)$. Thus the integral of $r(\tau)c(2L-\tau)^{p-1}$ over $[0,L]$ lies between $q^{p-1}(1-c(L))$ and $c(L)^{p-1}(1-c(L))$. Both bounds tend to $q^{p-1}(1-q)$. Inserting these two integrals into \eqref{sel:prefix-energy}, we find that $h_L$ tends to $e_\infty+\beta pq^{p-1}(1-q)-\beta(1-q^p)=h$. For part~\ref{item:fork-saturation}, part~\ref{item:fork-K-limit} and $d_L\ge q>0$ imply that $h_{K,L}/Q_{K,L}^{p/2}\to h_L/d_L^{p/2}$ as $K\to\infty$, and part~\ref{item:fork-L-limit} gives $h_L/d_L^{p/2}\to h/q^{p/2}$ as $L\to\infty$. By the equality case of \eqref{sel:center-amgm} in \cref{lem:center-energy}, the marginality hypothesis implies that $h/q^{p/2}=\Eth$. \end{proof} Part~\ref{item:fork-saturation} uses marginality, which is proved in \cref{sel:positive-marginality} from part~\ref{item:threshold-ceiling} of \cref{prop:threshold-inputs}. The proof of that part in \cref{sec:general-proof-inputs} uses only \cref{sel:finite-forks} and part~\ref{item:fork-K-limit}, and there is no circularity. \section{Adaptive disclosure and orthogonality to early queries} \label{sec:general-disclosure} The overlap penalty of \cref{sel:absolute-field-cost} applies to directions independent of $J$. At positive times, however, an earlier state $X_s$ depends on $J$. The goal of this section is to prove that an algorithm that saturates the threshold is almost orthogonal to every subspace explored by a fixed finite number of gradient queries. We condition $J$ on the information revealed by the queries. This is adaptive Gaussian conditioning, in the line of Bolthausen's iterative construction of solutions of the TAP equations \cite[Section~3]{bolthausen2014}. Bayati and Montanari used the same method in their analysis of approximate message passing \cite[arXiv version~4, Section~3.8, Lemma~11]{bayatimontanari2011}, and they credit it to Bolthausen. The underlying facts on Gaussian vectors and their orthogonal decompositions are classical; see \cite[Chapter~9, Theorem~9.1, pp.~127--128, and Theorems~9.8--9.9, pp.~131--132]{janson1997}. We use projection onto the span of the deterministic observations, which also permits linearly dependent observation families. El Alaoui, Montanari, and Sellke conditioned the disorder tensor on adaptive gradient observations in their analysis of message passing for spin glasses \cite[arXiv version~1, Appendix~A, Lemma~A.2, and Theorem~7(a)]{eams2021}, and Huang and Sellke extended this conditioning to multi-species spherical models \cite[arXiv version~1, Appendix~A, Lemma~A.2, and Theorem~3(a)]{hs2024amp}. The displayed inverse-Gram formulas in these AMP analyses use rank and conditioning hypotheses. Our projection-based disclosure below also covers dependent queries. Here we disclose all tensor blocks needed to determine gradients throughout the explored span. We combine this disclosure with the external-field threshold to prove forgetting of a fixed finite history. \paragraph{Outer parameters.} Let $\theta$ be a finite list of outer parameters with a prescribed order of limits; in \cref{sec:general-transfer}, $\theta=(n,\varepsilon,K,L)$, in the order \eqref{sel:aging-limit-order}. For nonnegative quantities $F_{N,\theta}$, we write $\lim_\theta\limsup_NF_{N,\theta}=0$ if the upper limit in $N$, followed by the upper limits in the outer parameters in their prescribed order, vanishes. We do not assume that any intermediate limit exists. \paragraph{Standing hypotheses.} For each $N$ and $\theta$, let $\zeta_{N,\theta}$ be a random variable independent of $J$, which contains all auxiliary randomness used both by the algorithms and by the disclosure procedure below. Its measurable space and law may depend on $\theta$; below we write $\zeta_N$ with $\theta$ suppressed. For each $N$ and $\theta$, let $\mathcal A_{N,\theta}=\mathcal A_{N,\theta}(J,\zeta_N)$ be an $L_{\mathrm{Lip},\theta}$-Lipschitz algorithm with seed $\zeta_N$. The constant $L_{\mathrm{Lip},\theta}$ is deterministic and independent of $N$, but it may be unbounded in the outer limits. We define the energy deficit \begin{equation}\label{sel:adaptive-deficit} \delta_\theta=\max\left\{0,\;\Eth-\liminf_{N\to\infty} \E\frac{H_N(\mathcal A_{N,\theta})}N\right\}. \end{equation} \Cref{sel:absolute-field-cost} implies that $\limsup_N\E H_N(\mathcal A_{N,\theta})/N\le\Eth$. The hypothesis $\lim_\theta\delta_\theta=0$ says exactly that the algorithms saturate the threshold in mean. For a subspace $W\subset\R^N$, we write \[ \mathcal T(W)=(W^\perp)^{\otimes p}\subset(\R^N)^{\otimes p}, \] and $P_V$ for the orthogonal projection onto a subspace $V$ of $\R^N$ or of $(\R^N)^{\otimes p}$. If $W\subset W'$, then $\mathcal T(W')\subset\mathcal T(W)$, and we write $\mathcal T(W)\ominus\mathcal T(W')$ for the orthogonal complement of $\mathcal T(W')$ in $\mathcal T(W)$. We choose orthonormal bases of measurable subspaces measurably, by Gram--Schmidt in a fixed order with deterministic fallbacks. \begin{lemma}[Adaptive tensor disclosure]\label{sel:adaptive-disclosure} Let $W_0\subset\R^N$ be a $\zeta_N$-measurable subspace, and let $\mathcal F_0$ be the $\sigma$-algebra generated by $\zeta_N$ and $P_{\mathcal T(W_0)^\perp}J$. For $j\ge0$, given $W_j$ and $\mathcal F_j$, let $u_j$ be an $\mathcal F_j$-measurable vector that is either zero or a unit vector in $W_j^\perp$, and set \[ W_{j+1}=W_j+\Span(u_j),\qquad \mathcal F_{j+1}=\sigma\Big(\mathcal F_j,\; P_{\mathcal T(W_j)\ominus\mathcal T(W_{j+1})}J\Big). \] Then, for every $j$: \begin{enumerate}[label=(\roman*)] \item\label{item:disclosure-gaussian} $P_{\mathcal T(W_j)^\perp}J$ is $\mathcal F_j$-measurable, and conditionally on $\mathcal F_j$, the tensor $P_{\mathcal T(W_j)}J$ is standard Gaussian on $\mathcal T(W_j)$; \item\label{item:disclosure-gradient} if $x_j\in W_j$ is $\mathcal F_j$-measurable, then $\nabla H_N(x_j)$ is $\mathcal F_j$-measurable. In particular, $u_j$ may be chosen as the normalized projection of $\nabla H_N(x_j)$ onto $W_j^\perp$, and as zero if that projection vanishes. \end{enumerate} \end{lemma} \begin{proof} The conditional Gaussian assertions are understood almost surely with respect to the disclosed history. A zero query discloses the zero tensor block and leaves the residual law unchanged. We prove part~\ref{item:disclosure-gaussian} by induction on $j$. For $j=0$, conditionally on $\zeta_N$, the tensor $J$ is standard Gaussian, because it is independent of $\zeta_N$, and the subspaces $\mathcal T(W_0)$ and $\mathcal T(W_0)^\perp$ are fixed. The projections of a standard Gaussian vector onto two complementary orthogonal subspaces are independent standard Gaussian vectors on them. Revealing $P_{\mathcal T(W_0)^\perp}J$ leaves $P_{\mathcal T(W_0)}J$ standard Gaussian on $\mathcal T(W_0)$. Suppose that part~\ref{item:disclosure-gaussian} holds for $j$. Conditionally on $\mathcal F_j$, the vector $u_j$ and the subspaces in the orthogonal decomposition \[ \mathcal T(W_j)=\mathcal T(W_{j+1})\oplus \big(\mathcal T(W_j)\ominus\mathcal T(W_{j+1})\big) \] are fixed, and $P_{\mathcal T(W_j)}J$ is standard Gaussian on $\mathcal T(W_j)$. Its projections onto the two summands are independent standard Gaussian vectors. Revealing the second leaves the first, which is $P_{\mathcal T(W_{j+1})}J$, standard Gaussian on $\mathcal T(W_{j+1})$. Since $\mathcal T(W_{j+1})^\perp=\mathcal T(W_j)^\perp\oplus (\mathcal T(W_j)\ominus\mathcal T(W_{j+1}))$, the projection $P_{\mathcal T(W_{j+1})^\perp}J$ is $\mathcal F_{j+1}$-measurable. This completes the induction. At each step, we choose the new direction before we reveal the corresponding part of $J$; this is what makes the induction work, and we never condition on a subspace that depends on undisclosed randomness. For part~\ref{item:disclosure-gradient}, let $x\in W_j$ and $y\in\R^N$. We have \[ \ip{\nabla H_N(x)}{y}=N^{-(p-1)/2}\Big\langle J,\;\sum_{i=1}^p x^{\otimes(i-1)}\otimes y\otimes x^{\otimes(p-i)}\Big\rangle. \] Since $p\ge2$, each tensor in the sum has at least one factor $x\in W_j$. It is orthogonal to every product tensor with all factors in $W_j^\perp$, and these product tensors span $\mathcal T(W_j)$. The right side is equal to $N^{-(p-1)/2}$ times the inner product of $P_{\mathcal T(W_j)^\perp}J$ with that tensor. By part~\ref{item:disclosure-gaussian}, it is $\mathcal F_j$-measurable when $x=x_j$ is. \end{proof} \begin{proposition}[Orthogonality to every fixed finite query span] \label{sel:adaptive-orthogonality} Assume the standing hypotheses and $\lim_\theta\delta_\theta=0$. Fix integers $k_0\ge0$ and $M\ge0$. For each $N$ and $\theta$, let $W_0\subset W_1\subset\dots\subset W_M$ be subspaces constructed as in \cref{sel:adaptive-disclosure} from the seed $\zeta_N$, with $\dim W_0\le k_0$. Then \begin{equation}\label{sel:finite-span-orthogonality} \lim_\theta\limsup_{N\to\infty} \E\frac{\norm{P_{W_M}\mathcal A_{N,\theta}}^2}N=0. \end{equation} In particular, if $Y_N\in W_M\cap B_N$ is a random vector, then $\lim_\theta\limsup_N\E|\ip{\mathcal A_{N,\theta}}{Y_N}/N|^2=0$. \end{proposition} \begin{proof} For $0\le j\le M$, we define \[ \psi_{j,\theta}=\limsup_{N\to\infty} \E\frac{\norm{P_{W_j}\mathcal A_{N,\theta}}^2}N. \] We bound $\psi_{0,\theta}$ directly and then show by induction that $\psi_{j+1,\theta}$ is controlled by $\psi_{j,\theta}$ and $\delta_\theta$. We use the constants of the threshold only at fixed $\theta$, before any outer limit, and the argument does not require $L_{\mathrm{Lip},\theta}$ to stay bounded. \paragraph{Step 1: the initial span.} Let $e_1,\dots,e_{k_0}$ be a $\zeta_N$-measurable orthonormal basis of $W_0$, completed by zero vectors if $\dim W_0k_0+M$. Every residual dimension below is then positive. We now show that $\mathcal A^\perp$ is the output of a Lipschitz algorithm for a pure $p$-spin model in dimension $N'=N-\dim W$, and that its energy has the same normalization up to a factor tending to one. We condition on $\mathcal F_j$. Then $W$ and $u_j$ are fixed, and we choose an $\mathcal F_j$-measurable orthonormal basis $f_1,\dots,f_{N'}$ of $W^\perp$. Let $J'\in(\R^{N'})^{\otimes p}$ be the coordinates of $P_{\mathcal T(W)}J$ in the basis $(f_{i_1}\otimes\dots\otimes f_{i_p})$. By \cref{sel:adaptive-disclosure}\ref{item:disclosure-gaussian}, $J'$ is a standard Gaussian vector in the $N'^p$-dimensional space $(\R^{N'})^{\otimes p}$, conditionally on $\mathcal F_j$. Let $y'\in\R^{N'}$ be the coordinates of $\mathcal A^\perp$ in the basis $(f_i)$, and set $x=\sqrt{N'/N}\,y'$. Then $\norm x\le\sqrt{N'}$, that is, $x\in B_{N'}$. Since $(\mathcal A^\perp)^{\otimes p}\in\mathcal T(W)$, only $P_{\mathcal T(W)}J$ contributes to $H_N(\mathcal A^\perp)$, and a change of normalization yields \begin{equation}\label{sel:residual-normalization} \frac{H_N(\mathcal A^\perp)}N =N^{-(p+1)/2}\ip{J'}{y'^{\otimes p}} =\sqrt{\frac{N'}N}\,\frac{H'_{N'}(x)}{N'}, \end{equation} where $H'_{N'}(x)=N'^{-(p-1)/2}\ip{J'}{x^{\otimes p}}$ is the pure $p$-spin Hamiltonian in dimension $N'$ with coefficient tensor $J'$. Indeed, $\ip{J'}{y'^{\otimes p}}=(N/N')^{p/2}\ip{J'}{x^{\otimes p}}$, and $N^{-(p+1)/2}(N/N')^{p/2}N'^{(p-1)/2}=(N'/N)^{1/2}N'^{-1}$. Moreover, $J$ is an affine isometric function of $J'$ given $\mathcal F_j$, and $\mathcal A^\perp\mapsto x$ is a contraction. Therefore, conditionally on $\mathcal F_j$, the map $J'\mapsto x$ is $L_{\mathrm{Lip},\theta}$-Lipschitz with values in $B_{N'}$. Let $u'\in\R^{N'}$ be the coordinates of $u_j$ when $u_j\ne0$, and the first coordinate vector otherwise. The direction $w'=\sqrt{N'}u'$ has norm $\sqrt{N'}$, and when $u_j\ne0$, \[ \frac{\ip{w'}{x}}{N'}=\frac{\ip{u'}{y'}}{\sqrt N} =\frac{\ip{u_j}{\mathcal A_{N,\theta}}}{\sqrt N}, \] where we used $u_j\in W^\perp$ in the last equality. \paragraph{Step 4: the penalty in the residual model.} Let $\alpha_j=\ip{u_j}{\mathcal A_{N,\theta}}/\sqrt N$, which is zero if $u_j=0$. The conditional law of $J'$ given $\mathcal F_j$ is standard Gaussian, and on each event $\{N'=n'\}$ the tensor $J'$ is independent of $\mathcal F_j$. We may therefore apply \eqref{eq:conditional-field-cost} in dimension $N'$, with $\mathcal F_j$ in the role of the seed, to the algorithm $J'\mapsto x$ and the direction $w'$. If $u_j\ne0$, its overlap term is $\bd|\alpha_j|^p$; if $u_j=0$, we drop its nonnegative overlap term. Since $N'\ge N-k_0-j$ and $\phi_N\ge0$, we obtain \[ \E\left[\frac{H'_{N'}(x)}{N'}+\bd|\alpha_j|^p\,\middle|\,\mathcal F_j\right] \le\Eth+\sup_{N''\ge N-k_0-j}\phi_{N''}(p,L_{\mathrm{Lip},\theta}). \] We use \eqref{sel:residual-normalization} to write \[ \frac{H_N(\mathcal A^\perp)}N+\bd|\alpha_j|^p =\sqrt{\frac{N'}N}\left(\frac{H'_{N'}(x)}{N'}+\bd|\alpha_j|^p\right) +\left(1-\sqrt{\frac{N'}N}\right)\bd|\alpha_j|^p. \] Here $\sqrt{N'/N}\le1$, $0\le1-\sqrt{N'/N}\le(k_0+j)/N$, and $|\alpha_j|\le1$. Taking expectations and letting $N\to\infty$, we find \[ \limsup_{N\to\infty}\left\{\E\frac{H_N(\mathcal A^\perp)}N +\bd\,\E|\alpha_j|^p\right\}\le\Eth. \] Combining this with \eqref{eq:residual-energy-lower}, we obtain $\bd\limsup_N\E|\alpha_j|^p\le\delta_\theta+pQ_*\sqrt{\psi_{j,\theta}}$. By Jensen's inequality as in Step~1, and since $\norm{P_{W_{j+1}}\mathcal A_{N,\theta}}^2=\norm{P_{W_j}\mathcal A_{N,\theta}}^2 +N|\alpha_j|^2$, we conclude that \begin{equation}\label{sel:adaptive-recursion} \psi_{j+1,\theta}\le\psi_{j,\theta} +\left(\frac{\delta_\theta+pQ_*\sqrt{\psi_{j,\theta}}}{\bd}\right)^{2/p}. \end{equation} \paragraph{Step 5: the outer limits.} The right sides of \eqref{sel:adaptive-initial-estimate} and \eqref{sel:adaptive-recursion} are continuous nondecreasing functions of $(\delta_\theta,\psi_{j,\theta})$ that vanish at $(0,0)$. Upper limits pass through such functions. Taking the outer limits in their prescribed order and inducting on $j\le M$, we find $\lim_\theta\psi_{j,\theta}=0$ for every $j\le M$. This is \eqref{sel:finite-span-orthogonality}. Finally, if $Y_N\in W_M\cap B_N$, then \[ \left|\frac{\ip{\mathcal A_{N,\theta}}{Y_N}}N\right|^2 =\left|\frac{\ip{P_{W_M}\mathcal A_{N,\theta}}{Y_N}}N\right|^2 \le\frac{\norm{P_{W_M}\mathcal A_{N,\theta}}^2}N, \] which proves the last assertion. \end{proof} \section{Transfer to a fixed waiting-time state}\label{sec:general-transfer} We now combine the forks with finite-query orthogonality on their fixed product probability spaces. The hypotheses of the next proposition are those of \cref{prop:threshold-inputs}\ref{item:threshold-forgetting}. Under them, the translated family $(C_T,R_T,\lambda_T)$ converges along every sequence $T_n\to\infty$ to one stationary cluster $(c,r,\ell)$. Indeed, by the precompactness in \cref{prop:stationary-clusters}, every subsequence of $(T_n)$ has a further subsequence along which the family converges, and the limit is a stationary cluster whose kernels are $(c,r)$. Its multiplier is determined by $r$: setting $\tau=0$ in \eqref{eq:stationary-cluster-response} and using $r(0)=1$, we find $\ell=-r'(0)$. All subsequential limits are therefore equal. In particular, \cref{sel:finite-forks} applies along every sequence $T_n\to\infty$. \begin{proposition}[Fixed-waiting-time decorrelation] \label{sel:general-fixed-wait} Fix $p\ge3$ and $\beta>0$. Suppose that the translated kernels converge along every sequence $T\to\infty$ to one stationary profile $(c,r)$, and that its plateau $q$ is positive and satisfies $\beta^2p(p-1)q^{p-2}(1-q)^2=1$. Then \[ \lim_{t\to\infty}C(t,s)=0\qquad\text{for every fixed }s\ge0. \] \end{proposition} \begin{proof} The proof is by contradiction. Recall from \eqref{eq:ck-physical-bounds} that $C\ge0$. Suppose that, for some $s\ge0$, there exist $\gamma>0$ and $t_n\to\infty$ such that $C(t_n,s)\ge\gamma$ for all $n$. The proof has three steps. We first express $C(t_n,s)$ through the center of a fork that ends at $t_n$. Then we show that the Lipschitz algorithms approximating the normalized center saturate the threshold in the order \eqref{sel:aging-limit-order}. Finally, on the fixed fork product law, we approximate $X_s$ by a finite-query Euler vector and use \cref{sel:adaptive-orthogonality} to force their overlap with $X_s$ to vanish, which contradicts the first step. \paragraph{Step 1: the overlap of the center.} Fix $K\ge1$ and $L\ge0$, and set $T_n=\max\{s+1,t_n-L\}$ for every $n$. Then $T_n\to\infty$, $T_n\ge s$, and for all sufficiently large $n$ we have $T_n+L=t_n$. Let $m_{N,n}$ be the center of the fork at time $T_n$ with $K$ branches of length $L$. For these sufficiently large $n$, the branches end at time $t_n$. The time $s$ lies in the common past, and each pair $(Y_b,X_s)$ has the law of $(X_{t_n},X_s)$. This implies that \begin{equation}\label{sel:fork-fixed-wait-overlap} \E\frac{\ip{m_{N,n}}{X_s}}N=\E C_N(t_n,s), \end{equation} and by \cref{prop:hard-bridge}, the right side tends to $C(t_n,s)$ as $N\to\infty$. In the limit in $n$, the endpoint of the fork is the time $t_n$ itself, so we do not need any regularity of $C(\cdot,s)$ in its first variable. \paragraph{Step 2: saturation of the threshold.} Let $\theta=(n,\varepsilon,K,L)$, where $\varepsilon=1/(m+1)$ and $K=k+1$ for $m,k\in\mathbb N$, with the order of \eqref{sel:aging-limit-order}, and let $\mathcal A_{N,\theta}=\mathcal A^\varepsilon_{N,T_n}$ be the algorithms of \cref{sel:finite-forks}\ref{item:fork-lipschitz} for this fork. They satisfy the standing hypotheses of \cref{sec:general-disclosure} on the fixed fork product law of \cref{sec:general-forks}, with seed $\zeta_N$ consisting of $X_0$, the $K$ full future Wiener paths, and the full trunk Wiener path $B$, jointly independent of $J$, and with a constant $L_{\mathrm{Lip},\theta}$ depending only on $p$, $\beta$, the fixed waiting time $s$, the horizon $T_n+L$, $K$, and $\varepsilon$. We write $Q=Q_{K,L}$, and recall that $q\le Q\le1$. Let \[ \Delta_{N,\theta}=\left(\frac1N\E\norm{\mathcal A_{N,\theta} -m_{N,n}/\sqrt Q}^2\right)^{1/2}. \] By \eqref{sel:fork-lipschitz} and the comparison of normalizations in \cref{sel:finite-forks}\ref{item:fork-lipschitz}, we have \begin{equation}\label{eq:transfer-approximation} \limsup_{n\to\infty}\limsup_{j\to\infty}\Delta_{N_j,\theta}\le\varepsilon. \end{equation} The vector $m_{N,n}/\sqrt Q$ has norm at most $\sqrt{N/q}$. Using \eqref{eq:general-energy-transfer} with $R=q^{-1/2}$ and the homogeneity of $H_N$, we find \[ \E\frac{H_N(\mathcal A_{N,\theta})}N \ge Q^{-p/2}\E\frac{H_N(m_{N,n})}N-C_q\Delta_{N,\theta},\qquad C_q=p\,q^{-(p-1)/2}Q_*. \] The constant $C_q$ does not depend on $n$, $\varepsilon$, $K$, or $L$; this is where we use $q>0$. Then \eqref{sel:fork-center-energy-limit} implies that $\lim_n\liminf_jQ^{-p/2}\E H_{N_j}(m_{N_j,n})/N_j=Q^{-p/2}h_{K,L}$. Together with \eqref{eq:transfer-approximation}, this gives \[ \limsup_{n\to\infty}\delta_\theta \le\max\left\{0,\;\Eth-\frac{h_{K,L}}{Q_{K,L}^{p/2}}+C_q\varepsilon\right\}. \] We let $\varepsilon\downarrow0$, then $K\to\infty$, then $L\to\infty$. By \eqref{sel:fork-saturation}, which applies because $q$ is marginal, we conclude that $\lim_\theta\delta_\theta=0$ in the order \eqref{sel:aging-limit-order}. \paragraph{Step 3: orthogonality on the fork product law and the contradiction.} We use the algorithms of Step~2 on precisely the fork seed and disorder product space just described, and take the limits in precisely the order \eqref{sel:aging-limit-order}, with $\varepsilon=1/(m+1)$. All disclosure histories below include the full fork seed. If $s=0$, then $X_0$ lies in the seed-measurable subspace $W_0=\Span(X_0)$, and \cref{sel:adaptive-orthogonality} with $k_0=1$ and no disclosure steps gives zero limiting squared overlap. For $s>0$, fix $M\ge1$ before any limit. We construct the Euler--Maruyama approximation \cite[Sections~4--5]{higham2001} of this fork's trunk state $X_s$ and prove the required normalized error directly. \subparagraph{The coefficients.} Let $\Psi_N=\Pi_{B_N}$ be the metric projection onto the ball of radius $\sqrt N$. We use $P_y=I-yy^{\mathsf T}/N$ for every $y\in\R^N$, as in \cref{sec:model}. For $x\in\R^N$, we define \[ b_N(x)=\beta P_{\Psi_N(x)}\nabla H_N(\Psi_N(x))-\frac{N-1}N\Psi_N(x),\qquad \sigma_N(x)=\sqrt2\,P_{\Psi_N(x)}. \] Since $\Psi_N(x)=x$ on $S_N$, these are the coefficients of \eqref{eq:hard-sde} on $S_N$, and \[ X_t=X_0+\int_0^tb_N(X_r)\,\dd r+\int_0^t\sigma_N(X_r)\,\dd B_r. \] Let $G_N=\{Q_N\le A_p\}$, and let $y=\Psi_N(x)$ and $z=\Psi_N(x')$, so that $\norm y,\norm z\le\sqrt N$ and $\norm{y-z}\le\norm{x-x'}$. The matrix $P_y$ has eigenvalues $1$ and $1-\norm y^2/N\in[0,1]$, which gives $\norm{P_y}_\op\le1$ and $\norm{P_y}_{\HS}^2\le N$. Moreover, \[ \norm{P_y-P_z}_{\HS}=\frac{\norm{yy^{\mathsf T}-zz^{\mathsf T}}_{\HS}}N \le\frac{(\norm y+\norm z)\norm{y-z}}N\le\frac{2\norm{x-x'}}{\sqrt N}. \] On $G_N$, \eqref{eq:general-ball-derivatives} with $R=2$ gives $\norm{\nabla H_N(y)}\le p2^{p-1}A_p\sqrt N$ and $\norm{\nabla H_N(y)-\nabla H_N(z)}\le p(p-1)2^{p-2}A_p\norm{y-z}$. Writing $P_y\nabla H_N(y)-P_z\nabla H_N(z) =P_y(\nabla H_N(y)-\nabla H_N(z))+(P_y-P_z)\nabla H_N(z)$, we obtain, on $G_N$ and for all $x,x'\in\R^N$, \begin{equation}\label{eq:euler-coefficient-bounds} \begin{gathered} \norm{b_N(x)-b_N(x')}\le K_b\norm{x-x'},\qquad \norm{\sigma_N(x)-\sigma_N(x')}_{\HS}\le\frac{2\sqrt2}{\sqrt N}\norm{x-x'},\\ \norm{b_N(x)}^2\le K_b'N,\qquad\norm{\sigma_N(x)}_{\HS}^2\le2N, \end{gathered} \end{equation} with $K_b=1+\beta pA_p2^{p-2}(3p+5)$ and $K_b'=(3\beta p2^{p-1}A_p+2)^2$. The Hilbert--Schmidt bound on $\sigma_N(x)-\sigma_N(x')$ is the reason that the error below does not grow with $N$: a bound of order one for each matrix entry would lose a factor of $N$. \subparagraph{The Euler--Maruyama scheme.} Let $t_i=is/M$ for $0\le i\le M$. We define $Z_0=X_0$ and \[ Z_{i+1}=Z_i+\frac sM\,b_N(Z_i)+\sigma_N(Z_i)\big(B_{t_{i+1}}-B_{t_i}\big), \qquad 0\le i0$. This completes the proof. \end{proof} \section{Proof of \texorpdfstring{\cref{prop:threshold-inputs}}{the threshold inputs}} \label{sec:general-proof-inputs} We now prove both parts of \cref{prop:threshold-inputs} for every $p\ge3$. \begin{proof}[Proof of \cref{prop:threshold-inputs}\ref{item:threshold-ceiling}] Let $T_n\to\infty$ be a sequence along which the translated kernels converge to a stationary cluster $(c,r,\ell)$ with plateau $q>0$, and fix $L\ge0$. For every $K\ge1$, \cref{sel:finite-forks} applies along $T_n$, and \eqref{sel:fork-center-threshold} gives \[ h_{K,L}\le\Eth\,Q_{K,L}^{p/2}. \] By \cref{lem:fork-parameter-limits}\ref{item:fork-K-limit}, we have $|Q_{K,L}-d_L|\le1/K$ and $|h_{K,L}-h_L|\le\beta p(p-1)/K$. The second bound is the dominated convergence argument made explicit: the difference $h_{K,L}-h_L$ is $\beta p$ times the integral of $r(\tau)$ times $k_{K,L}(\tau)^{p-1}-c(2L-\tau)^{p-1}$, which is at most $(p-1)/K$ in absolute value, and $\int_0^Lr\le1$. Letting $K\to\infty$ and using the continuity of $x\mapsto x^{p/2}$, we obtain \[ h_L\le\Eth\,d_L^{p/2}, \] which is \eqref{eq:threshold-input-ceiling}. \end{proof} Letting $L\to\infty$ in \eqref{eq:threshold-input-ceiling} and using \cref{lem:fork-parameter-limits}\ref{item:fork-L-limit}, we obtain \begin{equation}\label{sel:general-positive-upper-bound} h\le\Eth\,q^{p/2}, \end{equation} where $h$ is given by \eqref{sel:limiting-center-energy}. This is the form in which part~\ref{item:threshold-ceiling} enters the proof of \cref{sel:positive-marginality}. It holds for every stationary cluster with positive plateau, and its proof does not use marginality. \begin{proof}[Proof of \cref{prop:threshold-inputs}\ref{item:threshold-forgetting}] The hypotheses of part~\ref{item:threshold-forgetting} are those of \cref{sel:general-fixed-wait}, which gives $C(t,s)\to0$ as $t\to\infty$ for each fixed $s$. It remains to make the convergence uniform in $s\in[0,S]$. Let $k\ge1$ be an integer and let $s_i=i/k$ for $0\le i\le\lceil kS\rceil$. Every $s\in[0,S]$ is within $1/k$ of some $s_i$, and by the correlation modulus \eqref{eq:global-modulus} in the second variable, \[ |C(t,s)-C(t,s_i)|\le\frac{D_{p,\beta}}k+\sqrt{\frac2k} \] for all $t$. Since $C\ge0$, we obtain \[ \sup_{0\le s\le S}|C(t,s)|\le\max_{i}C(t,s_i)+\frac{D_{p,\beta}}k+\sqrt{\frac2k}. \] The maximum is over finitely many points, and it tends to zero as $t\to\infty$ by \cref{sel:general-fixed-wait}. Letting $t\to\infty$ and then $k\to\infty$ proves \eqref{eq:threshold-input-forgetting}. \end{proof}