Proof. Use the backward martingale \(F(s,X_s)=S_{d-s}f(X_s)\), whose initial value is \(S_df(x)\) and terminal value is \(f(X_d)\). A spin flip has increment \(-2x_i d_iF\), of magnitude at most \(2\sqrt{b(d-s)}\). Its instantaneous sum of squared jump sizes times rates is \[\sum_i\frac{w_i}{2}(2d_iF)^2\leq4b(d-s).\] The backward equation cancels the linear part of the exponential drift. The remaining jump compensator is controlled by the preceding square-jump bound and \(e^z-1-z\leq z^2e^{\left\lvert z\right\rvert}/2\). Integrating the resulting drift inequality and dividing by \(e^{\theta S_df(x)}\) gives (6). Expansion at \(\theta=0\) gives the variance assertion. This argument used the flip-rate bound and the assumed gradient bound; it used no restriction on the temperature. ◻

Proof. For the upper bound, use the full density relative to equilibrium. Starting from \(x\), the spins at time \(t\) are independent with means \(e^{-t}x_i\). Against uniform measure, \[1+\chi^2(S_t(x,\cdot),\mu)=(1+e^{-2t})^n.\] Thus total variation tends to zero uniformly in \(x\) at \(t=(1+\epsilon)\log n/2\). For the lower bound, a single statistic suffices. From the all-plus state, the normalized sum \(n^{-1/2}\sum_iX_i\) has mean \(\sqrt n\,e^{-t}\) and variance at most one; at equilibrium its mean is zero and its variance is one. At \(t=(1-\epsilon)\log n/2\), the mean tends to infinity. Chebyshev’s inequality gives total variation tending to one. ◻

Proof. We first extend the weak residual identity to expectations under \(Q\), then use first moments to select two consecutive small residuals, and finally compare the selected iterate with the stable root. The deterministic differentiation bounds are supplied by (OpenAI 2026, Lemmas 5.7, 6.1, 6.3, and 6.4). The conclusion [eq:stat-mean], including its square-root dependence on the energy, is the additional assertion proved in this argument.

Residual identities under the changed law.

The recursion uses Onsager corrections, as in Bolthausen’s TAP iteration (Bolthausen 2014). Post-iteration Gaussian comparison and local convexity are studied by Celentano (Celentano 2024); neither result substitutes for the buffered residual estimates used here. Define the primary fields by \[\begin{gathered} m^0=x,\quad b_0=0,\quad h^1=h+Mx,\quad m^l=\tanh h^l,\\ b_l=n^{-1}\sum_i(1-(m_i^l)^2),\qquad h^{l+1}=h+Mm^l-jb_lm^{l-1}, \end{gathered}\] and put \(R_l=m^{l-1}-m^l\). At any fixed depth the derivative matrices of \(h^l,m^l\) have bounded operator norm and \(\left\lVert db_l\right\rVert\le C_l n^{-1/2}\). The small-vector constructions in (OpenAI 2026, Definition 6.2) start with seeds of bounded Euclidean norm that may depend on \(M,h\) but are independent of the spin \(x\). Their sources are finite sums of smooth site coefficients times seeds or earlier auxiliary fields, and are linear in those auxiliary arguments. The coefficients and their derivatives through every fixed order used in the construction are uniformly bounded. Scalar parameters \(\theta\) satisfy \(\sup_x(|\theta(x)|+\left\lVert d\theta(x)\right\rVert)\le C\); they may include the averages \(b_l\). A source \(U\) is admissible from level \(p\) when every explicit primary field in its site coefficients has index at least \(p\). In particular, the seed \((1,\ldots,1)/\sqrt n\) permits the source \(m^l/\sqrt n\) at level \(l\). An ordinary construction starts with an empty auxiliary list. At stage \(r\), write \(w^r=U\) for the new source and \(y^r\) for the output of the operation below; the auxiliary sum runs over earlier pairs \((w^a,y^a)\), \(a<r\). The formal partial \(\partial_lU_i\) differentiates the explicit primary-field argument \(h_i^l\), holding the seeds, scalar parameters (including the averages \(b_l\)), and auxiliary arguments fixed. The partial \(\partial_{y^a}U_i\) differentiates the linear argument \(y_i^a\) with its coefficient held fixed. A partial in an absent argument is zero. The associated auxiliary operation is \[ U\longmapsto MU-j\sum_l\alpha_lm^{l-1}-j\sum_a\gamma_a w^a, \quad \alpha_l=n^{-1}\sum_i\partial_lU_i,\quad \gamma_a=n^{-1}\sum_i\partial_{y^a}U_i. \tag{15}\] These partials act on the displayed arguments of a recipe. They are distinct from the discrete derivative matrix \((\nabla U)_{ij}=d_jU_i\), which records the response of the whole recipe to spin flips. For every fixed construction the imported deterministic bounds are \[ \left\lVert U\right\rVert+\left\lVert \nabla U\right\rVert_{\mathrm{HS}}+\sum_i|d_iU_i|\le C, \quad |\alpha_l|+\left\lVert d\alpha_l\right\rVert\le Cn^{-1/2}, \quad |\gamma_a|+\left\lVert d\gamma_a\right\rVert\le C. \tag{16}\] They also hold for the local implicit construction used below, on its specified fixed flip buffers. These statements concern pointwise differences and have no dependence on the law under which they are tested.

Put \(D=D_P(Q)\) and, for a scalar test, let \[B(G)=\sup_x|G(x)|+\sup_x\left\lVert dG(x)\right\rVert.\] We claim that the weak residual bound becomes \[ |\mathbb E_Q G R_p^{\mathsf T}U| \le C B(G)(1+\sqrt D). \tag{17}\] For a local construction the test is supported on its small-residual region, and the supremum bounds are only needed on the corresponding buffer. In particular no conditional regularity of \(Q\) is assumed.

At the first level, the only change from the equilibrium residual argument is the mismatch between the \(Q\)-conditional mean and the \(P\)-conditional mean. Conditional covariance under \(Q\) gives \[\begin{align*} \mathbb E_Q\sum_i(x_i-m_i^1)GU_i ={}&\mathbb E_Q\sum_i(1-\bar m_i^2)d_i(GU_i)\\ &+\mathbb E_Q G\sum_i(\bar m_i-m_i^1)U_i. \end{align*}\] This remains true when a conditional law of \(Q\) is a point mass. The product rule \(d_i(GU_i)=Gd_iU_i+(d_iG)(U_i-2x_id_iU_i)\) and [eq:stat-small-bounds] bound the first term by \(CB(G)\). [eq:stat-conditional-means] and Cauchy–Schwarz bound the second by \(2C\left\lVert G\right\rVert_\infty\sqrt D\). This proves the base case.

For higher levels we reduce to lower residual indices by an exact cancellation. For a target admissible from level \(p+1\), set \[a_p=j(b_p-b_{p-1}),\qquad U'_i=\Phi(h_i^p;h_i^{p+1},a_p)U_i,\] where \[ \Phi(z;r,a)=\int_0^1 e^{a(1-u^2)/2} \frac{\cosh(u(z-r))}{\cosh z\cosh r}\,\,\mathrm du. \tag{18}\] It satisfies \([(z-r-a\tanh z)-a\partial_z]\Phi=\tanh z-\tanh r\). Adjoin the auxiliary field \(y'\) of \(U'\) using [eq:stat-auxiliary]. With \(\alpha_l,\gamma_a\) belonging to \(U'\), the recurrence gives \[\begin{align*} R_{p+1}^{\mathsf T}U ={}&R_p^{\mathsf T}y'+a_pR_p^{\mathsf T}U' -jb_{p-1}R_{p-1}^{\mathsf T}U' +j\sum_a\gamma_a R_p^{\mathsf T}w^a\\ &+j\sum_{l\ge p}\alpha_lR_p^{\mathsf T}m^{l-1} -a_p n\alpha_p. \tag{19}\end{align*}\] The \(R_{p-1}\) term is absent at \(p=1\). The \(l=p\) term cancels the last term because \[R_p^{\mathsf T}m^{p-1} =n(b_p-b_{p-1})-R_p^{\mathsf T}m^p.\] After this cancellation, the remaining \(l=p\) term is \(-j(\sqrt n\alpha_p)R_p^{\mathsf T}(m^p/\sqrt n)\). Its target \(m^p/\sqrt n\) is admissible from level \(p\) and has bounded norm. For \(l>p\), use the target \(m^{l-1}/\sqrt n\) in the same way. Thus every term has a lower residual index, a permitted target, and one of the bounded multipliers \(a_p,b_{p-1},\gamma_a,\sqrt n\alpha_l\). Their difference bounds in [eq:stat-small-bounds] and the product rule give \(B(\theta G)\le C B(G)\) for each such multiplier. Strong induction over finite admissible constructions therefore proves [eq:stat-Q-residual]. Each step adds finitely many operations and decreases the residual index, so a finite word length suffices. For a supported local test, a difference can be nonzero only on a one-flip neighborhood of its support; the imported buffers include every such neighborhood used in this finite induction.

Selecting two consecutive small residuals. The weak estimate now has the desired \(1+\sqrt D\) cost. We preserve that cost in the selection step by using first moments rather than second moments. Set \(S_l=R_l^{\mathsf T}m^l/\sqrt n\) and \(\psi(s)=s/\sqrt{1+s^2}\). The primary derivative bounds give \(\sup\left\lVert dS_l\right\rVert\le C_l\), while \(|\psi|,|\psi'|\le1\) and \(s\psi(s)\ge|s|-1\). Testing [eq:stat-Q-residual] with \(U=m^l/\sqrt n\) and \(G=\psi(S_l)\) yields \[ \mathbb E_Q|S_l|\le C_l(1+\sqrt D). \tag{20}\] The exact telescoping identity is \[\left\lVert R_l\right\rVert^2/n=b_l-b_{l-1}-2S_l/\sqrt n.\] For a fixed small \(\rho>0\), choose a fixed depth \(d_0\) with \(\lfloor d_0/2\rfloor\rho^2/4>2\), and then a fixed \(\varepsilon>0\) with \(2d_0\varepsilon<1\). If every \(|S_l|\le\varepsilon\sqrt n\), the sum of the squared residuals divided by \(n\) is less than two. Some consecutive pair consequently has both norms at most \(\rho\sqrt n/2\). Let \(C_k\) be smooth cutoffs equal to one on this latter condition and supported where both norms are at most \(\rho\sqrt n\). The primary derivative bounds give \(\left\lVert dC_k\right\rVert\le C_\rho/\sqrt n\). Combine the cutoffs in order so that earlier cutoffs reduce the weight available to later ones, and retain the unassigned deficit. Put \[\widehat C_k=C_k\prod_{l=2}^{k-1}(1-C_l),\qquad \Delta=1-\sum_{k=2}^{d_0}\widehat C_k.\] Markov’s inequality and [eq:stat-S-first-moment] give \[ Q(\Delta>0)\le C_\rho(1+\sqrt D)/\sqrt n. \tag{21}\]

Comparing the selected iterate with the stable root. Let \(r\) be the unique zero of [eq:stat-field-map], whose existence and inverse bound follow from (OpenAI 2026, Lemma 5.7), and put \(t_* =\tanh r\), \(q_*=n^{-1}\left\lVert t_*\right\rVert^2\). On the buffered region of \(C_k\), fix the primary fields and the root, and choose one of the two sources \(w_0\) specified below. The implicit construction of (OpenAI 2026, Lemma 6.3) supplies a vector \(w\), an auxiliary vector \(y^{\rm imp}\), and a scalar \(c_1\) solving \[\begin{gathered} a=j(1-b_{k-1}-q_*),\quad A_i=\Phi(h_i^k;r_i,a),\quad A=\mathop{\mathrm{diag}}(A_i),\quad \chi=n^{-1}\sum_iA_i,\\ w=w_0+A(y^{\rm imp}-jc_1t_*),\qquad y^{\rm imp}=Mw-j\chi w-jc_1m^{k-1},\qquad c_1=n^{-1}\sum_i\partial_k w_i. \end{gathered}\] Here \(w_0\) is either \(Ae\) for a fixed \(\left\lVert e\right\rVert\le1\), or \(n^{-1/2}\partial_r\Phi(h_i^k;r_i,a)\) coordinatewise. In \(\partial_k w_i\), the arguments \(y_i^{\rm imp},c_1,a\) are held fixed; in \(\partial_r\Phi\), \(a\) is held fixed. With this convention the displayed equations form a linear system for \((w,y^{\rm imp},c_1)\), whose unique solution is supplied by the cited lemma. That lemma also gives \[\left\lVert w\right\rVert+\left\lVert y^{\rm imp}\right\rVert+\sqrt n|c_1|\le C,\qquad I+j\chi A-A^{1/2}MA^{1/2}\succeq c_0I\] after decreasing the common positive inverse margin if necessary. The distinction between its bounds matters for selecting parameters: derivative constants may depend on the fixed depth, while size and inverse constants do not. We will first control the susceptibility average with the second seed, then test the mean in direction \(e\) with the first seed.

Define \[T_k(w)=\{h^k-h-(M-jb_{k-1}I)m^k\}^{\mathsf T}w -j(b_k-b_{k-1})nc_1.\] The algebra in (OpenAI 2026, Lemma 6.6) expresses this as \[R_k^{\mathsf T}y^{\rm imp} +j(\chi-b_{k-1})R_k^{\mathsf T}w -jb_{k-1}R_{k-1}^{\mathsf T}w-jc_1R_k^{\mathsf T}m^k.\] Thus the already proved local residual estimate gives \[ |\mathbb E_Q G T_k(w)|\le CB(G)(1+\sqrt D). \tag{22}\] This is the required arbitrary-law version of the terminal residual identity; the last term uses the multiplier \(\sqrt n c_1\).

To extract a mean comparison from this terminal residual, write \(p=m^k-t_*\) and \(L_* =\sqrt n\{\chi-(1-q_*)\}\). Subtract the root equation and substitute the displayed equation for \(Mw\) to obtain \[\begin{align*} T_k(w)={}&\sum_i[(h_i^k-r_i-am_i^k)-a\partial_k]w_{0,i}\\ &+j(1-q_*-\chi)p^{\mathsf T}w-jc_1R_k^{\mathsf T}p. \tag{23}\end{align*}\] The inverse bound and the two small residuals imply \(\left\lVert p\right\rVert\le C\rho\sqrt n\). Consequently the middle term has absolute value at most \(C_0\rho|L_*|\), with \(C_0\) independent of the selection depth. Also \(\sup\left\lVert dL_*\right\rVert\le C_k\). Because \(C_0\) is independent of depth, we can choose \(\rho\) with \(C_0\rho<1/2\) first, and only then choose \(d_0\) and \(\varepsilon\) as in the selection step above.

For \(w_{0,i}=n^{-1/2}\partial_r\Phi(h_i^k;r_i,a)\) the first sum in [eq:stat-implicit-comparison] is \(L_*\): differentiate the identity following [eq:stat-Phi] in \(r\), holding \(a\) fixed. Test with \(G=C_k\psi(L_*)\). Its \(B\)-norm is bounded. The last term is controlled by [eq:stat-Q-residual] with target \(p/\sqrt n\) and multiplier \(\sqrt n c_1\). Absorbing the middle term gives \[ (1-C_0\rho)\mathbb E_Q[C_k|L_*|]\le1+C_k'(1+\sqrt D). \tag{24}\] For \(w_0=Ae\), the first sum is \(p^{\mathsf T}e\). Test this time with \(\widehat C_k\) alone. Since \(\widehat C_k\le C_k\), [eq:stat-Q-terminal,eq:stat-L-first-moment] and the same residual bound imply \[|\mathbb E_Q\widehat C_k(m^k-t_*)^{\mathsf T}e| \le C_k'(1+\sqrt D).\] It remains to pass from the selected iterate back to the original spin. Sum [eq:stat-Q-residual] with constant target \(e\) to handle \(x-m^k=\sum_{l\le k}R_l\), and then sum over \(k\). The omitted part has absolute value at most \(2\sqrt n\,Q(\Delta>0)\le C(1+\sqrt D)\) by [eq:stat-selection]. Taking the supremum over unit \(e\) proves \(\left\lVert \mathbb E_Qx-t_*\right\rVert\le C(1+\sqrt D)\). Apply this also with \(Q=P\), when \(D=0\), and use the triangle inequality. ◻

Proof. We verify second-stage stability and its gap first. We then run the observation argument conditionally from each first-stage posterior to obtain its gap, and finally place all single-site cavities on one common disorder event.

Stability through the two stages. For the first stage, (OpenAI 2026, Proposition 5.1 and Corollary 5.2) give [eq:stat-stability], with positive slack and a test-independent stopping rule, throughout the first stage up to any prescribed fixed \(b\), outside an event of ordinary observation probability \(e^{-cn}\). Intersect its matrix event with the finite word events required in 6 and in the directional estimate (OpenAI 2026, Proposition 7.2).

On the second stage write \(u=1-r\) and use \(M=uJ\), \(j=u^2\beta^2\). The word bounds hold uniformly in \(u\in[0,1]\). In fact their inverse factor is exactly \[(I-uAJ+u^2\beta^2\chi_AA)^{-1} =(I-(uA)J+\beta^2\chi_{uA}(uA))^{-1}.\] The other word factors only acquire powers of \(u\). The fixed upper bound on \(A\) can thus be the same as for \(J\). All derivative and implicit construction estimates used above are uniform for these parameters: the recurrences have bounded coefficients, \(j\) remains bounded away from one, and the inverse margins are fixed. At \(u=0\) the measure is product; [eq:stat-conditional-means] gives its mean bound directly.

To obtain a margin throughout the second stage, use the large field created in the first one. Gaussian maximal bounds on the two fixed Brownian intervals imply, outside \(Ce^{-cn}\), that \[ \sup_{0\le r\le1}\left\lVert Y_b+Z_r-bX\right\rVert\le C\sqrt{bn} \tag{28}\] for \(b\ge1\), with a constant depending on \(K,k\). One elementary proof of the bound for \(B\) uses a \(1/2\)-net of the unit sphere and the scalar Brownian maximal inequality; the covariance norm of \(A_*^{1/2}W\) is at most \(K+k\). This also gives the asserted exponential probability. If \(y\) has residual at most \(\rho_0\sqrt n\) in [eq:stat-field-map], then \(\left\lVert y-h\right\rVert\le(\rho_0+\beta^2+K)\sqrt n\). Thus, by [eq:stat-large-field], at most \(C n/b\) coordinates have \(|y_i|<b/2\). Outside these coordinates \(V(y)_{ii}\le\operatorname{sech}^2(b/2)\). For a candidate \(A\) in [eq:stat-stability], an additional set of size at most \(\epsilon_A^2n/a^2\) has \(|A_{ii}-V(y)_{ii}|>a\).

The small principal submatrix bounds of (OpenAI 2026, Lemma 8.2) apply also to \(J\), because \(\left\lVert J-(\tanh J_{ij})\right\rVert\to0\) on that event. If \(S\) is the preceding exceptional set and \(A_{ii}\le2a\) off \(S\), they give \[\begin{align*} \left\lVert A^{1/2}uJA^{1/2}\right\rVert &\le a_+\left\lVert J_{SS}\right\rVert+2K\sqrt{2a_+a}+2Ka,\\ \frac{2u^2\beta^2}{n}\left\lVert A^{1/2}\tanh y\right\rVert^2 &\le2\beta^2\{a_+|S|/n+2a\}. \end{align*}\] Choose the principal-submatrix tolerance and \(a\) first so that the sum of these bounds is less than \(1/2\); choose an allowed exceptional fraction next, then \(b\) large and \(\epsilon_A\) small. The nonnegative term \(jb(y)A\) in [eq:stat-field-hessian] can be kept or dropped. This proves [eq:stat-stability] with \(\kappa=1/2\) on the second stage, after reducing the common margins if necessary. These choices are compatible with the first-stage theorem: choose this fixed \(b\) first and subsequently intersect and reduce its supplied margins.

The second-stage gap. The same large-field event also gives a gap at least \(1/2\) throughout this stage. The curvature bound must hold uniformly in the spin configuration. For that purpose combine [eq:stat-large-field] with \(\left\lVert uJx\right\rVert\le K\sqrt n\). Only a small fixed fraction of the fields \(h_i+(uJx)_i\) can be smaller than \(b/4\) in absolute value. Let \(v_i=\operatorname{sech}^2(h_i+(uJx)_i)\) and \(a_i=v_id_if\). The signed curvature inequality of (OpenAI 2026, Lemma 8.1) is \[\mathbb E_P(Hf)^2\ge(1-Cr_4)\mathcal E_P(f) -\mathbb E_P\{a^{\mathsf T}Ua+C|a|^{\mathsf T}Q_0|a|\},\] where \(H=-L_P\), \(U_{ij}=\tanh(uJ_{ij})\), \((Q_0)_{ij}=U_{ij}^2\), and \(r_4=\max_i\sum_jU_{ij}^4=o(1)\). On the weak-field set use the small principal-submatrix norms; on its complement use \(v_i\le\operatorname{sech}^2(b/4)\). If \(q_f(x)=\sum_i v_i(d_if)^2\), each quadratic cost is bounded by \([\varepsilon+K'(2\sqrt\eta+\eta)]q_f(x)\) with \(\eta=\operatorname{sech}^2(b/4)\) and a fixed \(K'\). Choosing the same tolerances sufficiently small proves \(\mathbb E_P(Hf)^2\ge\mathcal E_P(f)/2\). The finite-dimensional spectral theorem gives the gap \(1/2\). The matrix bounds are uniform in \(u\): \(\tanh(uJ)=uJ+o_{\mathrm{op}}(1)\), while the nonnegative matrix \((\tanh(uJ_{ij})^2)\) is dominated entrywise by \((J_{ij}^2)\).

The first-stage posterior gaps. The remaining task is to obtain a gap at every first-stage posterior. We do this by conditioning the observation experiment from that posterior onward. The zero-field gap theorem does not by itself apply to these random fields. Let \(E\) be the event of a first-stage stability failure or failure of the terminal gap at time \(b\). The preceding bounds and (OpenAI 2026, Proposition 8.3) allow a fixed \(a_0>0\) such that \(\mathbb P(E\mid J)\le e^{-2a_0 n}\). For the observation filtration set \(q_s=\mathbb P(E\mid\mathcal F_s,J)\). The nonnegative martingale maximal inequality gives \[ \mathbb P\{\sup_{s\le b}q_s>e^{-a_0n}\mid J\}\le e^{-a_0n}. \tag{29}\] Outside this event and \(E\), fix a time \(s<b\). Conditional on \(\mathcal F_s\), the remaining experiment is an ordinary white observation starting from its posterior \(P_s\), with horizon \(b-s\le b\). Its future bad probability is at most \(e^{-a_0n}\).

For every \(f\) with \(P_sf=0\) and \(P_sf^2=1\), set \(d=\mathcal E_{P_s}(f)\), and stop the remaining experiment at its next stability failure or at \(b\), denoting this time by \(\sigma\). Write \(N_t=P_tf^2\) and \(V_t=P_tf^2-(P_tf)^2\). Conditional variance decomposition gives \(\mathbb E[\mathcal E_{P_t}(f)\mid\mathcal F_s]\le d\). At every retained stable posterior, the directional estimate of (OpenAI 2026, Proposition 7.2) states, in squared form, \[\left\lVert \mathop{\mathrm{Cov}}_{P_t}(f,x)\right\rVert^2 \le C_{\rm d}\{\mathop{\mathrm{Var}}_{P_t}(f)+\mathcal E_{P_t}(f)\}.\] It uses the same stable-field implication and a sufficiently long finite set of word diagnostics. This homogeneous estimate, combined with the filtering variance identity, implies \[ \mathbb E[V_\sigma\mid\mathcal F_s] \ge v_s-(1-v_s)d, \qquad v_s=e^{-C_{\rm d}(b-s)}\ge v_0=e^{-C_{\rm d}b}>0. \tag{30}\] Indeed the stopped expected variance has derivative \(-\mathbb E[\mathbf 1_{t<\sigma}\left\lVert \mathop{\mathrm{Cov}}_{P_t}(f,x)\right\rVert^2\mid\mathcal F_s]\), which is at least \(-C_{\rm d}(\mathbb EV_{t\wedge\sigma}+d)\).

The variance lower bound must be compared with the terminal energy. To control the variance lost on bad paths, change the input law to \(f^2P_s\). The observation density is \(N_t\) and its posterior is \(Q_t=f^2P_t/N_t\). Before stopping, 6 and conditional variance decomposition give \[\mathbb E^{f}[\mathbf 1_{t<\sigma}\left\lVert \mathbb E_{Q_t}x-\mathbb E_{P_t}x\right\rVert^2]\le C(1+d), \qquad \mathbb E^{f}D_{P_t}(Q_t) =\mathbb E\mathcal E_{P_t}(|f|)\le d.\] By [eq:stat-observation-entropy], the stopped observation entropy is at most \(Cb(1+d)\). Relative entropy between event probabilities gives \[ \mathbb P^{f}(\mathrm{future\ bad}\mid\mathcal F_s) \le\frac{Cb(1+d)+\log2}{a_0n}. \tag{31}\] Here the bad event is measurable at \(\sigma\): it is the stopping event before \(b\), or the terminal gap failure at \(b\). Its ordinary conditional probability is exponentially small by [eq:stat-future-bad]. The changed probability in [eq:stat-changed-bad] is derived anew for each \(f\); it is not asserted to be exponentially small.

On good terminal paths \(V_b\le2\mathcal E_{P_b}(f)\); on bad paths \(V_\sigma\le N_\sigma\). The stopped density identity and [eq:stat-conditional-variance,eq:stat-changed-bad] consequently give, for \(d\le1\), \[v_0-(1-v_0)d\le2d+C/n.\] For all sufficiently large \(n\) this implies \(d\ge v_0/6\); for \(d>1\) the same lower bound is automatic. All constants are uniform in \(s\) and \(f\). A short remaining horizon improves \(v_s\) and decreases the entropy integrals, so the estimate stays uniform as \(s\uparrow b\). At \(s=b\) use the terminal gap directly. Alternatively one can first argue at rational times and then use continuity of the finite-state gap in the field.

One event for every cavity. We now transfer the matrix and observation estimates to all single-site cavities at once. This requires a common disorder event, rather than a union bound over unspecified \(o(1)\) disorder probabilities. Write \(J^{(i)}\) for a principal minor. Word diagnostics for this matrix are restrictions of those for \(J\): insert the complementary coordinate projection among diagonal factors, and set the missing entry of an inverse diagonal to zero. All normalized traces can be retained with denominator \(n\); equivalently the \((n-1)\)-site variance parameter is \(j_i=\beta^2(n-1)/n\). The identity \(j_i/(n-1)=\beta^2/n\) makes these the same contractions. Inserting these projections may increase the required word length by a fixed factor; choose that larger finite length in the full event. The event is already simultaneous in its diagonal parameters. The small-submatrix bounds pass directly to every minor.

For first-stage stability, take a candidate cavity vector \(y_{-i}\) and extend it by choosing \(y_i\) to solve the missing scalar equation \(F_{J,h}(y)_i=0\). This equation is continuous with a bounded nonlinear part and hence has a solution. On the retained coordinates the residual changes by a vector of norm at most \(\left\lVert J_{-i,i}\right\rVert+O(n^{-1/2})=O(1)\). Extend the candidate diagonal with \(A_{ii}=\operatorname{sech}^2 y_i\). Its Frobenius discrepancy is unchanged. The principal block of the full Hessian differs from the cavity Hessian by \(O(1/n)\) in norm: the rank coefficient is exactly \(2\beta^2/n\) in either dimension, and only the missing contribution to \(jb(y)\) changes. Start with smaller cavity residual and Frobenius thresholds and use the slack in the full stability event. It follows, uniformly in \(i\), that full stability implies cavity stability with common positive margins for large \(n\).

The comparison of sampling laws costs only \(\exp(C\sqrt n)\). Indeed the full marginal on \(x_{-i}\) is proportional to \(\mu^{(i)}(x_{-i})\cosh(J_i x_{-i})\), and \(|J_i x_{-i}|\le K\sqrt n\). More explicitly, the law obtained by sampling the cavity spin configuration and an independent uniform missing spin has density at most \(\exp(2K\sqrt n)\) relative to \(\mu\). Extend its Brownian observation by an independent coordinate. Thus a full exponential observation failure remains exponential for the cavity experiment. The second-stage large-field and curvature arguments apply directly to the minors with the same constants. A union bound over the \(n\) cavities now loses only a factor \(n\) from exponential observation probabilities. This proves the claimed simultaneous statement on one common disorder event. It also proves a common equilibrium gap by taking \(s=0\).

A test-independent stopping rule. To finish, all properties tested above depend only on the observed field and finite matrices. Stable-field failure is detected by the Lipschitz residual-distance stopping construction in (OpenAI 2026, Corollary 5.2); the gap is a continuous finite-matrix eigenvalue. Stopping at smaller stability margins or at a gap below half the established bound gives an observation stopping time independent of the test function, with the same exponential exceptional bound. ◻

Proof. We prove a Gaussian subset bound under the planted law, transfer it to stationary sampling using the partition function, and then convert the averaged bound to one at fixed disorder. The exponential slack will also make the cavity statement simultaneous. Let \(Z(J)\) be the partition function with counting measure on the cube. The annealed planted law has density \(e^{x^{\mathsf T}Jx/2}/\mathbb E_J Z(J)\) relative to Gaussian disorder times counting measure on \(x\). Under it \(X\) is uniform, and, conditionally on \(X\), the off-diagonal entries are independent Gaussian variables with mean \(\beta^2X_iX_j/n\) and variance \(\beta^2/n\). Moreover \[ \frac{\mathbb E_J Z(J)^2}{(\mathbb E_J Z(J))^2} =e^{-\beta^2/2}\mathbb E\exp\left(\frac{\beta^2 R_n^2}{2n}\right) \le(1-\beta^2)^{-1/2}, \tag{33}\] where \(R_n\) is the sum of \(n\) independent uniform signs. The last bound follows by introducing a standard Gaussian \(G\) and using \(\cosh t\le e^{t^2/2}\): \(\mathbb Ee^{\beta^2R_n^2/(2n)}=\mathbb E_G\cosh(\beta G/\sqrt n)^n\). The gradient of \(\log Z\) in the independent standardized Gaussian coordinates has norm at most \(C\sqrt n\). Gaussian concentration and Paley–Zygmund applied to [eq:stat-Z-moment] imply \[ \log Z(J)-\log\mathbb E_JZ(J)=o(n) \quad\hbox{in disorder probability}. \tag{34}\] For clarity, Paley–Zygmund gives a fixed positive probability of \(\log Z\ge\log\mathbb EZ-\log2\); concentration places its mean within \(O(\sqrt n)\) of this level. Concentration once more proves [eq:stat-Z-comparison].

We next estimate the fields under planting. At the observation endpoint \[h=(b+k)X+JX+B_b+A_*^{1/2}W_1.\] Under planting, conditionally on \(X\), \(JX\) is a Gaussian vector with bounded covariance norm and a mean of bounded coordinate magnitude. On \(\left\lVert J\right\rVert\le K\), the last observation noise has conditional covariance norm at most \(K+k\). Thus, for a fixed subset of size \(m\), Gaussian norm tails, followed by a union bound over at most \((en/m)^m\) subsets, give an annealed bound \(C_1e^{-c_1t}\) for violation of \[\sum_{i\in I}h_i^2\le C_2[m\log(en/m)+t]\] simultaneously in \(I\). This uses the deterministic inequality \(\left\lVert z_1+\cdots+z_4\right\rVert^2\le4\sum\left\lVert z_a\right\rVert^2\), so independence of \(JX\) and \(A_*^{1/2}W_1\) is unnecessary. Increasing \(C_2\) gives any fixed desired exponential rate in \(t\). Fix this increase once, before the quenched comparison below; it does not depend on \(\delta\).

To return to the quenched experiment, note that the density of the planted joint spin-disorder law relative to stationary sampling with the original disorder is \(Z(J)/\mathbb E_JZ(J)\). Restrict to [eq:stat-Z-comparison] with, for example, \(\log Z\ge\log\mathbb EZ-\delta n/10\). The preceding annealed bound, with ample exponential slack, therefore bounds the disorder average of the quenched failure probability at level \(s\) by \(e^{-3s}\) for \(s\ge\delta n\), once \(n\) is large. Markov’s inequality and a union bound over the integer levels \(s\ge\delta n\) give a disorder exceptional probability \(O(e^{-c\delta n})\) outside which the required conditional probability is at most \(e^{-s}\). For a real \(s\ge\delta n\ge1\), apply the integer estimate at \(\lceil s\rceil\) and double the threshold constant; then \(\lceil s\rceil\le2s\) and \(e^{-\lceil s\rceil}\le e^{-s}\). Only the good disorder event depends on \(\delta\).

For a cavity the same planted computation has parameter \(\beta\sqrt{(n-1)/n}<\beta\) and common constants. Alternatively extend its spin and observation experiment as in 7; the additional \(\exp(C\sqrt n)\) factor is absorbed by the exponential slack for fixed \(\delta\). Taking the union over cavities still gives probability tending to one. This argument avoids a union bound over unquantified convergence in probability in [eq:stat-Z-comparison]: the full partition lower bound transfers to the minors with an \(O(\sqrt n)\) error. More explicitly, \[2Z(J^{(i)})\le Z(J)\le2e^{K\sqrt n}Z(J^{(i)}),\qquad \log\frac{\mathbb EZ(J)}{2\mathbb EZ(J^{(i)})} =\frac{\beta^2(n-1)}{2n}.\] These inequalities transfer the same full lower bound to every minor at once; the remaining exponential Markov bounds tolerate the union over \(i\). ◻

Proof. We first bound the entropy of the observation law, then the conditional entropy left at its product endpoint. The latter bound contains a term involving the original entropy, which will be absorbed only at the end.

Entropy of the observations. Put \(D=D_\mu(\nu)\) and \(H=H_\mu(\nu)\), and change the input of the two-stage experiment from \(\mu\) to \(\nu\). Let \(P_t,Q_t\) denote the ordinary and changed posteriors, and let \(\sigma\) be the first loss of the retained stability conditions, capped at \(b+1\). By [lem:stat-mean,eq:stat-observation-entropy,eq:stat-energy-monotone], \[ H(\mathbb Q|_{\mathcal F_\sigma}\mid \mathbb P|_{\mathcal F_\sigma})\le C(1+D). \tag{36}\] The ordinary probability of \(\sigma<b+1\) is at most \(Ce^{-cn}\). Applying the binary entropy inequality to this stopped event gives \[\mathbb Q\{\sigma<b+1\}\le C(1+D)/n.\] Continue the observations even on this event. The squared difference of posterior means is always at most \(4n\), and the precision and horizon are bounded. Hence the contribution after stopping in [eq:stat-observation-entropy] is at most another \(C(1+D)\). The relative entropy of the complete observation laws is therefore at most \(C(1+D)\).

Entropy remaining at the product endpoint. The observation channel is the same for the two input laws. Thus the entropy chain rule on the joint experiment reads \[ H=H(\mathbb Q\mid\mathbb P) +\mathbb E_{\mathbb Q}H_{P_{b+1}}(Q_{b+1}). \tag{37}\] At the product endpoint, entropy tensorization gives \[H_{P_{b+1}}(Q_{b+1}) \le\sum_i\mathbb E_{(Q_{b+1})_{-i}} H_{(P_{b+1})_i}((Q_{b+1})_i).\] This version with all other coordinates conditioned follows, for example, by the entropy chain rule and convexity of relative entropy: conditioning on more coordinates can only increase the corresponding average conditional entropy relative to a fixed coordinate law.

For a two-point law \(p\) of least atom \(p_{\min}\) and any \(q\), \[ H_p(q)\le C(1+\log(1/p_{\min})) \left[1-\left(\sum\sqrt{pq}\right)^2\right]. \tag{38}\] Indeed for \(0\le z\le1/p_{\min}\), \(z\log z-z+1\le C(1+\log(1/p_{\min}))(\sqrt z-1)^2\): the ratio has a finite limit at \(z=1\), is bounded on \([0,4]\), and for \(z\ge4\) is at most \(C\log z\). Sum this inequality with weights \(p\) and use \(\sum p(\sqrt{q/p}-1)^2=2(1-\sum\sqrt{pq}) \le2[1-(\sum\sqrt{pq})^2]\). For the product law of logit \(h_i\), \(\log(1/p_{\min})\le\log2+2|h_i|\). Define the conditional energy contributions \[z_i=\mathbb E_{(Q_{b+1})_{-i}} \left[1-\left(\sum_{s=\pm1} \sqrt{(P_{b+1})_i(s)(Q_{b+1})_i(s)}\right)^2\right].\] They satisfy \(0\le z_i\le1\) and \(\mathbb E_{\mathbb Q}\sum_i z_i\le D\) by [eq:stat-energy-monotone]. Consequently \[ H\le C(1+D)+C\mathbb E_{\mathbb Q}\sum_i z_i|h_i|. \tag{39}\]

The weighted field sum and absorption. The variables \(z_i\) record where the energy is spent. We use subset tails to bound their field-weighted sum without replacing the field by its maximum over sites. By 8 there is a nonnegative excess \(R(h)\), at least \(\delta n\), such that simultaneously in all subsets \[\sum_{i\in I}h_i^2\le C\left[|I|\log\frac{en}{|I|}+R(h)\right], \qquad \mathbb P\{R(h)>\delta n+t\}\le e^{-t}\quad(t\ge0),\] after a fixed increase of \(C\). In particular \(\mathbb E_{\mathbb P}e^{R/2}\le C e^{\delta n/2}\). The entropy variational inequality on the joint spin-observation laws, whose relative entropy is exactly \(H\), gives \[ \mathbb E_{\mathbb Q}R\le C(\delta n+1+H). \tag{40}\] The same conclusion follows from the observation marginal by data processing. At this stage \(H\) is still unknown. Keeping it on the right side is legitimate; the square-root estimate below will allow us to absorb the resulting contribution.

Set \(z=\sum_i z_i\). Layer cake for \(I_t=\{i:z_i>t\}\) and concavity of \(u\mapsto u\log(en/u)\) on \([0,n]\) give \[\sum_i z_i h_i^2 =\int_0^1\sum_{i\in I_t}h_i^2\,\,\mathrm dt \le C\{z\log(en/z)+R\}.\] Thus, first by weighted Cauchy–Schwarz and then by Cauchy–Schwarz in the observation law and Jensen’s inequality, \[\begin{align*} \mathbb E_{\mathbb Q}\sum_i z_i|h_i| &\le C\mathbb E_{\mathbb Q}\sqrt z\sqrt{z\log(en/z)+R}\\ &\le C\sqrt D\sqrt{D\log(en/D)+\delta n+1+H}. \end{align*}\] Here \(u\log(en/u)\) is increasing on \([0,n]\), allowing \(\mathbb Ez\le D\) to be substituted. Insert this in [eq:stat-H-z]. The term \(C\sqrt{DH}\) is at most \(H/2+C'D\); the term \(C\sqrt{D(\delta n+1)}\) is at most \(C'D+C'(\delta n+1)\). Absorbing \(H/2\) proves [eq:stat-entropy]. All estimates and choices were uniform over \(\nu\) and the cavities. ◻

Proof. A defective logarithmic Sobolev inequality. We keep the dependence on the defect parameter explicit. For \(0<\eta<1/2\), the concave function \(u\sqrt{\log(en/u)}\) has at \(u=\eta n\) a tangent with slope at most \(\sqrt{\log(e/\eta)}\) and intercept at most \(\eta n/2\). Thus 9, with \(\delta=\eta\), implies, after normalizing \(f^2\) to a density, \[ \mathop{\mathrm{Ent}}_\mu(f^2)\le A_\eta\mathcal E_\mu(f) +B_\eta\mu(f^2), \quad A_\eta=C(1+\sqrt{\log(1/\eta)}),\quad B_\eta=C(1+\eta n). \tag{42}\] The energy of the normalized density square root lies in \([0,n]\), as required by 9. This also proves [eq:stat-defective-lsi] for signed \(f\), using \(\mathcal E_\mu(|f|)\le\mathcal E_\mu(f)\).

Hypercontractivity with the defect retained. Use the logarithmic-Sobolev-to-hyper­contractivity argument of Gross (Gross 1975), carrying the additive defect through the norm calculation. Let \(g_t=S_tf>0\), \(p(0)=2\), and \(p'=4(p-1)/A_\eta\). Reversibility gives

\[\frac{\,\mathrm d}{\,\mathrm dt}\log\left\lVert g_t\right\rVert_{p} =-\frac{\mathcal E_\mu(g_t^{p-1},g_t)}{\mu(g_t^p)} +\frac{p'}{p^2}\frac{\mathop{\mathrm{Ent}}_\mu(g_t^p)}{\mu(g_t^p)}.\] For \(z,u>0\), Cauchy–Schwarz on the interval between them gives \[(z^{p/2}-u^{p/2})^2 \le\frac{p^2}{4(p-1)}(z^{p-1}-u^{p-1})(z-u).\] Apply [eq:stat-defective-lsi] to \(g_t^{p/2}\) and use this inequality edge by edge in the Dirichlet form. The energy terms cancel, leaving \[\frac{\,\mathrm d}{\,\mathrm dt}\log\left\lVert g_t\right\rVert_{p(t)} \le B_\eta p'(t)/p(t)^2.\] Since \(p(t)=1+\exp(4t/A_\eta)\) and \(\int_0^t p'/p^2\le1/2\), positivity and approximation give \[ \left\lVert S_t\right\rVert_{2\to p(t)}\le e^{B_\eta/2}. \tag{43}\]

A pointwise kernel bound and the choice of defect. On \(\left\lVert J\right\rVert\le K\), every spin Hamiltonian has absolute value at most \(Kn/2\), so \(\mu_{\min}\ge e^{-(K+\log2)n}\). Consequently \[\left\lVert S_t\right\rVert_{2\to\infty} \le\exp\left(\frac{B_\eta}{2}+\frac{C n}{p(t)}\right).\] Self-adjointness and the semigroup property give \[ \sup_{x,y}\frac{S_{2t}(x,y)}{\mu(y)} =\left\lVert S_{2t}\right\rVert_{1\to\infty} \le\left\lVert S_t\right\rVert_{2\to\infty}^2 \le\exp\left(B_\eta+\frac{2Cn}{p(t)}\right). \tag{44}\] The equality is the elementary formula for the norm of a positive kernel with respect to \(\mu\).

We now choose the defect for the prescribed fixed time \(d\). Put \(t=d/2\) and choose \(\eta=\exp(-a d^{2/3})\), where \(a>0\) is a sufficiently small fixed constant. For \(d\) large, \(A_\eta\le C\sqrt a\,d^{1/3}\) after adjusting the lower threshold for \(d\). Hence \(p(d/2)^{-1}\le\exp(-c_1a^{-1/2}d^{2/3})\). Choose \(a\) so that \(c_1a^{-1/2}>a\). The logarithm in [eq:stat-kernel-bound] is then at most \(C+Cn e^{-a d^{2/3}}\). For this fixed \(d\) its constant term is absorbed once \(n\) is sufficiently large, proving [eq:stat-density]. All ingredients were common to the cavities, with their dimension \(n-1\); enlarging \(C\) gives the stated common bound. ◻

Proof. Intertwining. The exact product rule is \[d_i(fg)=f\,d_ig+g\,d_if-2x_i(d_if)(d_ig).\] Use \(Lf=\sum_j(m_j-x_j)d_jf\), \(d_i^2f=0\), and \(d_id_jf=d_jd_if\). Differentiation gives \[d_iLf=Ld_if-d_if+\sum_j a_{ij}d_jf -2x_i\sum_j a_{ij}d_jd_if.\] Uniqueness of the finite-dimensional evolution proves intertwining.

Coefficient bounds with the variance factors retained. For \(i\ne j\), the subtraction formula for \(\tanh\) gives \[\frac{a_{ij}}{v_j} =\frac{\tanh(2x_iJ_{ij})} {2x_i\{1-m_j\tanh(2x_iJ_{ij})\}} =J_{ij}+2x_i m_jJ_{ij}^2+O(|J_{ij}|^3).\] The remainder is uniform for \(|m_j|\le1\) and \(\max|J_{ij}|\le1/10\), including when the local conditional variance is small. Its maximum absolute row and column sums are \(o(1)\), because \(\sum_j|J_{ij}|^3\le\max|J_{ij}|\sum_jJ_{ij}^2\). The last condition in [eq:path-disorder] proves [eq:path-a-expansion] and the first bound in [eq:path-a-bounds]. Finally, \(|a_{ij}|\le C|J_{ij}|v_j\) and \(v_j=w_j(1+x_jm_j)\le2w_j\) give \(v_j^2/w_j\le2\). This proves the row bound. ◻

Proof. We first prove the energy inequality, then represent each row evolution by a likelihood change, and finally truncate the expansion in the multiplication term.

Energy control. Put \(q=K_tp\) and \(\mathcal D(q)=\sum_jw_j\left\lVert d_jq\right\rVert^2\). The jump identity gives \[(\partial_t-L)\left\lVert q\right\rVert^2 =-2\left\lVert q\right\rVert^2+2q^taq -4\sum_{i,j}q_ix_i a_{ij}d_jq_i-2\mathcal D(q).\] The middle first-order term in absolute value is at most \(C\left\lVert q\right\rVert\sqrt{\mathcal D(q)}\) by [eq:path-a-bounds]. Young’s inequality absorbs one copy of \(\mathcal D(q)\); the multiplication term costs \(2\left\lVert a\right\rVert\left\lVert q\right\rVert^2\). Integrating the resulting inequality along an ordinary trajectory proves [eq:path-rough-L2]. Retaining dissipation also gives \[\int_0^t\mathbb E_{\nu S_s}\sum_jw_j \left\lVert d_jK_{t-s}p\right\rVert^2\,\mathrm ds \le C_te^{Ct}\mathbb E_{\nu S_t}\left\lVert p\right\rVert^2.\]

Likelihood representation and moments. To control the expansion tails, represent the evolution without its multiplication term by a change of the ordinary update rules. In row \(i\) replace, at attempts at \(j\ne i\), the field \(H_j=(JX)_j\) by \(H_j-2X_iJ_{ij}\); attempts at \(i\) are unmodified. The corresponding row generator is \(L-2x_i\sum_ja_{ij}d_j\). If \(\ell_i[s,t]\) is its likelihood relative to ordinary attempts, its semigroup, including \(-I\), is \[(T_{t-s}p)_i(x)=e^{-(t-s)}\mathbb E_x[\ell_i[s,t]p_i(X_{t-s})].\] The site-\(i\) spin is part of the ordinary trajectory; only its update rule is left unchanged in this representation.

The required moments come from a one-attempt calculation. Let \(b_H(\sigma)=e^{\sigma H}/(2\cosh H)\), \(\sigma\in\{-1,1\}\), and \(r=b_{H+u}/b_H\). Taylor’s formula and \((\log\cosh)''\le1\) show, for every fixed real \(p\), that \[\mathbb E_{b_H}r^p =\exp\{\log\cosh(H+pu)-p\log\cosh(H+u) +(p-1)\log\cosh H\} \le e^{C_pu^2}.\] For a rate-one clock at \(j\), the compensator of the \(p\)-th likelihood moment is therefore at most \(C_pJ_{ij}^2\); the largest entry is small. The exponential martingale formula gives, conditionally on any past, \[ \mathbb E[\ell_i(I)^p\mid\text{past at the first interval}] \le \exp(C_p|I|), \tag{51}\] where \(I\) is any fixed union of intervals and \(|I|\) is its total length. Negative powers are allowed. All fields and starting states are covered by the same constants.

Truncating insertions and then likelihoods. Rowwise Cauchy–Schwarz gives \(\left\lVert T_hp(x)\right\rVert^2\le e^{Ch}S_h\left\lVert p\right\rVert^2(x)\). Writing \(A\) for multiplication by \(a\), the term with \(k\) insertions is \[ \int_{0<t_1<\cdots<t_k<t} T_{t_1}A T_{t_2-t_1}A\cdots A T_{t-t_k}p\, \,\mathrm dt_1\cdots\,\mathrm dt_k. \tag{52}\] Repeated rowwise Cauchy–Schwarz and the Markov property bound its \(L^2\) norm by \(e^{Ct}C^kt^k/k!\) times the terminal \(L^2\) norm. Consequently \[\sum_{k>At}e^{Ct}\frac{(Ct)^k}{k!} \le e^{Ct}\left(\frac{eC}{A}\right)^{At} (1-eC/A)^{-1},\] which is at most \(C_He^{-Ht}\) when \(A\) is sufficiently large. Once this insertion cutoff is fixed, only finitely many likelihood intervals remain in each term. For clipping, with any fixed \(r>0\), \[\mathbb E[\ell_i^2\mathbf 1_{\ell_i>e^{Bt}}\mid\text{past}] \le e^{-rBt}\mathbb E[\ell_i^{2+r}\mid\text{past}] \le e^{-rBt+C_{2+r}t}.\] Apply this estimate at the first clipped interval, use the ordinary second-moment bounds elsewhere, and sum over at most \(At+1\) intervals and the simplex. Increasing \(B\) after \(A\) proves the stated bound. For fixed \(t>0\) and a prescribed error, choose a finite insertion cutoff making the factorial-series tail sufficiently small. Then choose a likelihood cap controlling the finitely many retained terms, using the displayed moment estimate. This convergence does not require \(t\ge1\); that restriction was used only in the stated exponential-in-time tail formulation. Every estimate sums over columns, which proves the matrix version. ◻

Proof. Products. First assign independent column indices to every factor. Their rowwise tensor product is obtained from \(\bigotimes_lB_l\) by restricting its output to the diagonal copy of \(\mathbb R^n\), an operator of norm one. Next require all occurrences of a slot to have the same index. The map \[e_{j_I}\longmapsto\bigotimes_l e_{j_{S_l}}\] is an isometry, since the sets \(S_l\) cover \(I\). This proves [eq:path-tensor-product]. There is no absent, freely repeated column index that could contribute a dimension factor.

An entrywise square and arbitrary bounded coefficients. For the second assertion put \(A=|B_0|^2\) entrywise. Its maximum row sum and maximum column sum are at most \(b_0^2\), hence the rectangular Schur test gives \(\left\lVert A\right\rVert_{\mathrm{op}}\le b_0^2\). For any two matrices \(A,B\) with the same output rows, their rowwise tensor product satisfies \[\left\lVert A\otimes_{\rm row}B\right\rVert_{\mathrm{op}} \le\left\lVert A\right\rVert_{\mathrm{op}}\max_i\left\lVert \operatorname{row}_i(B)\right\rVert_2.\] For example, factor it as \(D_B(A\otimes I)\), where \((D_Bz)_i=\sum_b B_{ib}z_{i,b}\); the norm of \(D_B\) is the displayed maximum. Apply this repeatedly to \(A\) and the absolute values of the other factors, whose row Euclidean norms are at most \(b_l\). Restrict repeated column indices by the same isometry as before. Finally entrywise domination implies \(|u^tRv|\le C|u|^tP|v|\) for the nonnegative dominating matrix \(P\). Taking unit-vector suprema proves the assertion. ◻

Proof. For spin inputs and fixed labels, the tensors are explicit: a spin input in slot \(a\) has \(T_{\{a\}}=I\) and every other nonempty tensor zero, while a fixed label has all nonempty tensors zero. Linear maps and sums preserve the claimed bounds. The remaining task is closure under smooth componentwise functions. We use an exact corner expansion so that no compact-range assumption is imposed on a linear query.

First consider \(F_i=\phi(z_i)\). Fix \(I\), \(|I|=r\), and a derivative tuple \(j_I\). For every nonempty \(S\subseteq I\) put \[t_S=(-2)^{|S|}\left(\prod_{a\in S}x_{j_a}^{(a)}\right) (T_Sz)_{i,j_S}.\] If the indicated coordinate is flipped in each slot of \(B\subseteq I\), the exact cube identity is \[ z_i(x^B)=z_i(x)+\sum_{\varnothing\ne S\subseteq B}t_S. \tag{56}\] Temporarily regard the \(2^r-1\) increments as independent scalar variables. Repeated use of the fundamental theorem of calculus gives, for any set \(V\) of increments, \[ \phi\left(z+\sum_{S\in V}t_S\right) =\sum_{A\subseteq V}\left(\prod_{S\in A}t_S\right) \int_{[0,1]^A}\phi^{(|A|)} \left(z+\sum_{S\in A}\theta_St_S\right)\,\mathrm d\theta. \tag{57}\] The empty term is \(\phi(z)\). Apply this with \(V=\{S:\varnothing\ne S\subseteq B\}\) and take the alternating sum over \(B\subseteq I\) defining \(T_I\). A term survives exactly when \[ \bigcup_{S\in A}S=I. \tag{58}\] Indeed, otherwise summing over a missing slot cancels the term. The remaining normalization is \(2^{-r}\) and a product of column signs.

Every surviving monomial thus contains enough increments to cover all derivative slots. Its coefficient may still depend on the derivative indices, so the product estimate alone is not yet sufficient. Expand that integral coefficient once at \(z\): \[\int\phi^{(|A|)}\left(z+\sum_{S\in A}\theta_St_S\right)\,\mathrm d\theta =\phi^{(|A|)}(z)+\sum_{S\in A}t_SR_{A,S}(z,t), \qquad |R_{A,S}|\le\left\lVert \phi^{(|A|+1)}\right\rVert_\infty.\] This is an exact integral remainder. The first term is a bounded row coefficient times a product of tensors satisfying [eq:path-covering], so [eq:path-tensor-product] applies. The remainder acquires a second copy of an increment already present in its monomial. It is therefore dominated by a product with one squared tensor factor, and [eq:path-square-domination] applies, even though its coefficient depends on every derivative index. There are finitely many terms.

For a componentwise function of \(q\) scalar inputs \(z^1,\ldots,z^q\), introduce increments \(t_{b,S}\) for every channel \(b\) and nonempty \(S\subseteq I\). Apply the same anchored expansion in these \(q(2^r-1)\) formal variables. The surviving monomials again cover \(I\), and expanding each coefficient once repeats an increment already in that monomial. Bounded mixed scalar derivatives through order \(q(2^r-1)+1\) suffice at this gate. The same two estimates prove the claim. Induction over the fixed calculation completes the proof. ◻

Proof. We integrate one leaf at a time. The propagation bound controls the evolving derivative slot, while the mixed tensors retain the derivatives in every stored slot. After writing this as a finite linear recursion, we use one common truncation of each evolution operator.

Propagation with auxiliary derivative indices. The point-start bounds in 13 are \[\begin{align*} \left\lVert K_hq(x)\right\rVert&\le B_h\{S_h\left\lVert q\right\rVert^2(x)\}^{1/2},\\ \left\lVert (K_h-K_{h,M})q(x)\right\rVert &\le\varepsilon_{h,M}\{S_h\left\lVert q\right\rVert^2(x)\}^{1/2}, \qquad \varepsilon_{h,M}\longrightarrow0. \tag{61}\end{align*}\] They amplify to any auxiliary Euclidean space by applying them to each component and summing squares. In particular the other derivative indices of \(T_IF\) can all be retained as Hilbert coordinates without a dimension factor. The truncation is one common linear operator, not a choice depending on an output test vector.

Integrating a leaf and identifying its parent. Integrate a final observed slot \(m\) from its parent slot \(p\) over an interval of length \(h\). Let \(y\) denote all slots other than \(m\). Before identifying the parent and the evolving starting state, set \[G(y,z)=S_h[w\mapsto F(y,w)](z), \qquad H(y)=G(y,z)\big|_{z=y^{(p)}}.\] The two occurrences of the parent are independent arguments of \(G\). The copy identity at the start of this subsection gives \[ d_j^{(p)}H =\left[d_j^{(p)}G+d_j^zG -2y_j^{(p)}d_j^{(p)}d_j^zG\right]_{z=y^{(p)}}. \tag{62}\] All other frozen-slot differences commute with \(S_h\) and \(K_h\). For \(p\notin I\), the resulting tensor is simply \(S_hT_IF\) restricted to \(z=y^{(p)}\). If \(p\in I\) and \(I'=I\setminus\{p\}\), its three terms are respectively \[ S_hT_IF,\qquad K_hT_{I'\cup\{m\}}F,\qquad -2D_{y^{(p)}}\,\mathsf R_{j_m=j_p} K_hT_{I\cup\{m\}}F. \tag{63}\] The sign multiplier in the last term acts on the derivative index. In the last two terms \(K_h\) acts only on the slot-\(m\) derivative coordinate; all other indices are auxiliary Hilbert coordinates. The restriction \(\mathsf R_{j_m=j_p}\) is a coordinate restriction within that Hilbert space, hence has norm one. It never identifies an output row with an input column across the chosen flattening.

We can now estimate the three terms without changing the tensor flattening. Fix a deterministic output vector \(c\) and apply the transpose of each tensor to \(c\). Jensen for \(S_h\) and [eq:path-amplified-tail] give \[ C_I(H)\le C_I(F)+B_hC_{I'\cup\{m\}}(F) +2B_hC_{I\cup\{m\}}(F) \quad(p\in I), \tag{64}\] and \(C_I(H)\le C_I(F)\) otherwise. Taking the supremum over \(c\) after these deterministic inequalities proves the operator bounds. Only one difference per original slot occurs. Iterate this finite elimination along a path, or remove leaves of a finite causal tree. Other branches can be held as parameters during each leaf integration. If the original calculation does not use the joining state, adjoin it as a dummy slot. Its initial difference tensors are zero. Reversibility makes a stationary reverse edge another use of the same \(S_h\).

One truncation for the complete derivative matrix. The leaf elimination is a finite linear recursion on the family of mixed tensors. Open that recursion and replace each \(K_h\) by the same chosen \(K_{h,M}\) at every occurrence with that interval. Ordinary \(S_h\) expectations may remain exact. Each step has the bounded norm in [eq:path-copy-bound]; replacing one step costs at most the corresponding \(\varepsilon_{h,M}\) from [eq:path-amplified-tail]. Telescoping the finitely many steps gives [eq:path-conditional-op]. This approximates the derivative of the exact conditional expectation. It does not assume that \(A_M\) is the derivative of a separately approximated scalar message, and no operator supremum is moved through an expectation.

Weighted errors and recombination. Finally, \(\left\lVert D_uE\right\rVert_{\mathrm{HS}}^2=\sum_i|u_i|^2\left\lVert \operatorname{row}_i(E)\right\rVert^2\) proves [eq:path-row-weighted-error] pointwise. The recursion and the common truncation are linear in the endpoint observables; therefore actual and predictor terms can be recombined before the centered label is used. Opening their ordinary conditional expectations as causal branches is done only after these uniform norm errors have been removed. ◻

Proof. Forward evolution preserves the density bound. Reverse conditioning introduces denominators, so we first remove joining states where those denominators are too small. All forward marginals have relative density at most \(e^{o(n)}\), since \(S_t\) is a positive \(L^\infty(\mu)\) contraction. A reverse conditional edge from time \(t\) to \(s\) has, relative to the stationary reverse edge, the factor \(h_s(X_s)/h_t(X_t)\). Discard endpoints where \(h_t<e^{-\delta n}\). Their actual marginal probability is at most \(e^{-\delta n}\), and their polynomially weighted probability vanishes. The remaining joint density on a tree with \(r\) reverse edges is at most \(\exp\{o(n)+r\delta n\}\). Choose \(\delta<c/(2r+2)\) and compare probabilities. The same calculation after Fubini proves the integrated assertion. For a finite tree with more than one original marginal, apply this argument successively at each copying operation. ◻

Proof. We first compare absolute log densities on retained forward and reverse bins. We then retain the quadratic log-likelihood cost to obtain the sharper entropy estimate. In both arguments the exceptional events are discarded on the original forward marginals before forming the weighted conditional law.

The forward log-density bound. Write \(\Delta(u)=J(X_u-X_s)\) in a bin starting at \(s\), and let \(N_{i,b}\) be its attempt counts. On a fixed mesh, Poisson exponential bounds imply, with exponential accuracy, \[ \sup_{u\in b}\left\lVert X_u-X_s\right\rVert^2\le C hn, \qquad \sup_{u\in b}\left\lVert \Delta(u)\right\rVert^2\le C hn. \tag{67}\] The log ratio at an attempt at \(i\) is bounded by \(2|\Delta_i|\). For a bounded predictable integrand the compensated attempt sum obeys \[\mathbb P(M\ge a)\le \exp\!\left(-\frac{a^2}{2(V+ba/3)}\right),\] where \(V\) bounds its predictable bracket and \(b\) its jumps. Apply this after cutting \(|\Delta_i|\) at \(B\), then use \(\sum_i|\Delta_i|\le\sqrt n\left\lVert \Delta\right\rVert\) for its compensator. The bounded part costs \(O_T(\sqrt h)n\). For the remaining part, \(\sum_i\Delta_i^2\le Chn\) bounds the compensator by \(Chn/B\). Its individual jumps are \(O(\sqrt n)\), its bracket is \(O_T(n)\), and the displayed inequality makes a fixed positive multiple of \(n\) an event of superpolynomially small probability. This proves the forward density comparison.

Reversed bins. For a reversed bin separate the forward within-bin difference from \(J(X_{s+h}-X_s)\). The latter vector has norm \(O(\sqrt{hn})\). Its entries larger than \(B\) occupy at most \(Chn/B^2\) sites. Independently of how this set is selected, the sum of \(N_{i,b}^2\) over such a set is bounded by the sum of the largest \(Chn/B^2\) squared Poisson counts. For each fixed \(h\), truncating counts at a level \(K\) gives \[\sum_{i\in I}N_{i,b}^2 \le K^2|I|+\sum_iN_{i,b}^2\mathbf 1_{N_{i,b}>K} =o_B(hn)\] with superpolynomial accuracy, by first choosing \(K\) and then \(B\). Cauchy–Schwarz bounds the across-bin excess by \(o_B(hn)\). The bounded part has exponential stationary tails by reversibility. Discard the excess on each actual edge in its original forward direction, before applying 18 to the bounded part. Summing the finitely many bins proves the first assertion.

The entropy of the retained actual tree. For the entropy assertion, first compare \(Q\) with \(P\). Let \(c_n=o(n)\) bound the logarithms of the original forward marginal densities. Discard reverse joining states with marginal density below \(e^{-a_n}\), where \(a_n=o(n)\) and \(a_n/\log n\to\infty\). Their actual marginal probabilities are at most \(e^{-a_n}\) and remain negligible under the permitted polynomial weights. On the retained set, the reverse density factors \(h_s(X_s)/h_t(X_t)\) bound the total tree log density by \(C(c_n+a_n)\). Polynomial reweighting and retained event normalization add \(O(\log n)\). Thus \(H(Q\mid P)=o(n)\). The change of reference is exactly \[ H(Q\mid P_h)=H(Q\mid P) +\mathbb E_Q\log\frac{\,\mathrm dP}{\,\mathrm dP_h}. \tag{68}\] Changing to the frozen reference. The root laws cancel in the last ratio, leaving the root-directed within-bin refresh log ratios. The absolute estimate already proved would lose the desired factor \(h\). We instead keep the quadratic log-cosh remainder. Conditional on the old configuration, the Kullback–Leibler divergence between two refresh coins at fields \(H\) and \(H+u\) is at most \(u^2/2\). Consequently its predictable integral over all bins is at most \[C\int_0^T\left\lVert J(X_u-X_{\lfloor u\rfloor_h})\right\rVert^2\,\mathrm du \le C_Thn\] on [eq:path-bin-square]. For the centered log likelihood \(M\) with field differences cut at \(B\), stop when [eq:path-bin-square] fails. Its jumps are bounded by \(CB\) and its predictable bracket is at most \(C_T hn\): the conditional variance of a refresh log ratio is at most a constant times the squared field difference. The bounded-jump exponential martingale inequality therefore gives \[\log\mathbb E_P e^{\theta M}\le C_T\theta^2 hn, \qquad |\theta|B\le c.\] Apply the entropy inequality \(\mathbb E_QM\le\theta^{-1}\{H(Q\mid P)+\log\mathbb E_Pe^{\theta M}\}\) with fixed positive \(\theta\le c/(1+B)\). At fixed \(h,B\), the additional \(o(n)\) entropy costs \(o(n)\), and the centered term costs \(O_T(hn)\). The same estimate for \(-M\) gives the reverse bound. It remains to remove the field cutoff. For the unbounded excess use the preceding forward compensator bound and the reversed-bin count bound. At fixed \(h\) its normalized entropy contribution tends to zero with the cutoff; choose it below \(h\) before taking the next mesh limit. Conditioning on any retained event of polynomially significant probability adds \(O(\log n)\) entropy. All superpolynomial discards remain negligible under the permitted polynomial weights and polynomially significant conditioning. An arbitrary \(o(n)\)-entropy tilt would not have this property and is not being allowed here. The chain rule over the finite tree, with the same argument on every root-directed edge, proves [eq:path-mesh-entropy]. ◻

Proof. We prove a two-point entropy estimate and then tensorize it. The small minority mass enters through the factor \(\log(1/p)\); keeping that factor is what makes the subsequent mesh refinement converge.

One coordinate. For a nondegenerate Bernoulli coordinate let its minority mass be \(p\in(0,1/4]\), put \(L_p=\log(1/p)\), and set \(\Lambda_p(t)=\log(1-p+pe^t)-pt\). For \(|t|\le L_p/2\), \[\Lambda_p''(t) =\frac{p(1-p)e^t}{(1-p+pe^t)^2} \le C\sqrt p\le C/L_p.\] Since \(\Lambda_p(0)=\Lambda_p'(0)=0\), Taylor’s formula gives \(\Lambda_p(t)\le Ct^2/L_p\) in this range. Outside it, \(\Lambda_p(t)\le|t|\le2t^2/L_p\). Jensen also gives \(\Lambda_p(t)\ge0\).

If a two-point function \(G\) has values \(a,a+t\), with the second value on the minority atom, direct calculation yields \[\frac{\mathop{\mathrm{Ent}}_p(e^G)}{\mathbb E_p e^G} =t\Lambda_p'(t)-\Lambda_p(t).\] Convexity at \(t\), evaluated at \(2t\), gives \(t\Lambda_p'(t)\le\Lambda_p(2t)-\Lambda_p(t)\) for either sign of \(t\). Consequently \[ \frac{\mathop{\mathrm{Ent}}_p(e^G)}{\mathbb E_pe^G} \le\Lambda_p(2t)-2\Lambda_p(t) \le\frac{Ct^2}{\log(1/p)}. \tag{71}\] This obtains the entropy inequality by convexity, without differentiating a bound on \(\Lambda_p\). A deterministic coordinate contributes zero entropy.

Tensorization and the exponential-moment bound. Apply [eq:path-bernoulli-entropy] conditionally with \(G=\theta F\) and \(|t|=2|\theta d_iF|\). Entropy tensorization gives \[\mathop{\mathrm{Ent}}_\pi(e^{\theta F}) \le\frac{C\theta^2}{\log(1/h)} \mathbb E_\pi\left[e^{\theta F}\sum_i(d_iF)^2\right] \le\frac{CB^2\theta^2}{n\log(1/h)}\mathbb E_\pi e^{\theta F}.\] For \(\psi(\theta)=\log\mathbb E_\pi e^{\theta F}\) this bounds \(\theta\psi'(\theta)-\psi(\theta)\). Integrate the derivative of \(\psi(\theta)/\theta\) from zero to obtain [eq:path-product-mgf]; use \(-F\) for negative \(\theta\).

Changing the product law. Finally, the entropy variational inequality gives \[\theta(\mathbb E_QF-\mathbb E_\pi F) \le H(Q\mid\pi)+\frac{CB^2\theta^2}{n\log(1/h)}.\] Optimize over \(\theta>0\) and apply the same argument to \(-F\). This proves [eq:path-product-entropy]. ◻

Proof. We first compare one terminal test under the actual and autonomous frozen laws. This requires two telescopes with a common ordinary semigroup reference. We then treat stored states by backward leaf elimination and smooth each fixed simulation.

The actual-law telescope. Choose a regular \(h\)-mesh augmented by the finitely many prescribed observation times, so that \(0=t_0<\cdots<t_N=T\), \(\delta_b=t_b-t_{b-1}\le h\), and \(N\le C_T/h\). First consider one terminal test \(F\), with \(\sup_x\left\lVert dF(x)\right\rVert\le B/\sqrt n\), and put \[f_b=S_{T-t_b}F,\qquad \mathsf P_b(x,\cdot)=S_{\delta_b}(x,\cdot).\] The point-start bound in 13 gives \(\sup_x\left\lVert df_b(x)\right\rVert\le C_TB/\sqrt n\). Project \(Q\) and \(P_h\) to the mesh states and put \(q_b(\cdot\mid\mathcal F_{b-1}) =Q(X_{t_b}\in\cdot\mid\mathcal F_{b-1})\), where \(\mathcal F_{b-1}=\sigma(X_{t_0},\ldots,X_{t_{b-1}})\). The entropy chain rule and data processing give \[\sum_{b=1}^N\mathbb E_Q H\!\left(q_b(\cdot\mid\mathcal F_{b-1}) \mid\pi_b(X_{t_{b-1}},\cdot)\right) \le H(Q\mid P_h)\le C_Thn+o(n).\] The omitted initial-state entropy is nonnegative. Since \(f_{b-1}=\mathsf P_bf_b\), the exact telescope is \[ \mathbb E_QF(X_T)-\mathbb E_Q S_TF(X_0) =\sum_{b=1}^N\mathbb E_Q[(q_b-\mathsf P_b)f_b]. \tag{73}\] Both kernels on the right are evaluated at the actual joining history; \(\mathsf P_b\) depends on it only through \(X_{t_{b-1}}\). Insert \(\pi_b\) between them and apply 20 separately to \(q_b\) and \(\mathsf P_b\). The second entropy has the uniform bound \[\begin{align*} H(\mathsf P_b(x,\cdot)\mid\pi_b(x,\cdot)) &\le H(\text{ordinary bin path from }x \mid\text{frozen bin path from }x)\\ &\le C\int_0^{\delta_b}\mathbb E_x\left\lVert J(X_u-x)\right\rVert^2\,\mathrm du \le Cn\delta_b^2. \end{align*}\] The first inequality is data processing, the second is the refresh-coin log-cosh bound, and the last uses \(\mathbb E_x\left\lVert X_u-x\right\rVert^2\le4nu\) and the bound on \(\left\lVert J\right\rVert\). In particular it holds for every joining-state law. Writing \(H_b\) for the sum of these two conditional entropies yields \[\begin{align*} |\mathbb E_QF(X_T)-\mathbb E_Q S_TF(X_0)| &\le\frac{C_TB}{\sqrt{n\log(1/h)}} \sum_{b=1}^N\mathbb E_Q\sqrt{H_b}\\ &\le\frac{C_TB}{\sqrt{n\log(1/h)}} \sqrt{N\sum_{b=1}^N\mathbb E_QH_b} \le\frac{C_TB}{\sqrt{\log(1/h)}}+o_n(1). \end{align*}\] Here dimension tends to infinity at fixed \(h\).

The autonomous frozen-law telescope. Let \(\widehat P_h\) be the autonomous frozen simulation started from the same initial law \(Q_0\). The second, zero-tilt telescope is \[ \mathbb E_{\widehat P_h}F(X_T)-\mathbb E_{Q_0}S_TF(X_0) =\sum_{b=1}^N\mathbb E_{\widehat P_h} [(\pi_b-\mathsf P_b)f_b]. \tag{74}\] Its conditional kernels equal \(\pi_b\), so their entropy relative to \(\pi_b\) is zero. The ordinary-kernel entropies still sum to at most \(Cn\sum_b\delta_b^2\le C_Thn\), uniformly in \(Q_0\). Consequently this second difference is at most \(C_TB/\sqrt{\log(1/h)}\). Since \(\mathbb E_QS_TF(X_0)=\mathbb E_{Q_0}S_TF(X_0)\), the triangle inequality proves [eq:path-simulation-error], after enlarging \(C_T\), for the difference between the actual and frozen tests.

Stored states and admissible conditioning. Stored parameters may be included by writing the test as \(F(y,x)\) with the same derivative bound uniformly in \(y\). Disintegrate the reference tree at the joining state and the passive variables \(y\). The active future must retain its ordinary, respectively frozen, Markov kernels under that reference law. Past variables and other branches conditionally independent at the joining state have this property. Conditioning on a later endpoint of the active branch generally does not. The conditional entropy of the active future is bounded by the total tree entropy by the chain rule, so both telescopes apply after averaging over \((y,X_0)\), and the simulation retains their joint initial law. A backwards leaf elimination gives precisely these admissible disintegrations. For replicated predictions the other copies are passive and the test bound is uniform in their realized values.

Smoothing a fixed simulation. A fixed frozen simulation is a finite calculation using \(J\), independent site clocks and uniforms, and scalar threshold maps. Replace each threshold by a smooth transition in a strip of width \(\varepsilon\). Fresh uniforms put at most \(O(\varepsilon)\) mass in that strip. Induction over finitely many operations, using the bound on \(\left\lVert J\right\rVert\), shows that the normalized mean square error tends to zero. Derivative constants of the resulting simulation can depend on its fixed mesh and smoothing width.

Backward replacement and derivative control. Replace the last observed state first. The smooth simulation produces a new smooth test of earlier states. Fix that calculation and its finite derivative constants before choosing the accuracy at the next earlier replacement. At a branch, duplicate the branching state as an input to the child calculations. For replicated predictions, hold all other copies as passive parameters while replacing one copy; the conditional entropy chain rule bounds its cost. Thus its derivative constant depends only on its own currently fixed test, not on the accuracy subsequently selected for a passive copy. This backwards, nested construction avoids requiring constants uniform over all meshes.

For clarity, the derivative bound for a fixed smooth recipe follows without any probabilistic assertion. If a vector \(z\) has bounded half-difference matrix, then, off its diagonal, \[d_j\phi(z_i)=\phi'(z_i)d_jz_i +O(|d_jz_i|^2).\] The absolute row and column sums of the error are bounded by the squared operator norm of that matrix. Keep its diagonal exact. The Schur estimate in 14 and induction give a bounded derivative matrix. A normalized empirical inner product has difference norm \(O(n^{-1/2})\) by the product rule. The remaining assertion concerns square tails of linear queries. Test such a tail with a smooth excess function whose second derivative is bounded and whose first derivative is at most \(C(1+|z|)\). The same Taylor calculation, using bounded column square sums, gives a difference bound independent of the tail cutoff. For each fixed simulation Gaussian regression gives square integrability and vanishing square tails. The uniform comparison just proved then transfers this assertion to the original paths. ◻

Proof. We first transfer exponential estimates to a Gaussian experiment in which the initial spin is planted. Under the annealed one-replica weight, the initial spin \(\sigma\) is uniform and the off-diagonal disorder has the form \[J=\beta G+\frac{\beta^2}{n}\sigma\sigma^t +\text{a diagonal correction},\] where \(G\) is independent GOE. The correction from its Gaussian diagonal has norm tending to zero. Its mean shift on the diagonal has norm \(O(1/n)\). The Gibbs-to-planted comparison loses only \(e^{o(n)}\): indeed \[\frac{\mathbb EZ^2}{(\mathbb EZ)^2} \le C_\beta\mathbb E\exp\!\left(\frac{\beta^2R_n^2}{2n}\right) \le C'_\beta,\] where \(R_n\) is a sum of iid signs. Gaussian concentration of \(\log Z\) and Paley–Zygmund then give \(\log Z-\log\mathbb EZ=o(n)\) in probability. Thus an annealed exponential failure transfers to an exponentially small Gibbs failure on a high-probability disorder event. This calculation explains why the planting comparison remains valid throughout \(\beta<1\).

We can now describe exactly which randomness remains after a query. Let \(U\) have orthonormal columns, put \(P=I-UU^t\), and suppose \(Y=GU\) has been revealed. Symmetry gives \(U^tY=Y^tU\). The conditional matrix is \[ G=YU^t+UY^t-U(U^tY)U^t+PG'P, \tag{79}\] with \(G'\) a fresh GOE independent of the revealed sigma-field. To verify independence, test the GOE covariance \[\mathbb E[G_{ij}G_{kl}]=n^{-1} (\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})\] against a matrix supported in \(P\mathbb R^n\) and against each revealed column. The covariance is zero. Joint Gaussianity proves independence, and the displayed revealed part is the unique symmetric completion with those columns and zero \(P\)–\(P\) block.

Conditional Gaussian regression along iterative spin-glass calculations also underlies Bolthausen’s construction of TAP solutions (Bolthausen 2014, secs. 2–3 of the arXiv version). Here it is used for finite causal path simulations and their extra recipe queries; the required empirical and operator conclusions are proved separately. The same formula applies adaptively. Given previous answers, normalize the residual of the next query to a unit vector \(q\in P\mathbb R^n\). Its new perpendicular answer has law \(n^{-1/2}(P-qq^t)g\), and its component along \(q\) has law \(\sqrt{2/n}\,zq\), with \(g,z\) independent standard Gaussians. After this answer the remaining compression is again independent. Projection coefficients are empirical inner products of previous calculation vectors. Thus every fixed finite recursion reduces to iid site Gaussians and seeds, finitely many empirical scalar coefficients, and finitely many rank corrections.

It remains to control the finite recursion uniformly at a fixed accuracy. For bounded smooth tests, iid empirical concentration and Gaussian quadratic tails propagate through each successive query. If a query residual has length tending to zero, omit that query; the spectral bound makes the normalized square error vanish. Equivalently, fix a residual-length threshold, propagate its deterministic error through the finite recursion, and send the threshold to zero only after the dimension limit. Fresh uniform thresholds are continuity tests. They can therefore be smoothed and the smoothing subsequently removed without changing the limits. This proves deterministic empirical convergence with exponentially small failure at each fixed positive tolerance. ◻

Proof. We first construct the limiting inner product, then identify conditional means in its completion. Apply [prop:path-simulation,lem:path-regression] to successively fixed smooth simulations. The comparison error in [eq:path-simulation-error] makes their empirical limits Cauchy. If a deviation had polynomially significant probability, conditioning on it would cost only \(O(\log n)\) extra entropy. The same comparison would then force the original limit on that event, a contradiction. Apply this argument to the weight-normalized tilted laws to obtain the weighted typical assertions. The limiting Gram matrices are positive semidefinite; their quotient by the null space, followed by completion, defines \(\mathcal H\).

For the second step, let \(z\) be a path recipe and put \(m(X_0)=\mathbb E[z\mid X_0]\) on the stationary rooted tree. Replace the future by a fixed finite smooth simulation and average its seeds. An average of \(R\) independent seed copies has spatial \(L^2\) error at most \(C/R\) from that simulated conditional mean. At each fixed \(R\), it is a one-state recipe under the seed convention above. To compare it with the actual \(m\), expand the squared error into \(\left\lVert m\right\rVert^2\), its cross term with the simulation, and the simulation’s squared norm. Realize these three terms on two independent branches sharing \(X_0\), with the same nested approximation on both copies. 21, applied while the other branch is passive, controls each term and proves the conditional mean-square approximation. They also give \[n^{-1}\sum_i(z_i^{(1)}-m_i)(z_i^{(2)}-m_i)\longrightarrow0\] for distinct branches; here the \(z_i\) are written without the normalizing factor. Reversibility identifies stationary reverse predictors, and 18 plus the finite-simulation comparison gives the same pair centering on actual conditional trees.

The remaining assertion is orthogonality to a path recipe. Let \(u\) satisfy [eq:path-hidden] and use \(R\) centered replicas \(z^{(r)}-m\). Their Gram matrix tends, conditionally in the typical weighted sense, to a bounded diagonal matrix. Bessel’s inequality therefore gives \[\sum_{r=1}^R|u^t(z^{(r)}-m)|^2 \le \left\lVert u\right\rVert^2\left\lVert \operatorname{Gram}(z^{(1)}-m,\ldots,z^{(R)}-m)\right\rVert.\] All summands have the same expectation; divide by \(R\), take dimension large, and then send \(R\to\infty\). The \(m\) term vanishes by recipe approximation and [eq:path-hidden]. For a matrix word \(Q\), \(Q^tz\) is another path calculation with negligible square tails after clipping its linear outputs. The same argument applied to it proves the last assertion. ◻

Proof. Let \(c_j(x)=(1-x_j\tanh H_j)/2\) be the flip rate. Its logarithmic field derivatives are uniformly bounded: the first is \(-x_j-m_j\), and the next two are bounded functions of \(m_j\). For row \(i\) the field perturbation at \(j\) is \(u_{ij}=-2x_iJ_{ij}\) on its specified interval mask. The flip likelihood has logarithm \[\sum_j\int\log\frac{c_j^{(i)}}{c_j}\,\mathrm dN_j -\sum_j\int(c_j^{(i)}-c_j)\,\mathrm ds.\] Expand this logarithm in \(J_{ij}\). The linear term is a centered score, the quadratic term is its bracket/compensator contribution, and the remaining terms have absolute expectation at most \(C_T\sum_j|J_{ij}|^3=o(1)\), uniformly in \(i\). Higher moments are controlled by [eq:path-likelihood-moment]. Both the score integrands and the coefficients of \(J_{ij}^2\) are bounded smooth site functions. We may consequently cap the exponential at a fixed continuity value, or use a smooth cap, with arbitrarily small spatial square-mean loss.

Freeze the bounded integrands on short bins. A bin with two or more attempts at one site has probability \(O(h^2)\); summing its squared Poisson counts shows that ignoring all such bins costs \(O_T(h)\) in normalized mean square. On the remaining bins the spin values at the two ends determine whether a flip occurred. Bounded derivatives, [eq:path-bin-square], count truncation, and predictable rate averaging show that freezing the field-dependent integrands also has vanishing normalized mean-square cost. Multiplication by \(J\) preserves that cost.

The preceding freezing estimate still leaves the outside row-spin factor \(x_i(s)\). Freeze this factor at the beginning of its bin. The resulting error is a vector martingale, with row-\(i\) bracket bounded by \[C\int\mathbf 1_{\{x_i(s)\ne x_i(\lfloor s\rfloor_h)\}}\,\mathrm ds.\] The total bracket trace is \(O_T(hn)\) on the Poisson count event, while the operator norm of its bracket density is bounded by \(C\left\lVert J\right\rVert^2\). A vector jump is bounded in Euclidean norm. Apply the scalar bounded-jump inequality to its squared norm, stopped at a fixed multiple of \(\sqrt n\): its jumps are \(O(\sqrt n)\) and its bracket is \(O_T(n)\). For every fixed tolerance exceeding the \(O(h)\) bracket trace, failure has superpolynomially small probability. Thus the fraction of rows with a significant score error tends to zero with arbitrary polynomial accuracy. Bracket terms need only absolute rate averaging. Use a finer inner mesh when approximating integrands inside a bin whose outside factor has already been frozen.

After these error estimates, the retained score can be written as a finite calculation from observed spins. On a bin \([s,t]\), set \(X=X_s\), \(Y=X_t\), \(h=t-s\), and freeze \(m=m(X)\) and \(v=v(X)\). The first log-rate derivative is \(-X_j-m_j\); outside the negligible multiple-flip bins, \[\Delta N_j=(1-X_jY_j)/2, \qquad (-X_j-m_j)\Delta N_j =(Y_j-X_j-m_j+m_jX_jY_j)/2.\] Since \((-X_j-m_j)c_j=-X_jv_j/2\), the vector of frozen row scores is \[ D_XJ\{X-Y+m-mXY-hXv\}, \tag{80}\] where products inside braces are componentwise. In particular the mixed query \(J(mXY)\) is retained. It need not be a finite sum of sitewise products of single-time outputs. Quadratic bracket terms are finite calculations using \(J\circ J\) and bounded site factors; this matrix has bounded operator norm and bounded absolute row sums. Their values are therefore uniformly bounded at each fixed mesh.

For the cap of the exponential, choose a smooth bounded function \(\psi\) of the logarithmic score, with globally bounded derivatives, which agrees with the desired clipped exponential outside an arbitrarily small smoothing interval. An outside spin multiplying an unbounded query is handled by its two sign cases, not by assuming the map \((a,u)\mapsto\psi(au)\) has bounded derivatives for arbitrary bounded \(a\) and unbounded \(u\). Explicitly, \[ \psi(x_iU_i)=\sum_{\varepsilon=\pm1} \frac{1+\varepsilon x_i}{2}\,\psi(\varepsilon U_i). \tag{81}\] For finitely many bins, split the finite sign pattern of all the frozen outside row spins. Each summand is a bounded product of spin selectors and a smooth cap applied to a fixed signed linear combination of the mixed \(J\) queries and the bounded quadratic terms. All scalar coefficient bounds are global, independently of the query values. Products of already bounded factors have smooth bounded extensions. Thus 15 applies to every fixed capped calculation, including all its mixed temporal slots.

Fix the mesh, smoothing, sign decomposition, and cap before taking the dimension limit. When these choices are subsequently refined, their derivative constants may grow. The refinement removes the spatial approximation errors established above; it makes no claim that derivatives of those errors converge. Observation times can be strictly inside their bins, and can be chosen one-sidedly at endpoints. For predictors first take these fixed calculations, then their exact stationary conditional means. Conditional expectation contracts spatial \(L^2\); replicated pair tests and 21 give the one-state recipe approximation. When an exact predictor is differentiated, use its fixed multitime calculation on an added stationary causal branch and apply 16. The opened tensors are treated by 33. No score estimate on an artificial reverse branch with a tilted endpoint, and no derivative estimate for a tracial approximation error, is used. ◻

Proof. The norm proof has three stages: types of positive proportion, rare coordinate types, and a concentration estimate uniform in the type counts. We begin with the first stage. Every finite-dimensional Wick contraction uses the empirical trace \(\tau_n=n^{-1}\mathop{\mathrm{Tr}}\), so that internal returning pairs cancel exactly. During the moment calculation, \(p_a\) means the empirical type proportion and \(\tau\) still means the limiting state. Suppose first that \(p_a\ge\alpha>0\) for every type. If every limiting proportion is positive, this holds eventually after decreasing \(\alpha\). For a homogeneous Wick string \(W_F\) of degree \(k\), use the product-state norm \[\left\lVert F\right\rVert_2^2=\sum_{a_0,\ldots,a_k} p_{a_0}\cdots p_{a_k}|F(a_0,\ldots,a_k)|^2.\] For \(k=0\) the string is diagonal and has norm at most \(\alpha^{-1/2}\left\lVert F\right\rVert_2\), so the preliminary bound below is immediate. Assume henceforth in the moment calculation that \(k\ge1\). Let \(r\) be even and consider \(n^{-1}\mathbb E\mathop{\mathrm{Tr}}((W_F^*W_F)^{r/2})\). Gaussian pairings glue the \(rk\) sides of a polygon in either orientation. There are \(E=rk/2\) paired edges; if \(V\) is the number of free index vertices, set \(e=E-V+1\ge0\). A connected gluing contributes \(n^{-e}\) times a type average. The two GOE covariance terms are precisely the two possible band orientations, including their multiplicities for a loop.

Internal immediate untwisted returns cancel. Indeed selecting a specified pair of adjacent edges and applying their covariance gives exactly the trace contraction subtracted in the Wick definition. Inclusion and exclusion over disjoint pairs cancels every diagram with at least one such internal return. Hence every degree-one vertex of a surviving graph lies at a boundary of one of the \(r\) strings. There are at most \(r\) leaves, and the degree identity gives \[ \sum_v(\deg(v)-2)_+\le r+2e. \tag{84}\]

We need a diagram count whose loss grows polynomially with the word degree. Suppress every chain of degree-two vertices, keeping one vertex for a pure circle. By [eq:path-map-degree], the remaining core has \(s\le C(r+e)\) edges. Delete the \(e\) edges outside a spanning tree. Reversing local cyclic orders at vertices untwists all tree bands, and a rooted plane tree on \(s-e\) edges has at most \(4^s\) encodings. Reinsert the deleted bands in ordered corner slots. Their endpoints, order, and twist signs have at most \((Cs+C)^{Ce+C}\) choices. Subdividing the core edges and choosing the rooted boundary start adds a factor at most \[Crk\binom{rk+s}{s}.\] Using \(s\le C(r+e)\), assign the \(O(r)\) core edges to the factor \((C(1+k))^{Cr}\) and the additional \(O(e)\) edges to \((Crk)^{Ce+C}\). Thus the number of diagrams of excess \(e\) is at most \[ (C(1+k))^{Cr}(Crk)^{Ce+C}. \tag{85}\] Loops, parallel bands, and the pure-circle case are included by the same encoding. No factor exponential in \(rk\) remains: tree-band twists were removed by vertex reversal, and only the \(e\) extra bands carry independent twist choices.

The count must now be combined with a bound for each type average. Delete \(e\) nontree edges. At most \(2e\) strings cross these edges; bound each of them by \[\left\lVert F\right\rVert_\infty\le\alpha^{-(k+1)/2}\left\lVert F\right\rVert_2.\] Every remaining string is a simple tree path, since a repeated tree edge would force an internal immediate return. To count repeated type variables, let \(c_v\) be the number of remaining string occurrences at vertex \(v\), and let \(b_v\) count their endpoints there. Each internal occurrence uses two edge incidences and each endpoint uses one. As each paired edge has at most two occurrences, \(2c_v-b_v\le2\deg(v)\), using the degrees of the original graph. Thus [eq:path-map-degree] and \(\sum_v b_v\le2r\) give \[\sum_v(c_v-2)_+ \le\sum_v(\deg(v)-2)_++\tfrac12\sum_v b_v \le2r+2e.\] Replace each occurrence beyond two by a fresh independent type variable. For a nonnegative summand the comparison costs at most \(\alpha^{-1}\) per replacement: an equality constraint can increase a product-state integral by at most the inverse of its smallest atom. Each remaining variable occurs at most twice, never twice within one tensor. Successive coordinate Cauchy–Schwarz therefore bounds the product integral by one \(\left\lVert F\right\rVert_2\) per surviving string. Altogether a diagram is bounded by \[ n^{-e}\alpha^{-2r-(k+3)e}\left\lVert F\right\rVert_2^r. \tag{86}\]

We can now sum the diagram bounds. Choose an even \(r\) of order \(\log n\), keeping \(k,\alpha\) fixed. Equations (85) and (86) give \[n^{-1}\mathbb E\mathop{\mathrm{Tr}}((W_F^*W_F)^{r/2}) \le (C_\alpha(1+k))^{Cr}\left\lVert F\right\rVert_2^r (Crk)^C\sum_{e\ge0} \left\{\frac{(Crk)^C\alpha^{-(k+3)}}n\right\}^{e}.\] The sum is eventually at most two. Markov’s inequality for the trace power, increasing the constant in \(r/\log n\) if necessary, proves \[ \limsup\left\lVert W_F\right\rVert_{\mathrm{op}} \le C_\alpha(1+k)^C\left\lVert F\right\rVert_2. \tag{87}\] The strategy of obtaining a polynomial word-length norm bound and then removing its loss by taking powers has an antecedent in Haagerup (Haagerup 1979, Lemmas 1.3–1.4, pp. 283–286). His setting is free-group convolution; the Wick/GOE diagram bound, rare-type treatment, and adaptive-label passage here are proved locally. To turn this preliminary estimate into the sharp comparison, decompose a degree-\(k\) polynomial into homogeneous Wick strings. Complete noncrossing pairing identifies the sum of their squared tensor norms with the vacuum square norm. Cauchy–Schwarz and [eq:path-wick-preliminary] therefore bound the matrix norm by \(C_\alpha(1+k)^{C+1}\left\lVert P(s,D)\right\rVert\). Apply that bound to \((P^*P)^m\), whose degree is \(2mk\). Since \(\|(P^*P)^m\|=\|P\|^{2m}\) in both representations, it gives \[\limsup\|P(G_n,D)\|_{\mathrm{op}} \le \bigl\{C_\alpha(1+2mk)^{C+1}\bigr\}^{1/(2m)} \|P(s,D)\|.\] Letting \(m\to\infty\) removes the polynomial loss. For each fixed polynomial and each fixed power, there are only finitely many empirical contraction coefficients, and they converge to the coefficients formed with \(\tau\). The difference of the resulting matrix polynomials is \(o(1)\) in operator norm on every fixed spectral ball. Thus empirical and limiting Wick conventions have the same norm limit. Dimension is sent to infinity before the power is increased; no rate for \(\tau_n\to\tau\) is required. This proves [eq:path-fock-domination] when all types have positive proportion.

The positive-proportion case is complete. We next keep the coordinate space of the rare types instead of discarding it. Write this space as \(R_n\), with dimension \(m_n=o(n)\), and split \[G_n=\begin{pmatrix}B_n&V_n\\V_n^*&C_n\end{pmatrix} \quad\text{on }R_n^\perp\oplus R_n.\] The internal rare block has norm \(o(1)\). Conditional on the bulk block and its deterministic diagonals, \(V_n\) has independent \(N(0,1/n)\) entries. For every fixed bounded bulk word \(A_n\), Gaussian quadratic-form concentration, polarization, and a net of size \(\exp(O(m_n))\) imply \[ \left\lVert V_n^*A_nV_n-\tau(A)I_{R_n}\right\rVert_{\mathrm{op}}\longrightarrow0. \tag{88}\] Indeed a unit-vector quadratic form has variance \(O(1/n)\) and fixed accuracy tail \(e^{-cn}\), while the net entropy is \(o(n)\). The normalized bulk traces converge by ordinary Wick pairing. [eq:path-rare-gram] holds jointly for every finite list of \(A\), including \(A^*A'\).

The remaining task is to turn the rare-coordinate Gram comparisons into an operator-norm bound. Take a subsequence realizing a limiting norm witness and form its Hilbert ultraproduct. The closed linear span of \(\{A V\xi:A\text{ a bulk word},\ \xi\in R\}\) is a reducing subspace for the bulk word algebra, and [eq:path-rare-gram] identifies it isometrically with \(L^2(\mathscr A,\tau)\otimes R\) through \(A\Omega\otimes\xi\mapsto AV\xi\). Here \(\mathscr A\) is the bulk Gaussian/diagonal algebra, whose norm has just been controlled. On this subspace, \(V\xi=\Omega\otimes\xi\) and \(V^*(A\Omega\otimes\xi)=\tau(A)\xi\). The orthogonal bulk complement has no coupling to \(R\) and obeys the bulk norm bound. On the reducing sum of \(R\) and its bulk cyclic space, every return through a bulk word is exactly its vacuum expectation, while the diagonal on \(R\) retains its actual type value. This is the representation induced from that base in [eq:path-fock-space]. Its polynomial norm is bounded by the faithful-base Fock norm. The norm on the orthogonal sum of the two reducing pieces is the maximum of their restricted norms, proving the rare-type assertion.

Both the positive-proportion and rare-type comparisons are now proved. To obtain the stated exponential rate, project GOE onto the operator-norm ball of a fixed radius \(K>3\). The bare-GOE comparison already puts a median of \(\left\lVert G_n\right\rVert_{\mathrm{op}}\) below \(3\) for large \(n\). This norm is one-Lipschitz in Hilbert–Schmidt norm, so Gaussian concentration gives \[\mathbb P(\left\lVert G_n\right\rVert_{\mathrm{op}}>K)\le e^{-c_Kn},\qquad c_K>0.\] A word of degree \(k\) with diagonal bound \(M\) satisfies, by telescoping, \[\left\lVert P(G,D)-P(G',D)\right\rVert_{\mathrm{op}} \le L\left\lVert G-G'\right\rVert_{\mathrm{HS}},\] where \(L\) depends on \(k,M\), the coefficients, and the spectral cutoff, but not on \(\alpha\), the type counts, or any compression rank. The projection is nonexpansive in Hilbert–Schmidt norm. Gaussian concentration for the clipped polynomial, combined with its median comparison to the limiting target, therefore gives the original polynomial the failure bound \[2e^{-c_0na^2/L^2}+e^{-c_Kn}.\] For \(0<a\le1\), reducing the positive constant \(c\) absorbs the escape term and the prefactors into \(2e^{-cna^2/L^2}\) for sufficiently large \(n\). The preceding convergence puts the clipped median below the limiting norm target plus any fixed slack. These median comparisons are uniform over type counts: otherwise take a failing sequence, extract convergent proportions, and use the full-support or rare-type case just proved. One can realize this continuity on a fixed letter space \(\mathbb R^q\) with creation vector \((\sqrt{p_a})_a\) and a faithful base; creation changes in norm by at most \(\left\lVert \sqrt p-\sqrt{p'}\right\rVert_2\). This completes the proof. ◻

Proof. The parameters are chosen by comparing the entropy of all allowed changes with a fixed Gaussian failure rate. First fix the word family, caps, bin resolution, spectral cutoff, and operator slack \(a>0\). For one label, changing at most \(\delta n\) entries with at most \(q\) replacement choices gives \[ \log N_{n,\delta} \le n\{H(\delta)+\delta\log q\}+o(n), \qquad H(\delta)=-\delta\log\delta-(1-\delta)\log(1-\delta). \tag{89}\] For finitely many labels add these entropies. At fixed positive \(\delta\) this is a small positive multiple of \(n\), not \(o(n)\). Let \(c a^2/L^2\) be the Gaussian rate from 26. Choose \(\delta\) so that the sum in [eq:path-sparse-entropy], divided by \(n\), is below one quarter of that rate. The union bound then gives a common exponential event for every possible modification. Polynomially many type-count cases cost only \(o(n)\).

This common event is enough for tracial label approximations. Indeed, if \(n^{-1}\sum_i|d_i-\widetilde d_i|^2\le\eta^2\), then at most \(\delta n\) sites satisfy \(|d_i-\widetilde d_i|>\eta/\sqrt\delta\). Off those sites, binning and polynomial telescoping absorb the small uniform error. On those sites, the comparison for every modification has already been secured by the union bound.

Only after fixing these label approximations choose a different mesh for the path density. By 19, its specific log-density price \(\kappa\) can be made smaller than another quarter of the same Gaussian rate. A density tilt \(e^{\kappa n+o(n)}\) leaves a strictly positive exponential failure rate. Then take \(n\to\infty\). The minimum positive type proportion never enters the rate: it only enters the preliminary moment constant used to establish the median. Finally let the sparse fraction, density cost, and operator slack tend to zero in this nested order. ◻

Proof. We must pass from the independent Gaussian comparison to the matrix left by the path queries. Fix the finite label calculations first. After conditioning on the frozen path data, the spin inputs at each mesh time are fixed, and the likelihood sees \(J\) only through its images on those input columns. Retain an orthonormal basis of their nonvanishing residual directions, together with the input seed columns and the finitely many adaptive queries required by the fixed labels. Formula (79) leaves an independent compressed Gaussian matrix. The removed part is a bounded sum of dyads of input and innovation vectors, of rank at most twice the number of retained directions.

Small query residuals are handled before any matrix inverse is needed. At the fixed density mesh, process the queries in causal order and omit residuals shorter than a fixed threshold \(\theta\). Use the orthogonal projection of each omitted query when evaluating later fields. This still defines a probability scheme: every refresh uses a normalized Bernoulli probability depending on its past. In particular, no lower bound on the smallest eigenvalue of an arbitrarily long mesh Gram matrix is assumed. The omitted spin input has normalized length at most \(\theta\). The spectral bound, finite-bin count truncation and 19 bound the resulting log likelihood error by \(\kappa_\theta n\), with \(\kappa_\theta\to0\) at the fixed mesh. Subsequent label queries with small residual can be omitted using normalized square-mean continuity. Thus nearly singular regressions cause a small density error, not an inverse-Gram operator factor.

Before applying a full-GOE word comparison, augment the residual \(PG'P\) by its missing Gaussian blocks, independently of the actual data. Conditional on the revealed columns and fixed labels, this gives a full independent GOE \(G'\) on an enlarged probability space. The path-density change depends only on the compressed residual; the independent added blocks have density cost one. A full-GOE operator comparison is applied only to this completed matrix. In particular, no unrestricted polynomial norm domination for \(PG'P\) by the full-GOE Fock norm is asserted.

We now apply the comparison on that enlarged space. The GOE map from standard Gaussian coordinates has Hilbert–Schmidt Lipschitz scale \(\sqrt{2/n}\). For a Gaussian good set of probability at least \(1/2\), isoperimetry bounds the probability of Frobenius distance greater than \(r\) from the set by \(2e^{-cnr^2}\). Use as the good set the full-GOE event for the required fixed-word comparisons in 26. Under a pointwise density price \(e^{\kappa n+o(n)}\), taking \(r>C\sqrt\kappa+a\) leaves an exponential error. Telescoping a bounded polynomial changes its operator norm by at most \(Lr\). Both \(L\) and the Gaussian scale are independent of the number of queried directions; compressing a perturbation has norm at most its original norm. The countably many fixed approximations are taken in successive limits, not intersected as a dimension-dependent word family. Sparse label changes are covered by 27 on this same event.

Only the hidden word inner products remain to be compared with the actual compressed matrix. The discrepancy between the actual and completed matrices has range and corange in the span of the queried directions and finitely many added isotropic Gaussian columns. Expand each polynomial discrepancy as a sum of dyads. Its two vectors lie in the cyclic spaces of those directions under the actual bounded words and labels. The queried directions and their images are path vectors, so 24 makes them orthogonal to hidden endpoint vectors. For an added Gaussian column \(g\) of covariance at most \(I/n\), condition on all actual data. For any actual bounded word \(Q\) and actual endpoint vector \(u\), \[\mathbb E_g|u^tQg|^2\le n^{-1}\left\lVert Q^tu\right\rVert^2 \le Cn^{-1}\left\lVert u\right\rVert^2.\] The same estimate treats the finitely many added columns jointly and, by a further finite polynomial expansion, their word orbits. Thus the added cyclic directions vanish in hidden word inner products as well. The hidden vectors do not anticipate the added blocks: these blocks were sampled only after fixing the actual data. Removing them at this stage proves the comparison on the hidden complement that is required here, while retaining the rare diagonal base of the full-GOE model.

This gives a positive matrix-valued functional for every finite family of hidden vectors on the algebra generated by \(D\) and \(s\), bounded by the norm in [eq:path-fock-space]. To represent creation as well as the self-adjoint sum, lift the word inner products to the algebra containing \(\ell\). Use a norm-preserving Hahn–Banach extension of the positive functional from the unital algebra generated by \(D,s\) to the one generated by \(D,\ell\). Positivity follows from preservation of the norm and its value at the identity. Apply this to matrix amplifications to retain all the joint vector states, and use the Gelfand–Naimark–Segal (GNS) construction. The resulting lifted vectors have exactly the original joint word inner products, while \(\ell^*D\ell=\tau(D)I\) holds in the ambient representation. If a finite star has a permutation symmetry one may average the extension over that finite group and take direct sums. This proves precisely the one-tree assertion; it imposes no symmetry joining independent choices at two different endpoints. ◻

Proof. First prove the entry predictions, and then strengthen the resulting Hadamard estimate to an operator bound. For independent GOE and deterministic binned diagonals, each entry of a fixed word is Frobenius Lipschitz on a spectral cutoff, with a dimension-free constant. Gaussian concentration and a union bound over \(n^2\) entries therefore give uniform convergence to the entry expectations at every fixed positive accuracy. Conjugation by diagonal sign matrices makes all off-diagonal expectations zero. A fixed-degree Wick expansion of the diagonal entry leaves precisely its complete noncrossing contraction with the outer site type fixed. Every other pairing has at least one additional index identification and costs \(O(1/n)\), uniformly in the outer type, including a rare type. This proves [eq:path-entry-prediction]. The same expansion for the normalized trace averages the remaining outer diagonal and gives the vacuum trace rule.

For the off-diagonal Hadamard product, its expectation depends only on the two endpoint types. A Wick pairing connecting the two distinct fixed endpoints has at least one fewer free index than paired edges, so each mean entry is \(O(1/n)\). If \(E_a\) is a type projection, this mean matrix is, up to an \(O(1/n)\) diagonal correction, a sum of dyads \[\sum_{a,b}c_{ab}\,(E_a\mathbf 1/\sqrt n)(E_b\mathbf 1/\sqrt n)^t, \qquad |c_{ab}|\le C.\] It is visible. We must still prove that the centered product is small in operator norm; entrywise smallness alone would not prove it.

We obtain the operator bound by treating diffuse and large coordinates separately. Start with unit vectors \(u,v\) satisfying \(\max(\left\lVert u\right\rVert_\infty,\left\lVert v\right\rVert_\infty)\le\zeta\) and consider \(u^t(Q\circ R)v\). Perturbing \(G\) by \(\dot G\) gives terms of the form \[\left\langle D_uR D_v,\dot Q\right\rangle_{\mathrm{HS}} +\left\langle D_uQ D_v,\dot R\right\rangle_{\mathrm{HS}}.\] The word differentials are Hilbert–Schmidt bounded by \(C\left\lVert \dot G\right\rVert_{\mathrm{HS}}\). Moreover, \[\left\lVert D_uR D_v\right\rVert_{\mathrm{HS}} \le\left\lVert u\right\rVert_\infty\left\lVert R\right\rVert\left\lVert v\right\rVert\le C\zeta.\] Hence this bilinear statistic has Gaussian concentration rate \(\exp(-cn a^2/\zeta^2)\). Nets of the two unit balls at a fixed resolution have size \(\exp(C_an)\). Choose \(\zeta\) sufficiently small after the required accuracy \(a\). A union bound controls all diffuse unit tests.

To cover all unit vectors, separate the coordinates of magnitude greater than \(\zeta\). Each vector has at most \(\zeta^{-2}\) such coordinates. After removing the diagonal, \[\sum_{j\ne i}|Q_{ij}R_{ij}|^2 \le\max_{j\ne i}|Q_{ij}|^2\sum_j|R_{ij}|^2=o(1),\] and the same holds for columns. Thus the portions involving the finitely many large coordinates have norm \(o(1)\) uniformly in their locations. The finite-net approximation is legitimate because the Schur bound gives a uniform a priori operator norm. Combining sparse and diffuse parts proves that the centered off-diagonal Hadamard product is \(o_{\mathrm{op}}(1)\).

Visible dyads do not spoil these claims. A bounded path vector \(b\) with uniform spatial square tails has \(\max_i|b_i|\to0\), since \[\max_i|b_i|^2 \le M^2/n+\sum_i|b_i|^2\mathbf 1_{\{\sqrt n|b_i|>M\}},\] with dimension first and then \(M\to\infty\). Thus a visible dyad has negligible maximum entry. Its Hadamard product with a bounded matrix is \(D_bQD_c\) and has norm at most \(\left\lVert b\right\rVert_\infty\left\lVert Q\right\rVert\left\lVert c\right\rVert_\infty=o(1)\). Finite-rank approximation extends the argument to the visible closure.

Finally use [lem:path-compressed,lem:path-sparse] to transfer the uniform Gaussian comparisons to actual path labels. Polynomial norm differences tolerate the small Frobenius comparison error. Uniform entry predictions retain the actual outer diagonal on rare sites. The dyads of binned type indicators are limits of bounded path-recipe dyads by tracial approximation. This proves the stated Hadamard reduction and diagonal formula on admissible trees. ◻

Proof. We prove closure by induction over the recipe operations. The spin input satisfies \(E(x)=I\), and a query by \(J\) left-multiplies its difference matrix by \(J\). For the nonlinear step, assume closure for \(z\) and apply a bounded smooth scalar function \(\phi\). Off the diagonal, use \(z_i(x^j)=z_i(x)-2x_jd_jz_i\) in Taylor’s formula to obtain \[ d_j\phi(z_i) =\phi'(z_i)E(z)_{ij} -x_j\phi''(z_i)E(z)_{ij}^2+R_{ij}, \qquad |R_{ij}|\le C|E(z)_{ij}|^3. \tag{90}\] The remainder’s maximum row and column absolute sums tend to zero: \[\sum_j|R_{ij}| \le C\max_{k\ne l}|E(z)_{kl}|\sum_j|E(z)_{ij}|^2=o(1),\] and similarly for columns. Keep \(d_i\phi(z_i)\) exactly on the diagonal. The first two terms in [eq:path-difference-taylor] are diagonal multiplications of \(E(z)\) and its Schur square; they belong to the asserted closure by 29. Their operator norms are bounded, and their off-diagonal maxima vanish. The exact diagonal is a bounded site label, obtained from \(z_i\) and its already constructed own-spin difference. Its smooth limiting versions follow by the same recipe recursion. This completes induction over every finite calculation. Finite sums and products use the exact product rule and the bounded Schur estimate; no unbounded derivative coefficient appears.

For column flips define \(\mathsf F(Q)_{ij}=Q_{ij}(x^j)\). Starting at the right end of a word, one has the exact recursions \[\begin{align*} \mathsf F(JQ)&=J\mathsf F(Q),\\ \mathsf F(D_fQ) &=D_f\mathsf F(Q)-2\{E(f)\circ\mathsf F(Q)\}D_x. \end{align*}\] The first uses that \(J\) is independent of \(x\); the second is the identity \(f_i(x^j)=f_i(x)-2x_jd_jf_i(x)\) with its derivative index still tied to column \(j\). These recursions express every column-flip matrix through bounded path words and Schur insertions. They preserve the closure just proved. The conditional derivative of a fixed smooth multitime calculation requires the full mixed-slot recursion of [lem:path-mixed-tensors,lem:path-conditional-derivative]. After its uniform tails are removed, the equality-network reduction in 33 expresses every opened term through the matrix closure just described. Thus the dynamic assertion uses that additional argument; it is not inferred from first partial Jacobian bounds alone. ◻

Proof. Only the increments need polynomial approximation; the base query values may remain unbounded. In the corner expansion from 15, the formal increments satisfy \(|t_{b,S}|\le 2^{|S|}C_S(z^b)\) and hence range in a fixed compact box. Each remainder has the form \[ \left(\prod_{\alpha\in A}t_\alpha\right)t_\gamma R_{A,\gamma}(z,t),\qquad\gamma\in A. \tag{91}\] Here \(z\) consists of the undifferentiated calculation values at the output site. The integral formula for \(R\) and the global scalar derivative bounds give a uniform bound and a uniform modulus of continuity in \(t\), independently of \(z\in\mathbb R^q\).

Apply multivariate Bernstein approximation on the increment box, treating \(z\) as a parameter. Given \(\eta>0\), this gives a fixed finite polynomial \[P_\eta(z,t)=\sum_\nu b_\nu(z)t^\nu, \qquad \sup_{z,t}|R_{A,\gamma}(z,t)-P_\eta(z,t)|\le\eta.\] In the Bernstein basis the coefficients are values of \(R\) at fixed increment grid points. They are therefore bounded smooth functions of \(z\) with bounded derivatives of each required order. Conversion to the fixed monomial basis preserves these properties. This approximates only the increments, not the possibly unbounded \(z\).

The error still contains the repeated factor \(t_\gamma\) in [eq:path-increment-remainder]. The repeated-square estimate in 14 therefore bounds its flattened operator norm by \(C\eta\). Every retained monomial still covers the original set \(I\) of derivative slots. Carry out this operation at every nonlinear gate, and substitute the already constructed approximations of its input tensors. The product bound in [eq:path-tensor-product] makes each finite substitution continuous in those operator norms. Finite induction proves the asserted approximation.

We record the spatial structure of the resulting terms for the conditional derivative calculation. Unroll one term into a tree rooted at its output site. A linear gate gives a matrix edge. At a tensor monomial, take copies of the nonempty child tensors and join their output roots at an equality vertex labelled by \(b_\nu\). Each spin input supplies an identity edge ending in a derivative leaf; retain this edge even if it could immediately be contracted. A constant input has no nonempty derivative term. Factors \(x^{(a)}_{j_a}\) from the corner increments become bounded labels at the corresponding leaves. The covering property ensures that every required derivative slot has a leaf. Several occurrences of one slot remain several leaves with the same slot name, to be copied by equality tensors when the temporal indices are attached. Thus no coefficient depending on two unrelated vertex indices remains. ◻

Proof. We first bound a network contraction, then show how to reduce the network while controlling each error. Put \(H=\mathbb R^n\). For \(p,q\ge1\), the labelled equality gate \(E^d_{p,q}:H^{\otimes p}\to H^{\otimes q}\) sends \(e_a^{\otimes p}\) to \(d(a)e_a^{\otimes q}\) and annihilates nonconstant input tuples. Its norm is \(\left\lVert d\right\rVert_\infty\). Count the external input as an incoming wire at the source and the external output as an outgoing wire at the sink. Each vertex now has both an incoming and an outgoing wire. Executing vertices in topological order, with tensor products on the other live wires, gives the deterministic bound \[ \left\lVert Z_G\right\rVert_{\mathrm{op}} \le\prod_{e\in E(G)}\left\lVert A_e\right\rVert_{\mathrm{op}} \prod_{v\in V(G)}\left\lVert d_v\right\rVert_\infty. \tag{92}\] In particular, there is no free summation map with norm \(\sqrt n\).

Call an internal edge \(e:u\to v\) safe if deleting it preserves the unique source and sink and makes \(u,v\) incomparable in the remaining directed order. Both endpoints then retain positive incoming and outgoing degrees. Process their predecessors first and process \(u,v\) jointly. If their remaining degrees are \(p_u,q_u,p_v,q_v\ge1\), the joint gate with coefficient \(R\) is \[e_a^{\otimes p_u}\otimes e_b^{\otimes p_v} \longmapsto d_u(a)d_v(b)R_{ab}\, e_a^{\otimes q_u}\otimes e_b^{\otimes q_v},\] and is zero on all other input tuples. Both the displayed input family and output family are orthonormal. Its norm is at most \(\left\lVert d_u\right\rVert_\infty\left\lVert d_v\right\rVert_\infty\max_{a,b}|R_{ab}|\). Taking \(R\) to be the edge matrix restores the original network; taking its diagonal identifies \(a=b\) and contracts \(u,v\). Hence \[ \left\lVert Z_G-Z_{G/e}\right\rVert_{\mathrm{op}} \le\max_{a\ne b}|(A_e)_{ab}| \prod_{f\ne e}\left\lVert A_f\right\rVert_{\mathrm{op}} \prod_w\left\lVert d_w\right\rVert_\infty=o(1), \tag{93}\] with new vertex label \(d_ud_v\mathop{\mathrm{diag}}(A_e)\). Its diagonal can be replaced by \(d_ud_v\delta_{A_e}\) using [eq:path-network-bound]. Contraction cannot create a directed cycle: such a cycle would yield a path from \(u\) to \(v\) or from \(v\) to \(u\) after deletion. It preserves the unique source and sink.

The norm estimate justifies pinching a safe edge. To ensure that such an edge is available, first consolidate parallel edges by Hadamard products and suppress every nonterminal vertex of indegree and outdegree one, using matrix multiplication with its diagonal label. Continue until neither reduction applies. If any internal vertex remains, the earliest one in a topological order has just one incoming edge, from the source. It must consequently have at least two outgoing edges, or it would have been suppressed. Choose the latest internal vertex \(u\) having at least two outgoing edges, and let \(v\) be its earliest immediate successor. The successors are distinct, so \(v\) is internal. It has outdegree one by the choice of \(u\), and indegree at least two because suppression has ended. There is no alternate path \(u\to v\): its first edge would go to another successor of \(u\), later than \(v\), which cannot subsequently reach \(v\). Deleting \(u\to v\) therefore makes them incomparable and leaves positive incoming and outgoing degrees at both endpoints. This is a safe edge.

Pinch that edge and repeat the ordinary reductions. A pinching or series reduction decreases the number of vertices, and parallel consolidation decreases the number of edges. The procedure therefore ends after finitely many steps with one edge from source to sink. All newly formed edges belong to \(\mathcal A\), so their own entry predictions justify every subsequent pinching. Summing the finitely many errors proves the claim. If source and sink have already been identified, acyclicity leaves only a diagonal equality gate; this case is immediate. Finally \(\max_{ij}|E_{ij}|\le\left\lVert E\right\rVert_{\mathrm{op}}\) transfers the off-diagonal assertion through the accumulated errors. ◻

Proof. The first step removes errors that are uniform in the joining state. Use [eq:path-copy-bound,eq:path-amplified-tail] for the tensor approximations and the common \(K\) tails. At fixed accuracy, only finitely many terms and fixed finite-measure Duhamel integrals remain. Every \(K\) insertion history is a bounded finite path-matrix word, with a fixed cap on each likelihood label. Keep ordinary \(S\) expectations exact for now. On opening a later inner-product or replica test, they become ordinary causal branches and introduce no new spatial summation index.

We now identify the two trees whose gluing represents an opened term. The spatial recipe tree is the one constructed in 31. A second, temporal tree starts at the joining derivative coordinate: construct it by reading the finite conditioning recursion backwards. In [eq:path-copy-tensors], the memory term keeps its current slot \(p\). The active term replaces \(p\) by the newly introduced slot \(m\), through the \(K\) path-word edge. The mixed term forks into \(p\) and \(m\), puts the \(K\) edge on the latter branch, and puts the bounded sign \(-2x_p\) at the fork. The new slot \(m\) was absent from the current derivative subset. It therefore cannot reconnect to an existing temporal leaf. Repeating this construction gives a temporal tree with one terminal for each original derivative slot used in that term. Finite causal-tree conditioning has the same construction, with copies attached at the common joining vertex.

If a slot occurs several times in the recipe polynomial, append an equality fork to its temporal terminal, one outgoing branch for each occurrence. Glue the resulting temporal leaves to the recipe leaves in pairs, retaining the input-spin identity edges until this gluing is complete. Every leaf is paired. The covering property ensures that there is at least one leaf, and that both trees are connected. Orient the temporal tree away from the joining derivative index and the recipe tree toward its output site. A directed path can cross from the temporal tree into the recipe tree only at a paired leaf; it cannot cross back. Thus the glued network is acyclic, with its unique source the joining derivative coordinate and its unique sink the output site. Every vertex is on a path between them. The only free indices are these two external ones; all internal indices are ordinary equality contractions with no normalization factor.

The glued graph now has exactly the orientation required by 32. Its matrix algebra, including bounded visible dyads, is closed under ordinary products and transposes. Lemma 29 supplies the Hadamard reduction and entry predictions for each newly formed fixed expression. Applying the network lemma therefore reduces the opened term to the same path-matrix closure in operator norm. The bounded scalar coefficients, spin signs and diagonal predictions are path labels. Integrate its pointwise network bound over the fixed Duhamel parameter sets and use the integrated entry predictions. No worst auxiliary time or starting configuration is selected. The same argument applies to stationary reversed causal branches. Actual and predictor terms keep the same subdivisions and approximations before they are recombined and centered.

This proves closure for the opened histories used in the subsequent conditional tests. It does not assert a uniform-operator-norm approximation of an arbitrary conditional matrix average by finitely many sampled histories. Conditional expectations remain exact until those branches are opened. For a still-averaged off-diagonal maximum one may instead use conditional Jensen: \[\max_{i\ne j}|\mathbb E[Q_{ij}\mid X]| \le\mathbb E[\max_{i\ne j}|Q_{ij}|\mid X].\] The uniform errors were removed before this typical network step. In particular, no spatial square-mean error has been substituted for an operator error. ◻

Proof. We first separate the two planted starting directions, then propagate that separation through the finite simulations. Plant the samples \(\sigma\) and \(\sigma'\). Their overlap \(r=n^{-1}\sigma^t\sigma'\) has the independent-uniform-spin law tilted by \(\exp(\beta^2nr^2/2)\), up to a spin-independent constant. The Rademacher rate function satisfies \(I(r)\ge r^2/2\). Since \(\beta<1\), for every fixed \(\varepsilon>0\) the probability \(|r|>\varepsilon\) is at most \(e^{-c_\varepsilon n}\). Conditionally on the spins the Gaussian disorder has the two rank shifts \(\beta^2(\sigma\sigma^t+\sigma'\sigma'^t)/n\). The two initial site spins are therefore asymptotically independent, and the two planted directions are asymptotically orthogonal.

Perform the finite Gaussian recursion from 23 for both causal simulations. Each query needed for spin and field histories is odd under global spin inversion of its own root; a score query for likelihoods is also odd. Fresh refresh uniforms can be reflected under this inversion without changing their law. Deterministic cross-overlap limits of an odd query from each different tree change sign under either inversion, and hence vanish. The Gaussian regression innovations for the two trees have zero cross covariance and are independent. By induction their site recursions are independent, because their initial spins and site seeds are independent as well. Likelihood labels are bounded functions of the resulting histories; their computation needs no additional noncentered Gaussian query. Thus their joint limits also factor. This induction does not apply to an arbitrary shared noncentered query such as \(J\mathbf 1\), explaining the restriction in the statement.

Transfer this factorization to the actual stationary paths using the two-replica planting comparison and then [prop:path-simulation,lem:path-likelihood-approximation]. For a bounded label \(b\), represent the spatial mean of \(\mathbb E[b_i\mid J]^2\) by the empirical product on the two independent roots. The factorization makes its limit \((\tau b)^2\), proving the claimed self-averaging of the site means. ◻

Completion of the proof of 22. The required empirical space and endpoint predictions are supplied by 24, together with orthogonality of hidden vectors. Lemma 25 puts the likelihood labels in this construction with their stated tracial convention. The norm comparison then follows from [lem:path-fock-norm,lem:path-sparse,lem:path-compressed]: the representation is on one tree, retains the rare diagonal base, and uses the Gaussian rate before paying the entropy and density costs. The entry and Hadamard assertions come from 29; the difference and column-flip closure is 30, with the opened conditional terms just reduced to that closure. Finally, 34 proves the restricted two-root independence assertion.

Every construction used the fixed countable dense recipe family and successive fixed approximations. Diagonal extraction therefore gives common high-probability disorder events for any finite initial list of tests. Lemma 18 permits polynomial weights and actual conditional copies. All superpolynomial path discards were made separately on the original forward marginals. These observations give the typicality convention and the order of limits in the theorem. ◻

Proof. Consider first a refresh at another site. The logarithmic perturbation is a Bernoulli likelihood increment with field displacement \(\theta aJ_{ij}\). Its conditional exponential moment, at any fixed real order, is at most \(\exp(C_qJ_{ij}^2)\). Iteration over the attempts, followed by the Poisson exponential formula, gives \(e^{C_qd}\) because \(\sum_jJ_{ij}^2\le C\) and \(\max_j|J_{ij}|=o(1)\). The same calculation controls inverse moments. After its prescribed refresh the forced spin is predictable, so it does not change the calculation.

We next control the forced-coin factor. For \(f(y)=J_iy\), 13 supplies a fixed-horizon bound on \(\sup_y\left\lVert dS_sf(y)\right\rVert^2\). Applying 3 controls every fixed exponential moment of \(f(Y(s_*))-\mathbb Ef(Y(s_*))\). Combine this with [eq:row-forced-coin] and Hölder’s inequality. The resulting constant may grow rapidly with fixed \(d\). This step provides uniform integrability at fixed time, rather than a large-\(d\) rate.

For a positive \(V\) and \(A=\mathbb E[V\mid Z]\), Jensen gives \[ A^q\le\mathbb E[V^q\mid Z]\quad(q\ge1),\qquad A^{-q}\le\mathbb E[V^{-q}\mid Z]\quad(q>0). \tag{100}\] Choosing an order strictly above the desired one gives uniform integrability for the required powers of \(A\). To identify those powers by replicated tests, take copies \(V^{(1)},\ldots,V^{(b)}\) that are conditionally independent given \(Z\). They satisfy \[\mathbb E\left|b^{-1}\sum_{a=1}^bV^{(a)}-A\right|^2 \le b^{-1}\mathbb E[V^2].\] Cap the \(V^{(a)}\) first and pass to the dimension limit with the cap and \(b\) fixed. Then let \(b\to\infty\) and remove the cap. The higher moments in [eq:row-jensen-ui] justify this passage for positive and negative powers alike; in particular, a limiting \(A\) has no mass at zero. Hölder’s inequality in [eq:row-ratio-moment] transfers the conclusion to the terminal density ratio. ◻

Proof. Begin with a finite-dimensional list and condition on \(Z\). Diagonalize \[T=\mathbb E\left[\sum_\alpha u_\alpha\otimes u_\alpha\mid Z\right],\] where \(u_\alpha\) denotes a normalized list vector. Work first on the sets where the list norms are bounded. If \(\Pi_k\) projects on the first \(k\) eigenvectors, independence of two branches and \(\mathop{\mathrm{Tr}}T\le K\) give \[\mathbb E\left[\left|\left\langle (I-\Pi_k)u_\alpha,(I-\Pi_k)u'_\gamma\right\rangle\right|^2\Bigm|Z\right] \le\mathop{\mathrm{Tr}}[((I-\Pi_k)T)^2]\le\frac{K^2}{k+1}.\] Extract the laws of the finite coordinate lists together with their self-Grams, and then let \(k\to\infty\). Count tails and the spin-norm bounds remove the preliminary restriction on list norms. The resulting directing law on lists in \(\ell^2\) identifies the limiting cross-Grams. At every finite coordinate projection, the stated covariance difference is positive semidefinite, and this property persists in the limit. A common isonormal process on \(\ell^2\), plus independent Gaussians with the covariance difference, realizes exactly these Gaussian Gram arrays. The conditional-average assertion follows from the replicated approximation in 36. ◻

Proof. Suppose a support point had squared norm greater than \(\delta\). Continuity would give a sufficiently small ball about it in which every pair has inner product greater than \(\delta\). The ball has positive probability, so independence makes this pair event have positive probability, contradicting the hypothesis. A countable base of balls shows, by separability, that the law is carried by its support. The same argument applies to the conditional law for almost every directing variable. ◻

Proof. Fix the specified data. The full list is then Gaussian, and the orthogonality identities make each score independent of the earlier scores and of every field already used to choose its sign. Its variance is \(b_h\). It may remain correlated with fields at later coin times, which does not affect this chronological calculation. Generate the Gaussian list and the coins in time order: the sign \(a_h\) is measurable before revealing \(\widetilde Z_h\), so \[\mathbb E\left[\exp\{r a_h\widetilde Z_h-r^2b_h/2\} \mid\text{past}\right]=1.\] Their product has mean one. Using \(a_h^2=1\) and multiplying by \(\exp\{(r^2-r)\sum_hb_h/2\}\) proves the bounds. Conditioning on future self-Gram data is legitimate here because the needed Gaussian orthogonality identities hold pointwise for each value of those data. At no point are the scores themselves conditioned on future observed fields. ◻

Proof of 35. The prefix is now controlled at powers close to one. Conditional Hölder inequalities will put every fixed larger power on the short suffix and on the last-spin factor. First retain schedules with at most \(Kd\) refreshes. In the limiting representation, factor \[L_\theta F_\sigma=W_{\mathrm{pre}}U, \qquad A=\mathbb E[W_{\mathrm{pre}}U\mid\mathcal D,g],\] where the prefix ends at \(T=d-\sqrt d\) and \(U\) contains the suffix and the normalized last-spin factor, wherever that last refresh occurs. For each fixed \(q\), positive and negative moments of the suffix likelihood are at most \(e^{C_q\sqrt d}\). Apply the forward martingale calculation on the interval of length \(\sqrt d\), and then use the fixed-\(d\) moment passage proved above. The last-field factor has bounded moments independent of \(d\) in this limiting calculation: its centered Gaussian field has variance at most \(4\beta^2\) by [eq:row-gram-budgets], before the common and private parts are separated. Hölder therefore gives \[ \mathbb E[|U|^q+|U|^{-q}]\le 2e^{C_q(1+\sqrt d)}. \tag{109}\] The deterministic field means disappear from this bound through [eq:row-forced-coin].

For \(0<\eta<1\), conditional Hölder gives the two inequalities \[\begin{align*} A&\le \mathbb E[W_{\mathrm{pre}}^{1+\eta}\mid\mathcal D,g]^{1/(1+\eta)} \mathbb E[U^{(1+\eta)/\eta}\mid\mathcal D,g]^{\eta/(1+\eta)}, \tag{110}\\ \mathbb E[W_{\mathrm{pre}}^{1-\eta}\mid\mathcal D,g] &\le A^{1-\eta} \mathbb E[U^{-(1-\eta)/\eta}\mid\mathcal D,g]^\eta. \tag{111}\end{align*}\] For example, the second follows by writing \(W_{\mathrm{pre}}^{1-\eta} =(W_{\mathrm{pre}}U)^{1-\eta}U^{-(1-\eta)}\). This second inequality supplies the inverse-moment bound: inverse powers of \(A\) are controlled by inverse powers of the prefix conditional average together with positive powers of a suffix moment.

By 39, each corrected prefix conditional average with power \(1\pm\eta\) lies between \(e^{-C\eta d}\) and \(e^{C\eta d}\), uniformly in the branch data and common field forcing. Jensen permits integration over the directing law and also taking inverse powers. The small common score and covariance-coupling errors discussed above are handled by Hölder, with a power change smaller than \(\eta\). Combining [eq:row-holder-upper,eq:row-holder-lower,eq:row-suffix-cost] yields for every fixed \(p>0\) \[ \log\mathbb E[A^p+A^{-p}] \le C_p\eta d+C_{p,\eta}(1+\sqrt d)+o(d). \tag{112}\] This inequality concerns the moments after the dimension limit. Now send \(d\to\infty\) and then \(\eta\downarrow0\). The rough full-horizon constants from 36 served only to identify these limiting moments; they contribute no additional term to [eq:row-final-rate].

Hölder’s inequality in [eq:row-ratio-moment] proves the required bound for every retained schedule. We next remove the schedule conditioning. Under the ordinary path law, the tower property gives \[A_{i,x}^M(X_d) =\mathbb E_x[A_{i,x}^{M,\mathcal S}(X_d)\mid X_d].\] The conditional expectation averages the schedule with its ordinary conditional law given \(X_d\). For \(r\ge1\) or \(r<0\), conditional Jensen therefore decreases the moment when the schedule is no longer observed. Control the omitted schedules by a higher raw likelihood moment, Hölder’s inequality, and the Poisson estimate \[\mathbb P\{N_i(d)>Kd\}\le e^{-d(K\log K-K+1)}.\] The raw bounds are uniform conditional on the schedule. Taking \(K\) large after the desired moment order makes this contribution smaller than any prescribed \(e^{-Hd}\). The powers \(0\le r\le1\) again follow from Jensen. This completes the moment estimate.

It remains to obtain the stated exceptional sets, including for an integrated family of masks. Fix \(d\), the schedule-count cutoff, the moment orders, and the comparison accuracies. Every row concentration constant above is then uniform in the mask parameters and deterministic schedule times. If \(\mathcal B(J,x,\alpha)\) denotes a failure of one of the finitely many bounded tests, its row-and-start bound has the form \[\mathbb E_J\int\mu_J(\,\mathrm dx)\int \mathbf 1_{\mathcal B(J,x,\alpha)}\,\nu(\,\mathrm d\alpha) \le C e^{-cn}\] on the common good cavity events. Apply Fubini at this point, before choosing the quenched event. Markov’s inequality gives integrated Gibbs bad mass at most \(e^{-cn/2}\) outside an event of probability \(Ce^{-cn/2}\). If necessary, apply Markov once more to remove starts with excessive bad-schedule mass, and use higher rough moments to bound its contribution. The countable approximations are fixed successively, with arbitrary fixed slack, so their finite good events suffice before the dimension limit. Polynomial endpoint weights and an \(e^{o(n)}\) initial density preserve negligibility. At fixed times the remaining integrands are dominated by the rough moment bounds and the relevant mean-square vector budgets.

In a path expansion the order is therefore: fix the global time, hop cutoff, shapes, elementary caps, moment orders, and label bins; integrate over their deterministic simplex parameters; take dimension to infinity; then remove caps and approximation errors; finally increase the global time and decrease rate slack. Annulus and first-exceedance restrictions multiply a positive masked label by factors in \([0,1]\). Its positive moments are bounded by those of the unrestricted label; no inverse moment of a restricted label is used. A selected start or endpoint vector changes the weights in this integrated assertion and does not choose an exceptional mask. Likewise, good interior times can be chosen from a positive-measure set after an integrated estimate; a prescribed worst time is not being asserted to be good. ◻

Proof. For \(p\ge1\), condition on both endpoints of the span and then apply Jensen to the left endpoint’s conditional law given the right endpoint. The resulting integral over ordinary starts is bounded by 35. A higher raw moment and Hölder’s inequality make the exponentially small bad-start contribution negligible. The Markov property and stationarity make conditioning at a later endpoint another \(L^p\) contraction. At the left endpoint, the bound follows directly from \(\mathbb E_xa_i\le\mathbb E_x\ell_i^M=1\). Powers between zero and one follow by concavity. The raw bound is 13. ◻

Proof. The proof proceeds from a time-domain diagram identity to the transform bound, then identifies the covariance trace density and tests its edge. We first derive the diagram identity, making explicit the role of stationarity. Integration by parts and gradient intertwining yield \[ R(t)=\lim_{n\to\infty}\frac1n\mathop{\mathrm{Tr}}\mathbb E_\mu[D_vK_tI]. \tag{118}\] Here \(K_tI\) applies \(K_t\) to each constant coordinate vector. Insert the Duhamel expansion and use 22: visible finite-rank terms vanish in normalized trace, each hop contributes \(\beta sD_v\), and the remaining trace is the sum of complete noncrossing pairings. Each resulting vertex carries the product of its diagonal waiting likelihoods, its arrivals \(v\), and the bare factors \(e^{-h}\). These are nonnegative. The root initially carries \(v\), as in [eq:spin-gradient-trace].

The recurrence for \(P_k(s)\) removes exactly the adjacent returning contraction, making this polynomial the Wick string of \(k\) consecutive edges. Each of its edges must therefore pair into the Duhamel walk. Removing those \(k\) pairs exposes an ordered trunk of \(k+1\) completely paired rooted time intervals. Each interval starts with a \(v\) weight, including the interval at the original root, and each trunk edge contributes \(\beta\). Conversely, concatenate \(k+1\) rooted complete diagrams and adjoin the outside Wick string. This recovers one and only one pairing, so no multiplicity factor appears. After contraction, distinct vertices contribute scalar traces. Stationarity makes each interval’s trace depend only on its duration. The separation into scalar factors comes from the noncrossing contraction, rather than from temporal independence of intervals along one spin history. The spin/site self-averaging assertion in 22 is what identifies each such scalar factor with its stationary trace when it is represented by independent samples. It is used only for spin, field, and likelihood labels.

The simplex integral over the successive interval lengths therefore gives the convolution identity \[ R_k(t)=\beta^k R^{*(k+1)}(t). \tag{119}\] For the limiting argument, first keep time, Duhamel order, and likelihood caps fixed. Then 13 removes the caps and order cutoff in the matrix \(L^2\) comparison, establishing the identity at each fixed \(t\). Because the complete-diagram terms are nonnegative, we may also sum them monotonically when taking Laplace transforms.

To pass to transforms, we need a bound that is valid before the dimension limit and does not require a finite-dimensional resolvent at \(z\). Positivity supplies it: for a positive matrix \(B\) and a self-adjoint \(Q\), \(|\mathop{\mathrm{Tr}}(QB)|\le\left\lVert Q\right\rVert_{\mathrm{op}}\mathop{\mathrm{Tr}}B\). At finite \(n\) apply this to the positive matrix \(\int u e^{-tu}\mathsf M_n(\,\mathrm du)\) and to \(Q=P_k(J/\beta)\). The Wigner spectral bound and \(\sup_{[-2,2]}|P_k|=k+1\) show that, after the dimension limit, \[ |R_k(t)|\le(k+1)R(t). \tag{120}\] The same justification can be expressed through truncated integrals: integrate the finite-dimensional inequality over \(0\le t\le T\), take the dimension limit, and then send \(T\to\infty\). The right-hand side is integrable for \(z<\lambda_{\rm sp}\). Tonelli’s theorem applied to [eq:spin-trunk-convolution] now proves [eq:spin-trunk-transform] and yields \[(\beta\widehat R(z))^k\le k+1.\] Taking \(k\)th roots yields \(\beta\widehat R(z)\le1\). Given \(z<\lambda_{\rm sp}\), choose \(z'\) strictly between them. Since \(\widehat R\) is strictly increasing, \(\beta\widehat R(z)<\beta\widehat R(z')\le1\).

Integrating [eq:spin-trunk-convolution] at \(z=0\) gives \[\lim_{n\to\infty}\frac1n\mathop{\mathrm{Tr}}\left[ P_k(J/\beta)\mathbb E_\mu(xx^{\mathsf T})\right]=\beta^k.\] Once again, justify the integral by first stopping at fixed \(T\). The spectral gap bounds the omitted covariance tail by \(\left\lVert P_k(J/\beta)\right\rVert_{\mathrm{op}}e^{-\gamma T}\). To identify the resulting covariance density, use the orthonormality of the \(P_k\) for \(\rho_{\rm sc}\) and the generating function \[\sum_{k\ge0}\beta^kP_k(u)=\frac1{1-\beta u+\beta^2} \quad(-2\le u\le2);\] the series converges uniformly because \(\beta<1\) and \(|P_k(u)|\le k+1\). This proves [eq:spin-covariance-density]. The measure identified here is positive: at finite \(n\) it is the trace of the positive covariance matrix against the spectral projections of \(J/\beta\).

We next use this covariance density to bound the spin edge. Fix a real polynomial \(q\). Weighting \(\mathsf M_n\) by \(q(J/\beta)^2\) gives a nonnegative scalar spectral measure, dominated by \(\left\lVert q(J/\beta)\right\rVert_{\mathrm{op}}^2\rho_n\). All its limiting mass therefore lies in \([\lambda_{\rm sp},\infty)\), even though the finite-dimensional spin measures may put small masses below that edge. Its first moment satisfies \[\frac1n\mathop{\mathrm{Tr}}\bigl[q(J/\beta)^2D_{\mathbb E_\mu v}\bigr] \le\frac1n\mathop{\mathrm{Tr}}q(J/\beta)^2,\] because the energy matrix \(\int u\mathsf M_n(\,\mathrm du)\) is exactly \(D_{\mathbb E_\mu v}\). Use [eq:spin-covariance-density], and then the semicircle trace limit, to obtain \[ \lambda_{\rm sp} \int\frac{q(u)^2}{1-\beta u+\beta^2}\rho_{\rm sc}(\,\mathrm du) \le\int q(u)^2\rho_{\rm sc}(\,\mathrm du). \tag{121}\] Alternatively, discard scalar spectral mass below \(\lambda_{\rm sp}-\delta\) before sending \(\delta\downarrow0\). This gives the same inequality without requiring a uniform finite-\(n\) lower edge. Choose \(q_m(u)=((u+2)/4)^m\). After normalization, the measures \(q_m^2\rho_{\rm sc}\) concentrate at \(2\). Letting \(m\to\infty\) in [eq:spin-edge-test] gives \(\lambda_{\rm sp}\le1-2\beta+\beta^2\).

Finally, for \(0\le z<\lambda_{\rm sp}\), \[\beta^2\widehat R(z)<\beta <2\beta-\beta^2 \le1-\lambda_{\rm sp}<1-z.\] For \(z<0\), \(\widehat R(z)\le\widehat R(0)=1\) and \(\beta^2<1<1-z\). These prove the second renewal inequality and complete [eq:spin-conditions]. ◻

Proof. The Fock comparisons in this proof first remove the terminal visible component, as in the endpoint projection argument used below for the predictor caps. Fix a finite family of endpoint recipes with nonsingular limiting Gram matrix, and let \(Q_m\) be their actual orthogonal projection on the corresponding Gram-good event, and zero off that event. Both \(Q_m\) and \(I-Q_m\) are contractions. For every fixed capped path word \(W\), Cauchy–Schwarz bounds the removed pairing by \(\|Q_mW^tu\|_2\|p\|_2\), which tends to zero at fixed \(m\) by Assertion (ii) of 22 and the fixed inverse Gram bound. The weighted typicality convention removes the exceptional Gram event. Exhaust the endpoint recipe completion, omitting null directions, and choose \(m=m(n)\to\infty\) sufficiently slowly that these errors vanish for every fixed word and \((I-Q_m)p\) is hidden for every fixed endpoint recipe. Its second-moment and polynomial bounds follow from contraction. The projection depends only on the shared endpoint and its fixed seeds, so permutations of the replicated branches preserve it. This projection is used only for the Fock bilinear comparisons; we retain the original exact \(p\) for the Hessian estimates that follow.

Work on one finite tree and use the cut expansion behind [eq:hidden-cut-bound]. Every cut away from the replaced factor puts that factor on a pure creation or pure annihilation side. Its trace two-norm then bounds the error, multiplied by the fixed bounds on the other factors and the two endpoint norms. Only the cut at the replaced factor needs another argument. At that cut, at least one distinguished centered factor lies on the pure side leading to its own centering endpoint.

Attach \(b\) conditionally independent copies at this endpoint. Lift this single replicated star to the Fock representation, and symmetrize by taking a direct sum over permutations of the copies. Repeatedly applying \(l^*D_al=\tau(a)I\) expresses inner products of pure-side strings as products of pair traces. For different copies, one factor is the pair trace of the centered label and has zero limit. With approximated predictors, bound its absolute value by \(\eta\), tending to zero with the approximation. Write \(C_\alpha\) for the pure-side creation operator on copy \(\alpha\), oriented toward the shared endpoint. The Gram bounds are \[\left\lVert C_\alpha^*C_\alpha\right\rVert\leq C,\qquad \left\lVert C_\alpha^*C_\gamma\right\rVert\leq C\eta\quad(\alpha\ne\gamma),\qquad \left\lVert [C_1\ \cdots\ C_b]\right\rVert^2\leq C(1+(b-1)\eta).\] If \(v\) denotes the vector at the shared endpoint, symmetry of the lifted star gives \[\mathbb E\left\lVert C_1^*v\right\rVert^2 =\frac1b\sum_{\alpha=1}^b\mathbb E\left\lVert C_\alpha^*v\right\rVert^2 \leq C(b^{-1}+\eta)\mathbb E\left\lVert v\right\rVert^2.\] The vector at the opposite end may depend on its whole branch; Cauchy–Schwarz now bounds the squared bilinear contraction by \[C\bigl(b^{-1}+\eta\bigr)\left\lVert u\right\rVert_2^2\left\lVert p\right\rVert_2^2.\] The fixed caps bound the diagonal Gram blocks. Apply Cauchy–Schwarz and take the limits \(n\to\infty\), \(\eta\to0\), and \(b\to\infty\) in that order to obtain zero. All cuts for this replacement are treated on the same star. The argument makes no simultaneous choice of creation operators on stars rooted at different endpoints.

A terminal insertion with small trace support has the same two cases. Every nonterminal cut pays its trace two-norm; at the terminal cut, the left distinguished factor lies on its centered pure side, so the replication argument applies. Iterate for finitely many replacements. When the replaced factor is an approximated distinguished label, keep its exact forward or backward subtraction throughout this step and use the other distinguished factor at the exceptional cut.

The word comparison retains every rare diagonal’s action on the base. Small trace is used on a pure side, and centering treats the remaining cut; neither step uses the false implication \(\tau(c^2)\to0\Rightarrow\left\lVert D_c\right\rVert_{\mathrm{op}}\to0\). ◻

Proof. First fix a tensor-network approximation from 31. Before opening paths, use the mixed-tensor budget to bound its operator errors, and remove those errors in the prescribed successive approximation limit. The temporal tree off the marked path consists of disjoint subtrees ending at derivative leaves of the chosen label. Each attachment has at least one such leaf. The corresponding maps are bounded tensor products and compositions of capped path matrices and equality copies; every equality copy has norm at most its fixed scalar label bound. Pull their adjoints into the row tensor of the label circuit. Repeated occurrences of a derivative index are handled by the norm-one equality embedding. The resulting tensor \(U_i\) on the \(r\) attachment indices satisfies \[ \sup_i\sum_{a_1,\ldots,a_r}|U_i(a_1,\ldots,a_r)|^2\leq C^2. \tag{135}\] Use the operator norm of each flattened mixed tensor to bound each of its rows in Euclidean norm. The off-path maps act on disjoint groups of leaves, so their tensor-product operator norms give the stated estimate without a dimension factor.

Suppress vertices on the marked path with no label attachment. The matrices between its remaining consecutive attachments are bounded path matrices \(Q_1,\ldots,Q_{r-1}\). For \(r\geq2\), put \[ L_i(a_1,a_r)=\sum_{a_2,\ldots,a_{r-1}} U_i(a_1,\ldots,a_r)\prod_{\ell=1}^{r-1} (Q_\ell)_{a_\ell a_{\ell+1}}. \tag{136}\] Then \(\Phi(M)_{i,b}=\sum_a M_{i,a}L_i(a,b)\). If \(r=1\), this formula means \(L_i(a,b)=\mathbf 1_{a=b}U_i(a)\).

The next step is to make every \(Q_\ell\) in [eq:hidden-marked-kernel] diagonal. Control one replacement at a time: fix an edge \(e\) and put \(Q_e^{\rm off}\) there. With the endpoints fixed, Cauchy–Schwarz bounds the squared kernel by \[\left(\sum_{a_2,\ldots,a_{r-1}}|U_i(a_1,\ldots,a_r)|^2\right) \left(B_1\cdots B_e^{\rm off}\cdots B_{r-1}\right)_{a_1a_r},\] where \(B_\ell=|Q_\ell|^{\circ2}\) entrywise. Every row sum and every column sum of \(B_\ell\) is at most \(\left\lVert Q_\ell\right\rVert_{\mathrm{op}}^2\). Writing \(\eta_e=\max_{a\ne b}|(Q_e)_{ab}|\), we have \(B_e^{\rm off}\leq\eta_e^2\mathbf 1\mathbf 1^t\) entrywise. The second factor is therefore at most \[\eta_e^2\prod_{\ell\ne e}\left\lVert Q_\ell\right\rVert_{\mathrm{op}}^2.\] Sum over the endpoints and use [eq:hidden-marked-row-tensor]. This gives \[\sup_i\left\lVert L_i^{(e,\rm off)}\right\rVert_{\mathrm{HS}} \leq C\eta_e\prod_{\ell\ne e}\left\lVert Q_\ell\right\rVert_{\mathrm{op}}.\] The estimate remains valid after any other edges have been made diagonal. Telescope over the finitely many edges and apply the entry rule in 22 to obtain \(\sup_i\left\lVert L_i-L_i^{\rm diag}\right\rVert_{\mathrm{op}}=o(1)\). Its row action costs at most \(o(1)\left\lVert M\right\rVert_{\mathrm{HS}}\).

After these replacements, diagonal marked-path edges force all attachment indices to agree. The remaining action takes the form \(M\circ E\), where \[E_{i,a}=U_i(a,\ldots,a) \prod_{\ell=1}^{r-1}(Q_\ell)_{aa}.\] This \(E\) contains the entire label derivative subnetwork, including its memory copies. It is the two-terminal network obtained by tying all temporal attachment indices to the common source \(a\), with the label output \(i\) as sink. Every derivative leaf is paired. [lem:path-directed-network,prop:path-opened-derivative] place it in the bounded path-matrix closure. The diagonal factors above may first be replaced by their uniformly accurate diagonal predictions. Consequently \(\max_i|E_{ii}|\leq C\) and \(\max_{i\ne a}|E_{i,a}|=o(1)\). The estimate \[\left\lVert M\circ E\right\rVert_{\mathrm{HS}}^2 \leq C^2\left\lVert \mathop{\mathrm{diag}}(M)\right\rVert^2 +\max_{i\ne a}|E_{i,a}|^2\left\lVert M\right\rVert_{\mathrm{HS}}^2\] proves [eq:hidden-marked-spine-bound]. Here the entry estimates are used on fixed typical finite trees, followed by their integrated comparison. Their scope does not include uniformity over arbitrary states. ◻

Proof of 43. Fix a summand of [eq:hidden-double-centered-error], with distinguished labels \(a=a_r-a_r^0\) and \(b=a_q-a_q^1\), where \(r<q\). By 44, approximate its path labels by fixed bounded smooth multitime circuits, with every observed state inside the label’s time span. These circuits may contain mixed queries such as \(J[f(X_s)\odot X_r]\); no representation as a sum of separated single-time products is required. The frozen-score construction of 25 puts its smooth caps in the class of 15; in particular, outside spins multiplying an unbounded query are handled by its finite sign-pattern decomposition. Before estimating derivatives, fix the circuit, its cap, and every observation time. Keep both distinguished vertices exactly conditionally centered for each approximant. Approximate the nonspecial endpoint factors by fixed smooth one-state functions before the derivative limits, and perform the field cutoff above. Its threshold and the suffix approximants remain fixed throughout the derivative comparisons.

For the exact right predictor, use a conditional path representation instead of requiring a regular one-state recipe. Represent its value by an independent ordinary stationary-kernel path from \(X_t\), reading the multitime evaluations in reverse order. Differentiate by propagating the full mixed-slot recursion down this added causal branch. Its tensor and path-network conclusions are those of [lem:path-conditional-derivative,prop:path-opened-derivative]. Keep the exact conditional subtraction before opening any of these derivatives. The conditional estimates [eq:hidden-conditional-op,eq:hidden-row-weighted-error] apply there as well. The error in replacing a tracial predictor by this exact prediction is handled before derivatives are opened, by 44.

Split the Wick string at \(a\) and \(b\) using [eq:hidden-wick-split]. The subtractions involving \(\tau(a)\) or \(\tau(b)\) vanish; errors from approximate trace centering are controlled in the unsplit Wick form by 44. Absorb the outer strings into the endpoint vectors. The left vector remains hidden by 22; the right vector has bounded mean-square unweighted Hessian by [eq:hidden-cutoff-hessian] and [eq:hidden-matrix-product-rule]. It remains to prove \[ \mathbb Eu(X_0)^tD_a QD_b p(X_t)\longrightarrow0, \tag{137}\] where \(Q\) is a positive-length Wick string with identity end labels. The left centered label \(a\) will give a martingale covariance. If its paired derivative reaches \(p\), the backward centering of \(b\) will cancel the term. If the derivative first hits a tail label, the nonempty Wick prefix will supply the cancellation instead.

To prove the remaining bilinear limit, use conditional martingale covariance up to the last evaluation in \(a\). For scalar future observables \(F,G\), it reads \[ \mathop{\mathrm{Cov}}_x(F,G) =2\mathbb E_x\int\sum_j w_j(X_s) d_jF_s(X_s)\,d_jG_s(X_s)\,\mathrm ds, \tag{138}\] where \(F_s,G_s\) are the conditional future expectations with their already observed arguments held fixed. The integration stops once \(F\) is fully observed. To check the factor, a flip of \(j\) has rate \(w_j/2\) and changes a scalar martingale by \(-2x_jd_jF_s\); its product bracket is \(2w_jd_jF_s d_jG_s\). Since \(a\) has exact forward conditional mean zero, [eq:hidden-covariance-target] is the corresponding covariance, with \(u(X_0)\) fixed under the initial conditioning.

[eq:hidden-martingale-covariance] introduces a derivative index in addition to the sites of the Wick string. Propagate the two derivatives by the mixed-slot recursion and \(dS_h=K_hd\), including every exact product and memory-copy correction. First conditionally average the factor differentiating \(a\) and remove its truncation with [eq:hidden-row-weighted-error]. On the other side, each terminal tensor term contains \(p\) or \(dp\) exactly once, because \(p\) has only the terminal state slot. The bounded mixed label tensors and the Hilbert-space amplification of 13 therefore propagate its unweighted mean-square Hilbert–Schmidt budget. Tensor and propagation errors are removed in these norms before paths are opened. For the retained histories use 33 to reduce their complete derivative subnetworks to bounded path matrices. The two conditional future expectations in the covariance can then be represented by independent causal futures from their joining state.

We now split the retained terms according to the derivative’s first destination: either it reaches \(p\) before differentiating any tail label, or it first differentiates such a label. The two cases use different cancellations.

The derivative reaches \(p\) without differentiating a tail label. Expand each terminal mixed tensor by the exact product rule, and classify a term by whether any difference falls on a tail label. For a label \(c\) and a remaining factor \(q\), we use \[d_j(cq)=c\,d_jq+q\,d_jc-2x_j(d_jc)(d_jq).\] The term that bypasses \(c\) retains its unshifted value; classify the mixed correction with the first-hit terms. If no tail label is differentiated, the only differentiated state slot is the terminal slot of \(p\). Any extra frozen-slot branch would differentiate a label, as would a same-slot correction containing both a label difference and \(dp\); both belong to the second case. Thus the history in the present case is unbranched and follows the common \(K\) propagation through the entire observation grid. With the derivative as column index, its term is a finite sum or integral of \[ \mathbb E\left\langle dp(X_t),D_bQ^tD_u A\right\rangle_{\mathrm{HS}}, \tag{139}\] where \(A\) is a bounded path matrix containing the derivative propagation, the differentiated \(a\)-side, and the bounded rate diagonal. Condition first on \(X_t\), so the terminal Hessian is fixed when we apply Cauchy–Schwarz. We must estimate the conditional matrix \[H(X_t)=\mathbb E[D_bQ^tD_u A\mid X_t].\] Take two conditionally independent copies of the entire path system given \(X_t\). Directly expanding its Hilbert–Schmidt square gives \[ \mathbb E\left\lVert H(X_t)\right\rVert_{\mathrm{HS}}^2 =\mathbb E\,u_1^t\left[ (Q_1D_{b_1b_2}Q_2^t)\circ(A_1A_2^t) \right]u_2. \tag{140}\] Reversibility and the tree comparison identify this system as a causal tree rooted at \(X_t\). Each conditional copy shares that one root; we do not introduce a new bridge conditioned on two endpoints.

Apply the Hadamard rule in 22. Operator-small errors and visible dyads have zero bilinear limit between these hidden vectors. The remaining diagonal is the product of the two diagonal predictions. The prediction of \(Q_1D_{b_1b_2}Q_2^t\) is zero. Indeed, the Wick strings admit no internal return, so the two equal-length strings must pair fully across each other. Their innermost contraction is \(\tau(b_1b_2)\). Under the stationary kernel, the two copies are conditionally independent and \(\mathbb E[b_{1,i}b_{2,i}\mid X_t]=0\) at every site. The zero limiting pair trace transfers to the actual, possibly tilted, reversed tree by 22; exact finite-dimensional stationary centering is not asserted for that actual reversed kernel. Thus the right-hand side of [eq:hidden-hadamard-pair] tends to zero, and the Hessian budget proves that [eq:hidden-hessian-ending] vanishes.

This cancellation requires common subdivisions and identical truncated propagation kernels for the actual and predictor terms. Keep every evaluation time in the subdivision, even when the factor evaluated there is absent from a particular term. In the present case no derivative falls on \(b\), so its actual and predictor pieces have the identical matrix \(A\). Recombine them into the exactly centered \(b\) before using [eq:hidden-hadamard-pair]. Independent predictor simulation branches may then be used in the two copies. Separate truncations followed by separate Hadamard estimates would not establish the vanishing of \(\tau(b_1b_2)\).

A tail label is differentiated. Stop at the first such vertex from the left, including a product-rule term that differentiates the label while also continuing to the right. If an earlier contraction has made this vertex scalar, its coefficient is a deterministic limiting trace from the word calculus, not a state-dependent empirical trace, and is not differentiated. The remaining prefix retains its Wick inclusion–exclusion.

The stopping rule must be fixed before opening derivatives of the later suffix. The surviving actual labels have ordered spans with disjoint interiors: an adjacent return removes the interior excursion and merges the two neighboring labels, whose convex span contains no surviving vertex span in its interior. Repeating returns preserves this property. The endpoint predictions in the double-centered string occur only in its terminal suffix, apart from the initial predictions already absorbed into \(u\). If the first differentiated actual label \(c_i\) has span ending at \(v\), every later actual label is observed after \(v\), and every later endpoint predictor is a function of \(X_t\). For each fixed spatial suffix, form its exact conditional value \[ q_i(z)=\mathbb E[\text{later spatial suffix ending in }p\mid X_v=z]. \tag{141}\] The Markov property makes the local observable exactly \(c_i(\text{path up to }v)q_i(X_v)\). Spatial matrix contractions in that suffix are included in \(q\); they need not be diagonal.

Before reopening the suffix, we need its \(L^2\) value and mean-square Hilbert–Schmidt derivative to be bounded. Prove both by working backwards from the bounds for \(p,dp\), using the bounded mixed label tensors and exact product and copy rules at each ordered span. For the empty derivative-index set, this means multiplication by the bounded label \(c\), with bound \(C_{\varnothing}=\left\lVert c\right\rVert_\infty\). The single terminal state slot contributes the value or the first difference of the current suffix once. Their tensor Hilbert–Schmidt norms are bounded by the row tensor bounds of the labels times the corresponding suffix norm. The Hilbert-space amplification of \(S,K\) and their common tails propagates these bounds to the preceding span, while a spatial matrix costs its bounded operator norm. For a terminal predictor, the ordinary one-state product rule uses its bounded conditional derivative operator. This finite backward induction uses no second difference of \(p\) in its original state slot and does not require \(q\) to be an exact gradient.

Now apply the local temporal recursion to \(c_iq_i\), retaining \(q\) and \(dq\) as unopened terminal objects. A term that differentiates \(c\) either terminates in its label tensor, leaving \(q\) as a value, or has exactly one marked leaf carrying \(dq\). Every other derivative leaf belongs to this single label \(c\), including all its mixed-slot and memory-copy leaves. Thus the latter term has precisely the marked history in 45, illustrated in Figure 3. The later differentiated labels remain inside \(dq\) until after this local estimate. Conditioning in this order keeps all side leaves at the same spatial output site, which a simultaneous expansion of the later labels would not ensure.

For a first hit on a terminal predictor \(g(X_t)\), apply the exact one-state product rule to \(D_gq\) before opening the auxiliary causal branch representing \(dg\). Its terminating term uses the whole two-terminal matrix \(dg\); its continuing correction is the one-attachment case of 45. Actual and predictor pieces of \(b\) can be treated separately in this first-hit case, since the cancellation below uses the Wick prefix, not a later centering. This completes the stopping rule for every possible first differentiated vertex.

Up to bounded row multipliers at that vertex, and with an unbranched temporal segment before its first label attachment absorbed into \(A\), the incoming matrix is \[ M=P^tD_uA, \tag{142}\] where \(P\) is a positive-length Wick prefix ending in identity. Its Hilbert–Schmidt norm is bounded by \(\left\lVert P\right\rVert_{\mathrm{op}}\left\lVert A\right\rVert_{\mathrm{op}}\left\lVert u\right\rVert\), whereas its diagonal satisfies the exact identity \[ \mathop{\mathrm{diag}}(M)=(P\circ A)^tu. \tag{143}\] The diagonal prediction of \(P\) is zero, since it is a nonempty Wick string. The Hadamard rule and hiddenness hence show that \(\mathbb E\left\lVert \mathop{\mathrm{diag}}(M)\right\rVert^2\to0\).

For a continuing term, use 45 on the whole derivative subnetwork of \(c\), including more than its first Jacobian. This gives \[ \mathbb E\left\lVert \Phi(M)\right\rVert_{\mathrm{HS}}^2 \leq C\mathbb E\left\lVert \mathop{\mathrm{diag}}(M)\right\rVert^2 +o(1)\mathbb E\left\lVert M\right\rVert_{\mathrm{HS}}^2 \longrightarrow0 \tag{144}\] in the typical weighted comparison. The exact \(-2x_j\) factors in the product and copy rules are bounded labels on the equality forks of this history. An unbranched segment after the last label attachment is a subsequent bounded propagation. Pairing with \(dq\) at the stopping boundary now gives zero by Cauchy–Schwarz and its Hilbert–Schmidt budget. Equivalently, later suffix operations may then be opened; their bounded Hilbert–Schmidt maps preserve this zero estimate.

If the derivative terminates at its first differentiated vertex, let \(E\) be the matrix of the complete terminating label subnetwork. All its mixed leaves and memory copies have been contracted. By 33 this is a bounded path matrix. Contract its incoming index against \(M\). In indices, the vector left at that vertex is \[\sum_j M_{ij}E_{ij} =\sum_h P_{hi}u_h(AE^t)_{hi} =\bigl[(P\circ(AE^t))^tu\bigr]_i.\] It tends to zero by the same Hadamard rule, again because the prediction of \(P\) is zero. Bounded row or column labels can be included in this identity without changing its conclusion. Pair with the bounded \(L^2\) value of \(q\) from [eq:hidden-stopped-suffix]. The terminating and continuing cases together cover every term and correction supplied by the exact product and copy rules.

The finitely many deterministic observation times may be separated within their prescribed spans before taking limits. Integration times coinciding with them have zero Lebesgue measure. The ordinary rough bounds control the limits and all discarded propagation tails. We have proved [eq:hidden-covariance-target]; summing [eq:hidden-double-centered-error] proves [eq:hidden-averaging]. ◻

Completion of the proof of 42. Normalize the limiting endpoint second moments to one, using the rough comparison if an endpoint norm vanishes. For each fixed \(t\), first truncate the hops and elementary likelihoods. Then apply 43, remove the likelihood annuli, and sum with the positive renewal estimate just proved. It gives, for every \(z'<\lambda_{\rm sp}\), \[\limsup_{n\to\infty}|\mathbb Eu^tK_t p| \leq C_{z',\epsilon} e^{-z't+O(\epsilon)t+o(t)}+O(e^{-Ht}).\] The derivative and field-cutoff errors were removed before the large-time limit. Their constants therefore impose no growth condition on [eq:hidden-energy-hypothesis]. Choose \(z'>z\), then the predictor cap parameter \(\epsilon\) sufficiently small and the tail rate \(H\) sufficiently large. The result is bounded by \(C_ze^{-zt}\), enlarging \(C_z\) on a bounded interval of times using 13. Restoring the endpoint norms proves [eq:hidden-propagation].

Finally, the exceptional starting-state sets in the row estimates have exponentially small integrated mass for the deterministic simplex masks. Fubini and the polynomial endpoint bounds extend the proof to the integrated comparisons in the statement. Every replicated diagram used above is a causal tree; no step calls for a uniform estimate on an adaptively selected bridge or mask. ◻

Proof. We first establish local boundedness in each norm, and then smoothing in the ordinary norm. Conditional prediction and the path expansion in 22 preserve the visible space. In the ordinary norm, 13 gives the bound and removes likelihood tails in stationary square mean. For the weighted bound, integrate the square evolution against \(\mu\) and retain the weights at both edge corners. The principal integration-by-parts term absorbs a fixed multiple of \[\mathbb E_\mu\sum_{i,j}v_iv_j\left\lvert d_jp_i\right\rvert^2.\] Here both corners of an edge are retained. The elementary bounds \[v_i(x^j)/v_i(x)\in[e^{-C|J_{ij}|},e^{C|J_{ij}|}], \qquad |d_jv_i(x)|\le C|J_{ij}|v_i(x), \qquad \sup_i\sum_jJ_{ij}^2\le C\] control the terms differentiating the weight. Cauchy–Schwarz and Young’s inequality absorb the first-order term using \(\sum_j a_{ij}^2/v_j\le C\). The multiplication term is bounded by \(\left\lVert aD_v^{-1}\right\rVert\le C\), the comparable edge weights, and \(v\le1\). Consequently \[\frac{\,\mathrm d}{\,\mathrm dt}\left\lVert K_tp\right\rVert_v^2 \le C\left\lVert K_tp\right\rVert_v^2 -c\mathbb E_\mu\sum_{i,j}v_iv_j|d_j(K_tp)_i|^2.\] This proves the extension to the weighted completion. Strong continuity holds on smooth recipes by their difference bounds and then on both completions by density and local boundedness.

Analyticity follows by treating the first-order and multiplication terms as a relatively bounded perturbation of componentwise heat flow. Their uniform relative bound is \[ \left\lVert (\mathcal G^{(n)}-L^{(n)})p\right\rVert_0 \le C\bigl(\left\lVert (-L^{(n)})^{1/2}p\right\rVert_0+\left\lVert p\right\rVert_0\bigr). \tag{162}\] For example, square the first-order term row by row and use \(\sum_j a_{ij}^2/v_j\le C\); its sum is bounded by \(C\mathfrak d(p)\). The remaining term has bounded operator norm. If \(z\) is on either ray bounding a sector strictly larger than the negative real half-line, the spectral theorem gives \[\left\lVert (-L^{(n)})^{1/2}(z-L^{(n)})^{-1}\right\rVert \le C|z|^{-1/2},\qquad \left\lVert (z-L^{(n)})^{-1}\right\rVert\le C|z|^{-1}.\] At sufficiently large \(|z|\), independently of \(n\), the resolvent for \(\mathcal G^{(n)}\) is therefore given by a convergent Neumann series. Contour inversion gives uniform analytic bounds. Moreover, [eq:exact-relative] and the scalar inequality \(\sqrt u\le\varepsilon u+C_\varepsilon\) show that the graph norms of \(L^{(n)}\) and \(\mathcal G^{(n)}\) are uniformly equivalent after adding a fixed multiple of the identity. This proves [eq:exact-graph]. Passing first on the dense recipe class and then by the uniform bounds gives the assertions in the limiting spaces. ◻

Proof. On the generating span, the path limits make every inner product and fixed-time matrix element deterministic. Self-adjointness comes from finite-dimensional exactness, through the identity \[\left\langle df,K_tdg\right\rangle_v =\left\langle f,(-L)S_tg\right\rangle_{L^2(\mu)}.\] The right side is symmetric in \(f,g\), is nonnegative when \(f=g\), and has operator norm at most \(e^{-\gamma t}\) in the gradient norm. These properties pass to the weighted closure. The spectral theorem then gives the generator and its lower bound. The embedding properties follow from [eq:exact-metrics] and the definition of the closures; the final spin-edge bound is 41. ◻

Proof. It suffices to show that the component orthogonal to \(\mathcal X_v\) vanishes. Subtract the weighted orthogonal projection and call the remainder \(q\). It remains weighted-visible and a weighted limit of exact gradients, while its pairing with every generating orbit is zero. Take centered potentials \(G_n\) representing this exact limit. The gap gives \(\left\lVert G_n\right\rVert_{L^2(\mu)}^2\le\gamma^{-1}\left\lVert dG_n\right\rVert_v^2\). For every fixed test potential \(f=f_{B,\delta}\), \[ \left\langle G_n,f\right\rangle_{L^2(\mu)} =\int_0^\infty\left\langle dG_n,K_tdf\right\rangle_v\,\mathrm dt. \tag{166}\] The exponential gap bound gives a uniform integrable tail; the limit is zero by orthogonality.

The same potential pairing has a local form that detects \(q\) as the short time interval shrinks. Reversibility and conditional martingale covariance give the exact identity \[\begin{align*} \left\langle G_n,f_{B,\delta}\right\rangle_{L^2(\mu)} =\frac2\delta\int_0^\delta \mathbb E_\mu\!\left[ B(X_0)^{\mathsf T}(dS_{\delta-s}x)(X_s) D_{w(X_s)}dS_{\delta-s}G_n(X_s)\right]\,\mathrm ds. \tag{167}\end{align*}\] Here the derivative matrix has output as row. The factor \(2\) is the quadratic-variation factor for flip rate \(w_j/2\) and spin increment \(2\). The derivative matrix \(dS_rx\) differs from the identity by \(O(r)\) in operator norm: integrate \(dS_rx-I=\int_0^r dS_uLx\,\mathrm du\) and use \(\left\lVert dLx\right\rVert\le C\) together with the pointwise rough gradient bound. Also \(\mathbb E_\mu\left\lVert B(X_s)-B(X_0)\right\rVert^2=O_B(s)\).

The passage \(\delta\downarrow0\) is justified in the weighted norm, without comparing the two completions in reverse. For every own-spin independent vector \(p\), \[ \mathbb E_\mu\left\lVert D_wp\right\rVert^2 =\mathbb E_\mu\sum_i\mathbb E_\mu[w_i^2\mid x_{-i}]p_i^2 =\mathbb E_\mu\sum_i v_ip_i^2, \qquad \mathbb E_\mu[w_i\mid x_{-i}]=v_i. \tag{168}\] Thus \(D_w\) is an isometry from the closure of exact gradients with their weighted norm into ordinary square mean. In [eq:exact-covariance-potential] one may first replace \(G_n\) by \(S_\varepsilon G_n\), shrink \(\delta\), and then shrink \(\varepsilon\), using strong continuity on exact gradients. Equivalently one passes in the displayed covariance using [eq:exact-w-isometry]. It follows from the zero pairing that \(\left\langle B,D_wq\right\rangle_0=0\) for every \(B\).

To conclude from these zero pairings, we must place \(D_wq\) in the ordinary visible space. Cut off the coordinates where \(v_i<\varepsilon\). On the retained set a weighted-visible approximation is an ordinary visible approximation, and multiplication by \(w\) is an allowed bounded label. On the removed set [eq:exact-w-isometry] and own-spin independence identify its squared norm with the corresponding weighted mass of \(q\). That mass tends to zero for a fixed element of \(\mathcal H_v\), by density of bounded recipes. Hence \(D_wq\in\mathcal H\). Its orthogonality to all recipes makes it zero, and the isometry makes \(q=0\). ◻

Proof. We test the claimed transport first on finite recipe projections. Truncate the path expansion of \(K_{t-s}\) at fixed order and likelihood level. Between finitely many testing recipes, its matrix elements concentrate at their deterministic stationary values by 22. To remove the endpoint complement, apply the adjoint bounded path word to a testing recipe and project at time \(t\). Conditional prediction on replicated trees makes the remaining squared contraction arbitrarily small. This squared estimate is obtained before multiplying by any scalar test; Cauchy–Schwarz therefore permits arbitrary bounded-\(L^2\) scalar coefficients, including past-dependent tests. Polynomial pointwise bounds and the exponential exceptional-set estimates justify discards. The operator-small path tails then remove the truncations. Increasing the finite basis gives [eq:exact-transport], since conditional expectation is a contraction. Apply [eq:exact-graph] to the first tensor factor for the last assertion. ◻

Proof. We obtain the two budgets from different estimates on the same path. The rough square estimate supplies the time-integrated first line and \(\left\lVert p_s(x)\right\rVert^2\le C\mathbb E[\left\lVert p_b(X_b)\right\rVert^2\mid X_s=x]\). Tilt the entire path by \(g(X_b)/\mathbb Eg(X_b)\), where \(g=1+\left\lVert p_b\right\rVert^2\). Its entropy relative to the stationary path is \(o(n)\): the original relative log density is \(o(n)\) and the extra log density is \(O(\log n)\). The tilt depends on the forward terminal state, so it preserves the original reverse transitions. Those transitions have rates \(w_jr_{s,j}/2\), whereas stationary reverse rates are \(w_j/2\). The jump-process entropy chain rule and \[(\sqrt r-1)^2\le r\log r-r+1\qquad(r\ge0)\] give an integrated Hellinger budget \(o(n)\) under the tilted path. The preceding conditional bound on \(p_s\) converts it into the time integral of [eq:exact-reverse-budget]. Markov’s inequality selects a positive-measure set of times where the Hessian budget is bounded and the reverse budget is \(o(n)\). Countably many fixed typicality tests can be imposed successively there by their integrated failure bounds and extraction. Auxiliary-time exceptional sets cost their square-mean bounds, rather than a polynomial supremum times their measure. ◻

Proof. The conclusion concerns the square mean of each random coefficient, so we multiply an adjoint test by a second visible contraction. Write \(p=p_s\), and take \(e=p^{\mathsf T}B\) with a fixed smooth scaled recipe \(B\). The flip product rule and [eq:exact-hessian-budget] give \(\mathbb E\sum_jw_j|d_je|^2\le C_B\). In [eq:exact-nonstationary-ibp] use \[F_j=e\,U_j,\qquad U_j=\frac1{\sqrt n}\sum_i p_i(v_iA_{ij})(x^j).\] The elementary column estimate needed here is \[ \mathbb E\sum_jw_j(1+\sqrt{r_j})^2|U_j|^2 \le\frac{C_A}{n} \left(\mathbb E\left\lVert p\right\rVert^2+\mathbb E\sum_jw_j\left\lVert d_jp\right\rVert^2\right). \tag{176}\] For the term without \(r_j\), this is the bounded operator estimate for the column-flipped test matrix. For the term with \(r_j\), change \(x\) to \(x^j\) using detailed balance. Replace \(p_i(x^j)\) by \(p_i(x)-2x_jd_jp_i(x)\); the first part has the same operator bound, and bounded individual column norms bound the second part by the displayed derivative budget. Local comparability of \(v\) at the two corners supplies the same constants for the weighted labels.

Factor \(1-r_j=(1-\sqrt{r_j})(1+\sqrt{r_j})\) in the remainder. Since \(|e|\le C_B\left\lVert p\right\rVert\) and \(|x_j-m_j|\le2\), [eq:exact-reverse-budget,eq:exact-column-budget] make that remainder \(o(1)\) by Cauchy–Schwarz: its two squared budgets are \(o(n)\) and \(O(n^{-1})\). Derivatives falling on \(e\) cost \(O(n^{-1/2})\), by its Dirichlet bound and the same column estimate. Derivatives falling on \(p\) cancel exactly, since \(d_jp_i=d_ip_j\) and \(A\) is antisymmetric. The remaining identity is \[\mathbb E\bigl[(p^{\mathsf T}B)(p^{\mathsf T}B_A)\bigr]=o(1).\] Approximate \(B_A\) in the visible norm by allowed recipes \(B\) and use Bessel’s inequality and the squared contraction comparison before removing that approximation. The result is \(\left\lVert \left\langle q_s,B_A\right\rangle_{\mathcal H}\right\rVert_{\mathcal S_s}^2=0\). The diagonal argument is identical, with the cancellation supplied by \(d_ip_i=0\). It uses the stationary adjoint of the stated diagonal functional and the same \(n^{-1/2}\) normalization. ◻

Proof. The approximation has two stages: smoothing in form norm, followed by finite recipe approximation. By [prop:exact-transport,lem:exact-smoothing], \(q\) lies in the graph domain of componentwise \(L\). First replace it by \(S_\varepsilon q\). The spectral theorem makes this approximation strong in the norm \(\left\lVert q\right\rVert_0^2+\mathfrak d(q)\). For fixed \(\varepsilon>0\), approximate the input \(q\) by finite smooth recipes \(q_k\). Since \[\left\lVert (-L)^{1/2}S_\varepsilon(q-q_k)\right\rVert_0 \le C_\varepsilon\left\lVert q-q_k\right\rVert_0,\] the second approximation is also strong in form norm. This norm controls both weighted derivative quantities displayed in the statement. These operations apply to the deterministic vector \(q\); the scalar Hilbert direction used to obtain it has not been differentiated.

For fixed \(\varepsilon,k\), the derivative of the one-state recipe \(q_k\) is \(A/\sqrt n\) in the smooth word, Hadamard, and column-flip closure. Bounded finite-rank errors disappear in normalized Hilbert–Schmidt norm. Propagate this terminal derivative by the direct single-future identity \(dS_\varepsilon=K_\varepsilon d\), separately for each component. The common columnwise path tail bound approximates this operation in Hilbert–Schmidt square mean, because the fixed input derivative has bounded Hilbert–Schmidt norm.

The remaining conditional expected path matrices lie in the same strong tracial closure. Choose their bounded labels as fixed smooth multitime circuits in the class of 15, with likelihood caps and any finite spin-sign decomposition already fixed. Then simulate the finitely many transitions with the nested accuracies of 22. Conditional pair-trace tests give the squared error. Averaging \(r\) independent simulation seeds has conditional Monte Carlo error at most \(C/r\) in normalized Hilbert–Schmidt square mean. Each fixed simulated calculation is a one-state word matrix. Only strong Hilbert–Schmidt approximation is required in this step: we neither differentiate a spatial-square-mean approximation of a label nor promote its error to an operator error. When an actual derivative of a fixed multitime circuit is needed elsewhere, its operator truncation and subsequent matrix reduction are instead the separate conclusions of [lem:path-conditional-derivative,prop:path-opened-derivative]. Thus the derivative approximation here is strong, rather than only a list of converging individual traces. Diagonal extraction is performed before taking closure and is a contraction for the relevant diagonal norm; its convergence also follows directly from the form approximation.

The derivative matrices were strongly approximated by \(A/\sqrt n\) with \(A\) in the word closure. Multiplication on the two sides by \(D_{\sqrt v}\) is a contraction in Hilbert–Schmidt norm. The weighted curl is therefore strongly approximated by \(D_{\sqrt v}(A-A^{\mathsf T})D_{\sqrt v}/\sqrt n\). Since the word closure is linear and transpose-stable, these are precisely weighted antisymmetric test matrices of the type used in [eq:exact-curl-adjoint]. [eq:exact-curl-adjoint] and adjoint annihilation force its inner product with every element of this dense antisymmetric test class to be zero, hence its norm is zero. The diagonal tests do the same for the own-spin derivative. The strong form approximants then give [eq:exact-vanishing-curl]. Applying the argument to all scalar Hilbert directions, or to a countable orthonormal basis of their separable span, proves the corresponding tensor assertion in square mean. ◻

Proof. The first assertion is the conditional-variance identity \[\left\lVert Q-\overline Q\right\rVert_v^2 =\mathbb E_\mu\sum_i v_i^2|d_iQ_i|^2\le O(Q).\] For the curl estimate, work on one conditional two-spin law before summing over pairs. Condition on the spins outside \(i,j\) and write the pair law as \(p_{ab}\), \(a,b\in\{-1,1\}\), with coupling \(K=J_{ij}\). Delete \(K\) and normalize to obtain a product law \(q=q_i\otimes q_j\). Put \(m=\mathbb E_q a\), \(m'=\mathbb E_q b\), \(A=1-m^2\), and \(B=1-(m')^2\). Uniformly in the fields, \[\frac{p_{ab}}{q_{ab}},\quad\frac{v_i(b)}A, \quad\frac{v_j(a)}B\ \in[e^{-2|K|},e^{2|K|}].\] Write \(u_b=d_iQ_i(b)\) and \(t_a=d_jQ_j(a)\). Affinity in one spin gives \[(P_iQ_i)(b)=\mathbb E_{q_i}Q_i(a,b)+(m_i(b)-m)u_b.\] Consequently the averaged curl has the exact representation \[ d_j\overline Q_i-d_i\overline Q_j=C_0+E_i-E_j, \qquad C_0=\mathbb E_q[d_jQ_i-d_iQ_j], \tag{179}\] where \[E_i=\tfrac12\{(m_i(+)-m)u_+-(m_i(-)-m)u_-\}\] and \(E_j\) has the symmetric definition. In particular all three quantities in [eq:exact-average-difference] are independent of the two spins. With \(W=\mathbb E_p[v_i(b)v_j(a)]\), comparison and Jensen give \[WC_0^2\le e^{12|K|}\mathbb E_p[v_iv_j|d_jQ_i-d_iQ_j|^2].\] The log derivative of \(\operatorname{sech}^2\) has absolute value at most \(2\), so \(|m_i(\pm)-m|\le|K|e^{2|K|}A\). Thus \[E_i^2\le\tfrac12K^2e^{4|K|}A^2(u_+^2+u_-^2), \qquad B(u_+^2+u_-^2)\le4\mathbb E_{q_j}u_b^2.\] The last inequality retains the two opposite corners even when one has very small probability. Combining it with \(W\le e^{6|K|}AB\), \(A^3\le A\), and the comparison of \(p\,v_i\) with \(qA\), gives \[WE_i^2\le2K^2e^{14|K|}\mathbb E_p[v_i|d_iQ_i|^2].\] Use the analogous bound for \(E_j\), square [eq:exact-average-difference] with factor \(3\), average over the other spins, and sum ordered pairs. The two error sums together cost \(12e^{14\delta_J}R_JO(Q)\), proving the claim. No lower bound on an individual conditional variance, and no unweighted derivative bound, is used in this removal step. ◻

Proof. For a full-support posterior \(P\) define \[M(P)=\min_g\mathbb E_P\sum_i v_{P,i}(Q_i-d_ig)^2, \qquad r=Q-dg_P.\] Center the minimizer under \(P\) to fix its additive constant. On the finite cube the Dirichlet form is positive definite modulo constants, so the minimizer is unique and the following differentiations are legitimate. We estimate its field derivative, then its observation drift, and finally the curl energy at the product endpoint.

The normal equation and its derivative. Since \(r_i\) is own-spin independent, stationarity on a single edge gives \[ 0=\mathbb E_P\sum_i v_{P,i}r_i d_i\phi =\mathbb E_P\phi\sum_i(x_i-m_{P,i})r_i \quad\text{for every }\phi. \tag{181}\] Full support therefore implies \(\sum_i(x_i-m_{P,i})r_i=0\) pointwise. Freeze the minimizer and vary the external field in coordinate \(j\). The derivative of the normal functional against \(\phi\) is exactly \[ F_j(\phi)=-\mathbb E_P\phi v_{P,j}r_j. \tag{182}\] Indeed the derivative of the measure multiplies the pointwise zero in [eq:exact-normal]; the remaining derivative is \(\partial_{h_j}m_{P,i}=\mathbf 1_{i=j}v_{P,j}\). Testing [eq:exact-normal] with \(\phi=x_j\) also gives \(\mathbb E_Pv_{P,j}r_j=0\). If the heat-bath gap of \(P\) is at least \(\gamma\), then the dual norm relative to \(\mathcal E_P\) satisfies \[ \sum_j\left\lVert F_j\right\rVert_{\mathcal E_P^*}^2 \le\gamma^{-1}\sum_j\mathbb E_P(v_{P,j}r_j)^2 \le\gamma^{-1}M(P). \tag{183}\] This uses Cauchy–Schwarz after centering \(\phi\), followed by the posterior Poincaré inequality.

The drift of the minimum. Let \(A\succeq0\) be the precision per unit time on one segment of the observation. In innovation coordinates the posterior masses satisfy \[\,\mathrm dP(x)=P(x)(x-\mathbb E_Px)^{\mathsf T}A^{1/2}\,\mathrm dB.\] This is the finite-state filtering identity; it follows by the quotient rule for likelihood weights, as in (OpenAI 2026, Equation (2.1)), with \(A^{1/2}\) inserted for the stated precision. For a fixed \(i\) and rest configuration \(z\), write \(a=P(z,+)\), \(b=P(z,-)\), and \[c_i(z)=\frac{4ab}{a+b},\qquad v_i(z)=\frac{4ab}{(a+b)^2}.\] The edge contribution to the quadratic cost for frozen \(r\) is \(c_i(z)r_i(z)^2\). The Hessian of \(c(a,b)=4ab/(a+b)\) is \[D^2c(a,b)=-\frac8{(a+b)^3} \begin{pmatrix}b^2&-ab\\-ab&a^2\end{pmatrix}.\] The difference of the innovation vectors of the two corners is \(2A^{1/2}e_i\). Itô’s formula therefore gives precisely \[ \operatorname{drift}(c_i)=-A_{ii}c_iv_i. \tag{184}\] Summing frozen residual costs bounds this drift below by \(-\left\lVert A\right\rVert M(P)\).

The drift of the optimized cost also includes motion of the minimizer. This contribution is nonpositive and must be bounded from below. In coordinates on potentials modulo constants, write the cost near its minimum as \(M+\mathcal E_P(u)-2F(u)\). For an infinitesimal innovation, minimization subtracts the squared dual norm of the first variation \(F\). Thus this contribution is \[-\sum_\alpha\left\lVert \sum_j(A^{1/2})_{j\alpha}F_j \right\rVert_{\mathcal E_P^*}^{\!2} \ge-\left\lVert A\right\rVert\,\gamma^{-1}M(P),\] by [eq:exact-normal-dual]. This is also obtained directly by Itô differentiating the inverse of the positive Dirichlet matrix. Together with [eq:exact-edge-drift], it proves \[ \operatorname{drift}M(P_t) \ge-\left\lVert A_t\right\rVert(1+\gamma^{-1})M(P_t) \tag{185}\] until the posterior-gap stopping time.

The product endpoint. At a product law write \(\sigma_i^2=v_i\) and \(\chi_i=(x_i-m_i)/\sigma_i\). Expand the own-independent \(Q_i\) in the orthonormal products \(\chi_T\). For each nonempty \(S\subset[n]\), put \(b_{i,S}=\sigma_i\widehat Q_i(S\setminus\{i\})\), \(i\in S\). The weighted cost on this support is the squared distance of \((b_{i,S})_{i\in S}\) from the constant vector, since a potential coefficient \(\widehat g(S)\) contributes that same value to each \(b_{i,S}\). For \(|S|=1\) this distance is zero. For \(|S|\ge2\), \[\sum_{i\in S}|b_{i,S}-\overline b_S|^2 =\frac1{2|S|}\sum_{i,j\in S}|b_{i,S}-b_{j,S}|^2 \le\frac14\sum_{i,j\in S}|b_{i,S}-b_{j,S}|^2.\] The ordered-pair sum is exactly the weighted curl energy on \(S\). Consequently \[ M(P_{\rm prod})\le\frac14\mathbb E_{P_{\rm prod}} \sum_{i,j}v_iv_j|d_jQ_i-d_iQ_j|^2. \tag{186}\]

Transporting the curl energy. Fix \(i\ne j\) and the remaining spins \(z\). The curl \(c_{ij}(z)=d_jQ_i-d_iQ_j\) is independent of both \(x_i,x_j\). For the four joint masses \(p_{ab}=P(z,a,b)\) define \[H(p)=\left(\sum_{a,b}p_{ab}^{-1}\right)^{-1}.\] This function is homogeneous and concave on the positive cone, since \[ H(p)=\inf\left\{\sum_{a,b}p_{ab}u_{ab}^2: \sum_{a,b}u_{ab}=1\right\}. \tag{187}\] Extend it continuously to the boundary. If the pair has no coupling, \(\sum_{a,b}p_{ab}v_i(b)v_j(a)=16H(p)\). Deleting a coupling \(\theta_{ij}\) changes each conditional pair mass and each conditional variance by factors \(e^{O(|\theta_{ij}|)}\), uniformly in the external fields and \(z\). Hence, throughout the observation path where \(|\theta_{ij}|\le|J_{ij}|\), \[ e^{-C|J_{ij}|}16H(p) \le\sum_{a,b}p_{ab}v_i(b)v_j(a) \le e^{C|J_{ij}|}16H(p). \tag{188}\] This concave function of the four joint masses supplies the required transport estimate. Each joint posterior mass is a martingale, so concavity in [eq:exact-harmonic-concavity] implies \(\mathbb EH(p_t)\le H(p_0)\). Multiply by \(c_{ij}(z)^2\) and sum over \(z,i,j\). Using [eq:exact-pair-weight] at the initial point and the product endpoint shows that expected terminal curl energy is at most \(e^{C\max|J_{ij}|}\) times its initial value.

Finally stop before the posterior gap fails, using the test-independent stopping rule in 7. The horizon and precision norm are fixed. Equation (185) makes \(e^{Ct}M(P_t)\) a submartingale up to this stopping time. Its expected terminal value is bounded by [eq:exact-product-hodge] on good paths. On stopped paths use \(M(P)\le\mathbb E_P\sum_i Q_i^2\le\sup_x\sum_iQ_i(x)^2\), a polynomial bound. Their probability is exponentially small. More explicitly, let \(T\) be the fixed observation horizon, let \(\tau\) be the stopping time, and write \[\mathcal C(P)=\mathbb E_P\sum_{i,j}v_{P,i}v_{P,j} |d_jQ_i-d_iQ_j|^2.\] Using the full observation posterior \(P_T\) also on the exceptional paths, optional stopping and nonnegativity give \[M(\mu) \le e^{CT}\left\{\tfrac14\mathbb E\mathcal C(P_T) +n^{C_Q}\mathbb P(\tau<T)\right\} \le C'\mathcal C(\mu)+o(1).\] The first inequality uses the product endpoint on paths that reach \(T\) without stopping; the second is the concave curl-energy comparison. This proves [eq:exact-hodge-bound]. All operations can first be localized where posterior masses are bounded away from zero; bounded finite-cube costs and the displayed inequalities then remove the localization. ◻

Proof. Apply the preceding results to a deterministic vector obtained by contracting \(q_s\) with an arbitrary fixed scalar Hilbert direction. First, 53 gives strong approximants with vanishing curl and own-spin derivative. 54 makes the approximants own-spin independent, preserving their weighted limit and vanishing curl. Apply 55 separately to each fixed polynomial approximant, letting dimension tend to infinity before improving the approximation. Their weighted distance to exact gradients tends to zero. 49 puts the limiting deterministic vector in \(\mathcal X_v\). Orthogonality to \(\mathcal X_v^\perp\) after contraction against every scalar direction gives [eq:exact-closedness]. ◻

Proof. The weighted estimate alone cannot be inverted through \(\iota\). Instead, separate the gradient extension into its diagonal differential part \[(\mathcal Dp)_i=(L-I)p_i-2x_i\sum_j a_{ij}d_jp_i\] and the multiplication operator \(ap\). Uniformly in \(x,i\), \[ \sum_j\frac{a_{ij}^2}{v_j} \le\sum_jJ_{ij}^2(1+o(1))\le\beta^2+o(1). \tag{191}\] The first inequality follows by the one-variable mean-value formula for \(\tanh\), with error uniform because \(\max|J_{ij}|\to0\); the second is uniform concentration of row square sums. Integration against stationarity and completion of the square give \[\left\langle p,\mathcal Dp\right\rangle_0 \le-\left\lVert p\right\rVert_0^2-\mathfrak d(p) +2\sqrt{(\beta^2+o(1))\left\lVert p\right\rVert_0^2\mathfrak d(p)} \le-(1-\beta^2-o(1))\left\lVert p\right\rVert_0^2.\] Thus its limiting semigroup \(T_t^{\rm diag}\) has norm at most \(e^{-(1-\beta^2)t}\). The other part maps weighted to unweighted norm: \[\left\lVert ap\right\rVert_0\le\left\lVert aD_v^{-1}\right\rVert\left\lVert D_vp\right\rVert_0\le C\left\lVert p\right\rVert_v.\] Duhamel’s formula consequently gives \[\left\lVert K_tq\right\rVert_0 \le e^{-(1-\beta^2)t}\left\lVert q\right\rVert_0 +C\int_0^t e^{-(1-\beta^2)(t-s)}e^{-as}\left\lVert q\right\rVert_v\,\mathrm ds.\] Since \(\left\lVert q\right\rVert_v\le\left\lVert q\right\rVert_0\), the convolution gives [eq:exact-unweighted-decay], including equality of the two exponents by taking strict slack. Tensoring and conditional expectation do not increase any norm used here. Finally \(\lambda\le(1-\beta)^2<1-\beta^2\) for \(0<\beta<1\). ◻

Proof. There are two steps. First, the hidden estimate bounds the essential norm in the ordinary exact space. Second, the dense intertwining map identifies the resulting finite-dimensional spectral subspaces in the weighted completion.

Essential norm. For every fixed deterministic \(u\in\mathcal H\) and bounded recipe \(B\), the finite-dimensional contractions \(u^{\mathsf T}B\) concentrate at \(\left\langle u,B\right\rangle_{\mathcal H}\). This first holds for a fixed finite recipe; for a dense approximation \(u'\) use \[\left\lVert (u-u')^{\mathsf T}B\right\rVert_{L^2(\mu)} \le\sup_x\left\lVert B(x)\right\rVert\left\lVert u-u'\right\rVert_0.\] It need not hold for an arbitrary random scalar times a visible vector.

We turn this concentration into the squared orthogonality required by the hidden theorem. Let \(u_k\) be a weakly null unit sequence in \(\mathcal X\), and approximate each \(u_k\) within \(1/k\) by a finite exact test combination. At dimension \(n_k\) impose concentration for its contractions with the first \(k\) recipes, its norm, and the operator pairing currently being tested. Take \(n_k\) sufficiently large that the finite pointwise bounds of this approximant are bounded by a fixed power of \(n_k\). A sufficiently slow diagonal sequence \(k=k(n)\to\infty\) then has \[\mathbb E_\mu|u_{k(n)}^{\mathsf T}B|^2\longrightarrow0 \quad\text{for every fixed recipe }B.\] Thus the representatives are hidden in the squared-contraction sense of 42; weak convergence by itself was not used as a substitute for this property.

To see that this controls the essential norm, take finite-rank orthogonal projections \(P_m^0\uparrow I\) on \(\mathcal X\). The Hilbert-space identity \[\left\lVert K_T\right\rVert_{\rm ess}=\lim_m\left\lVert (I-P_m^0)K_T\right\rVert\] follows because \(P_m^0K_T\) is finite rank and every compact operator is uniformly small after applying \(I-P_m^0\). If the claimed essential-rate bound failed, choose unit \(u_m\in\operatorname{ran}(I-P_m^0)\) and unit inputs \(p_m\) witnessing the excess. The outputs are weakly null, so the previous diagonal construction applies to these arbitrary witnesses; approximate the inputs too by finite exact test combinations while preserving the pairing. Split \(K_T=K_{T-1}K_1\) for \(T>1\). 46 supplies a uniform stationary Hessian budget for the exact inputs \(K_1p_m\). Since their derivatives are independent of the differentiating spin, their \(w_j\)-weighted budget equals their \(v_j\)-weighted budget. Apply 42 to the hidden outputs and these smoothed inputs. Here the hidden theorem imports the multitime conditional derivative argument: its fixed smooth circuits use the mixed-slot hierarchy, common tensor truncations, and opened network reduction of [lem:path-mixed-tensors,lem:path-conditional-derivative,prop:path-opened-derivative]. Its path entry comparisons retain the typical integrated-parameter convention of 22; no supremum over auxiliary Duhamel times is added here. The bounded \(K_1\) factor and loss of one time unit disappear after dividing the logarithm by \(T\). This proves [eq:exact-essential-rate] for all possible norm witnesses.

The dense embedding and its spectral projections. Fix \(0<\rho<\lambda_{\rm sp}\) and choose \(T\) so large that the essential norm of \(T_X:=K_T|_{\mathcal X}\) is smaller than \(e^{-\rho T}\). Choose \(r>e^{-\rho T}\) away from the spectrum, with \(r<1\). The usual resolvent argument for a finite-rank perturbation of an operator of norm smaller than \(r\) gives a Riesz decomposition \[\mathcal X=E_r\oplus F_r,\] where \(E_r\) is finite dimensional, contains the spectrum outside \(|z|=r\), and the spectral radius of \(T_X|_{F_r}\) is at most \(r\). For clarity, a finite-rank approximation with smaller remainder norm makes the resolvent outside that norm the inverse of a finite-dimensional analytic matrix; its zeros are isolated and have finite multiplicity. On a compact annulus \(|z|\ge r\) there are finitely many such zeros. This supplies precisely the decomposition just used.

Let \(T_v=K_T|_{\mathcal X_v}\). It is positive self-adjoint and \(\iota T_X=T_v\iota\). Injectivity implies that every generalized eigenvector in \(E_r\) embeds as a generalized eigenvector of \(T_v\). Self-adjointness removes all nontrivial Jordan chains and forces these eigenvalues to be real and positive. Write \(W_r=\iota E_r\). For \(f\in F_r\) and \(r'>r\), the spectral-radius formula yields \[\left\lVert T_v^k\iota f\right\rVert_v \le\left\lVert \iota\right\rVert\,C_{r'}(r')^k\left\lVert f\right\rVert_0.\] If the \(T_v\)-spectral measure of \(\iota f\) charged \((r',\infty)\), its \(k\)th moment in this inequality would grow faster than \((r')^k\). The spectral theorem therefore forces that measure into \([0,r']\); let \(r'\downarrow r\). Thus \(\iota F_r\) is orthogonal to \(W_r\). On the dense image \(\iota\mathcal X\) the Riesz decomposition is already an orthogonal decomposition into \(W_r\) and the low spectral subspace. It follows that \[\mathcal X_v=W_r\mathbin{\oplus^{\perp}} \overline{\iota F_r}, \qquad \Pi_{W_r}\iota=\iota P_{E_r}.\] Density now excludes additional weighted spectral mass outside \([0,r]\). The displayed identity proves agreement of the projections through \(\iota\); no boundedness of an ordinary Riesz projection in the weaker norm was assumed.

Given any rate strictly below \(\lambda_{\rm sp}\), choose \(\rho\) above that rate and then \(r\) between the essential disk and its corresponding semigroup eigenvalue. The preceding argument accounts for all weighted spectrum below that rate by finitely many eigenvectors in \(\mathcal X\). If the bottom \(\lambda\) is strictly below \(\lambda_{\rm sp}\), it is therefore attained and isolated. Its complement has weighted generator spectrum bounded below by \(\lambda+\eta\) for some \(\eta>0\). The inverse \(E_v\to E\) is bounded because these spaces are finite dimensional, so its composition with \(\Pi_v\iota\) gives the asserted projection for vectors of \(\mathcal H\) whose image is weighted-exact. Applying 57 to their complementary parts proves the last assertion. ◻

Proof. Extend \(g\) multilinearly to the cube: let \(\overline g(z)\), \(z\in[-1,1]^n\), be its expectation under the product law with coordinate means \(z_i\). Then \(\partial_i\overline g(z)=\mathbb E_z d_ig\), since \(d_ig\) does not depend on its own spin. Jensen’s inequality gives \[\left\lVert \nabla\overline g(z)\right\rVert^2 \le\mathbb E_z\sum_i(d_ig)^2 \le\sup_x\left\lVert dg(x)\right\rVert^2.\] Integrate along the Euclidean segment joining two vertices, whose length is at most \(2\sqrt n\). ◻

Proof. We will find a late time with bounded gradient energy, propagate that bound back to time zero, and then control oscillation. Relabelling \(\epsilon\), it suffices to work at \(T=(1+2\epsilon)t_n\). Fix \(|f|\le1\) and a starting point. The backward martingale \(S_{T-s}f(X_s)\) has jump bracket density \(2\sum_iw_i p_{s,i}^2\). Hence \[ \int_0^T\mathbb E\sum_iw_i p_{s,i}^2\,\mathrm ds\le\tfrac12. \tag{206}\] On the interval \(I=[(1+\epsilon)t_n,T]\), whose length is \(\epsilon t_n\), [eq:cutoff-field-tail] applies. On its good field event, \[w_i\ge\frac{2}{1+e^{2|(JX_s)_i|}}\ge n^{-2\delta}.\] On the complementary event \(\left\lVert p_s\right\rVert^2\le n\), since \(|f|\le1\). There is thus a time \(s\in I\) at which \[ N_s^2\le\frac{n^{2\delta}}{2\epsilon t_n}+n^{1-R}. \tag{207}\] Apply [eq:cutoff-propagation] between \(0\) and this time. Uniformly over the starting point and \(f\), \[\left\lVert dS_Tf(x)\right\rVert^2 \le C n^{-(1+\epsilon)\rho/\lambda+2\delta} +C n^{1-R}+C n^{-H}.\] Choose \[0<2\delta<(1+\epsilon)\rho/\lambda-1, \qquad R>3,\quad H>2.\] This choice makes the squared gradient \(o(n^{-1})\), exactly the scale needed to offset the cube diameter in 62. Hence \(\operatorname{osc}(S_Tf)=o(1)\) uniformly for \(|f|\le1\). Since \(\mu S_Tf=\mu f\), the variational formula for total variation proves the result. ◻

Proof. Let \(\sigma_n\) be the probability measure characterized by \[\frac1n\sum_i\left\langle x_i,S_tx_i\right\rangle_\mu =\int e^{-rt}\sigma_n(\,\mathrm dr).\] Its fixed-test weak convergence to the spin spectral law is supplied by 41. For each fixed \(\delta>0\), the definition of its support edge supplies \(c_\delta>0\) such that, with disorder probability tending to one, \[\sigma_n([0,\lambda+\delta])\ge c_\delta.\] One obtains this statement by testing a nonnegative continuous function supported below \(\lambda+\delta\) with positive limiting integral. It follows for every \(t\ge0\) that \[ \mathbb E_\mu[X_0^tX_t]\ge c_\delta n e^{-(\lambda+\delta)t}. \tag{208}\] The fixed spectral-mass bound remains valid for every time in this inequality. It therefore permits logarithmic times without taking a dimension-dependent Laplace-transform limit.

Averaging over the initial equilibrium spin selects a configuration \(x\) for which \(\mathbb E_x[x^tX_t]\) is at least the value in [eq:cutoff-spin-overlap]. The unit linear test \(f_x(y)=n^{-1/2}x^ty\) detects this remaining overlap. Its equilibrium mean is zero and its transition mean is at least \(c_\delta\sqrt n e^{-(\lambda+\delta)t}\). Its transition variance is bounded by [eq:cutoff-linear-mgf]; its equilibrium variance is at most the inverse gap, by 2. At \(t=(1-\epsilon)t_n\), choose, for example, \(0<\delta<\lambda\epsilon/4\). The displayed mean diverges as a positive power of \(n\). Chebyshev’s inequality applied at half this mean gives total variation tending to one. ◻

Proof. We may choose \(D>2\). First put the block at prescribed relative times using [eq:cutoff-restart]. In the terminal alternative take \(\ell=D\) and \(g=S_{T-b}f\). In the subsequent-block alternative take \(\ell=2D\) and \(g=S_{T-b-D}f\). In both cases the restart origin is \(a=b-D\). The new initial law is the old marginal at \(a\), with the same upper density bound and common trajectory Poincaré constant. The gradient and recipe contractions are unchanged at matching times; in particular \(\widehat A_r=A_{a+r}\). The terminal alternative is the same condition on \(\widehat p_D\), while the other alternative gives \(\widehat N_{2D}\le R\widehat N_D\). For the limiting argument denote the restarted process again by \(X\) and use \(a=0,b=D\); retain also \(2D\) in the second alternative. Normalize \(N_D=1\).

We first establish exactness at \(D\), then control the mean and fluctuation of the slow coefficient, and finally recover a uniform finite-dimensional estimate. In the terminal alternative, \(q_b\) lies in the visible exact sector by assumption. In the subsequent-block alternative, the budget \(N_{2D}\le R\) allows 56 at a good time selected in a fixed subinterval compactly contained in \((D,2D)\). Both adjacent time separations thus stay positive after extraction. Transport back to the prescribed time \(D\) by 50 gives the same conclusion at \(b=D\). The value of the finite budget \(R\) has served this purpose and is not used again.

Retain \(0,D\) and the good time \(s\in[D-1,D-1/2]\) in the same scalar extraction, together with \(2D\) and the later good time when that alternative is used. If \(c_r\) denotes the random slow coefficient of \(q_r\), projection and 50 give \[ c_a=e^{-\lambda(s-a)}\mathbb E[c_s\mid X_a],\qquad \mathbb Ec_a=e^{-\lambda(b-a)}\mathbb Ec_b. \tag{210}\] This transports the mean at the desired rate. To align the vector itself in unweighted square mean, we must also make the centered coefficient small.

Choose a good time \(s\in[b-1,b-1/2]\) as in [eq:cutoff-good-time]. Approximate one slow column by a coarse smooth scaled recipe \(B\). The scalar function \[F_s(x)=B(x)^tD_{v(x)}p_s(x)\] approximates the corresponding coefficient \(c_s\) in scalar \(L^2\) to error at most \(C\delta\), where \(\delta\) is the unweighted approximation accuracy. Indeed multiplication by \(D_v\) is a contraction in the ordinary norm, and [eq:exact-contraction-continuity] extends the contraction with \(p_s\) from fixed recipes to the slow column. The bound uses typicality under the evolving law; stationary mean-square approximation alone would not give it.

The trajectory Poincaré inequality will control this fluctuation once we bound the gradient of \(F_s\). Set \(c_i=v_iB_i\); the exact product rule gives \[d_jF_s=\sum_i c_i(x^j)d_jp_{s,i} +\sum_i p_{s,i}(x)d_jc_i.\] The columns of \(B\) have entries bounded by \(C_B/\sqrt n\), and the matrix \((d_jc_i)_{ij}\) has operator norm at most \(C_B/\sqrt n\). For the first term, mixed-gradient symmetry changes \(d_jp_{s,i}\) to \(d_ip_{s,j}\). Also \(v_i(x^j)\le e^{C|J_{ij}|}v_i(x)\) and \(v_i\le2w_i\), so \[\sum_i\frac{c_i(x^j)^2}{w_i(x)}\le C_B.\] Cauchy–Schwarz, followed by summation over \(j\), proves \[ \mathbb E\left\lVert dF_s\right\rVert^2 \le C_B\left( \mathbb E\sum_iw_i\left\lVert d_ip_s\right\rVert^2+n^{-1}\mathbb E\left\lVert p_s\right\rVert^2\right) \le C_B. \tag{211}\] The same estimate applies to every one of the finitely many columns.

Before choosing these recipes or \(D\), freeze once and for all a moderate-rate instance of [eq:cutoff-propagation], say with \(\rho_0=\lambda/2\), and its constant \(C_{\rho_0}\). The Poincaré bound at time \(a\), this instance of propagation applied to the scalar function \(F_s\), and [eq:cutoff-coefficient-derivative] give \[ \mathop{\mathrm{Var}}\bigl(\mathbb E[F_s(X_s)\mid X_a]\bigr) \le C_P C_{\rho_0}C_B e^{-2\rho_0(s-a)}+o(1). \tag{212}\] The coefficient approximation costs \(C\delta\) in \(L^2\) before transport. By [eq:cutoff-slow-transport], both this error and the square root of [eq:cutoff-coefficient-fluctuation] carry the factor \(e^{-\lambda(s-a)}\). Relative to \(e^{-\lambda D}\), their cost is bounded by \[C\delta+C(C_PC_{\rho_0}C_B)^{1/2}e^{-\rho_0(D-1)}.\] In particular the approximation error does not acquire a factor \(e^{\lambda D}\).

The hidden part and the exact fast complement, propagated from \(s\), cost at most \(C e^{-(\lambda+\kappa)(D-1)}\) for some \(\kappa>0\), by [thm:hidden-propagation,prop:exact-isolation,lem:exact-upgrade]. The latter also applies when the visible vector is only in \(\mathcal H\) with weighted image in the fast part of \(\mathcal X_v\). After division by \(e^{-\lambda D}\), this fast-sector error is bounded by \(C e^{\lambda+\kappa}e^{-\kappa D}\). First choose \(\delta\) sufficiently small in terms of \(\eta\) and \(C_E\), and fix the coarse recipes and \(C_B\). Now choose \(D\) large so that both normalized exponential errors, \[C(C_PC_{\rho_0}C_B)^{1/2}e^{-\rho_0(D-1)} \quad\text{and}\quad C e^{\lambda+\kappa}e^{-\kappa D},\] are smaller than the required fixed fraction of \(\eta\). Restoring the common factor \(e^{-\lambda D}\) proves the limiting mean and fluctuation estimate with exact slow coefficients and reconstruction.

The block length is now fixed, so we may choose the finer recipes \(\widetilde E\) without altering the fluctuation estimate. At this \(D\), an accuracy smaller than \(c\eta e^{-(\lambda+C)D}\) makes all changes in the reported \(A_a\), \(A_b\), and the reconstruction smaller than the remaining fraction of \(\eta e^{-\lambda D}\). Here \(e^{CD}\) is a rough bound for any endpoint norm needed in this comparison. The finer derivative constants have not replaced \(C_B\) in [eq:cutoff-coefficient-fluctuation].

Finally use strict slack, the deterministic good-disorder selection in the proof of 60, and normalization by \(N_b\). The condition \(N_b\ge n^{-Q}\) preserves a fixed polynomial envelope; choose every Poincaré and propagation remainder with an exponent larger than \(2Q+H+2\) before using it. For a violating sequence, perform the one-block or two-block restart above before extraction. The normalized gradients, actual conditional laws, common Poincaré bound, and endpoint ratio meet the same hypotheses at the prescribed relative endpoints; any parameter selected from the family is fixed conditional on the disorder before the path is sampled. The resulting fixed-interval sequence would contradict the limiting estimate. Returning by [eq:cutoff-restart] proves [eq:cutoff-alignment] uniformly at the original block positions, without a rate for any fixed-time \(o(1)\). This includes the parameter selected from the preparation family below. ◻

Proof. Both the added field and its difference matrix are small. Scaled recipe calculus gives \[ \sup_{z,x}\left\lVert g^z(x)\right\rVert_\infty+ \sup_{z,x}\left\lVert (d_i g_j^z(x))_{ij}\right\rVert_\mathrm{op} \le C_{\widetilde E}n^{-\alpha}. \tag{217}\] At each refresh attempt, including an attempt that leaves the spin unchanged, the ratio of the perturbed outcome probability to the ordinary outcome probability is at most \(e^{2\varepsilon_n}\). On the common attempt construction the path likelihood is therefore at most \(e^{2\varepsilon_n N_h}\), with \(N_h\sim\operatorname{Poisson}(nh)\) under the ordinary process. Started at \(\mu\), the ordinary endpoint has law \(\mu\) conditional on \(N_h=k\), because its embedded update kernel preserves \(\mu\). Thus \(X_h\) and \(N_h\) are independent under this stationary reference law, and for every endpoint \(y\), \[\frac{\mu T_h^z(y)}{\mu(y)} \le\mathbb E_\mu[e^{2\varepsilon_n N_h}\mid X_h=y] =\exp\{nh(e^{2\varepsilon_n}-1)\}.\] Averaging the attempt likelihood therefore proves the endpoint density bound [eq:cutoff-preparation-density].

For the variance estimate, we use gradients of the perturbed semigroup. Differentiate [eq:cutoff-preparation-generator]. The exact gradient extension has coefficients \(a_{ij}^z=d_im_j^z\), including the diagonal \(a_{ii}^z\). Its formula is \[(L^z-I)p_i-2x_i\sum_j a_{ij}^z d_jp_i+\sum_j a_{ij}^z p_j.\] The diagonal \(a_{ii}^z p_i\) is an \(o(1)\) zero-order term; the own-spin derivative of an exact gradient is zero. Taylor expansion of \(\tanh\), with [eq:cutoff-small-perturbation] and the bounded row and column square sums, gives common bounds \[\left\lVert a^z\right\rVert_\mathrm{op}\le C,\qquad \max_i\sum_j\frac{|a_{ij}^z|^2}{w_j^z}\le C, \qquad w_j^z=1-x_jm_j^z.\] For example the leading matrix is the field-difference matrix times the diagonal variance; \(v_j^z\le2w_j^z\) controls the displayed weighted row sum. The quadratic Taylor remainders are bounded Schur products of matrices with bounded row and column square sums. Once the right side of [eq:cutoff-small-perturbation] is at most one, the square-evolution proof of 13 therefore has a common constant \(C_*\). In particular \[ \left\lVert dT_s^zf\right\rVert^2\le e^{C_*s}T_s^z\left\lVert df\right\rVert^2. \tag{218}\] Thus the gradient estimate remains valid for this possibly nonreversible preparation: it was derived from the stated generator and its exact gradients, without requiring a Gibbs invariant law.

Its jump bracket is at most \(4\left\lVert dT_{h-s}^zf\right\rVert^2\). The conditional martingale variance formula and [eq:cutoff-preparation-gradient] give \[\mathop{\mathrm{Var}}_x(f(X_h)) \le4\frac{e^{C_*h}-1}{C_*}\,T_h^z\left\lVert df\right\rVert^2(x).\] The other term in total variance is \[\mathop{\mathrm{Var}}_\mu(T_h^zf) \le C_\mu\mu\left\lVert dT_h^zf\right\rVert^2 \le C_\mu e^{C_*h}\mu T_h^z\left\lVert df\right\rVert^2.\] Their sum is [eq:cutoff-preparation-poincare]. ◻

Proof. The cone compares the full gradient norm with its mean slow coefficient. Assume \(N_b\le K|A_b|\) at the end of a block. [eq:cutoff-alignment] gives \[\begin{align*} |A_a|&\ge(1-K\eta)e^{-\lambda D}|A_b|,\\ N_a&\le C_E'|A_a|+\eta e^{-\lambda D}N_b \le\left(C_E'+\frac{K\eta}{1-K\eta}\right)|A_a|, \end{align*}\] where \(C_E'\) is arbitrarily close to \(C_E\) after fine approximation. Choose \(K\eta<1/4\) and then \(\eta\) small enough that the last coefficient is at most \(K\), and \(K\eta/(1-K\eta)\le\epsilon_0\). Thus the cone is preserved. At the terminal time it holds by [eq:cutoff-terminal-cone]. At each step, apply the fixed-interval restart in the proof of 65 to the ordinary trajectory with initial block law \(\nu_zS_a\). Its upper density bound follows from [eq:cutoff-preparation-density,eq:cutoff-restart-density], and its common Poincaré bound is the one fixed above. This restarts only the ordinary part of the evolution, leaving the preparation \(T_h^z\) unchanged. At subsequent blocks the required forward norm ratio follows from the preceding cone step: \[\frac{N_{b+D}}{N_b} \le\frac{K C_Ae^{\lambda D}}{1-K\eta}, \qquad |A_b|\le C_A N_b.\] This ratio is finite for the already chosen block \(D\). Its only role in 65 is to establish exactness at the block endpoint, so its dependence on \(D\) does not create a circular choice.

After \(k=t/D\) steps, \[|A_0|\ge c_0(1-K\eta)^k e^{-\lambda t} =c_0e^{-[\lambda-D^{-1}\log(1-K\eta)]t}.\] Take \(D\ge1\) and \(\eta\le\xi_1/(2K)\), so that the extra exponent is at most \(\xi_1\). To verify the inverse-power cutoff throughout the iteration, choose \[Q>A(\lambda+\xi_1)+2.\] The lower bound for \(|A|\), and \(|A|\le C_A N\), then keep \(N\) above \(n^{-Q}\) for all the \(O(\log n)\) blocks. Choose the inverse-power accuracies used in the supporting Poincaré estimates larger than \(2Q+H+2\). No new union bound over fixed-time limiting errors is needed: [eq:cutoff-alignment] holds on one event uniformly over the functions, parameter values, and comparison times in question. The residual estimate in [eq:cutoff-cone-output] follows from the final cone step. ◻

Proof. We estimate the Jacobian in each unit output direction. For \(|u|=1\), put \(F=\sum_j u_jF_j\) and apply [eq:cutoff-cone-output] at the exit from preparation. Differentiation on the finite state space and Duhamel’s formula give the row vector \[ u^tD\mathcal M_t(z) =\int_0^h\mathbb E_{\mu T_s^z} \left[(dT_{h-s}^zS_tF)^tD_{v^z}\widetilde E\right]\,\mathrm ds, \qquad v_i^z=1-(m_i^z)^2. \tag{221}\] Indeed \(\partial_{z_k}L^zf= \sum_i v_i^z\widetilde E_{ik}d_if\), also when the target depends on the old spin.

Write \(p_0=\widetilde E A_0+r\) under \(\nu_z\). The rough vector extension behind [eq:cutoff-preparation-gradient] propagates \(r\) back through preparation at cost at most \(e^{C_*h/2}\) in mean square. Its contribution to [eq:cutoff-duhamel-jacobian] is therefore \(O(h\epsilon_0|A_0|)\).

For the main vector, uniformly over \(0\le s\le h\), \[ \left(\mathbb E_{\mu T_s^z} \left\lVert K_{h-s}^z\widetilde E-\widetilde E\right\rVert^2\right)^{1/2} \le\omega_{\widetilde E}(h)+o_n(1), \qquad \omega_{\widetilde E}(h)\longrightarrow0. \tag{222}\] Here the norm is the sum over the fixed number of columns. To verify this estimate, the diagonal likelihood representation has row moments bounded by \(e^{C h}\), and its likelihood difference from one has squared mean \(O(h)\), by the likelihood martingale bracket and the bounded row square sums. The entries of a fixed scaled recipe are bounded by \(C/\sqrt n\), so this diagonal error costs \(O(h)\) after summation. The part with at least one inserted coefficient matrix costs \(O(h)\) in norm by its Duhamel expansion and the common rough bound. Finally the ordinary recipe value changes by \(o_h(1)\) in spatial mean square along the short path: its smooth Euclidean extension has Lipschitz constant \(C_{\widetilde E}/\sqrt n\), while the expected number of changed spins is at most \(nh\). These estimates also apply to the perturbed rates, using [eq:cutoff-small-perturbation]; the own-spin zero-order correction is \(o_n(1)\). They prove [eq:cutoff-short-continuity] without a regularity assertion for the exact eigenvectors.

By [eq:cutoff-preparation-density], every marginal in this calculation has upper relative log-density \(o(n)\). The one-state typicality comparisons, the chosen fine accuracy, and \(\left\lVert v^z-v\right\rVert_\infty=o(1)\) therefore give \[\mathbb E_{\mu T_s^z}[\widetilde E^tD_{v^z}\widetilde E] =I+O(\delta_{\rm fine})+o_n(1)\] uniformly in \(s,z\). Combining this with [eq:cutoff-duhamel-jacobian,eq:cutoff-short-continuity] yields \[\left|u^tD\mathcal M_t(z)-hA_0^t\right| \le Ch\bigl(\epsilon_0+\delta_{\rm fine} +\omega_{\widetilde E}(h)+o_n(1)\bigr)|A_0|.\] Choose \(\epsilon_0\) and the fine accuracy first, and then choose a positive fixed \(h\) so that the factor in parentheses is small. [eq:cutoff-cone-output] now proves [eq:cutoff-jacobian], uniformly for every unit output vector.

To pass from the Jacobian estimate to a ball of attainable means, we lift straight line segments by the inverse Jacobian. This argument requires only the proved singular-value bound, not global injectivity. For a target vector \(v\) with \(|v|<a_tR_n\), solve the inverse-Jacobian ordinary differential equation \[z'(r)=(D\mathcal M_t(z(r)))^{-1}v,\qquad z(0)=0, \qquad 0\le r\le1.\] The right side is smooth and \(|z'(r)|\le |v|/a_t<R_n\). The solution stays in a compact subball of the parameter ball and therefore exists to time one. Differentiation gives \(\mathcal M_t(z(r))=\mathcal M_t(0)+rv\). At \(z=0\) the preparation is ordinary stationary evolution, so \(\mathcal M_t(0)=(\mu F_j)_j\). This proves the assertion. ◻

Proof. Fix \(0<\epsilon<1\), take \(0<\alpha<\epsilon/8\) and \(0<\xi_1<\lambda\epsilon/4\), and round \((1-\epsilon)t_n\) up to an integer multiple \(t\) of \(D\). The radius in 68 is at least \[c h e^{-(\lambda+\xi_1)D} n^{\,1/2-\alpha-(1-\epsilon)(\lambda+\xi_1)/(2\lambda)}.\] The exponent is positive, so the ball of attainable means has diverging radius. In particular, one may choose a preparation for which the mean of \(F_1\) differs from \(\mu F_1\) by a quantity tending to infinity. The variances of \(F_1\) under \(\nu_zS_t\) and \(\mu\) are bounded: use the common trajectory Poincaré estimate and its bounded pointwise gradient for the first, and the equilibrium gap for the second. Chebyshev’s inequality implies \(\left\lVert \nu_zS_t-\mu\right\rVert_{\mathop{\mathrm{TV}}}\to1\). Since \(\nu_z\) is a probability law on starting configurations, convexity bounds this distance by \(\sup_x\left\lVert S_t(x,\cdot)-\mu\right\rVert_{\mathop{\mathrm{TV}}}\). Monotonicity then gives the same lower bound at the unrounded, earlier time \((1-\epsilon)t_n\). ◻

Proof. The upper bound follows from 63. For the lower bound, 64 applies when \(\lambda=\lambda_{\rm sp}\), and 69 applies when \(\lambda<\lambda_{\rm sp}\). These cases exhaust the definition of \(\lambda\). ◻

Proof. We prove both directions of clock transfer. Write \(d_c(t)\) and \(d_d(k)\) for the continuous and discrete distances; their kernels satisfy the exact Poissonization identity \(S_t=e^{nt(P-I)}\). For \(k=\lfloor(1-\epsilon)nt_n\rfloor\) and \(T=(1-\epsilon/2)t_n\), convexity and discrete-time contraction give \[d_c(T)\le\mathbb P\{\operatorname{Poisson}(nT)<k\}+d_d(k).\] The probability is \(O((nt_n)^{-1})\), while \(d_c(T)\to1\), proving the discrete lower bound.

For the upper bound, a uniform holding probability permits the local Poisson-to-binomial lemma. Restrict to the high-probability event \(\left\lVert J\right\rVert_\mathrm{op}\le K\). At every configuration, \(n^{-1}\sum_i(Jx)_i^2\le K^2\), so at least half of the fields have absolute value at most \(\sqrt2K\). The probability of keeping a selected spin is at least \((1+e^{2|(Jx)_i|})^{-1}\). Consequently

\[ P(x,x)\ge h_0:=\frac1{2(1+e^{2\sqrt2K})}>0. \tag{224}\] Choose \(\vartheta=1-h_0/2\in(0,1)\) and write \[P=(1-\vartheta)I+\vartheta R, \qquad S_t=e^{\vartheta nt(R-I)}.\] Here \(R\) is a Markov kernel preserving \(\mu\). Set \(M=\lceil(1+\epsilon)nt_n\rceil\). For every \(x\) and \(|f|\le1\), put \(u_j=R^jf(x)-\mu f\), so \(|u_j|\le2\). For each fixed real \(y\), \[\mathbb Eu_{\operatorname{Poisson}(\vartheta M+y\sqrt M)} =S_{M/n+y\sqrt M/(\vartheta n)}f(x)-\mu f.\] The time correction is \(O_y(\sqrt{\log n/n})\), and hence this time exceeds \((1+\epsilon/2)t_n\) for large \(n\). The absolute value of the display is bounded by \(2d_c((1+\epsilon/2)t_n)\to0\), uniformly over \(x,f\).

We have verified every shifted Poisson average required by 4, uniformly over starts and tests. Apply that uniform transfer: its assumptions are a fixed \(\vartheta\in(0,1)\), \(M\to\infty\), bounded arrays, and vanishing Poisson averages for every fixed shift \(y\), all just verified. It gives \[\mathbb Eu_{\operatorname{Binomial}(M,\vartheta)} =P^Mf(x)-\mu f\longrightarrow0\] uniformly over the test family. Taking half its supremum proves \(d_d(M)\to0\). For convergence in disorder probability, the proof of that lemma uses a fixed finite signed approximation by shifted Gaussian densities; its finitely many Poisson averages are all bounded by the same random quantity \(2d_c((1+\epsilon/2)t_n)\). Thus it applies on the present high-probability events without any additional convergence rate. ◻

Proof. For the concentration estimate, we use the Maurey–Pisier Gaussian rotation and double-Jensen argument; see Giraud (Giraud 2021, Exercise 1.6.7, equation (1.10), parts A–B). Rotation between two independent Gaussian vectors expresses the increment as an integral of directional derivatives. First take \(F\) smooth with \(\|\nabla F\|\le D\). Let \(Z'\) be an independent copy of \(Z\) and define the rotation by \[Z_\theta=Z\cos\theta+Z'\sin\theta, \qquad V_\theta=-Z\sin\theta+Z'\cos\theta, \qquad 0\le\theta\le\pi/2.\] For a fixed angle, \(Z_\theta\) and \(V_\theta\) are independent standard Gaussian vectors. This independence lets us evaluate the exponential moment after applying Jensen’s inequality, first in \(Z'\) and then with respect to the uniform measure on \([0,\pi/2]\): \[\begin{align*} \mathbb Ee^{s(F(Z)-\mathbb EF(Z))} &\le \mathbb Ee^{s(F(Z)-F(Z'))}\\ &\le \frac2\pi\int_0^{\pi/2} \mathbb E\exp\left(-\frac{\pi s}{2} \nabla F(Z_\theta)\cdot V_\theta\right)\,d\theta \le e^{\pi^2D^2s^2/8}. \end{align*}\] To pass to a general Lipschitz \(F\), smooth by convolution and use dominated convergence. Linear growth gives the Gaussian exponential integrability required for this passage. Optimization of the exponential Markov bound then gives the tail estimate. The case \(D=0\) follows directly because the function is constant.

For the comparison statement, we interpolate independent realizations of the two processes and show that the expected maximum increases along the interpolation. For \(\lambda>0\), replace the maximum by the smooth function \[\Phi_\lambda(z)=\lambda^{-1}\log\sum_{a\in I}e^{\lambda z_a}.\] Gaussian integration by parts gives \[\frac{d}{dt}\mathbb E\Phi_\lambda \bigl(\sqrt{1-t}\,X+\sqrt t\,Y\bigr) =\frac12\sum_{a,b}B_{ab}\,\mathbb E(\partial_{ab}\Phi_\lambda), \qquad B=\mathop{\mathrm{Cov}}(Y)-\mathop{\mathrm{Cov}}(X).\] The Hessian has zero row sums and nonpositive off-diagonal entries. Consequently the derivative can be expressed using only increment variances. With \(\Delta_{ab}=B_{aa}+B_{bb}-2B_{ab}\) denoting their difference, the right side becomes \[-\frac14\sum_{a,b}\Delta_{ab}\,\mathbb E(\partial_{ab}\Phi_\lambda)\ge0.\] The limit \(\lambda\to\infty\) gives the finite-set comparison. Exhausting a dense set by finite subsets and using continuity and dominated convergence gives the compact-set version. In the applications below, every such process is a linear function of a finite Gaussian vector, which ensures integrability of its supremum. ◻

Proof. We prove the operator-norm estimate first and then control the row sums and individual entries. The case \(\beta=0\) is deterministic. For \(\beta>0\), add independent diagonal entries of variance \(2\beta^2/n\) to \(J\). The resulting centered Gaussian orthogonal ensemble (GOE) matrix \(W\) has a covariance that permits a direct Gaussian comparison: for unit vectors \(p,q\), \[\mathbb E[(p^{\mathsf T}Wp)(q^{\mathsf T}Wq)] =\frac{2\beta^2}{n}(p^{\mathsf T}q)^2.\] With \(\rho=p^{\mathsf T}q\), the increment variance of \(p^{\mathsf T}Wp\) is \(4\beta^2(1-\rho^2)/n\). Bound it by \(8\beta^2(1-\rho)/n\) and compare with the linear process \((2\beta/\sqrt n)p^{\mathsf T}z\), where \(z\) is a standard Gaussian vector. Lemma 73 yields \[\mathbb E\lambda_{\max}(W) \le\frac{2\beta}{\sqrt n}\mathbb E\|z\|\le2\beta.\] In the independent standard Gaussian coordinates defining \(W\), the quadratic form \(p^{\mathsf T}Wp\) is Lipschitz with constant \(\sqrt2\beta/\sqrt n\), uniformly over unit vectors. The supremum inherits this constant. Concentration for \(\lambda_{\max}(W)\) and \(\lambda_{\max}(-W)\) therefore bounds \(\|W\|\) by \(2\beta+\delta_n/2\) with probability tending to one. The largest absolute diagonal entry is at most \(\delta_n/2\) with probability tending to one as well. Removing that diagonal proves the first estimate in Equation (227).

For a standard Gaussian \(g\), \[\mathbb Ee^{\theta(g^2-1)}=e^{-\theta}(1-2\theta)^{-1/2} \le e^{C\theta^2},\qquad |\theta|\le1/4.\] Apply the exponential Markov bound to the \(n-1\) independent squares in one row. The probability that their weighted sum exceeds \(\beta^2+\delta_n\) is at most \(\exp(-c_\beta n\delta_n^2)\) for large \(n\). A union bound over the rows gives the second estimate; Gaussian tails and a union bound over entries give the third. The choice \(n\delta_n^2=\sqrt n\) makes all the failure probabilities vanish. The cubic row sums needed for the two-spin remainder are now controlled by the last two estimates together: \[\max_i\sum_j|J_{ij}|^3 \le \left(\max_{i,j}|J_{ij}|\right) \left(\max_i\sum_jJ_{ij}^2\right) \le\delta_n(\beta^2+\delta_n).\] ◻

Proof. The proof has three steps: factor out the two conditional variances, identify the nonnegative part of the expansion, and absorb the cubic remainder into the one-site energies. Condition on all spins outside \(i,j\), and denote the remaining spins by \(x,y\). Their probabilities are \[\frac{(1+ax)(1+by)(1+txy)}{4(1+abt)}, \qquad |a|,|b|<1,\] where \(a,b\) are the hyperbolic tangents of the external fields left after conditioning. Set \(Z_0=1+abt\). The corresponding conditional means and variances take the form \[m_i(y)=\frac{a+ty}{1+aty},\qquad v_i(y)=\frac{(1-a^2)(1-t^2)}{(1+aty)^2},\] and the analogous expressions with \(a,x\) and \(b,y\) interchanged. There are real numbers \(p,q,r\) such that \[d_i f=p+r(y-b),\qquad d_j f=q+r(x-a).\] Indeed, expand the restriction of \(f\) in the basis \(1,x,y,xy\) and denote the coefficient of \(xy\) by \(r\). This isolates the common second-difference coefficient from the two first differences.

Let \(\mathbb E_0\) denote uniform expectation over the four pairs \((x,y)\). We now combine the mixed projection energy with the correction term in the claimed inequality. The identities \[x-m_i(y)=\frac{(x-a)(1-txy)}{1+aty}, \qquad (1+ax)(x-a)=x(1-a^2)\] and their counterparts for \(y\) give the exact factorization \[\begin{align*} &\nu[\mathsf D_i f\mathsf D_j f\mid X_{-\{i,j\}}] +\nu[(t+2t^2m_i m_j)v_i v_jd_i f d_j f\mid X_{-\{i,j\}}] =P_0Q,\\ &P_0=\frac{(1-a^2)(1-b^2)(1-t^2)}{Z_0},\qquad Q=\mathbb E_0\bigl([p+r(y-b)][q+r(x-a)]C(t;x,y)\bigr), \end{align*}\] where \[\begin{align*} C(t;x,y) ={}&\frac{xy-t}{(1+aty)(1+btx)}\\ &+(t+2t^2m_i(y)m_j(x)) \frac{(1-t^2)(1+ax)(1+by)(1+txy)} {(1+aty)^2(1+btx)^2}. \end{align*}\] Only the expansion through second order will contribute to the leading term. Expanding at \(t=0\) and reducing by \(x^2=y^2=1\) gives \[ C(t;x,y) =xy\left[1+ab t+(1-a^2-b^2+2a^2b^2)t^2\right] +R(t;x,y). \tag{229}\] The cancellation can be checked directly: before collecting terms, the coefficient of \(t^2\) is \[d+xy(d^2-abxy)+(1+ax)(1+by)(xy-2d+2ab), \qquad d=ay+bx.\] Collecting terms gives the coefficient in Equation (229), with the coefficients of \(1,x,y\) cancelling. Thus the displayed expansion contributes through the following single identity: \[\mathbb E_0\bigl(xy[p+r(y-b)][q+r(x-a)]\bigr)=r^2.\]

It remains to control the remainder without excluding nearly deterministic spins; we therefore allow the limiting values \(|a|=|b|=1\) in this estimate. On the complex disk \(|t|\le1/2\), the denominators \(1+aty\) and \(1+btx\) have modulus at least \(1/2\), while \(|m_i(y)|,|m_j(x)|\le3\). The first term of \(C\) is bounded in modulus by \(6\). In the second term, \(t+2t^2m_i m_j\) is bounded by \(5\); including its remaining factors gives a bound of \(600\). Hence \(|C(t;x,y)|\le606\) throughout the disk, and Cauchy’s coefficient estimate yields \[|R(t;x,y)|\le606\sum_{k\ge3}(2|t|)^k \le9696|t|^3,\qquad |t|\le1/4.\] Since \[|[p+r(y-b)][q+r(x-a)]|\le p^2+q^2+8r^2\] and \(1-a^2-b^2+2a^2b^2=a^2b^2+(1-a^2)(1-b^2)\ge0\), we obtain \[Q\ge\bigl(1-|t|-8\cdot9696|t|^3\bigr)r^2 -9696|t|^3(p^2+q^2).\] For \(|t|\le1/100\), the coefficient of \(r^2\) exceeds \(1/2\).

The leading term is now nonnegative. To finish, the negative cubic term must be charged to the conditional energies while retaining the variance factors. Write \(A_0=1-a^2\) and \(B_0=1-b^2\). Summing over \(x\) first gives the exact energy formula \[\nu[v_i(d_i f)^2\mid X_{-\{i,j\}}] =\frac{A_0(1-t^2)}{2Z_0} \sum_{y=\pm1}\frac{1+by}{1+aty}[p+r(y-b)]^2.\] For \(|t|\le1/4\), the denominators lie between \(3/4\) and \(5/4\), so \[\begin{align*} \nu[v_i(d_i f)^2\mid X_{-\{i,j\}}] &\ge\frac35 A_0(p^2+B_0r^2),\\ \nu[v_j(d_j f)^2\mid X_{-\{i,j\}}] &\ge\frac35 B_0(q^2+A_0r^2),\qquad P_0\le\frac43A_0B_0. \end{align*}\] These bounds make \(P_0p^2\) and \(P_0q^2\) at most \(20/9\) times their respective conditional energies. Multiply the estimate for \(Q\) by \(P_0\) and use \(9696\cdot20/9<22000\) to obtain the conditional inequality. Averaging over the spins on which we conditioned gives Equation (228). The estimates are homogeneous in \(p,q,r\), and throughout the argument neither these coefficients nor the external fields were bounded. ◻

Proof. We use the pair estimate to compare the squared generator with its Dirichlet form. Fix \(\nu=\nu_{s,\theta}\) and define the zero-diagonal matrix by \(T_{ij}=\tanh(sJ_{ij})\). On \(\mathcal E_n\), all entries are small enough for Lemma 75 to apply simultaneously. At each configuration put \(z_i=v_i d_i f\) and \(z'_i=m_i z_i\). Summation over ordered pairs gives \[\begin{align*} \left\|\sum_i\mathsf D_i f\right\|_{L^2(\nu)}^2 \ge{}&\sum_i\|\mathsf D_i f\|_{L^2(\nu)}^2 -\nu[z^{\mathsf T}Tz+2(z')^{\mathsf T}(T\circ T)z']\\ &-2C_0\left(\max_i\sum_j|T_{ij}|^3\right) \sum_i\|\mathsf D_i f\|_{L^2(\nu)}^2, \end{align*}\] where \(\circ\) denotes entrywise multiplication. The field-dependent factors remain inside the vectors, so the interaction matrices can be bounded by their operator norms. With \(M=\max(\|T\|,2\|T\circ T\|)\), the sum of the two quadratic forms is at most \[M\sum_i v_i^2(1+m_i^2)(d_i f)^2 \le M\sum_i v_i(d_i f)^2,\] because \(v_i^2(1+m_i^2)=v_i(1-m_i^4)\le v_i\).

We next control these operator norms using only the disorder event. The Taylor estimate \(|\tanh u-u|\le C|u|^3\), together with the maximum absolute row-sum bound for symmetric matrices, gives \[\|T-sJ\|\le C\max_i\sum_j|J_{ij}|^3=o(1).\] Also \(\|T\circ T\|\le\max_i\sum_jT_{ij}^2\le\beta^2+o(1)\). The error bounds depend only on the deterministic tolerances defining the event; they are uniform over \(s,\theta\) and \(J\in\mathcal E_n\). The fixed assumption \(\beta<1/2\) therefore gives \[M\le\max(2\beta,2\beta^2)+o(1)=2\beta+o(1).\] Thus, for \(A=-L_\nu=\sum_i\mathsf D_i\) and all sufficiently large \(n\), \[\|Af\|_2^2\ge\kappa_\beta\langle f,Af\rangle_\nu.\] To read this as a spectral gap, use that \(A\) is self-adjoint and nonnegative. Every one-site conditional law has full support, so \(\mathsf D_i f=0\) for all \(i\) forces \(f\) to be constant along every edge of the cube. Thus the kernel consists exactly of constants. Applied to a nonconstant eigenfunction, the last inequality bounds its eigenvalue below by \(\kappa_\beta\). This proves the gap and variance assertions. The covariance conclusion is the linear-function case \(f(x)=c^{\mathsf T}x\): \[c^{\mathsf T}\mathop{\mathrm{Cov}}_\nu(X)c \le\kappa_\beta^{-1}\sum_i\nu[v_i]c_i^2 \le\kappa_\beta^{-1}\|c\|^2.\] ◻

Proof. We interpolate from the Gibbs measure to product posteriors, compare their expected entropy with the initial entropy, and then remove the observation from the one-site terms. For large \(n\), Equation (227) ensures \(\|J\|<1\). Choose \(K=J+2I\); then \(K\) is positive definite and satisfies \(\|K\|<3\). Keep it fixed throughout the argument for the realized disorder. Given \(X\sim\mu\) and an independent standard Gaussian vector \(z\), use the observation, for \(0<u\le1\), \[Y_u=uK^{1/2}X+\sqrt u\,z.\] The posterior law \(\nu_{u,y}=\operatorname{Law}(X\mid Y_u=y)\) is \[ \nu_{u,y}(x)\propto\mu(x) \exp\left(y^{\mathsf T}K^{1/2}x-\frac u2x^{\mathsf T}Kx\right). \tag{233}\] Discarding the diagonal term, which is constant on the cube, leaves interaction matrix \((1-u)J\). The observation creates an external field, and a further linear tilt changes only that field. Thus both uniformities in Proposition 76 apply: every posterior and every such tilt has covariance norm at most \(C=\kappa_\beta^{-1}\). At \(u=1\), the interaction vanishes and the posterior is a product measure.

Derivative of the expected posterior entropy. To quantify the entropy lost to the observation, set \(H(u)=\mathbb E[\mathop{\mathrm{Ent}}_{\nu_{u,Y_u}}(f)]\). The derivative identity we need is \[ H'(u)=-\frac12\mathbb E\left[ \frac{\|K^{1/2}\mathop{\mathrm{Cov}}_{\nu_{u,Y_u}}(X,f)\|^2} {\nu_{u,Y_u}[f]}\right]. \tag{234}\] The covariance with \(f\) here is a vector. We verify the identity using the finite Gaussian mixture defining the observation. Let \(\phi_u\) be the centered Gaussian density with covariance \(uI\), and define \[\begin{align*} q(u,y)&=\sum_x\mu(x) e^{y^{\mathsf T}K^{1/2}x-u x^{\mathsf T}Kx/2},\\ p(u,y)&=\sum_x\mu(x)f(x) e^{y^{\mathsf T}K^{1/2}x-u x^{\mathsf T}Kx/2}. \end{align*}\] These functions give the density of \(Y_u\) as \(q\phi_u\) and the posterior expectation as \(p/q=\nu_{u,y}[f]\). Their finite exponential sums show directly that both \(p\) and \(q\) solve the backward heat equation: \[(\partial_u+\tfrac12\Delta_y)p=0,\qquad (\partial_u+\tfrac12\Delta_y)q=0.\] Writing \(r=p/q\), the chain rule gives the exact identity \[(\partial_u+\tfrac12\Delta_y)\bigl(p\log(p/q)\bigr) =\frac12\frac{q\|\nabla_y r\|^2}{r}.\] Since \(\partial_u\phi_u=\tfrac12\Delta_y\phi_u\), differentiation and integration by parts yield \[\frac{d}{du}\int p\log(p/q)\,\phi_u\,dy =\frac12\int\frac{q\|\nabla_y r\|^2}{r}\phi_u\,dy.\] Differentiation and integration by parts are legitimate: the spin sums are finite, \(f\) is bounded above and below by positive constants, and the remaining dependence on \(y\) grows at most exponentially in \(\|y\|\). Gaussian tails therefore control all terms. To identify the derivative as the claimed covariance expression, use \[H(u)=\mu[f\log f]-\int p\log(p/q)\phi_u\,dy, \qquad \nabla_y r=K^{1/2}\mathop{\mathrm{Cov}}_{\nu_{u,y}}(X,f),\] which proves Equation (234). At the initial endpoint \(u\downarrow0\), the observations tend to zero in probability and the posterior tends to \(\mu\). Since the entropy functional is bounded on this finite space, \(H(u)\to\mathop{\mathrm{Ent}}_\mu(f)\), giving the initial value for the comparison.

Covariance controlled by entropy. We next bound the derivative by the expected entropy itself. Let \(\nu\) be any posterior, take a unit vector \(a\), and set \(Z=a^{\mathsf T}(X-\nu[X])\). The second derivative of the log moment-generating function of \(Z\) is the variance of \(a^{\mathsf T}X\) under a linear tilt of \(\nu\). Applying the uniform covariance estimate to these tilts and integrating twice gives \[\log\nu[e^{\lambda Z}]\le\frac C2\lambda^2, \qquad \lambda\in\mathbb R.\] The change of measure \(\widehat\nu=f\nu/\nu[f]\) connects this exponential-moment bound to entropy. Apply Jensen’s inequality to the logarithm under \(\widehat\nu\) to obtain \[\lambda\widehat\nu[Z] \le\frac{\mathop{\mathrm{Ent}}_\nu(f)}{\nu[f]}+ \log\nu[e^{\lambda Z}] \le\frac{\mathop{\mathrm{Ent}}_\nu(f)}{\nu[f]}+\frac C2\lambda^2.\] Optimize in \(\lambda\) and take \(a\) parallel to \(\mathop{\mathrm{Cov}}_\nu(X,f)\) when this vector is nonzero. This yields \[ \|\mathop{\mathrm{Cov}}_\nu(X,f)\|^2\le2C\nu[f]\mathop{\mathrm{Ent}}_\nu(f). \tag{235}\] Equations (234) and (235) imply \(H'(u)\ge-C\|K\|H(u)\). Integrating from zero to one gives \[ \mathop{\mathrm{Ent}}_\mu(f)\le e^{C\|K\|}H(1)\le e^{3C}H(1). \tag{236}\] We have now compared the initial entropy with the entropy at the product endpoint. The comparison used covariance control at every interaction scale: controlling linear tilts only at the original interaction \(J\) would leave the quadratic change in Equation (233) untreated. It remains to express the endpoint entropy in terms of the original one-site energies.

Removing the observation at the product endpoint. For a product probability measure \(\pi=\bigotimes_i\pi_i\), entropy tensorizes: \[\mathop{\mathrm{Ent}}_\pi(f)\le\sum_i\pi[\mathop{\mathrm{Ent}}_{\pi_i}(f)],\] where in the \(i\)th summand the other coordinates are fixed inside the one-coordinate entropy. To prove tensorization, separate one coordinate using the entropy chain rule and apply induction to the remaining product. For a fixed base measure, \(f\mapsto\mathop{\mathrm{Ent}}(f)\) is convex; hence averaging the separated coordinate can only decrease the remaining one-coordinate entropies. The required convexity follows from joint convexity of \((a,b)\mapsto a\log(a/b)\) with \(a=f(x)\) and \(b=\pi[f]\).

Apply tensorization to the product posterior at \(u=1\). We must then average under the joint law of \((X,Y_1)\), retaining the dependence between the observation and the spins. For this step put \(F=f(X)\), \(\varphi(z)=z\log z\), \[\mathcal G_i=\sigma(X_{-i}),\qquad \mathcal H_i=\sigma(X_{-i},Y_1),\qquad e_{\mathcal A}(F)=\mathbb E[\varphi(F)\mid\mathcal A] -\varphi(\mathbb E[F\mid\mathcal A]).\] Conditional Jensen and the tower property give \[\begin{align*} \mathbb E[e_{\mathcal H_i}(F)\mid\mathcal G_i] &=\mathbb E[\varphi(F)\mid\mathcal G_i] -\mathbb E[\varphi(\mathbb E[F\mid\mathcal H_i])\mid\mathcal G_i]\\ &\le\mathbb E[\varphi(F)\mid\mathcal G_i] -\varphi(\mathbb E[F\mid\mathcal G_i]) =e_{\mathcal G_i}(F). \end{align*}\] Thus the additional conditioning on the observation decreases the averaged conditional entropy, even though the observation and the complementary spins are dependent. Combining this fact with the posterior tensorization gives \[H(1)\le\sum_i\mathbb Ee_{\mathcal H_i}(F) \le\sum_i\mathbb Ee_{\mathcal G_i}(F) =\sum_i\mu\bigl[\mathop{\mathrm{Ent}}_{\mu(\cdot\mid X_{-i})}(f)\bigr].\] We can now compare each original one-coordinate entropy with the corresponding heat-bath energy. Concavity of the logarithm gives \[\mathop{\mathrm{Ent}}(f)\le\mathop{\mathrm{Cov}}(f,\log f),\] because the difference of the right and left sides is \(\mathbb Ef\,[\log\mathbb Ef-\mathbb E\log f]\ge0\). Moreover, summing these conditional covariances gives \[\sum_i\mu[\mathop{\mathrm{Cov}}(f,\log f\mid X_{-i})] =\sum_i\mu[f\mathsf D_i\log f] =\mu[f(-L)\log f].\] The product-endpoint estimate is therefore bounded by the entropy dissipation of the original generator. Substituting into Equation (236) proves Equation (231), with the permissible choice \(C_1=e^{3/\kappa_\beta}\).

Entropy decay and mixing. For a point start, it remains to bound the initial relative entropy \(\log(1/\mu(x))\) uniformly in the starting point. The identity \(\|x\|^2=n\) gives \[\left|\frac12x^{\mathsf T}Jx\right|\le\frac n2\|J\|, \qquad \log\frac1{\mu(x)}\le n(\log2+\|J\|)\le n(\log2+1).\] Reversibility lets us evolve the density \(f_t=d\rho_{t,x}/d\mu\) using the same semigroup \(S_t\). For \(t>0\) the density is strictly positive, so the entropy inequality applies to its derivative: \[\frac{d}{dt}\mathop{\mathrm{Ent}}_\mu(f_t) =-\mu[f_t(-L)\log f_t] \le-C_1^{-1}\mathop{\mathrm{Ent}}_\mu(f_t).\] Integrating this differential inequality and using continuity at zero establishes Equation (232), with \(C_2=\log2+1\).

To finish, coarsen the measure to an event and its complement; relative entropy then dominates the resulting binary relative entropy. For fixed \(q\in(0,1)\), this binary entropy, viewed as a function of \(p\), has value and derivative zero at \(p=q\) and second derivative \(1/[p(1-p)]\ge4\). It is consequently at least \(2(p-q)^2\). Choosing an event that attains total variation and using the entropy decay just proved yields \[d_c(t)\le\sqrt{\frac{C_2n}{2}}\,e^{-t/(2C_1)},\] which proves the stated mixing upper bound. ◻

Proof. We first establish the unweighted operator estimate, then insert the continuous kernel \(G\) by approximation. The diagonal estimate will require a separate argument that removes one row from the fields. For the operator estimate, truncate \(X_n=nJ_{ij}^2\) at a fixed level \(T\). For a sufficiently small fixed \(t>0\), \[\sup_n\mathbb Ee^{t(X_n-T)_+}\longrightarrow1 \quad\text{as }T\longrightarrow\infty.\] For uniform dominated convergence, use \(|\beta Z+\beta^2/\sqrt n|\le\beta|Z|+\beta^2\); the square of the right side has an exponential moment for a sufficiently small parameter. We may therefore choose \(T\) so that both the mean tail and the logarithm of the displayed moment are as small as required. Exponential Markov bounds the probability that a row average of \((nJ_{ij}^2-T)_+\) exceeds the prescribed accuracy by \(e^{-cn}\). Taking the union over the \(n\) rows retains a positive exponential rate. Symmetry and entrywise nonnegativity bound the tail matrix’s operator norm by its largest row sum, and the same estimate controls its expectation.

For a fixed unit vector, the quadratic form of the centered truncated matrix is a sum of independent bounded variables with coefficients \(2p_ip_j/n\), \(i<j\). To bound its moment generating function, recall that a centered variable whose range has length \(a\) has logarithmic moment generating function at most \(a^2s^2/8\). The second derivative is a variance under a tilted law and is at most \(a^2/4\); integrating twice gives this bound. Since \(\sum_{i<j}(2p_ip_j)^2\le2\), the present quadratic form has logarithmic moment generating function at most \(CT^2s^2/n^2\), and hence a tail at fixed accuracy bounded by \(2e^{-c n^2/T^2}\). To obtain an operator estimate, use a \(1/4\)-net of the unit sphere with at most \(9^n\) points. This cardinality follows by comparing volumes of disjoint balls of radius \(1/8\). For a symmetric matrix \(A\), twice the maximum of \(|p^{\mathsf T}Ap|\) on this net bounds \(\|A\|\). The union bound is therefore absorbed by the quadratic exponential rate. It remains to restore the mean and diagonal: \[\mathbb EJ_{ij}^2=\frac{\beta^2}{n}+\frac{\beta^4}{n^2} \quad(i\ne j),\] so the resulting mean correction and missing diagonal have norm \(O_\beta(1/n)\). Together with the truncation and net bounds, this proves Equation (259).

We now insert \(G\). Uniform continuity gives an approximation on a finite rectangular partition of \([0,2]^2\), of the form \(\sum_{a,d}g_{ad}\mathbf 1_{I_a}(w_i)\mathbf 1_{I_d}(w_j)\). Multiplication of \(p\) by the corresponding coordinate masks reduces each summand to Equation (259). The number of summands and their coefficients are fixed. For the remaining kernel error \(R_{ij}\), use \[\left|\sum_{i,j}E_{ij}R_{ij}p_ip_j\right| \le\max_{i,j}|R_{ij}|\, |p|^{\mathsf T}|E|\,|p| \le\max_{i,j}|R_{ij}|\max_i\sum_j|E_{ij}|.\] The row-sum bound in Equation (255) controls this error, since \(\sum_j|E_{ij}|\le\sum_jJ_{ij}^2+\beta^2\). This proves Equation (260). On the operator and row-sum events, the argument is deterministic for every choice of masks and residual coefficients. Thus these weights may depend on the whole interaction matrix.

For the diagonal estimate we must recover independence before applying the weighted tail bound. Fix a row \(i\) and condition on \[\mathcal F_i=\sigma\bigl(J_{kl}:k,l\ne i;\ \zeta_1,\ldots,\zeta_n\bigr).\] This sigma-field leaves the entries of the off-diagonal row \((J_{ij})_{j\ne i}\) independent. Remove that row from the other fields by defining \[H_j^{(-i)}=\zeta_j+\sum_{k\ne i,j}J_{jk},\qquad j\ne i.\] The fields so obtained are \(\mathcal F_i\)-measurable, with \(H_j-H_j^{(-i)}=J_{ij}\). Keep \(a\in[0,2]\) deterministic for now and set \(c_j(a)=F(a,w(H_j^{(-i)}))\). Under the conditioning these weights are fixed and lie in \([0,32/27]\), so Equation (253) gives a bound \(e^{-cn}\) for the event \[\left|\sum_{j\ne i}c_j(a) \left(J_{ij}^2-\frac{\beta^2}{n}\right)\right|>u,\] for all large \(n\), uniformly in \(\mathcal F_i\) and \(a\); the mean correction is at most \((32/27)\beta^4/n\). Take a union first over a fixed finite mesh of \(a\) values and then over the \(n\) rows. Uniform continuity of \(F\) and the bound on \(\sum_j|E_{ij}|\) extend the estimate to all \(a\in[0,2]\). Next restore the original fields. On the maximum-entry event, \(|H_j-H_j^{(-i)}|\) is uniformly small. The Lipschitz property of \(w\) and uniform continuity of \(F\) bound the resulting error by a small multiple of the same row sum. Only after these simultaneous estimates do we set \(a=w_i\). Conditioning on \(w_i\) at the outset would condition on a row sum and destroy the independence used above. The mesh, continuity tolerances and entry bound are chosen before \(n\); their failures therefore retain positive exponential rates. This completes Equation (261). ◻

Proof. Separate the bin-constant space \(\mathcal C\) from the zero-sum space \(\mathcal U_a\) in each bin. Then \(\mathcal C\) is orthogonal to \(\bigoplus_a\mathcal U_a\), and \(\overline A\) preserves each space. Since \(Qu=u\) and \(Q\overline A u=\overline A u\), the process expands as \[\begin{align*} p^{\mathsf T}QWQ\overline A p ={}&C_{b}+ \sum_a u_a^{\mathsf T}WQ(v_aI+\overline A)p_c +u^{\mathsf T}W\overline A u,\\ C_b={}&p_c^{\mathsf T}QWQ\overline A p_c. \end{align*}\] For these fixed bin parameters, \(C_b\) is constant in the residual index and has conditional mean zero. The linear residual term uses the \(\mathcal U\)–\(\mathcal C\) block of the GOE, while the quadratic term uses the \(\mathcal U\)–\(\mathcal U\) block. Their independence follows from Equation (257).

For two indices \(u,u'\), set \[s_a=\widehat\pi_a r_a^2, \qquad q_a=u_a^{\mathsf T}u'_a, \qquad V_a=\mathop{\mathrm{Var}}_{d\sim\widehat\pi}((v_a+v_d)b_d).\] The linear residual covariance is \[ \frac{\beta^2}{n}\sum_a q_a V_a. \tag{272}\] Here \(Q(v_aI+\overline A)p_c\) lies in \(\mathcal C\) and has squared norm \(V_a\). Orthogonality to the residual spaces makes the other covariance contraction vanish. Applying the same covariance identity to the quadratic residual term gives \[ \frac{\beta^2}{2n}\sum_{a,d}(v_a+v_d)^2q_aq_d. \tag{273}\] This follows from \[\frac{\beta^2}{n} \bigl((u^{\mathsf T}u') ((\overline A u)^{\mathsf T}\overline A u') +(u^{\mathsf T}\overline A u') ((\overline A u)^{\mathsf T}u')\bigr)\] and symmetrization in the bin indices.

Cauchy–Schwarz gives \(-s_a\le q_a\le s_a\), and hence \[ s_as_d-q_aq_d \le s_a(s_d-q_d)+s_d(s_a-q_a), \tag{274}\] because the right side minus the left side is \((s_a-q_a)(s_d-q_d)\ge0\). The inequality therefore holds also for negative overlaps. Combining it with Equations (272) and (273) bounds the increment variance of the residual process by \[\frac{2\beta^2}{n}\sum_a(s_a-q_a) \left(V_a+\sum_d(v_a+v_d)^2s_d\right).\] This is exactly the increment variance of the linear process \[\frac\beta{\sqrt n}\sum_a \sqrt{V_a+\sum_d(v_a+v_d)^2s_d}\ z_a^{\mathsf T}u_a,\] with independent standard Gaussian vectors \(z_a\) in \(\mathcal U_a\). Lemma 73 therefore bounds the expected supremum of the residual process by that of the linear process. We can maximize the latter separately in each residual space: each maximum is the corresponding Gaussian norm multiplied by \(\sqrt{s_a}\). Using \(\mathbb E\|z_a\|\le\sqrt{\dim\mathcal U_a} \le\sqrt{n\widehat\pi_a}\), we obtain Equation (271). Adding \(C_b\) leaves this expectation unchanged because its conditional mean is zero and the bin parameters remain fixed.

Concentration must, however, include \(C_b\). For a unit vector \(p\), the full GOE functional has symmetric coefficient matrix \[\tfrac12\bigl((Qp)(Q\overline A p)^{\mathsf T} +(Q\overline A p)(Qp)^{\mathsf T}\bigr),\] whose Frobenius norm is at most one. Consequently its coefficient norm in independent standard Gaussian coordinates is at most \(\sqrt2\beta/\sqrt n\). The supremum inherits this Lipschitz constant, proving the assertion for the full process, including \(C_b\). ◻

Proof of Proposition 80. We assemble the estimates with five fixed error allowances. Given \(\delta\) from the hypothesis, put \(a=\delta/10\). First require an error at most \(a\) in Equation (262). Next choose the field partition and tolerances to make each error in Equations (267) and (269) at most \(a\). Take concentration error \(a\) in Equation (275) and choose \(\eta\) so that \(2K_{\rm form}\eta\le a\). Every choice precedes the limit in \(n\). The rescaling in Equation (268) makes the limiting functions admissible for the hypothesis. Functional approximation and concentration therefore bound each net value by \(1-\delta+2a\). The two form errors and extension from the net add the remaining \(3a\), giving, for every unit vector, \[M_J(p)\le1-\delta+5a=1-\frac\delta2.\]

To verify that this simultaneous assertion retains an exponential rate, collect the failure probabilities after fixing all parameters. Their sum has the form \[\begin{align*} &C n^2e^{-c_1n}+2nN_we^{-c_2n} +Ck e^{-c_3n}+2N_{\rm par}e^{-c_4n} +2\,9^ne^{-c_5n^2},\\ &\hspace{25mm}N_w<\infty,\qquad N_{\rm par}\le(1+2/\eta)^{2k}. \end{align*}\] Each \(c_i\) is positive and depends only on \(\beta\) and the fixed choices. The initial terms account for the entry, diagonal, norm, empirical-tail and common-shift events; \(N_w\) counts the deterministic row-parameter mesh. Only the last term uses an exponentially large sphere net, and its rate quadratic in \(n\) absorbs \(9^n\). The other prefactors are fixed or polynomial in \(n\). Hence the sum is at most \(e^{-\kappa n}\) for some \(\kappa>0\) and all sufficiently large \(n\). Finally the event in Equation (251) does not involve the auxiliary diagonals, so its probability on the enlarged space equals that on the planted marginal. ◻

Proof. Let \(Z_J\) be the original partition function and \[Z_{\mathcal B,J}=\sum_x e^{x^{\mathsf T}Jx/2}\mathbf 1_{\{J^x\in\mathcal A_n\}}.\] The original disorder law is invariant under sign conjugation. Each summand therefore reduces to the all-positive configuration, and the planted density in Equation (246) gives \[ \mathbb EZ_{\mathcal B,J} =2^n\mathbb E\left[e^{\sum_{i<j}J_{ij}}\mathbf 1_{\mathcal A_n}\right] =\mathbb EZ_J\,\widehat\mathbb P_n(\mathcal A_n) \le e^{-\kappa n}\mathbb EZ_J. \tag{277}\] Here \(\mathbb EZ_J=2^n\exp(\beta^2(n-1)/4)\).

We next bound the denominator \(Z_J\) from below on the exponential scale. Gaussian integration, with an independent uniform sign vector \((\varepsilon_i)_{i=1}^n\), gives the exact second-moment identity \[ \frac{\mathbb EZ_J^2}{(\mathbb EZ_J)^2} =e^{-\beta^2/2}\mathbb E_{\varepsilon} \exp\left\{\frac{\beta^2}{2n} \left(\sum_i\varepsilon_i\right)^2\right\} \le\frac{e^{-\beta^2/2}}{\sqrt{1-\beta^2}}=:C_\beta. \tag{278}\] For the inequality, introduce a standard Gaussian \(Z\) to linearize the square. The sign average becomes \(\cosh(\beta Z/\sqrt n)^n\le e^{\beta^2Z^2/2}\), with expectation \((1-\beta^2)^{-1/2}\). Separate the event \(\{Z_J\ge\mathbb EZ_J/2\}\) and apply Cauchy–Schwarz to obtain \[\mathbb P_n(Z_J\ge\mathbb EZ_J/2)\ge\frac1{4C_\beta}.\]

We upgrade this fixed positive probability using concentration. Each derivative of \(\log Z_J\) in an independent standard disorder Gaussian has magnitude at most \(\beta/\sqrt n\), giving Lipschitz constant at most \(\beta\sqrt{(n-1)/2}\). The preceding lower event and Lemma 73 force \[\mathbb E\log Z_J\ge\log\mathbb EZ_J-C\sqrt n.\] Otherwise \(\log\mathbb EZ_J-\log2\) would exceed the mean by a sufficiently large multiple of \(\sqrt n\), and upper-tail concentration would make the preceding event less likely than its fixed lower bound. Lower-tail concentration now gives, for all large \(n\), \[ \mathbb P_n\left(\log Z_J<\log\mathbb EZ_J-\frac{\kappa n}4\right) \le e^{-c n}. \tag{279}\] Constants here may depend on the fixed \(\beta\) and \(\kappa\).

By Equation (277) and Markov’s inequality, \[\mathbb P_n\left(Z_{\mathcal B,J}> e^{-3\kappa n/4}\mathbb EZ_J\right)\le e^{-\kappa n/4}.\] On the complement of this event and the event in Equation (279), divide the numerator bound by the partition-function lower bound to obtain Equation (276). A smaller positive exponent absorbs the sum of the two failure probabilities. ◻

Proof. We separate one rank-one term from the kernel and bound the remainder by its diagonal. Put \(q=(w-3/4)/(w+3/4)\). A finite field has \(w>0\) and \(|q|<1\), so expansion as a geometric series gives \[K_0(h,h_*)=\frac32 g(h)g(h_*)\sum_{j=0}^{\infty}q(h)^j q(h_*)^j.\] The \(j=0\) term will give \(\frac32\langle bg\rangle^2\). For the remaining terms set \(\phi_j=\sqrt{3/2}\,gq^j\), \(j\geq1\). Their diagonal sum is \[ \rho:=\sum_{j\geq1}\phi_j^2 =\frac{(1+m)v}{2}\,q^2 =\frac{(1+m)v}{2}-\frac32g^2\geq0. \tag{284}\] The bound \((1+m)v=(1-m)(1+m)^2\leq32/27\) implies \(\rho\leq16/27\). Moreover \(\langle\rho\rangle=R\), proving that \(R\geq0\). To justify the series for an arbitrary signed \(b\in L^2\), apply Cauchy–Schwarz: \[\sum_{j\geq1}\langle|b\phi_j|\rangle^2 \leq\langle b^2\rangle\sum_{j\geq1}\langle\phi_j^2\rangle =R\langle b^2\rangle.\] This proves absolute summability of the signed double-integral series, so its sum equals \(\sum_{j\geq1}\langle b\phi_j\rangle^2\). The same estimate bounds that sum by \(R\langle b^2\rangle\). Restoring the \(j=0\) term proves the kernel bound. ◻

Proof. We first reduce the temperature interval to its upper endpoint and then bound the dependence on the field. The Gaussian field has density \[p_\beta(h)=\frac1{\sqrt{2\pi}\beta} \exp\left(-\frac{\beta^2}{2}+h-\frac{h^2}{2\beta^2}\right).\] The logarithmic derivative of \(\beta^2p_\beta(h)\) is \(\beta^{-1}-\beta+h^2\beta^{-3}>0\) for \(\beta<1\). Thus, for fixed \(y\), positivity of the integrands bounds the corresponding integrals in \(D\) and \(k\) by those at \(\beta=1/2\). At that endpoint, \(D(h)=d(m(h))\).

It remains to bound the endpoint expression uniformly in \(-1\leq y\leq1\). Convexity of \(k\) as a function of \(y^2\) gives \(k(y)\leq k(0)+(k(1)-k(0))y^2\). To treat \(d\), take \(A=2-m>1\) and use the identity \[\frac1{A-y}=\frac1A+\frac{y}{A^2} +\frac{y^2}{A^2(A-y)}\] which, together with \(A-y\geq A-1\), gives, for every \(-1\leq y\leq1\), \[d(y)\leq d(0)+d'(0)y+ y^2\bigl(d(1)-d(0)-d'(0)\bigr).\] After adding \(\Lambda{}\), the quadratic bounds give the following upper bound for \(\Lambda{}+d+k\): \[ \max\{.45+d(0)+k(0),\ .70+k(1)+d(1)-d'(0)\} +|d'(0)-.06|. \tag{288}\] We evaluate this bound using slightly enlarged intervals: replace the enclosures in Equation (285) by radius \(.0003\) about the respective centers \[.4277,\quad .0899,\quad .0926,\quad .0523,\quad .2289.\] The two entries of the maximum are then at most \(.9709\) and \(.9674\), and the absolute-value term is at most \(.0080\). Their sum proves Equation (287). ◻

Proof. We reduce the inequality to a quadratic form in the constant and centered parts of \(b\). Write \(b=A+b_0\), with \(\langle b_0\rangle=0\), and set \(H_0=\langle b_0h\rangle\), \(B_0=\langle b_0v\rangle\), \(G_0=\langle b_0g\rangle\). Replace the kernel term by Lemma 85, and call the resulting upper form \(\mathcal C_\beta^+\). Equation (281) makes its purely constant part \(\beta^2(a+\tfrac32G^2+R-a^2/l)A^2\). Define the two slacks \[ \begin{split} U_0&=l+.018+\beta^2\left(\frac{a^2}{l}-a-\frac32G^2\right) -\beta^2R,\\ U_1&=.4644-\frac{\beta s_v}{2}+\beta^2\langle mv\rangle -\beta^2\left(\frac32s_g^2+R\right). \end{split} \tag{292}\] Expanding the upper form around \(A+b_0\) and subtracting the desired upper bound gives \[ \begin{split} \mathcal C_\beta^+(b)-\langle(\Lambda{}+.018)b^2\rangle ={}&-U_0A^2+A\langle b_0F\rangle+H_0B_0 +\frac32\beta^2G_0^2\\ &+\beta^2R\langle b_0^2\rangle -\frac{\beta^2B_0^2}{4l} -\langle(\Lambda{}+.018)b_0^2\rangle. \end{split} \tag{293}\] The coefficient of the mixed term can be checked directly. The first two terms in Equation (289) give \((v+a)h+2\beta^2mv\); the negative square gives \(-\beta^2av/l\); and the kernel bound gives \(3\beta^2Gg\). Subtracting \(2\langle \Lambda{}b_0\rangle\) contributes \(.50v+.12m\), up to a constant whose pairing with \(b_0\) vanishes. These contributions sum to \(F\) from Equation (286).

We next bound the terms quadratic in \(b_0\). On the centered space, \[H_0B_0\leq\frac{\beta s_v+\mathop{\mathrm{Cov}}(h,v)}2\langle b_0^2\rangle =\left(\frac{\beta s_v}{2}-\beta^2\langle mv\rangle\right) \langle b_0^2\rangle.\] For this inequality, the symmetric rank-two operator associated with \(u=h-\langle h\rangle\) and \(z=v-a\) has largest eigenvalue \((\|u\|_2\|z\|_2+\langle uz\rangle)/2\). Together with \(G_0^2\leq s_g^2\langle b_0^2\rangle\) and \(\Lambda{}+.018\geq.4644\), this controls all centered quadratic terms. Discarding the negative \(B_0^2\) term from Equation (293) yields \[ \mathcal C_\beta(b)-\langle(\Lambda{}+.018)b^2\rangle \leq-U_0A^2-U_1\langle b_0^2\rangle+A\langle b_0F\rangle. \tag{294}\] By Cauchy–Schwarz, the mixed term is at most \(s_F|A|\|b_0\|_2\). Thus positivity of \(U_0,U_1\) and the bound \(s_F<2\sqrt{U_0U_1}\) will finish the proof. We verify these inequalities on the entire temperature interval, using the full radius \(.001\) of Table 1.

First we establish the monotonicities needed between table entries. The density of \(|h|\) is proportional to \(\beta^{-1}e^{-\beta^2/2-h^2/(2\beta^2)}\cosh h\) on \(h\geq0\); its ratio at a larger and a smaller \(\beta\) is increasing in \(h\). To see the implication for expectations, let \(\ell\) be this ratio and \(f\) a decreasing function. For independent \(X,X'\) under the smaller-\(\beta\) law, \[\mathop{\mathrm{Cov}}(f(X),\ell(X)) =\frac12\mathbb E[(f(X)-f(X'))(\ell(X)-\ell(X'))]\leq0.\] Since \(\mathbb E\ell(X)=1\), the expectation of \(f\) decreases as \(\beta\) increases. In particular \(a\) decreases. For \(G\), average \(g\) over the conditional signs and write \(u=\tanh^2|h|\); the resulting function is \[\frac{(1-u)(1.75+u)}{1.75^2-u}.\] Its derivative, \(4(64u^2-392u-35)/(16u-49)^2<0\) for \(0\leq u\leq1\), proves that \(G\) decreases as well. The pointwise increase of \(\beta^2p_\beta\) from Lemma 87 gives the other needed monotonicities: both \(\beta^2\langle mv\rangle=\beta^2\langle m^2v\rangle\) and \(\beta^2R\) increase. For \(\beta^2s_v^2\) and \(\beta^2s_g^2\), write each weighted variance as the infimum over constant centers of the integral of a nonnegative squared function against \(\beta^2p_\beta\). Each such integral increases, so the infimum does too.

Now take successive temperatures \(j<z\) from Table 1. Enclose \(a\) on this interval by taking \(a_-\) to be the center at \(z\) minus \(.001\), and \(a_+\) the center at \(j\) plus \(.001\). Define \(G_-,G_+\) in the same way, and set \(l_-=.64-.19a_+\), \(l_+=.64-.19a_-\). For any endpoint quantity \(X_t\), write \(X_t^+\) and \(X_t^-\) for its table center plus and minus \(.001\). The monotonicities above give the following lower bounds for the slacks throughout \([j,z]\): \[ \begin{split} \underline U_0={}&l_-+.018 +j^2\left(\frac{a_-^2}{l_+}-a_--\frac32G_+^2\right)-z^2R_z^+,\\ \underline U_1={}&.4644-\frac{z(s_v)_z^+}{2} +j^2\langle mv\rangle_j^- -z^2\left(\frac32((s_g)_z^+)^2+R_z^+\right). \end{split} \tag{295}\] For the first bound, use that \(a^2/l-a\) increases in \(a\) and decreases in \(l\) on \(.795\leq a\leq.891\), \(.45\leq l\leq.489\). The parenthesized coefficient in that line is positive on each interval; its explicit lower bounds are given below.

To control the mixed coefficient, compare its standard deviation to the value at the right endpoint. For \(j\leq\beta\leq z\), the density ratio satisfies \[\sup_h\frac{p_\beta(h)}{p_z(h)} =\frac z\beta\exp\left(\frac{z^2-\beta^2}{2}\right) \leq\frac zj\exp\left(\frac{z^2-j^2}{2}\right).\] Choose a constant center before changing measure. This bounds the standard deviation of any function by its standard deviation under \(p_z\) times the square root of the displayed factor. Next compare the coefficients of \(F\) at \(\beta\) and \(z\), and apply the triangle inequality for the centered \(L^2(p_z)\) seminorm. With \(s_h=z\) under \(p_z\), this gives \[ \begin{split} \overline s_F={}& \sqrt{\frac zj\exp\left(\frac{z^2-j^2}{2}\right)} \biggl[(s_F)_z^++(a_+-a_-)z +2(z^2-j^2)(s_{mv})_z^+\\ &\hspace{15mm} +\left(\frac{z^2a_+}{l_-}-\frac{j^2a_-}{l_+}\right)(s_v)_z^+ +3(z^2G_+-j^2G_-)(s_g)_z^+\biggr]. \end{split} \tag{296}\] Substituting the full enclosures from Table 1 gives the interval bounds in Table 2. Appendix 17 specifies their directed-arithmetic verification. In the same interval order, the parenthesized coefficients in the first line of Equation (295) exceed \(.229787,.152558,.111290,.074675\), confirming the positivity used above.

The table therefore provides uniform margins \(U_0>.513\), \(U_1>.426\), \(s_F<.920\), and \(2\sqrt{U_0U_1}>.930\). These margins absorb the mixed term: \[s_F|A|\|b_0\|_2 \leq\frac{92}{93}\bigl(U_0A^2+U_1\|b_0\|_2^2\bigr).\] Substitution in Equation (294) makes its right side at most \(-(.426/93)\|b\|_2^2\), and in particular nonpositive. This proves Equation (291) for every \(b\in L^2\). ◻

Proof of Proposition 84. We can now assemble the two bounds. Insert the coherent-form estimate of Lemma 88 into Equation (290), and then apply the diagonal envelope of Lemma 87: \[\mathcal V_\beta(b,r) \leq\langle(D+k(m)+\Lambda{})(b^2+r^2)\rangle+.018\langle b^2\rangle \leq.9789+.018=.9969.\] The argument applies to the full function class in the proposition. The coefficients \(h,hv\) belong to \(L^2\), all other coefficient functions are bounded, and Lemma 85 already justifies the kernel integral for signed, unbounded \(b\). The quantity under the last radical in Equation (250) is at most \(4\langle b^2+r^2\rangle=4\), making its pairing with \(r\) finite. Thus Fubini for the nonnegative quadratic integrals and every use of Cauchy–Schwarz are valid without an infinity-norm bound on either function. The estimate consequently also covers concentration on arbitrarily rare field sets. ◻

Proof. Let \(G(r,x)=U(S_r f)(x)\). Equation (298) gives \(-\partial_rG+L G+VG\ge0\). Following the chain forward while running the semigroup parameter backward therefore gives nonnegative drift for \[\exp\!\left(\int_0^u V(X_v)\,dv\right)G(d-u,X_u), \qquad 0\le u\le s,\] so the finite state space and bounded time interval justify taking expectations. This proves Equation (299). For the rough envelope, take \(s=d=r\) and bound the exponential weight using \(V\le A\). This gives Equation (300). ◻

Proof. If \(\mathcal B\) is empty there is nothing to prove. Otherwise we first enlarge it by a small fixed fraction of the coordinates and show that the enlargement still has exponentially small Gibbs mass. Let \(d_H\) denote Hamming distance and, for \(0<\delta<1/2\), set \[\mathcal B^{(\delta)} =\{x:d_H(x,\mathcal B)\le\lfloor\delta n\rfloor\}.\] If \(d_H(x,y)\le\delta n\), then \[\left|\frac{x^{\mathsf T}Jx-y^{\mathsf T}Jy}{2}\right| =\frac12\left|(x-y)^{\mathsf T}J(x+y)\right| \le2K\sqrt\delta\,n.\] Thus the energy change within the neighborhood is controlled by the operator norm of the interaction. Counting the neighbors of each \(y\in\mathcal B\), with multiplicity, gives \[ \mu(\mathcal B^{(\delta)}) \le \mu(\mathcal B)e^{2K\sqrt\delta\,n} \sum_{j\le\delta n}\binom nj \le \exp\!\left[-n\bigl(\kappa-2K\sqrt\delta-h(\delta)\bigr)\right], \tag{302}\] where \(h(\delta)=-\delta\log\delta-(1-\delta)\log(1-\delta)\). The binomial bound follows by inserting \(z=\delta/(1-\delta)<1\) into \((1+z)^n\) and using \(z^j\ge z^{\delta n}\) for \(j\le\delta n\). Choose a fixed \(\delta\) so small that \(2K\sqrt\delta+h(\delta)\le\kappa/2\). Equation (302) then bounds the neighborhood mass by \(e^{-\kappa n/2}\).

We next use this small equilibrium mass to bound the probability of the neighborhood after burn-in. For probability measures \(\nu,\mu\) and an event of probabilities \(p,q\), respectively, conditioning on the event and its complement gives \[\mathop{\mathrm{Ent}}_\mu\!\left(\frac{\nu}{\mu}\right) \ge p\log\frac pq+(1-p)\log\frac{1-p}{1-q} \ge p\log\frac1q-\log2.\] Apply this inequality to \(\nu=S_{d_0}(x,\cdot)\) and the neighborhood, choosing the burn-in time to be \[d_0=1+C_L\log\max\{1,8C_E/\kappa\}.\] By Equation (297), for sufficiently large \(n\), \[\mathbb P_x(X_{d_0}\in\mathcal B^{(\delta)}) \le\frac{2C_E}{\kappa}e^{-d_0/C_L} +\frac{2\log2}{\kappa n} \le\frac12.\]

At least half the law is now outside the neighborhood. To strengthen this fact to avoidance of the original set, consider \(g(x)=d_H(x,\mathcal B)/\sqrt n\). Its half-differences have magnitude at most \(1/(2\sqrt n)\), so \(U(g)\le1/4\). Outside \(\mathcal B^{(\delta)}\) the distance is greater than \(\delta n\). The integer rounding in the neighborhood definition therefore gives \[S_{d_0}g(x)\ge\frac{\delta\sqrt n}{2}.\] Equation (300) permits the envelope \(b(r)=e^{Ar}/4\) for \(0\le r\le d_0\). Put \(Q=e^{Ad_0/2}\). Then \(B\le Q^2/4\) and \(I\le d_0Q^2/4\) in Lemma 3. Taking \(\theta=-1/Q\) in Lemma 3 yields \[\mathbb P_x(X_{d_0}\in\mathcal B) \le\exp\!\left(\frac{e d_0}{2} -\frac{\delta\sqrt n}{2Q}\right) \le e^{-c_{\mathrm a}\sqrt n}, \qquad c_{\mathrm a}=\frac{\delta}{4Q},\] after increasing the fixed minimum dimension. The event on the left is exactly \(\{g(X_{d_0})=0\}\), so the negative exponential moment controls precisely the required lower tail. This proves avoidance at the burn-in time. To obtain the same bound at every deterministic \(u\ge d_0\), apply the Markov property to the last burn-in interval: \[\mathbb P_x(X_u\in\mathcal B) =\mathbb E_x\!\left[\mathbb P_{X_{u-d_0}}(X_{d_0}\in\mathcal B)\right] \le q_n.\] Every choice depended only on the constants collected at the start of this section. The estimate is therefore uniform in the starting point and in the disorder on the good event. ◻

Proof. Measure only the exceptional occupation after burn-in: let \(Z_n=\int_{d_0}^{T_n}\mathbf 1_{\mathcal B}(X_u)\,du\) when \(T_n\ge d_0\), and put \(Z_n=0\) otherwise. Integrating Proposition 90 and applying Markov’s inequality gives \[\mathbb P_x(Z_n>1)\le\mathbb E_x Z_n\le T_nq_n.\] On \(\{Z_n\le1\}\), the total bad-set occupation up to any \(s\le T_n\) is at most \(d_0+1\). Therefore \[\exp\!\left(\int_0^s V(X_u)\,du\right) \le C_{\mathrm{dyn}}e^{-\kappa s} \quad\hbox{on }\{Z_n\le1\}.\] The rough bound \(e^{AT_n}\) applies to the exponential weight on the complementary event. Split the expectation in Lemma 89 accordingly. For occupation at most one, use the displayed weight bound and enlarge the expectation of the nonnegative endpoint quantity to the whole space. For larger occupation, use its supremum and the probability bound above. These two contributions are exactly the terms in Equation (304); occupation and endpoint gradient need not be independent. This also covers \(s<d_0\), since the occupation up to \(s\) is then at most \(d_0\). Finally, \(e^{-c_{\mathrm a}\sqrt n}\) dominates every fixed power of \(n\), which proves the stated decay of the error. ◻

Proof. For a linear observable, \(U(\ell_a)=\|a\|^2\), so the endpoint gradient in the propagation estimate is constant. Taking \(s=d=r\) in Equation (304) gives the deterministic envelope \[\|U(S_r\ell_a)\|_\infty \le b(r):=L^2\bigl(C_{\mathrm{dyn}}e^{-\kappa r}+r_n\bigr), \qquad 0\le r\le T_n.\] For sufficiently large \(n\), both \(r_n\le1\) and \(T_nr_n\le1\). Consequently \(\sup b\le B_L\) and \(\int_0^{T_n}b(r)\,dr\le I_L\). These are the two quantities needed by Lemma 3, which gives both the transient exponential-moment bound and the variance bound.

The equilibrium variance uses the gap input instead. Applying its Dirichlet-form inequality gives \[\mathop{\mathrm{Var}}_\mu(\ell_a) \le\gamma^{-1}\mu\!\left[\ell_a(-L)\ell_a\right] =\gamma^{-1}\sum_i a_i^2\mu(v_i) \le\gamma^{-1}\|a\|^2.\] The energy identity follows by conditional expectation: \(\mu(x_i m_i)=\mu(m_i^2)\), so the coordinate energy is \(\mu(v_i)\). Spin inversion in the zero-field model supplies the zero mean in Equation (306). Since the gradient envelope holds on the entire coefficient ball, none of these bounds requires \(a\) to be chosen before the disorder or the starting configuration. ◻

Proof. Write \(\sigma_i(x)=x_i\). Spin inversion gives \(\mu(\sigma_i)=0\) and \(\mu(\sigma_i^2)=1\), while \[\mu\!\left[\sigma_i(-L)\sigma_i\right]=\mu(v_i)\le1.\] Use the spectral decomposition of the finite-dimensional self-adjoint operator \(-L\). The squared coefficients of \(\sigma_i\) form nonnegative weights summing to one, and the preceding energy is their mean eigenvalue. Jensen’s inequality for \(\lambda\mapsto e^{-d\lambda}\) therefore yields \[ \mu\!\left[\sigma_i S_d\sigma_i\right] \ge\exp\!\left(-d\mu[\sigma_i(-L)\sigma_i]\right) \ge e^{-d}. \tag{309}\] Sum these correlations over \(i\). Since their sum is an average over the initial state under \(\mu\), some configuration \(x=x(d,J)\) satisfies \[S_d\ell_{x/\sqrt n}(x) =\frac1{\sqrt n}\sum_i x_i S_d\sigma_i(x) \ge\sqrt n e^{-d}.\] The coefficient vector \(x/\sqrt n\) has norm one, so Proposition 92, with \(D_*=1\) and \(L=1\), applies even to this choice. Let \(V_*=4I_1\). The displayed observable has transient variance at most \(V_*\), equilibrium mean zero, and equilibrium variance at most \(\gamma^{-1}\). Let its transient mean be \(m\ge\sqrt n e^{-d}\). The event that the observable exceeds \(m/2\) separates the transient law from equilibrium. Applying Chebyshev’s inequality under both laws gives, for \(0\le d\le\log n\), \[d_c(d)\ge1-\frac{4(V_*+\gamma^{-1})}{m^2} \ge1-\frac{4(V_*+\gamma^{-1})e^{2d}}{n}.\] Taking \(d=(\log n)/4\) proves Equation (307).

For the upper bound, the entropy-to-total-variation estimate proved in Proposition 77, applied to Equation (297), gives \[d_c(t)^2\le\frac12 C_E n e^{-t/C_L}.\] Hence \(t_n^{\rm med}\le C_L\log(2C_E n)\) for sufficiently large \(n\). Together with Equation (307), this proves Equation (308). It remains to justify the exact median identity. The function \(d_c\) is continuous, as a maximum of finitely many continuous transition-law distances, and the entropy bound shows that it tends to zero. Since \(d_c(0)=1-\min_x\mu(x)\ge1-2^{-n}>1/2\), its first crossing is attained and satisfies \(d_c(t_n^{\rm med})=1/2\). ◻

Proof. Take the common part of the measures, \(\rho(x)=\min\{\nu(x),\pi(x)\}\), whose mass is \(q=1-\|\nu-\pi\|_{\mathrm{TV}}\ge\varepsilon\). We compare each mean to the mean under the normalized common part. Since \(\rho\le\nu\), Cauchy–Schwarz gives \[\left|\nu H-\frac{\rho H}{q}\right| =\frac1q\left|\sum_x\rho(x)(H(x)-\nu H)\right| \le\sqrt{\frac{\mathop{\mathrm{Var}}_\nu H}{q}}.\] The same calculation for \(\pi\) bounds the other difference from the common mean. The triangle inequality, followed by the lower bound on the common mass, proves the assertion. ◻

Proof. First control the means. Contraction of total variation implies that the law \(\nu_{x,z}\) of \(X_z\) from any point \(x\) has common mass at least \(\varepsilon\) with \(\mu\) for every \(z\ge t\). Proposition 92 bounds \(\mathop{\mathrm{Var}}_{\nu_{x,z}}h_i\) uniformly in \(i,x,z\), and the equilibrium variance is at most \(K^2/\gamma\). Lemma 94 therefore gives a constant \(B_\varepsilon\) such that \[ |\mathbb E_x h_i(X_z)|\le B_\varepsilon \qquad(t\le z\le D_*\log n). \tag{313}\]

Next control the deviations from these means. For every fixed \(\theta>0\), Proposition 92 bounds both centered exponential moments by a common constant \(M_{\theta,K}\). Apply the exponential bound to both tails of each field and then take a union bound over the coordinates. Together with Equation (313), this gives \[ \mathbb P_x\left[\max_i|h_i(X_z)|>\delta\log n\right] \le 2M_{\theta,K}e^{\theta B_\varepsilon} n^{1-\theta\delta}. \tag{314}\] For a prescribed \(p>0\), first fix \(\theta\) with \(\theta\delta>p+1\), and then let \(n\) tend to infinity. Define \(\tau_n\) as the supremum of the displayed probabilities over the indicated parameters and good disorders. The same sequence therefore has every asserted inverse-power decay rate. The uniformity is over deterministic times. In the smoothing argument we will integrate the resulting expectation bounds, so a simultaneous field event along the whole sample path is unnecessary. ◻

Proof. Fix a starting point \(x\) and a real test function with \(\|f\|_\infty\le1\). We first express the transient variance as an integrated martingale energy. Set \(T=t+s\) and \[F_u=S_{T-u}f,\qquad G(u)=\mathbb E_x\Gamma(F_u)(X_u),\qquad 0\le u\le T.\] Here \(F_u(X_u)\) is the backward martingale with terminal value \(f(X_T)\). Differentiating its second moment and using \((\partial_u+L)F_u=0\) gives the exact identity \[ \mathop{\mathrm{Var}}_x(F_r(X_r))=2\int_0^rG(u)\,du, \qquad 0\le r\le T. \tag{315}\] Since \(\|F_u\|_\infty\le1\), this implies \[ \int_0^T G(u)\,du\le\frac12, \qquad \mathop{\mathrm{Var}}_x(S_sf(X_t))=2\int_0^tG(u)\,du. \tag{316}\]

The total energy is thus bounded. To show that little of it lies before the observation time, fix deterministic \(u\le t\) and \(z\in[t+s/2,t+s]\). Apply Equation (304) starting at \(X_u\), with duration \(T-u\) and comparison time \(z-u\). These parameters lie within the fixed horizon, and the estimate holds for every possible value of \(X_u\) with the same constants. Integrating against its law and using the Markov property yields \[\mathbb E_x U(F_u)(X_u) \le C_{\mathrm{dyn}}e^{-\kappa(z-u)} \mathbb E_x U(F_z)(X_z)+n r_n.\] The bound on the remainder uses \(U(F_z)\le n\): each half-difference of a function bounded by one has magnitude at most one. Combining this with \(\Gamma\le2U\) and the separation \(z-u\ge s/2\) gives \[ G(u)\le 2C_{\mathrm{dyn}}n^{-\kappa\alpha/2} \mathbb E_x U(F_z)(X_z)+2n r_n. \tag{317}\]

We still need to replace the late unweighted gradient by the late martingale energy. The field-tail estimate supplies the required lower bound on the heat-bath weights. Choose a fixed \(\delta\) with \(0<\delta<\kappa\alpha/4\), and let \(\tau_n\) be the sequence in Lemma 95. At every configuration, \[w_i=1-x_i\tanh h_i \ge 1-|\tanh h_i| =\frac{2}{1+e^{2|h_i|}}\ge e^{-2|h_i|}.\] On the event \(\max_i|h_i(X_z)|\le\delta\log n\) this gives \(U(F_z)(X_z)\le n^{2\delta}\Gamma(F_z)(X_z)\). On the complementary event use \(U(F_z)\le n\). These are pointwise bounds, so they remain valid even though the fields and \(F_z\) depend on the same configuration. Taking expectations gives \[ \mathbb E_xU(F_z)(X_z)\le n^{2\delta}G(z)+n\tau_n. \tag{318}\] Define the positive exponent and the deterministic remainder \[\xi=\frac{\kappa\alpha}{2}-2\delta>0, \qquad R_n=2n r_n+2C_{\mathrm{dyn}}n\tau_n.\] Equations (317) and (318) imply \[ G(u)\le 2C_{\mathrm{dyn}}n^{-\xi}G(z)+R_n \quad\text{for }u\le t,\quad t+s/2\le z\le t+s. \tag{319}\] The choice of the field threshold leaves a strictly positive decay exponent, and \(R_n\) decreases faster than every inverse power of \(n\). We have therefore compared each early energy to every energy in the late interval, with uniform error. Average Equation (319) over the interval of \(z\) of length \(s/2\), and then integrate over \(0\le u\le t\). The total energy budget in Equation (316) gives \[ \mathop{\mathrm{Var}}_x(S_sf(X_t)) \le 4C_{\mathrm{dyn}}\frac{t}{s}n^{-\xi}+2tR_n \le 4C_{\mathrm{dyn}}\frac{D_*}{\alpha}n^{-\xi} +2D_*\log n\,R_n=:V_n. \tag{320}\] Thus \(V_n\to0\) uniformly in \(f,x,t\) and in the good disorder, proving the required transient variance bound. The integrations concern deterministic expectations of finite-state semigroup expressions; their measurability and nonnegativity justify interchanging the two integrals by Tonelli’s Theorem. No random observation time or union over a continuum of field events enters this step.

It remains to control the equilibrium variance and compare the two means. The uniform spectral gap from Proposition 76 gives \(\frac{d}{dr}\mathop{\mathrm{Var}}_\mu(S_rf)\le-2\gamma\mathop{\mathrm{Var}}_\mu(S_rf)\). Integrating yields \[ \mathop{\mathrm{Var}}_\mu(S_sf)\le e^{-2\gamma s}\mathop{\mathrm{Var}}_\mu(f) \le n^{-2\gamma\alpha}. \tag{321}\] Both variance bounds now apply to the same observable. Use Lemma 94 for the law of \(X_t\) from \(x\) and for \(\mu\), with observable \(S_sf\). The common-mass hypothesis is exactly the assumed distance bound at the observation time, and the two expectations are \(S_{t+s}f(x)\) and \(\mu f\). Hence \[|S_{t+s}f(x)-\mu f| \le \varepsilon^{-1/2} \bigl(\sqrt{V_n}+n^{-\gamma\alpha}\bigr).\] Taking one half of the supremum over \(x\) and \(\|f\|_\infty\le1\) proves Proposition 96. ◻

Proof. Proposition 93 gives fixed constants \(c,C>0\) such that \(c\log n\le t_n^{\rm med}\le C\log n\) for all sufficiently large \(n\), and \(d_c(t_n^{\rm med})=1/2\). Fix \(b\in(0,1)\) and set \(\alpha=bc/2\). Fix the horizon constant \(D_*>(1+b)C\) so that both applications of positive-lag smoothing lie within its range. First apply Proposition 96 at \(t=t_n^{\rm med}\). This gives \(d_c(t_n^{\rm med}+\alpha\log n)\to0\). Since \(\alpha\log n\le bt_n^{\rm med}/2\), monotonicity yields the upper conclusion.

For the lower side, argue by contradiction while preserving uniformity in the disorder. If the conclusion failed uniformly, there would be a subsequence of dimensions, a good disorder at each of those dimensions, and a fixed \(\varepsilon_0>0\) such that \(d_c((1-b)t_n^{\rm med})\le1-\varepsilon_0\). Proposition 96 would then give vanishing distance at time \[(1-b)t_n^{\rm med}+\alpha\log n \le t_n^{\rm med}-\frac{bc}{2}\log n<t_n^{\rm med}.\] This contradicts the definition of \(t_n^{\rm med}\), since the distance at every time strictly below \(t_n^{\rm med}\) exceeds \(1/2\). ◻

Proof. The exact clock relation is \(S_t=e^{nt(P-I)}\), so continuous time averages discrete update attempts against a Poisson law. For the lower bound, put \(k=\lfloor(1-b)nt_n^{\rm med}\rfloor\) and \(T=(1-b/2)t_n^{\rm med}\), and let \(N\sim\operatorname{Poisson}(nT)\). Convexity of total variation and monotonicity of the discrete distance give \[ d_c(T)\le\sum_{j\ge0}\mathbb P(N=j)d_d(j) \le\mathbb P(N<k)+d_d(k). \tag{324}\] The Poisson mean exceeds \(k\) by at least \((b/2)nt_n^{\rm med}\). Chebyshev’s inequality therefore implies \(\mathbb P(N<k)=O((nt_n^{\rm med})^{-1})=o(1)\) uniformly. Theorem 97, with parameter \(b/2\), gives \(d_c(T)\to1\); Equation (324) proves the discrete lower bound.

For the upper bound, we must check the fixed holding condition used by the clock-comparison lemma. The operator-norm bound implies \[\frac1n\sum_i h_i(x)^2=\frac{\|Jx\|^2}{n}\le K^2.\] At least half the fields have magnitude at most \(\sqrt2K\). When spin \(i\) is selected, its chance of being retained is \(e^{x_ih_i}/(2\cosh h_i)\ge(1+e^{2|h_i|})^{-1}\). Averaging this lower bound over the selected coordinate gives \[ P(x,x)\ge h_0:=\frac{1}{2(1+e^{2\sqrt2K})}>0. \tag{325}\] Choose \(q=1-h_0/2\in(0,1)\) once for all \(n\) and all good disorders, and write \[P=(1-q)I+q R, \qquad R=\frac{P-(1-q)I}{q}.\] Subtracting this fixed holding component leaves a Markov kernel \(R\) with stationary measure \(\mu\). In terms of its powers, the two clocks become \[ S_t=e^{q nt(R-I)},\qquad P^M=\sum_{j=0}^M\binom Mjq^j(1-q)^{M-j}R^j. \tag{326}\]

We now verify convergence of every fixed shifted Poisson average, as required by Lemma 4. Set \(M=\lceil(1+b)nt_n^{\rm med}\rceil\). For any \(x\in\Omega_n\) and any real function \(f\) with \(\|f\|_\infty\le1\), define \(u_j=(R^jf)(x)-\mu f\), so that \(|u_j|\le2\). For fixed \(y\in\mathbb R\), Equation (326) identifies the Poisson average with mean \(m=q M+y\sqrt M\) as \[\mathbb E\bigl[u_{\operatorname{Poisson}(m)}\bigr] =S_{m/(q n)}f(x)-\mu f.\] Uniformly over the good disorders, \(t_n^{\rm med}\) is between fixed positive multiples of \(\log n\). Hence \[\frac{m}{q n} =\frac Mn+\frac{y\sqrt M}{q n} \ge(1+b/2)t_n^{\rm med}\] for all sufficiently large \(n\): the correction is \(O_y(\sqrt{\log n/n})\), whereas \((b/2)t_n^{\rm med}\) is of order \(\log n\). Theorem 97 and monotonicity bound this Poisson average in absolute value by \(2d_c((1+b/2)t_n^{\rm med})=o(1)\), uniformly over \(x\), \(f\), and the good disorder. Also \(M\) tends uniformly to infinity. All hypotheses of Lemma 4 are now uniform over the indicated family, so its binomial average tends uniformly to zero. Equation (326) identifies that average with \(P^Mf(x)-\mu f\). Taking one half of the supremum over \(x\in\Omega_n\) and real \(f\) with \(\|f\|_\infty\le1\) proves the upper bound in Equation (323). ◻

Completion of the proof of Theorem 72. It remains to express discrete cutoff as the asserted ratio of mixing times and then return to disorder probability. Fix \(\varepsilon\in(0,1/2)\) and \(\eta>0\). Choose once \[0<b<\frac{\eta}{2+\eta}, \qquad\text{so that}\qquad \frac{1+b}{1-b}<1+\eta.\] Proposition 98 implies, for all sufficiently large \(n\) and every disorder in \(\mathcal G_n\), \[t_{n,\beta,J}(1-\varepsilon)> \lfloor(1-b)nt_n^{\rm med}\rfloor, \qquad t_{n,\beta,J}(\varepsilon)\le \lceil(1+b)nt_n^{\rm med}\rceil.\] The first bound ensures a positive denominator and controls the integer rounding in the ratio. Hence \[\frac{t_{n,\beta,J}(\varepsilon)} {t_{n,\beta,J}(1-\varepsilon)} \le\frac{\lceil(1+b)nt_n^{\rm med}\rceil} {\lfloor(1-b)nt_n^{\rm med}\rfloor+1} \le\frac{1+b}{1-b}+\frac{1}{(1-b)nt_n^{\rm med}}<1+\eta\] for all sufficiently large \(n\), uniformly on \(\mathcal G_n\). The last step uses \(t_n^{\rm med}\ge c\log n\). Since \(\mathbb P_J(\mathcal G_n)\to1\), the probability of the event in Theorem 72 is at most \(\mathbb P_J(\mathcal G_n^c)\) for large \(n\), and therefore tends to zero. This last step uses only the probability of the good event: all preceding estimates hold for each fixed disorder, uniformly over the starting state and the test function. No coupling of disorders across dimensions is needed. ◻

Proof of Lemma 86. We now combine the finite enclosures with the transfer estimates. Every quantity in Table 3 is within \(.00065\) of its center in Table 1. Among the standard deviations, the largest transfer error is Equation (341); the three standard deviations not involving \(F\) require only Equation (338). For the first moments \(a,G,\langle mv\rangle,R\), the bounded-integrand error is at most \(e_a\), since Equation (284) bounds the integrand of \(R\) by one as well. Thus \[.00065+.000165=.000815<.001\] establishes every claimed interval in Table 1. To finish the five additional endpoint bounds, observe that every scaled real integrand in Equation (334) is bounded by one, so its normalization and quadrature error is at most \(e_a\). Adding this to the finite enclosures in Equation (335) proves the five strict intervals in Equation (285). All numerical comparisons used in this proof follow from the directed operations specified above. ◻