Proof. We use the eigenvalue-only event of (OpenAI 2026c, Proposition 3.1). For every deterministic \(L_n\to\infty\) with \(L_n\le\log n\), that proposition gives positive denominators and, uniformly for \(L_nn^{-2/3}\le u\le M\) with fixed \(M\), \[ \begin{split} m_1(u):=\frac1n\sum_i(d_i+u)^{-1} &=\frac{2+u-\sqrt{u(4+u)}}2+o(\sqrt u),\\ m_2(u):=\frac1n\sum_i(d_i+u)^{-2} &\le C_Mu^{-1/2},\\ \lambda_j&=2+o(L_nn^{-2/3})\quad(1\le j\le3), \qquad \|W\|\le3 . \end{split} \tag{37}\] The errors have deterministic uniform bounds on an event of probability at least \(1-Ce^{-cL_n^{1/4}}\).

For (36), take \(\mathcal S_n^{\rm glob}\) to be the separate source event with \(L_n=\sqrt{\log n}\). Its probability tends to one, and the edge and norm assertions follow directly from (37). For every fixed \(u>1\), the first resolvent converges uniformly there to \[\frac{2+u-\sqrt{u(4+u)}}2 =\int\frac{\rho_{\rm sc}(\,\mathrm dx)}{2+u-x}.\] The norm bound makes the empirical spectral measures a compact family. Every weak subsequential limit therefore has the displayed transform for every \(u>1\), and uniqueness of the Stieltjes transform identifies it with \(\rho_{\rm sc}\). A contradiction by subsequences gives this weak convergence uniformly over the source event. Intersecting with this event does not change the tight-threshold convention.

Let \(K_n\to\infty\) be arbitrary and set \(L_n=\min\{K_n^{1/3},\sqrt{\log n}\}\), changing finitely many initial values if necessary. For \(np^3\ge K_n\), \(L_nn^{-2/3}=o(p^2)\). Thus, for every fixed \(v>0\), \[m_1(vp^2)=1-\sqrt v\,p+O_v(p^2)+o_v(p)\] uniformly in the scale range, and \(d_1/p^2\ge-o(1)\). For \(u\ge cp^2\), the counted denominators satisfy \(0<d_i+u\le2u\). Consequently \[\#\{d_i\le u\}\le4u^2\sum_i(d_i+u)^{-2}\le Cnu^{3/2}.\] Summing on dyadic bands gives the second bound in (35); the last band above \(u=4\) costs \(O(p)\) by \(\|W\|\le3\), and obeys the same bound.

For fixed \(v,w>0\), the transform difference is exactly \[ \int\left(\frac1{x+v}-\frac1{x+w}\right)\nu_p(\mathrm dx) =\frac{m_1(vp^2)-m_1(wp^2)}p \longrightarrow\sqrt w-\sqrt v. \tag{38}\] To identify the limit including the tail, put \(\omega_p(\mathrm dx)=(1+x)^{-2}\nu_p(\mathrm dx)\). These finite measures have bounded mass, the tail bound (35), and negative support in an interval shrinking to zero. Every weak limit \(\omega\) is therefore supported on \([0,\infty)\). Taking \(w=1\) in (38) gives \[\int\frac{1+x}{x+v}\,\omega(\mathrm dx)=\frac1{1+\sqrt v}.\] Letting \(v\to1\) gives \(\omega([0,\infty))=1/2\), and hence \[\int\frac{\omega(\mathrm dx)}{x+v} =\frac1{2(1+\sqrt v)^2}.\] Uniqueness of the Stieltjes transform identifies \(\omega(\mathrm dx)=\pi^{-1}\sqrt x(1+x)^{-2}\mathrm dx\). The continuous distribution function of the corresponding limit of \(\nu_p\) gives uniformity in endpoints on compact intervals. Failure of uniformity over \(p\) would provide a sequence of scales to which this same compactness argument applies, a contradiction.

The clipped second trace has limit \[\frac p n\mathop{\mathrm{Tr}}R_{p,0}^2 =\int(\max\{x,1\})^{-2}\nu_p(\mathrm dx) \longrightarrow \frac1\pi\left(\int_0^1\sqrt x\,\mathrm dx+ \int_1^\infty x^{-3/2}\,\mathrm dx\right) =\frac8{3\pi}.\] For the first trace, subtract an ordinary resolvent before integrating: \[\frac{n-\mathop{\mathrm{Tr}}R_{p,0}}{np} =\frac{1-m_1(p^2)}p+ \int\left(\frac1{x+1}-\frac1{\max\{x,1\}}\right)\nu_p(\mathrm dx) \longrightarrow\frac4{3\pi}.\] The difference kernel is \(O(x^{-2})\). Its limiting integral is \(4/(3\pi)-1\), obtained by substituting \(x=t^2\) on each of \((0,1)\) and \((1,\infty)\).

These assertions hold for every arbitrarily slowly divergent \(K_n\). If (34) failed, one could choose \(K_j\to\infty\) and \(n_j\to\infty\) with failure probability bounded away from zero and embed \(K_{n_j}=K_j\) in a divergent deterministic sequence. The preceding uniform conclusion contradicts that subsequence. ◻

Proof. Put \(t=u/p^2\) and \(\mathfrak d_p(t)=(n-\mathop{\mathrm{Tr}}R_{p,tp^2})/(np)\). For \(y=\|h\|/(p^2M_p)\), the saddle equation becomes \[ \mathfrak d_p(t)=\frac{y^2}{(1+t)^2},\qquad t>-1,\qquad \mathfrak d'_p(t)=\frac p n\mathop{\mathrm{Tr}}R_{p,tp^2}^2. \tag{40}\] Lemma 4 gives \(\mathfrak d_p(0)\) arbitrarily close to \(b_0\), and \[\mathfrak d'_p(t)\ge \frac{\dim\mathcal H_p/k_p}{(1+t)^2}.\] It follows by integration that \(\mathfrak d_p(-1+\kappa)<0\) for a sufficiently small fixed \(\kappa>0\); no saddle can lie there or below. On \(t\ge0\), \(\mathfrak d_p(t)\ge b_0/2\), so (40) bounds \(t\) above when \(y\le R\). The right side of (17), viewed as a function of its parameter, is strictly decreasing and continuous, tends to infinity at \(-p^2\), and tends to zero at infinity. This also proves existence and uniqueness. Denominator comparison gives \(\mathop{\mathrm{Tr}}\widehat R^2\asymp n/p\), \(\|m\|^2\le C_Rnp\), and \(\|\widehat R\|\le C_Rp^{-2}\), hence \(\sigma^2\asymp n/p\). For a sufficiently small fixed \(s_0\), both \(\mathfrak d_p(t)\) and \(\mathfrak d'_p(t)\) are bounded above and below on \(|t|\le s_0\). Differentiating \(\ell_p(tp^2)=p^2M_p(1+t)\sqrt{\mathfrak d_p(t)}\) proves the length and derivative bounds.

We record the density argument used here and later. For independent \(Y_i=a_i+\sqrt{\upsilon_i}G_i\), suppose \[\max_i\upsilon_i\le Cp^{-2},\qquad \sigma^2:=2\sum_i\upsilon_i^2+4\sum_i\upsilon_i a_i^2 \asymp k_pp^{-4},\] and at least \(ck_p\) variances are at least \(cp^{-2}\). The Gaussian quadratic formula, as in (OpenAI 2026b, sec. 15.3, Equation (15.5)), gives \[ \left|\mathbb Ee^{it\sum_iY_i^2}\right| =\prod_i(1+4t^2\upsilon_i^2)^{-1/4} \exp\left(-\sum_i \frac{2t^2\upsilon_i a_i^2}{1+4t^2\upsilon_i^2}\right). \tag{41}\] Expansion of the centered log characteristic function through order two gives \[ \log\mathbb Ee^{it(\|Y\|^2-\mathbb E\|Y\|^2)/\sigma} =-t^2/2+O(k_p^{-1/2}|t|^3) \quad (|t|\le c\sqrt{k_p}). \tag{42}\] Indeed its remainder is at most \[C|t/\sigma|^3\sum_i(\upsilon_i^3+\upsilon_i^2a_i^2) \le Ck_p^{-1/2}|t|^3.\] The band in (41) bounds the modulus in these units by \((1+ct^2/k_p)^{-c'k_p}\). Its real logarithm is at most \(-c_0t^2\) up to a small fixed multiple of \(\sqrt{k_p}\), and its integral outside that interval is exponentially small in \(k_p\). Fourier inversion and (42) give \[ \sup_z\left|\sigma f_Y(\mathbb E\|Y\|^2+\sigma z) -(2\pi)^{-1/2}e^{-z^2/2}\right|\le Ck_p^{-1/2}. \tag{43}\] At the saddle \(\mathbb E\|Y\|^2=n\), proving the lower and upper density bounds. At any mean, the same characteristic majorant gives the upper bound \(Cp^2/\sqrt{k_p}\).

For replicas let \(T_t=\mathop{\mathrm{diag}}(t_1,\ldots,t_{m_0})\) and \(H_i=C_i^{1/2}T_tC_i^{1/2}\). The modulus of the row characteristic function is at most \(\det(I+4H_i^2)^{-1/4}\); its extra factor from a nonzero mean is at most one. The singular values of \(H_i\) are at least \(cu^{-1}\) times those of \(T_t\), including when the entries of \(T_t\) have different signs. Independence of the \(L\) rows bounds the joint characteristic function by \(\prod_j(1+ct_j^2/u^2)^{-L/4}\). Its integral proves (39). This is the argument of (OpenAI 2026b, sec. 15.3, Equation (15.6)); it explains the independence requirement on the spectral rows. ◻

Proof. We give the scale check in the polar argument of (OpenAI 2026b, sec. 15.3, Equations (15.5)–(15.8)). For an interaction \(A\), write \(R_{A,u}=(2+u-A)^{-1}\) and \[q_{A,u}(h)=c_n+\tfrac n2(2+u)+\tfrac12\log\det R_{A,u} +\tfrac12h^{\mathsf T}R_{A,u}h,\qquad c_n=\log\!\left(2^{n/2}\Gamma(n/2)n^{1-n/2}\right).\] Coarea in squared radius gives, exactly, \[ \phi_A^{\rm sp}(h)=q_{A,u}(h)+\log f_{A,u,h}(n),\qquad \partial_uq_{A,u}(h) =\tfrac12\bigl(n-\mathop{\mathrm{Tr}}R_{A,u}-\|R_{A,u}h\|^2\bigr), \tag{45}\] where \(f_{A,u,h}\) is the density of \(\|N(R_{A,u}h,R_{A,u})\|^2\). In the capped case take \(A=W_p\) and start at its saddle \(u=\alpha\). For \(u\ge p^2\), \[n-\mathop{\mathrm{Tr}}R_{p,u} \le Cn\sqrt u+Ck_pp^2/u^2\le Cn\sqrt u.\] The first term is (37); the second compares the clipped active denominators with \(d_i+u\), using \(\dim\mathcal H_p\le Ck_p\). For \(u\) in a bounded range in \(p^2\) units, the upper bound is \(C_Rnp\). Integrating (45) yields \[ q_{W_p,u}(h)-q_{W_p,\alpha}(h)\le Cnu^{3/2}+C_Rk_p, \tag{46}\] where the first constant is independent of the field radius.

For a rowwise quadratic test \(\mathcal T\) of \(m_0\) replicas and \(\lambda>0\), let \(f_{\mathcal Q}^{u,\lambda}\) be the norm density under the normalized Gaussian tilt \(e^{\lambda\mathcal T}\) of \(m_0\) independent copies of \(N(R_{p,u}h,R_{p,u})\). Exponential Markov and polar coarea give \[ \begin{split} (\mu_{p,h}^{\rm sp})^{\otimes m_0}\{\mathcal T\ge b\} \le{}& e^{m_0(q_{W_p,u}(h)-q_{W_p,\alpha}(h))-\lambda b} \mathbb E_u e^{\lambda\mathcal T} \frac{f_{\mathcal Q}^{u,\lambda}(n\mathbf1)} {f_{W_p,\alpha,h}(n)^{m_0}} . \end{split} \tag{47}\] This compares norm densities before and after the normalized tilt; it does not divide by the probability of a shrinking shell.

For a pair take \(\mathcal T=\pm X^{\mathsf T}X'\), \(b=nw\), \(u=\iota w^2\), and \(\lambda=c'u\), with \(c'>0\) a sufficiently small absolute constant. We will choose \(\iota>0\) small and then the lower multiple of \(w/p\) large, so that \(u\gg p^2\). The Gaussian formula gives \[ \log\mathbb E_u e^{\lambda\mathcal T} \le C\lambda^2\mathop{\mathrm{Tr}}R_{p,u}^2+C\lambda\|R_{p,u}h\|^2 \le Cnu^{3/2}+C_Rk_p. \tag{48}\] Indeed \(\mathop{\mathrm{Tr}}R_{p,u}^2\le Cn/\sqrt u\) and \(\|R_{p,u}h\|^2\le C_Rnp^5/u^2\); the pair quadratic has zero first trace in the centered Gaussian formula. For \(\mathcal T=\|P_pX\|^2\), \(b=nw\), its extra first trace costs only \(Ck_p\). The choice of \(c'\) preserves positive precision and covariance comparable to \(u^{-1}I_{m_0}\) on \(ck_p\) active rows. Lemma 5 bounds the density ratio in (47) by \(C_{m_0}(u/p^2)^{m_0}\).

For uncapped zero field start at \(u=p^2\). The resolvent gives \(n-\mathop{\mathrm{Tr}}(d+p^2)^{-1}\asymp np\), a leading variance comparable to \(p^{-2}\), and variance \(O(n/p)\) for the other squares. Chebyshev’s inequality puts the residual squared length to be supplied by that coordinate in \([cnp,Cnp]\) with probability \(1-C/k_p\). On this interval its Gaussian-square density is at least \(c\sqrt{p/n}e^{-Ck_p}\). Convolution gives the same lower density scale, with this extra exponential factor. The uncapped resolvent also gives (46), and the tilted upper density has the same band bound.

The logarithm of the resulting probability is at most \[-c'\iota nw^3+C\iota^{3/2}nw^3+C_Rk_p +O_{m_0}(\log(u/p^2)).\] Choose \(\iota\) so that \(C\sqrt\iota<c'/4\), and then \(w/p\) larger than a sufficiently large fixed constant. Since \(nw^3=k_p(w/p)^3\), the last two terms are absorbed by another \(c'\iota nw^3/4\) when \(K\) is large. This proves the pair bound. For projection put \(w=v^2p\le1\); then \(nw^3=k_pv^6\). For \(v>p^{-1/2}\) the event is empty because \(\|X\|=\sqrt n\). ◻

Proof. Fix errors in Lemma 4 for which the number of points with \(d_i\le p^2\) is between \(c np^3\) and \(Cnp^3\), and apply it at the countable scales \(np^3=2^jK\). For any finite list, finite edge-coordinate convergence transfers the inequalities, with slightly enlarged constants, to counts of limiting points below \((2^jK)^{2/3}\). A fixed small buffer in the endpoints may be used; the limiting process has no point at any of these countably many fixed endpoints. Increase the finite list and use continuity of probability for decreasing events. The uniform finite-\(n\) event shows that all the limiting inequalities hold with probability arbitrarily close to one when \(K\) is large. Letting that failure probability tend to zero shows that almost surely, after a finite threshold, \[cL^{3/2}\le\#\{i:g_i\le L\}\le CL^{3/2}\] on dyadic \(L\), and hence on all large \(L\) by monotonicity. Invert these inequalities, add \(b\), and adjust the bounds for the finitely many remaining positive numbers \(g_i+b\). The least integer threshold and rational bounds give measurable choices. ◻

Proof. We derive this form for cells centered at lattice points from the two-replica calculations in (OpenAI 2026b, sec. 15.4, Equation (15.10)) and (OpenAI 2026d, proof of Lemma 3.2). Set \[I(r)=\tfrac12\bigl((1+r)\log(1+r)+(1-r)\log(1-r)\bigr), \qquad J(r)=-\tfrac12\log(1-r^2).\] Uniform Stirling bounds on \(|r|\le\rho_0\) put \(p_n(r)\) between fixed multiples of \(n^{-1/2}e^{-nI(r)}\). The same bounds for the gamma ratio, and the bounded logarithmic oscillation of \(f_n\) on \(I_r\), put its cell integral between fixed multiples of \(n^{-1/2}e^{-nJ(r)}\). Since \(J(r)-I(r)=O(r^4)\), this proves (50). The assumed oscillation of the positive angular weight gives (51) by integration. The final assertion follows directly from the cell endpoints. ◻

Proof. We adapt the two-replica calculation of (OpenAI 2026d, Lemma 3.2, Equations (3.3)–(3.4)), whose stated observation range starts at \(r_n\). The estimate below the source range is proved here. Conditional on \(\rho=n^{-1}X^{\mathsf T}X'\), the Haar angular integral for a cube pair is the spherical angular integral. On the ordered orthonormal sum-and-difference frame \((q,q')\), its exponent is \[ \Psi_{p,h}(s;q,q')= \frac n2\bigl((1+s)q^{\mathsf T}W_pq +(1-s){q'}^{\mathsf T}W_pq'\bigr) +\sqrt{2n(1+s)}\,h^{\mathsf T}q. \tag{53}\] On a fixed \(|s|\le2\rho_0<1\), the spectral norm bound and the field bound \(\|h\|\le R\sqrt n\,p^{5/2}\) give \(|\partial_s\Psi_{p,h}|\le C_Rn\). Thus the positive angular weight \(e^{\Psi_{p,h}}\) has a fixed logarithmic oscillation on every width-\(2/n\) cell, as required in (51). Retain \(|\rho|\le C'r_n\), choosing \(C'\) larger than the lower tail constant in Lemma 6 times \(C_1\). On dyadic bands between \(C'r_n\) and a small fixed \(\rho_0\), the weighted cell comparison costs \(e^{C+Cn\rho^4}\), while Lemma 6 gives \(Ce^{-cn\rho^3}\). The spherical band endpoints move by at most \(1/n\), negligible compared with \(r_n\). Decreasing \(\rho_0\) makes the latter pay the former and leaves \(Ce^{-cnr_n^3}\). At overlaps bounded away from zero, the global spectral law, edge limit, and norm bound in (36) supply the global inputs for the uncapped zero-field angular comparison of (OpenAI 2026b, sec. 15.4). Indeed, for every fixed \(w>2\), they give \[\frac1n\log\det(wI-W)\longrightarrow \int\log(w-x)\rho_{\rm sc}(\,\mathrm dx);\] interlacing gives the same per-site limit for every orthogonal-hyperplane compression. The cited deficit-filling lower bound then gives the full critical free-energy normalization by taking \(n\to\infty\) at fixed \(w>2\), followed by \(w\downarrow2\). For clarity, the normalized zero-field contribution being bounded is \[\mathcal M_n(\rho_0;\boldsymbol\lambda) =E_U\!\left[L_\circ^2\, \mu_W^{\otimes2}\{|n^{-1}X^{\mathsf T}X'|\ge\rho_0\}\right].\] Here \(\boldsymbol\lambda=(\lambda_1,\ldots,\lambda_n)\) is fixed, \(W=U\mathop{\mathrm{diag}}(\boldsymbol\lambda)U^{\mathsf T}\), and \(\mu_W\) is the zero-field cube Gibbs law for this \(W\). The comparison in (OpenAI 2026b, sec. 15.4, “Zero field and overlaps bounded away from zero”), together with the lower normalization in its Section 15.3, gives \(\mathcal M_n(\rho_0;\boldsymbol\lambda)\le \exp\{-\kappa_{\rho_0}n+o(n)\}\) for a fixed \(\kappa_{\rho_0}>0\), along the retained spectral sequences with the preceding per-site errors uniform. This is a consequence of that proof; the norm bound supplies continuity at the endpoints. Changing to \(W_p,h\) changes weights and log normalizers by \(O_R(np^2+np^{5/2})=o(n)\), so that region still contributes \(e^{-cn}\).

On retained cells of width \(2/n\), Stirling’s formula for the binomial overlap mass and the spherical density proportional to \((1-\rho^2)^{(n-3)/2}\) gives relative error \[ O(nr_n^4+r_n+n^{-1}). \tag{54}\] This is the cell estimate in the cited replica proof, now with cutoff \(C'r_n\) at every \(p\). More precisely than the fixed bound above, the oscillation of (53) on one cell is at most \[C\bigl(|q^{\mathsf T}W_pq-{q'}^{\mathsf T}W_pq'|+p^{5/2}\bigr),\] which is uniformly bounded. The normalized spherical integral of this expression on retained cells is \(O(r_n)\). Indeed, \[q^{\mathsf T}W_pq-{q'}^{\mathsf T}W_pq' =\frac{2X^{\mathsf T}W_pX' -\rho(X^{\mathsf T}W_pX+{X'}^{\mathsf T}W_pX')} {n(1-\rho^2)}.\] The diagonal terms cost \(O(r_n)\). For the cross term, tilt a Gaussian pair at the cap saddle by \(\pm cp^2X^{\mathsf T}W_pX'\), with \(c\) small enough to preserve a fixed precision margin. The Gaussian logarithmic moment is \(O_R(k_p)\), since \(\mathop{\mathrm{Tr}}R_{p,\alpha}^2=O(n/p)\) and \(\|R_{p,\alpha}h\|^2=O_R(np)\). The replica density bound (39) costs a fixed factor. Hence \[\Pr_{\rm sp}\{|X^{\mathsf T}W_pX'|>t\} \le C e^{C_Rk_p-cp^2t}.\] Integrating from a sufficiently large multiple of \(np\) gives \(\mathbb E_{\rm sp}|X^{\mathsf T}W_pX'|\le C_Rnp\le C_Rnr_n\). This full probability integral also bounds its unnormalized restriction to retained cells.

The bounded oscillation permits the mean value theorem for exponentials. After integration and summation over disjoint cells, the error is the preceding integrated \(O(r_n)\); no supremum over frames is taken. Slightly enlarged boundary cells are paid by the pair tail. Thus \[E_UL_p(h)^2 \le1+C_R\{nr_n^4+r_n+n^{-1}+e^{-cnr_n^3}\}.\] Use \(E_UL_p(h)=1\), \(nr_n^4=n^{-1/3+4\cdot10^{-4}}\), and \(nr_n^3=n^{3\cdot10^{-4}}\) to obtain (52). ◻

Proof. On dyadic \(p\) and \(v\), the spherical tail and (21) give \[E_U[\mathbf 1_{\{L_\circ\ge1/2\}} \mu\{\|P_pX\|>vM_p\}] \le C e^{-c k_pv^6}.\] Markov’s inequality with half the exponent has total failure at most \[\sum_{j,l\ge0}C e^{-cK\,8^j64^l}\le C e^{-c'K}.\] The discarded \(L_\circ<1/2\) event has probability \(o(1)\) by (19). Monotonicity of \(P_p\), and a fixed-factor change in \(M_p\), extend the uncapped result to every \(p,v\).

The posterior tails follow deterministically. For \(q\ge p\) of bounded ratio, let \(B_\theta\) lie between \(B_p\) and \(B_q\). It is positive, supported on \(\mathcal H_q\), and has norm at most \(Cq^2\). Restricting the uncapped normalizer to \(\|P_qX\|\le VM_q\) gives \[\mu e^{-X^{\mathsf T}B_\theta X/2}\ge e^{-Ck_q}\] for a fixed large \(V\). The zero-field capped law is even, so its normalizer for a linear field is at least one. On the shell \(\|P_qX\|\asymp vM_q\), a field of norm at most \(Rq^2M_q\) costs at most \(Rk_qv\). This and the denominator cost are absorbed by the uncapped \(e^{-c k_qv^6}\) tail for large \(v\). This proves the posterior tail, including the interpolated assertion. The spherical argument is identical. The first two projection moments under all these posteriors are consequently bounded by \(C_RM_q\) and \(C_RM_q^2\).

For nearby scales and fields, integrating first derivatives of the log partition function gives \[ |\Delta\phi|\le C_Rk_q\left( |\Delta\log p|+\frac{\|\Delta h\|}{q^2M_q}\right), \tag{55}\] and likewise for \(\phi^{\rm sp}\). Indeed, \(\|B_q-B_p\|\le Cq^2|\log(q/p)|\) on \(\mathcal H_q\); its derivative is controlled by the second projection moment and the field derivative by the first. Fields in the smaller space are compared in \(\mathcal H_q\).

Cover \(K\le k_p\le C_0\sqrt{\log n}\), with fixed \(C_0\) large enough to overlap the source cutoff. Net scales from above and their spectral field balls at resolution \(\eta/(C_R(1+C_0\sqrt{\log n}))\) in the units of (55). Conditional on the spectrum, these are deterministic nets. Since \(\dim\mathcal H_p\le Ck_p\), their total size is \[\exp\{O_{\eta,R}(\sqrt{\log n}\log\log n)\}=n^{o(1)}.\] At a net point (52) and Chebyshev give \(\Pr_U(|\log L_p(h)|>\eta/2)\le C_\eta n^{-1/4}\). The union bound and (55) yield \[ \sup_{\substack{K\le k_p\le C_0\sqrt{\log n}\\ \|h\|\le Rp^2M_p}} |\phi_p(h)-\phi_p^{\rm sp}(h)|\le\eta \tag{56}\] with failure \(n^{-1/4+o(1)}\), in addition to the projection event. For \(k_p\ge c\sqrt{\log n}\), the cap and interpolation results of (OpenAI 2026d, Proposition 2.6, Equation (2.11), and Lemma 2.7) apply as stated, since \(C_1r_n\) is eventually below their fixed upper cutoff. Together these prove the value comparison at every scale.

For the mean, use this comparison on a larger ball. For unit \(v\in\mathcal H_p\), freeze \(\alpha=\alpha_p(h)\). The polar formula and Lemma 5 give \[\log\mu_{p,h}^{\rm sp} e^{\pm t v^{\mathsf T}(X-R_{p,\alpha}h)} \le\tfrac12t^2v^{\mathsf T}R_{p,\alpha}v+C.\] The cube bound has the additional \(2\epsilon_vk_p\). Jensen with \(t=\theta p^2M_p\), in the direction of the projected mean discrepancy, gives \[\frac{\|P_pm_p(h)-R_{p,\alpha}h\|}{M_p} \le C_R\theta+\frac{2\epsilon_v+C/k_p}{\theta}.\] Choose \(\theta\) from the required mean tolerance, then impose a smaller value tolerance and increase \(K\).

It remains to prove the Hessian assertion on the low range. Put \(\widehat R=R_{p,\alpha_p(h)}\), \(m=\widehat Rh\), and \(\sigma^2=2\mathop{\mathrm{Tr}}\widehat R^2+4m^{\mathsf T}\widehat Rm\). For a unit \(v\in\mathcal H_p\) and \(|\theta|\le1\), the shift \(h+\theta pv\) fits a fixed larger ball because \(p/(p^2M_p)=k_p^{-1/2}\). Freezing \(\alpha\), the exact polar formula is \[\log\mu_{p,h}^{\rm sp} e^{\theta p v^{\mathsf T}(X-m)} =\tfrac12\theta^2p^2v^{\mathsf T}\widehat Rv+ \log\frac{f_{\alpha,h+\theta pv}(n)}{f_{\alpha,h}(n)}.\] The shifted mean \(m_\theta=m+\theta p\widehat Rv\) satisfies \[\frac{\sigma_\theta^2}{\sigma^2}=1+O_R(k_p^{-1/2}),\qquad \frac{n-\mathbb E_\theta\|Y\|^2}{\sigma_\theta} =-\frac{2\theta p\langle m,\widehat Rv\rangle}{\sigma} +O_R(k_p^{-1/2}).\] The leading term is uniformly bounded. The density estimate (43), at this argument and at zero, gives \[ \begin{split} \log\mu_{p,h}^{\rm sp} e^{\theta p v^{\mathsf T}(X-m)} &=\tfrac12\theta^2v^{\mathsf T}\Gamma_{p,h}v+O_R(k_p^{-1/2}),\\ \Gamma_{p,h} &=p^2\left(\widehat R|_{\mathcal H_p} -\frac4{\sigma^2}(\widehat Rm)\otimes(\widehat Rm)\right). \end{split} \tag{57}\] This is uniform in \(h,v,\theta\). Cauchy–Schwarz and \(\sigma^2\ge4m^{\mathsf T}\widehat Rm\) imply \(0\preceq\Gamma_{p,h}\preceq C_RI\). Equation (56) changes the log moment by at most \(2\eta\), so the cube error is at most \(\delta=C_RK^{-1/2}+2\eta\).

To justify differentiating this approximation, suppose a real \(Z\) obeys \[\sup_{|\theta|\le1} \left|\log\mathbb Ee^{\theta Z}-a\theta^2/2\right|\le\delta,\qquad 0\le a\le C,\quad 0<\delta\le1/16.\] The values at \(\pm1\) bound \(\mathbb Ee^{|Z|}\), and hence bound the fourth derivative of \(\psi(\theta)=\log\mathbb Ee^{\theta Z}\) on \([-1/2,1/2]\). Taylor’s formula gives \[\left|\psi''(0) -\frac{\psi(t)+\psi(-t)-2\psi(0)}{t^2}\right|\le C_Ct^2.\] The assumed error changes this quotient by at most \(2\delta/t^2\). With \(t=\delta^{1/4}\), this proves \(|\mathop{\mathrm{Var}}Z-a|\le C_C\sqrt\delta\). Applying it to \(Z=pv^{\mathsf T}(X-m)\), for every unit \(v\), yields \[ \|p^2K_p(h)-\Gamma_{p,h}\| \le C_R\sqrt{C_RK^{-1/2}+2\eta}. \tag{58}\]

On the annulus put \(t=\alpha/p^2\), \(A=(n-\mathop{\mathrm{Tr}}\widehat R)/(np)\), \(B=(p/n)\mathop{\mathrm{Tr}}\widehat R^2\), and \(e=h/\|h\|\). Since \(\widehat R|_{\mathcal H_p}=p^{-2}(1+t)^{-1}I\) and \(m\) is parallel to \(e\), \[\langle e,\Gamma_{p,h}e\rangle=\frac{B}{(1+t)B+2A}.\] The trace estimates and denominator comparison give \(A=b_0+O(\epsilon_{\rm sp}+s_0)\) and \(B=2b_0+O(\epsilon_{\rm sp}+s_0)\), so this is \(1/2+O(\epsilon_{\rm sp}+s_0)\). All other quadratic directions are bounded above by \((1-s_0)^{-1}\). Choose the spectral accuracy and \(s_0\) small, then \(\eta\) small and \(K\) large in (58). This proves the annular curvature and radial deficit. On the entire ball, the saddle gap and the same estimate give \(C_Rp^{-2}\), with \(C_R\) fixed before the annular tolerance. The source curvature assertion covers the higher range. ◻

Proof. By (19), under \(\mathbf P\) the spectral spin is spherical. Its Gaussian representation is that of (OpenAI 2026d, Proposition 2.3). The annulus calculation in (OpenAI 2026d, Lemma 3.1, Equation (3.1)) is stated for the source observation range beginning at \(r_n\); the tight-threshold dyadic estimates below extend that calculation to the scales used here. Set \(D_i'=d_i+b_nn^{-2/3}=n^{-2/3}(g_{i,n}+b_n)\). Before conditioning, let \(Y_i\) be independent \(N(0,(D_i')^{-1})\); conditioning on \(\sum_iY_i^2=n\) gives the spherical law. The definition of \(b_n\) gives \(\min_iD_i'\ge n^{-2/3}\), while (59) supplies the fixed leading variance lower bounds. The density of \(n^{-2/3}\sum_iY_i^2\) at \(n^{1/3}\) is exactly \(q_n(D_n)\), and hence is bounded below on the chosen spectral set. The same set and the active count give \[ \sum_{d_i\le p^2}(D_i')^{-1} =n^{2/3}\sum_{d_i\le p^2}\ell_{i,n} \le Cn^{2/3}(1+k_p^{1/3})\le Cnp. \tag{62}\] Indeed the first \(J_\chi\) terms are bounded by one, and the remaining terms are bounded by \(C_\chi i^{-2/3}\).

Let \(\vartheta=p^2+b_nn^{-2/3}\) and, before conditioning, write \(h_{p,i}=B_{p,i}Y_i+B_{p,i}^{1/2}Z_i\). The quadratic form obtained from \(\|h_p\|^2-\vartheta^2\|Y\|^2\) has mean \(-\vartheta^2\mathop{\mathrm{Tr}}R_{p,b_nn^{-2/3}}\). On an active coordinate the coefficient of the standardized \(Y_i^2\) term is \[\frac{B_{p,i}^2-\vartheta^2}{D_i'}=D_i'-2\vartheta=O(p^2);\] the cross coefficient is \(O(p^3/\sqrt{D_i'})\). Equation (62) and the capped second trace bound therefore show that the matrix of this quadratic form, in the standardized \((Y,Z)\) variables, satisfies \[\|\mathsf A_p\|_{\rm F}^2\le Cp^4k_p,\qquad \|\mathsf A_p\|\le Cp^2k_p^{1/3}.\] The Gaussian quadratic-form moment formula for its centered version bounds a deviation of \(p^2k_p^{9/10}\) by \[C\exp\left[-c\min\{k_p^{4/5},k_p^{17/30}\}\right].\] This remains true after norm conditioning, with a constant loss. Indeed use the same exponential tilt, with parameter at most one eighth of \(\|\mathsf A_p\|^{-1}\). Its joint covariance and its \(Y\)-marginal covariance stay comparable to the originals. A fixed number of the leading marginal variances bounds the density of \(n^{-2/3}\|Y\|^2\) from above by a constant, through the characteristic formula (41). The original density at \(n^{1/3}\) is bounded below on the chosen spectral set. The polar density ratio therefore costs a constant.

After conditioning, the deterministic target for \(\|h_p\|^2\) is \(\ell_p(b_nn^{-2/3})^2\asymp p^2k_p\). The preceding error is a relative \(O(k_p^{-1/10})\). It first places the inferred saddle in a fixed annulus. On that annulus \(\mathrm d(\ell_p(u)^2)/\mathrm du\asymp k_p\). Hence \[|\alpha_p(h_p)-b_nn^{-2/3}|\le Cp^2k_p^{-1/10}.\] Since \(b_nn^{-2/3}/p^2=O(k_p^{-2/3})\), this proves the fixed-scale version of (60).

For a whole interval use a mesh of spacing \(k_q^{-3}\) in \(\log p\). Its size is polynomial in \(k_q\). Except with \(Ce^{-ck_q}\) probability, the spherical projection at the upper endpoint is at most \(CM_q\), by Lemma 6. The signal increments on a mesh gap are then at most \(Cq^2M_qk_q^{-3}\). The noise increments have covariance norm \(O(q^2k_q^{-3})\). A constant-resolution net of the sphere in \(O(k_q)\) dimensions and the Gaussian maximal inequality show that their norm on every gap is less than \(q^2M_qk_q^{-1/10}\), except with \(Ce^{-ck_q}\) probability. The target length is Lipschitz in the same scaled units: between activations differentiate the capped trace, using the active count and second trace; the target is continuous at activations. These observations interpolate the fixed-scale bound. The polynomial mesh size is absorbed by decreasing the positive exponent, proving (60).

For a fixed \(\zeta\), it suffices by inclusion to treat \(\zeta\) below a fixed small annulus width. The stronger exponent follows from a different Gaussian reference, following the source-range fixed saddle test in (OpenAI 2026b, sec. 19.2, Equation (19.9) and its proof). Fix a small \(\xi>0\), to be chosen after \(\zeta\), and let \[R_\xi=(2+\xi p^2-W)^{-1},\qquad Y\sim N(0,R_\xi),\qquad h_p=B_pY+\beta_p .\] Increase \(K\) so that \(d_1\ge-\xi p^2/2\). The vector \(a_{p,\xi}=R_{p,\xi p^2}h_p\) is centered Gaussian with covariance \[C_{p,\xi}=R_\xi-R_{p,\xi p^2}.\] The resolvent and clipped traces give \[\left|\mathop{\mathrm{Tr}}C_{p,\xi}-(n-\mathop{\mathrm{Tr}}R_{p,0})\right| \le C\sqrt\xi\,np+o_K(np),\qquad \|C_{p,\xi}\|\le C(\xi p^2)^{-1},\qquad \mathop{\mathrm{Tr}}C_{p,\xi}^2\le Cn/(\sqrt\xi\,p).\] For a fixed multiple of \(np\), exponential Markov with both signs of a sufficiently small multiple of \(\xi p^2\) therefore gives a deviation cost \(e^{-c_\zeta\xi k_p}\), once \(\xi\) is sufficiently small. The centered logarithmic moment costs \(O(\xi^{3/2}k_p)\), whereas the Markov gain is a fixed multiple of \(\zeta\xi k_p\).

We spell out the norm-conditioning cost. The deficit \(n-\mathop{\mathrm{Tr}}R_\xi\) is between fixed multiples of \(\sqrt\xi\,np\), after increasing \(K\) for this fixed \(\xi\). A leading variance is comparable to \((\xi p^2)^{-1}\); the other squared norms have variance at most \(Cn/(\sqrt\xi\,p)\). Chebyshev places the residual length in a fixed interval comparable to \(\sqrt\xi\,np\) when \(K\gg\xi^{-3/2}\). On this interval the leading Gaussian-square density is at least \[C_\xi^{-1}\sqrt{p/n}\exp(-C\xi^{3/2}k_p).\] This is a lower bound for the original norm density by convolution. Under the quadratic tilt used above, the marginal covariance of \(Y\) retains \(ck_p\) independent spectral rows with covariance comparable to \(p^{-2}\); the tilt keeps the joint covariance comparable. Equation (39) bounds its norm density by \(C_\xi\sqrt{p/n}\). Thus conditioning costs at most \(C_\xi e^{C\xi^{3/2}k_p}\).

Since \(R_{p,0}h_p=(1+\xi)R_{p,\xi p^2}h_p\), compare the saddle length at the two boundaries \(\alpha=\pm\zeta p^2\). The derivative bounds in (40), and monotonicity beyond those boundaries, put \(\{|\alpha_p(h_p)|>\zeta p^2\}\) inside the relaxed squared-length deviation just bounded. Choose \(\xi\) so that its \(O(\sqrt\xi)\) target error fits the \(\zeta\) tolerance and \(C\xi^{3/2}<c_\zeta\xi/2\). The conditioned deviation then has probability \(C_\zeta e^{-c_\zeta'k_p}\). The same \(k_q^{-3}\) mesh proves (61).

Finally take \(q_j=2^j(K/n)^{1/3}\). By (21) and Markov, the ordinary whole-interval probability is at most \(Ce^{-c k_{q_j}}\) outside disorder probability \(Ce^{-c k_{q_j}}\), after decreasing \(c\). Their sum is \[\sum_{j\ge0}Ce^{-cK8^j}\le Ce^{-c'K}.\] Use (19) for \(L_\circ<1/2\) and increase \(K\). Every scale lies in one such interval and \(k_p\asymp k_{q_j}\) there, proving (27) simultaneously in \(p\). ◻

Proof. Put \(t=p^2\), and include \(U\) in the observation filtration, so that every disorder event is measurable at time zero. The exact likelihood identity (20) yields \[ \mathrm d\log L_{\sqrt t} =a_{\sqrt t}\cdot\mathrm d\mathcal B_t +\tfrac12\|a_{\sqrt t}\|^2\,\mathrm dt, \tag{65}\] where \(\mathcal B\) is the continuous cube innovation martingale with \(\mathrm d\langle\mathcal B\rangle_t=P_{\sqrt t}\,\mathrm dt\); the coefficients lie in the active space. This is the exact filtering calculation in (OpenAI 2026d, proof of Lemma 3.5, after Equation (3.7)); it follows as well by differentiating the posterior partition function, including the interaction change \(-\mathrm dB_{\sqrt t}\). There is no jump in \(\log L\) at activation.

At a deterministic endpoint and for \(H>0\), (20) gives \[\mathbf P(\log L_p<-H)\le e^{-H}.\] For a fixed \(R_0\) containing the path annulus, put \(\mathcal U_p=\{\widetilde h:\|\widetilde h\|\le R_0p^2M_p\}\). Lemma 9 gives \[\mathbf P(\log L_p>H,h_p\in\mathcal U_p) \le e^{-H}\int_{\mathcal U_p}Q^{\rm sp}(\mathrm d\widetilde h)\, E_U L_p(\widetilde h)^2 \le (1+Cn^{-1/4})e^{-H}.\] The field in the inner expectation is deterministic. Lemma 11 excludes the complementary endpoint fields with probability \(Ce^{-c k_q^{a_{\rm an}}}\).

On an enlarged interval write \(\mathcal J_I=\int\|a_p\|^2\,\mathrm d(p^2)\) and \(M=\int a_p\cdot\mathrm d\mathcal B_{p^2}\). On the two endpoint events just given, \(M+\mathcal J_I/2\le2H\). If \(\mathcal J_I\ge8H\), then \(M\le-\mathcal J_I/4\). The positive exponential supermartingale \(\exp(-M/4-\mathcal J_I/32)\) bounds this event by \(e^{-H/4}\). Taking \(H=Ck_q^{1/2}\) proves (63).

Between activations the two filtering equations give, with \(\Delta_t=K_{\sqrt t}-K_{\sqrt t}^{\rm sp}\), \[\mathrm da_{\sqrt t} =\Delta_t\,\mathrm d\mathcal B_t -K_{\sqrt t}^{\rm sp}a_{\sqrt t}\,\mathrm dt .\] Choose a smooth nonnegative cutoff \(\chi(t)\), equal to one on the target interval, zero at the ends of a fixed enlargement, and with \(|t\chi'(t)|\le C\). The integrated Itô identity is \[ \begin{split} \int\chi t\|\Delta_t\|_{\rm F}^2\,\mathrm dt ={}&-\int(\chi+t\chi')\|a_{\sqrt t}\|^2\,\mathrm dt +2\int\chi t\langle a_{\sqrt t}, K_{\sqrt t}^{\rm sp}a_{\sqrt t}\rangle\,\mathrm dt\\ &-\mathcal M -\sum_j\chi(t_j)t_j \|\Delta P_j(m_{\sqrt{t_j}}-m_{\sqrt{t_j}}^{\rm sp})\|^2,\\ \mathcal M={}&2\int\chi t \langle a_{\sqrt t},\Delta_t\,\mathrm d\mathcal B_t\rangle . \end{split} \tag{66}\] Here \(\Delta P_j\) is the jump of \(P_{\sqrt t}\) at the activation time \(t_j\). The sum has the favorable sign. On \(\mathcal G\) and the whole-path annulus event, the coarse bounds give \(\|tK_{\sqrt t}\|+\|tK_{\sqrt t}^{\rm sp}\|\le C\). Thus the left side on the target interval is bounded by \(C\mathcal J_I+C|\mathcal M|\).

For concentration, stop only the stochastic integral at the first annulus failure, and set its integrand to zero on \(\mathcal G^c\). The latter event is measurable at time zero. The stopped bracket is at most \(C\mathcal J_I\), and the stopped integral agrees with \(\mathcal M\) on \(\mathcal G\) and the whole-path annulus event. The exponential bracket inequality, (63), and (60) consequently bound the target integral of \(\|p^2(K_p-K_p^{\rm sp})\|_{\rm F}^2\,\mathrm d\log p\) by \(Ck_q^{1/2}\), with the required exceptional probability intersected with \(\mathcal G\). The full identity (66) was not stopped, so it has no terminal boundary term.

On the shrinking annulus, rotational invariance inside the flat cap gives one spherical transverse covariance eigenvalue \(\kappa_\perp\) and one radial eigenvalue. The frozen linear moment bound at shifts \(\pm pe\) gives \[|e^{\mathsf T}(m_p^{\rm sp}-R_{p,\alpha}h_p)|\le C/p.\] Since \(\kappa_\perp=(e^{\mathsf T}m_p^{\rm sp})/\|h_p\|\) and \(\|h_p\|\asymp p^2M_p\), \[|p^2\kappa_\perp-1| \le C(k_p^{-1/10}+k_p^{-1/2}).\] The scaled radial covariance is bounded by the spherical version of (58). There are \(O(k_p)\) transverse dimensions and one radial dimension, so \[\frac{\|p^2K_p^{\rm sp}-I\|_{\rm F}^2}{k_p} \le C(k_p^{-1/5}+k_p^{-1}).\] Combine this with the preceding cube–sphere covariance estimate, divide by \(k_q\asymp k_p\), and decrease \(a_1>0\) if necessary. This proves (64). ◻

Proof. We verify the deterministic extension proof of (OpenAI 2026d, Lemma 4.3), following (OpenAI 2026b, Proposition 16.1), with the new lower threshold. On \(\{|\alpha_p|\le3s_{\rm aux}p^2/4\}\) take the infimum, over \(z\), of \[\phi_p(z)+\nabla\phi_p(z)\cdot(y-z) +\frac{1+\eta}{2p^2}\|y-z\|^2.\] Choose \(s_{\rm aux}\ll\eta,\gamma^2\). The length bounds in Lemma 5 give a fixed \(\beta_{\rm buf}>0\) such that a segment of length at most \(\beta_{\rm buf}p^2M_p\) starting in this smaller annulus remains in the full annulus. For such pairs the supporting inequality follows by integrating the coarse Hessian bound. For \(\|y-z\|\ge\beta_{\rm buf}p^2M_p\), freeze the spherical saddle at \(z\). The polar linear-moment bound gives the spherical quadratic support with Hessian \((p^2+\alpha_p(z))^{-1}I\), its saddle gradient, and a bounded norm-density error. The static value and mean estimates add \(2\epsilon_vk_p+\epsilon_gM_p\|y-z\|\). Relative to \(\|y-z\|^2/(2p^2)\), the total residual is bounded by \[Cs_{\rm aux}+\frac{4\epsilon_v+C/k_p}{\beta_{\rm buf}^2} +\frac{2\epsilon_g}{\beta_{\rm buf}} .\] Choose \(s_{\rm aux}\), then \(\beta_{\rm buf}\), then \(\epsilon_v,\epsilon_g\) so that this is less than \(\eta\), and finally increase \(K\). Every support lies above \(\phi_p\) on the smaller annulus. Its infimum is finite and semiconcave with the stated upper Hessian, and it agrees there by choosing \(z=y\). The length derivative gives \(\mathop{\mathrm{dist}}(\mathcal A_p,(\mathcal A_p^+)^c)\ge c_{\rm buf}p^2M_p\).

For the transition tails freeze the child saddle \(\alpha=\alpha_q(h_q)\), and use as reference the Gaussian transition tilted by the parent frozen quadratic \(q_{W_p,\alpha}\). Writing \(R_u=R_{u,\alpha}\), Gaussian convolution gives \[ R_ph_p\sim N(R_qh_q,R_q-R_p),\qquad R_q^{-1}=R_p^{-1}-\Delta. \tag{69}\] The child saddle equation shows that \(\mathbb E\|h_p\|^2=\ell_p(\alpha)^2\), exactly. The field covariance has norm at most \(C\gamma^{-2}p^2\). Gaussian concentration of its norm therefore gives \[\Pr_{\rm ref}\{|\|h_p\|-\ell_p(\alpha)|>vp^2M_p\} \le C e^{-c\gamma^2k_pv^2}, \qquad v\ge C/(\gamma\sqrt{k_p}).\] The reference radius lies strictly inside \(\mathcal A_p\): \(|\alpha|\le s_{\rm aux}q^2\le4s_{\rm aux}\gamma^2p^2\). The length derivative gives a fixed positive gap in units \(p^2M_p\) to the boundary of \(\mathcal A_p\).

Choose \(z_0\in\mathcal H_p\) with \(\|z_0\|=\ell_p(\alpha)\) and calibrate the reference quadratic by \[q^\sharp(y)=q_{W_p,\alpha}(y)+\log f_{W_p,\alpha,z_0}(n).\] The active covariance is scalar, so the added density depends only on \(\|z_0\|\); the additive constant leaves the reference transition unchanged. On bounded scaled radii, the noncentral upper norm-density bound and the static value comparison give \(\phi_p(y)-q^\sharp(y)\le\epsilon k_p+O(1)\), with any prescribed fixed \(\epsilon>0\). Outside a sufficiently large bounded radius, the zero-field projection tail gives \[\phi_p(y)-\phi_p(0) \le Ck_p\left(1+ \left(\frac{\|y\|}{p^2M_p}\right)^{6/5}\right).\] This follows by maximizing the exponent \(uv-cv^6\) after integrating the tail against a linear field of scaled length \(u\). Its growth is strictly subquadratic. For the extension choose the support centered at the point of radius \(\ell_p(\alpha)\) in the direction of \(y\). At distance \(vp^2M_p\), its excess over the reference is at most \[\epsilon k_p(1+v)+C(\eta+s_{\rm aux})k_pv^2+O(1).\] This uses the static mean error for the linear term and the difference of the two quadratic coefficients for the last term. Choose \(\eta+s_{\rm aux}\ll\gamma^2\), and then \(\epsilon\) small compared with the squared fixed gap to \(\mathcal A_p^c\). Outside \(\mathcal A_p\) these errors are smaller than the reference Gaussian deviation cost.

The normalizing integrals relative to the reference are at least \(e^{-\epsilon'k_p}\) for any needed fixed \(\epsilon'>0\). To see this, integrate over a thin fixed shell about the reference radius, contained in \(\mathcal A_p\). Its reference probability is bounded below when \(K\) is large. On that shell the frozen density formula (45) bounds the off-saddle logarithmic density loss by \(C\tau^2k_p+O(1)\) when the shell width is \(\tau\) in scaled units: the first parameter derivative vanishes at the saddle and the second is one quarter of the norm-square variance \(O(n/p)\). The static lower value error adds only \(\epsilon_vk_p\). Take \(\tau\) and \(\epsilon_v\) sufficiently small. Both potentials agree on this shell. Dividing the preceding upper tail integrals by this lower bound proves the stated disagreement tails. Polynomial moments are absorbed by the same quadratic reference tail and the subquadratic bound at large radii. ◻

Proof. Fix a deterministic nonactivation scale \(p\) on the retained spectral set. For \(h\in\mathcal H_p\), write \(D_p(h)=p^2K_p(h)-I_{\mathcal H_p}\). Define a Borel predicate \(\mathcal V_p(W_p,h)\) to be false unless \(h\in\mathcal H_p\), the saddle \(\alpha_p(h)\) exists, and \(h\ne0\). Otherwise, with \(e=h/\|h\|\), it requires \[ |\alpha_p(h)|\le s_{\rm aux}p^2,\qquad p^2K_p(h)\preceq(1+\epsilon_{\rm aux})I_{\mathcal H_p},\qquad p^2\langle e,K_p(h)e\rangle\le0.9. \tag{71}\] This condition contains no restriction to \(\mathcal G\). At a nonactivation scale, \(P_p\) is the spectral projection of \(W_p\) at its flat eigenvalue \(2-p^2\), and \(R_{p,u}=(2I-W_p+uI)^{-1}\). Thus the predicate, the saddle, and the covariance used here are determined by \((p,W_p,h)\). Their defining formulas are jointly Borel away from activation scales. In the rest of the proof these objects are evaluated at \(h=h_p\), and that argument is suppressed.

On \(\mathcal V_p\), use blocks relative to \(\mathcal H_p=\operatorname{span}\{e\}\oplus(\mathcal H_p\cap e^\perp)\), put \(z=(D_p)_{\perp e}\), and define on \(\mathcal H_p\cap e^\perp\) \[ E_p=\left((D_p)_{\perp\perp} +\frac{zz^{\mathsf T}}{|(D_p)_{ee}|}\right)_+, \qquad \widehat s_p=\|(D_p)_+\|. \tag{72}\] Extend \(E_p\) by zero on the radial direction and on \(\mathcal H_p^\perp\). Off \(\mathcal V_p\), set both \(E_p\) and \(\widehat s_p\) to zero, before any restriction to \(\mathcal G\). Covariance positivity and (71) give \(-I\preceq D_p\preceq\epsilon_{\rm aux}I\) and \((D_p)_{ee}\le-0.1\) on \(\mathcal V_p\). Maximizing the quadratic form of \(D_p\) over its radial coordinate, and then taking the positive part on \(e^\perp\), gives \[ D_p\preceq E_p,\qquad \|E_p\|\le C\widehat s_p,\qquad \frac{\mathop{\mathrm{Tr}}E_p}{k_p} \le C\frac{\|D_p\|_{\rm F}}{\sqrt{k_p}}+\frac C{k_p} \tag{73}\] on \(\mathcal V_p\), with \(D_p\preceq E_p\) understood on \(\mathcal H_p\). For the norm bound, the maximizing radial coordinate of a unit transverse vector is bounded because the radial deficit is fixed and \(D_p\) is bounded; compare the resulting vector with \(D_p\preceq\widehat s_pI\). For the trace, the rank-one correction has bounded trace, and \(\mathop{\mathrm{Tr}}(((D_p)_{\perp\perp})_+)\le \sqrt{\dim\mathcal H_p}\|D_p\|_{\rm F}\); the displayed trace bound also uses \(\dim\mathcal H_p\le Ck_p\) from the retained spectral set. These are the algebraic bounds of (OpenAI 2026d, Lemma 4.4, Equation (4.4)). In particular \(\widehat s_p\le\epsilon_{\rm aux}\) and \(\|E_p\|\le C\epsilon_{\rm aux}\) everywhere, including on disorder failures. Include \(C\epsilon_{\rm aux}\ll\gamma^2\) in the initial auxiliary choice.

Apply Brascamp–Lieb’s covariance inequality (Brascamp and Lieb 1976) to the transition with \(\bar\phi_p\). After whitening, its Hessian has a fixed positive margin because \(\Delta\preceq(1-x^2)p^2I\) and \(\eta\ll\gamma^2\). If \(\bar K\) is the extended Hessian, the covariance is at most the transition expectation of \[\Delta^{1/2} (I-\Delta^{1/2}\bar K\Delta^{1/2})^{-1}\Delta^{1/2}.\] Mollify the semiconcave extension to apply the smooth inequality and pass to the limit using the positive margin. Lemma 13 replaces the extended Hessian and transition by \(K_p\) and the true transition on agreement. Its polynomially weighted tails control first and second moments on the complement. In field units the covariance error is \(O(p^4M_p^2e^{-c k_p})\); after (68) and multiplication by \(q^2\), the remaining polynomial factor in \(k_p\) is absorbed by decreasing the exponent.

On \(\mathcal G\), the agreement annulus lies in \(\mathcal V_p\), so (73) gives \(K_p\preceq p^{-2}(I+E_p)\). Put \(A=I-\Delta/p^2\) and \(S=p^{-2}\Delta^{1/2}E_p\Delta^{1/2}\). Then \(A\succeq x^2I\) and \(\|A^{-1/2}SA^{-1/2}\|\ll1\), so \[(A-S)^{-1}\preceq A^{-1}+2A^{-1}SA^{-1}.\] The baseline \(A^{-1}\), substituted in (68), is exactly \(q^{-2}I_{\mathcal H_q}\). For the correction, the two outer factors \(\Delta A^{-1}\) are the scalar \(\Delta_q/x^2\) on \(\mathcal H_q\). Thus, after multiplication by \(q^2\), the first insertion is at most \(2x^{-2}P_qE_pP_q\). We obtain on \(\mathcal G\), for a child in its annulus, \[ \widehat s_q \le Cx^{-2}\mathbf E[\|P_qE_pP_q\|\mid W,\mathcal F_q] +C_\gamma e^{-c_1k_p}. \tag{74}\] Only \(\Delta_q\) is inverted outside the covariance in (68); directions with arbitrarily small new parent precision cause no loss.

We next compress \(E_p\) before imposing \(\mathcal G\). The assertion is trivial when \(E_p=0\); otherwise \(e\) is defined by \(\mathcal V_p\). The retained spectral set supplies \(ck_p\le\dim\mathcal H_p\le Ck_p\) and \(\dim\mathcal H_q\le Cx^3k_p\); after increasing the threshold, \(\dim(\mathcal H_p\cap e^\perp)\) is also comparable to \(k_p\). Condition under \(\mathbf P\) on the spectrum, \(W_p\) in physical coordinates, the physical field \(h_p\), and its spectral coordinates. Equation (20) makes the remaining rotation in \(e^\perp\cap\mathcal H_p\) exactly Haar. The operator \(E_p\) is fixed under this conditioning: its definition uses only these parent data. For a uniformly rotating unit vector \(v\) in this space, the Gaussian representation of the uniform sphere gives \[\mathbf P\left( v^{\mathsf T}E_pv> \frac{C(\mathop{\mathrm{Tr}}E_p+t\|E_p\|)}{k_p} \,\middle|\,\text{parent data}\right) \le C(e^{-t}+e^{-c_0k_p}).\] Indeed the numerator is a positive Gaussian quadratic form, and the Gaussian squared norm in the denominator is at least a fixed fraction of the dimension except with \(e^{-c_0k_p}\) probability. A fixed-resolution net chosen in the spectral child unit sphere before the remaining Haar rotation has at most \(\exp(C\dim\mathcal H_q)\le\exp(Cx^3k_p)\) points. Project each rotated net vector onto \(e^\perp\). Conditional on the parent data, it has a fixed norm at most one and a uniformly rotating direction. With \(t=C_2x^3k_p\), a sufficiently large \(C_2\) pays for the net, and a sufficiently small \(\gamma\) pays for the second exponential term. The net estimate for a positive quadratic form gives \[ \|P_qE_pP_q\| \le C\left(\frac{\mathop{\mathrm{Tr}}E_p}{k_p}+x^3\|E_p\|\right) \tag{75}\] outside a set of unconditional \(\mathbf P\)-probability \(Ce^{-c_\gamma k_p}\). This is the compression argument of (OpenAI 2026d, Lemma 4.5, Equation (4.6)); no restriction to \(\mathcal G\) was made in the Haar law.

Average (74) over \(I_q=[q/(2\gamma),q/\gamma]\), with normalized logarithmic measure. On \(\mathcal G\) outside the failure event in (64), (73) and Cauchy–Schwarz give \[\operatorname{Av}_{p\in I_q}\frac{\mathop{\mathrm{Tr}}E_p}{k_p} \le C_\gamma k_q^{-b},\qquad b=1/10.\] Here \(\operatorname{Av}\) denotes that normalized logarithmic integral. On its failure the same averaged trace is bounded, because \(E_p\) is bounded everywhere and \(\dim\mathcal H_p\le Ck_p\). Its contribution on that failure, with \(\mathbf 1_{\mathcal G}\) inserted, has expectation at most \(Ce^{-c k_q^{a_1}}\). The failures of (75) are charged in the same way by the bounded \(\|E_p\|\). These estimates are applied to the whole parent-scale integral before its conditional expectation. For every nonnegative error cost \(Y\), \[ \mathbf E[\mathbf 1_{\mathcal G}\mathbf E(Y\mid W,\mathcal F_q)] =\mathbf E[\mathbf 1_{\mathcal G}Y], \tag{76}\] because \(\mathcal G\) is measurable from \(W\).

The norm term in (75) has coefficient \(Cx^{-2}x^3\le C\gamma\). The initial choice of \(\gamma\) makes this coefficient \(\theta<1/2\). We have proved, on \(\mathcal G\), \[ \begin{split} \widehat s_q&\le \theta\operatorname{Av}_{p\in I_q} \mathbf E[\widehat s_p\mid W,\mathcal F_q] +C_\gamma k_q^{-b}+\operatorname{err}_q,\\ \operatorname{err}_q&\ge0,\qquad \mathbf E[\mathbf 1_{\mathcal G}\operatorname{err}_q] \le C e^{-c k_q^\beta} \end{split} \tag{77}\] for some \(\beta>0\). Off the child annulus the left side is zero. The bounded definition of \(E_p\) is essential here: no cost polynomial in \(n\) is charged on a failed parent interval.

Use a barrier \(B k_q^{-\alpha}\), with \(0<\alpha\le b\) and \(B\) large. Since \(k_p=k_qx^{-3}\), \[\theta\operatorname{Av}_{p\in I_q}B k_p^{-\alpha} \le\theta(2\gamma)^{3\alpha}B k_q^{-\alpha}.\] It absorbs the deterministic error in (77). Taking positive parts, applying conditional Jensen, and using (76), we obtain \[A(q):=\mathbf E[\mathbf 1_{\mathcal G} (\widehat s_q-Bk_q^{-\alpha})_+] \le\theta\operatorname{Av}_{p\in I_q}A(p)+Ce^{-c k_q^\beta}.\] For each deterministic nonactivation scale \(r_n\le p\le C_\gamma r_n\), use (OpenAI 2026d, Proposition 4.2, Equation (4.2)) within its stated range. Under the biased law \(\mathbf P\), it gives \(p^2K_p\preceq(1+C(p^{a_0}+k_p^{-a_0}))I\) outside a set whose intersection with the full event \(\mathcal G_n^{\rm FUA}\) has probability at most \(Ce^{-n^{\xi_0}}\). Since \(\mathcal G\subseteq\mathcal G_n^{\rm FUA}\), the same failure bound holds after intersection with \(\mathcal G\). Here \(pk_p=np^4\to0\), so \(p^{a_0}\le k_p^{-a_0}\), and \(k_p\asymp n^{3\cdot10^{-4}}\). By taking \(\alpha\le a_0\) and a sufficiently small positive terminal exponential power, the same bound for \(A(p)\) is \(Ce^{-c k_p^{\beta_0}}\).

Every parent generation multiplies \(k\) by at least \(\Lambda=(2\gamma)^{-3}>1\). Iteration, including the terminal bound, gives for a suitably decreased \(\beta_*>0\) \[A(q)\le C\sum_{j\ge0}\theta^j e^{-c(\Lambda^jk_q)^{\beta_*}} \le C e^{-c'k_q^{\beta_*}}.\] Markov’s inequality at the barrier costs only a factor \(k_q^\alpha\), absorbed by the exponential. On \(\mathcal G\), (71) holds throughout the full auxiliary annulus, and (60) removes the annulus truncation. Decrease to a common positive exponent in the barrier and probability. This proves (70). ◻

Proof of Proposition 3. Lemmas 4 and 5 prove the spectral and saddle assertions. Lemmas 6, 10, and 11 prove the static, curvature, projection, and ordinary annulus assertions. Assign a fixed fraction of the requested failure probability to each of their spectral and disorder restrictions, and take their finite intersection. The length bounds give a deterministic \(R_{\rm ann}\) containing the full annulus; include a further cube-cap request for the static comparisons on that ball, retaining the previously chosen width \(s_0\). We select one additional event, independent of \(f\), for the variance.

In the reduction above take \(\eta_p=Ck_p^{-a}\) and let \(\mathcal N\) be the jointly measurable field set where \(p^2K_p(h)\npreceq(1+Ck_p^{-a})I_{\mathcal H_p}\). Set its indicator to one at activation scales, which have zero scale measure for each \(W\). At every deterministic nonactivation scale its evaluation at \(h_p\) is the failure event in (70). Use \(Z_p\) and the common cost \(\rho_p\) from (30); both are independent of \(f\).

Under \(\mathbf P\), conditional Jensen and the spherical identity give \[\mathbf E[\mu_{p,h_p}(Z_p)^2] \le\mathbf E[\mu_{p,h_p}(Z_p^2)] =\mathbf E[Z_p^2]\le C,\] by the uncapped spherical projection tail. Cauchy–Schwarz and (70), absorbing \(k_p\) in the exponential, imply \[ E_U[L_\circ\mathbf 1_{\mathcal G}\rho_p]\le C e^{-c k_p^a}. \tag{78}\] The left side is an ordinary path expectation averaged with the Haar bias. Take \(q_j=2^j(K/n)^{1/3}\) up to \(r_n\). Fubini, (78), and \(L_\circ\ge1/2\) bound the expected ordinary cost on \([q_j,2q_j]\), intersected with \(\mathcal G\), by \(Ce^{-c k_{q_j}^a}\). Markov and \[\sum_{j\ge0}e^{-c(K8^j)^a}\le C e^{-c'K^a}\] give an event of arbitrarily large limiting disorder probability on which, simultaneously in \(j\), \[ \int_{q_j}^{\min\{2q_j,r_n\}}\rho_p\,\mathrm d\log p \le C e^{-c k_{q_j}^a}. \tag{79}\] Increase \(K\) and use (19) for the single discarded event. This event is independent of \(f\).

The conditional-variance estimate of (OpenAI 2026d, Proposition 4.1, Equation (4.1)) holds on an ordinary event of arbitrarily large limiting probability, simultaneously for all real cube functions and scales in its stated interval \([r_n,p_*]\). Its remainder is \(Ce^{-n^{\xi_0}}\|f\|_\infty^2\). We use it only at \(r_n\), for \(0\le f\le1\): \[V_{r_n}(f)\le Cr_n^{-2}\mathcal E(f)+Ce^{-n^{\xi_0}}.\] Intersect this event with the preceding ones. In (33) with terminal scale \(r_n\), its amplified energy is \(Cp^{-2}\mathcal E(f)\). Its amplified remainder is bounded by \(Ce^{-c k_p^a}\) after decreasing \(a>0\), since \(k_p\le k_{r_n}=n^{3\cdot10^{-4}}\). For the other errors use (79). The first partial dyadic interval has \(k_{q_j}\asymp k_p\); the rest increase geometrically. Hence \[\sum_{q_j\gtrsim p}C(q_j/p)^2e^{-c k_{q_j}^a} \le C e^{-c'k_p^a}.\] Applying (33) proves the first inequality of (28). It extends to activation points by continuity of \(V_p\) in \(B_p\).

For the second inequality impose the additional fixed annulus estimate with \(\zeta_*=\min\{\zeta,s_0/2\}\). On that annulus the coarse covariance and (31) contribute \(CV_p\). For its complement, Cauchy–Schwarz and conditional Jensen give \[\begin{split} k_pQ_\mu[\mathbf 1_{\{|\alpha_p(h_p)|>\zeta_* p^2\}}\mu_{p,h_p}(Z_p)] &\le k_pQ_\mu\{|\alpha_p(h_p)|>\zeta_* p^2\}^{1/2} \mu(Z_p^2)^{1/2}\\ &\le C e^{-c k_p}. \end{split}\] Here \(\mu(Z_p^2)\le C\) holds at every scale on the chosen uncapped cube projection event, and the ordinary annulus estimate is also simultaneous in scale. Weakening the exponent if needed proves the second inequality of (28), on the same event for every \(f\). ◻

Proof. Fix first one instance of Proposition 3, with \(\epsilon_{\rm sp}\le1\), \(K\ge1\), and a variance-remainder exponent \(0<a\le1\), decreasing the exponent if necessary. Its constants are fixed on a disorder event of arbitrarily high limiting probability. Choose \(M>1\) sufficiently large that \[Ma>1,\qquad \frac{2M}{3}>\frac53-\kappa .\] We retain this variance estimate when imposing further cap tolerances below; intersection with finitely many additional events does not change its already chosen exponent \(a\). Choose \(\delta>0\) so small that \(M\delta/3<10^{-4}\). This leaves every fixed factor buffer about the scales below within \(r_n=n^{-1/3+10^{-4}}\) for large \(n\). Use the same fixed small \(\gamma>0\) as in the covariance iteration of Section 3. Choose the bridge tolerance \(\epsilon>0\) after \(M\), with \(2(M-1)\epsilon/3<\kappa\).

For a small fixed \(c_0>0\), to be chosen using only the projection-tail constants, define \[k_{p_1}=np_1^3=c_0L,\qquad k_R=nR^3=L^M.\] Increasing \(K_0\) puts these scales above the threshold of Proposition 3. Suppose, for a contradiction, that \(T_n\mathcal E(f)/N<L^{5/3-\kappa}\). If \(F_R(h)=\mu_{R,h}[f]\) and \(V_R=Q_\mu[\mathop{\mathrm{Var}}(f\mid h_R)]\), that proposition gives \[ \frac{V_R}{N} \le C L^{5/3-\kappa-2M/3} +C\exp\{L-cL^{Ma}\}=o_{K_0\to\infty}(1). \tag{88}\] The estimate is uniform in \(n,L,f\) in the present range, because \(R^2T_n=L^{2M/3}\).

Consider the joint ordinary law \(Q_\mu\) of the spin and its observation path through \(R\). Change this law to \(Q_{\rm spin}\) or \(Q_{\rm obs}\) by the respective densities \[\Lambda^{\rm spin}=\frac{f^2}{N}, \qquad \Lambda^{\rm obs}=\frac{F_R(h_R)^2}{A}, \qquad A=Q_\mu[F_R^2]=N-V_R .\] For \(K_0\) large, \(A\ge N/2\), so both densities are at most \(2e^L\). For any of these laws, \(Q|_p\) denotes restriction to \(\sigma(h_q:0\le q\le p)\), always excluding the spin. Define, for \(p<R\), \[ H(p)=D(Q_{\rm obs}\Vert Q_\mu) -D(Q_{\rm obs}|_p\Vert Q_\mu|_p). \tag{89}\] The first entropy is on the full joint law. Since its likelihood is measurable from \(h_R\), it equals the relative entropy on the terminal observation sigma-field. Both terms are finite by the likelihood bound, and the conditional entropy chain rule identifies \(H(p)\) with the expected entropy of the terminal observation tilt remaining after the prefix through \(p\). To compare the two tilts, note that \(Q_\mu[fF_R]=A\), and hence \[Q_\mu\left[\left(\frac f{\sqrt N} -\frac{F_R}{\sqrt A}\right)^2\right] =2\left(1-\sqrt{\frac AN}\right)\le\frac{2V_R}{N}.\] Cauchy–Schwarz bounds the \(L^1(Q_\mu)\) distance between the two densities by \(2\sqrt{2V_R/N}\). This bound persists after projection to any part of the observation path. If two projected densities \(u,v\le2e^L\) have \(L^1\) distance \(d\), then \[ \left|Q_\mu[u\log u]-Q_\mu[v\log v]\right| \le (1+L+\log2)d+C. \tag{90}\] Indeed \(x\log x\) has derivative bounded by \(1+L+\log2\) on \([1,2e^L]\), while its absolute value is bounded on \([0,1]\). It follows from (88) that the two relative entropies, both before and after any projection, differ by \(o(L)\), uniformly as \(K_0\) increases.

We bound the information revealed at \(p_1\) under the spin tilt. The field \(h_{p_1}\) is a sufficient statistic for the path through that scale. Let \(Q_0\) be the output of the same Gaussian channel with zero signal. Convexity of relative entropy and evenness of \(\mu\) give \[D(Q_{\rm spin}|_{p_1}\Vert Q_0) \le\frac12\frac{\mu[f^2X^{\mathsf T}B_{p_1}X]}N, \qquad \log\frac{\,\mathrm d(Q_\mu|_{p_1})}{\,\mathrm dQ_0}(h) \ge-\frac12\mu[X^{\mathsf T}B_{p_1}X].\] These formulas are on the range of \(B_{p_1}\), so zero precision directions cause no difficulty. Subtraction bounds \(D(Q_{\rm spin}|_{p_1}\Vert Q_\mu|_{p_1})\) by half the sum of the two quadratic moments. The spectral bound \(d_1\ge-\epsilon_{\rm sp}p_1^2\) gives \(B_{p_1}\preceq(1+\epsilon_{\rm sp})p_1^2P_{p_1} \preceq2p_1^2P_{p_1}\). The projection tail in Proposition 3, integrated against a density at most \(e^L\), gives \[\frac{\mu[f^2\|P_{p_1}X\|^2]}{NM_{p_1}^2} +\frac{\mu[\|P_{p_1}X\|^2]}{M_{p_1}^2} \le C\left(1+(L/k_{p_1})^{1/3}\right).\] For example, integrate \(\min\{1,e^LCe^{-c k_{p_1}v^6}\}\) in the variable \(v^2\). The information is therefore at most \[Ck_{p_1}\left(1+(L/k_{p_1})^{1/3}\right) \le C(c_0+c_0^{2/3})L .\] This is the proportional-cap argument in (OpenAI 2026b, Equation (19.12)). Choose \(c_0\) so that this last bound is less than \(L/8\). The full spin-tilt entropy is at least \(L/2\). The two continuity comparisons in (90), combined into one uniform \(o(L)\), therefore give \[ \begin{split} H(p_1)&\ge D(Q_{\rm spin}\Vert Q_\mu) -D(Q_{\rm spin}|_{p_1}\Vert Q_\mu|_{p_1})-o(L)\\ &\ge L/2-L/8-o(L)\ge cL. \end{split} \tag{91}\]

We follow that remaining entropy up to a fixed fraction of \(R\). For \(p<R\), let \(\mathsf T_{p,R}^h\) be the ordinary conditional law of \(h_R\) given \(h_p=h\). It has density proportional to \[e^{\phi_R(y)}\,\mathcal N(h,B_R-B_p)(\,\mathrm dy),\] where the Gaussian is restricted to the range of its covariance. Let \(\widetilde{\mathsf T}_{p,R}^h\) be its tilt by \(F_R(y)^2\), and put \[A_p(h)=\mathsf T_{p,R}^h[F_R^2],\qquad l_p(h)=D(\widetilde{\mathsf T}_{p,R}^h\Vert\mathsf T_{p,R}^h).\] Observation sufficiency gives the prefix density \(\,\mathrm d(Q_{\rm obs}|_p)/\,\mathrm d(Q_\mu|_p)=A_p(h_p)/A\). The conditional chain rule in (89) now reads \[ H(p)=\frac1A Q_\mu\!\left[ \mathop{\mathrm{Ent}}_{\mathsf T_{p,R}^{h_p}}(F_R^2)\right] =Q_{\rm obs}[l_p(h_p)]. \tag{92}\]

Write \(\Delta=B_R-B_p\). On the currently active subspace \(\mathcal H_p\), \(\Delta=(R^2-p^2)I\). Differentiating its Gaussian convolution therefore gives \[\nabla_{\mathcal H_p}\log A_p(h)=\delta m_h,\qquad \delta m_h=\widetilde{\mathsf T}_{p,R}^h[S] -\mathsf T_{p,R}^h[S],\qquad S=\frac{P_p(h_R-h)}{R^2-p^2}.\] In the parameter \(p^2\), the likelihood martingale \(A_p(h_p)/A\) has innovation coefficient \((A_p/A)\delta m_h\). The Gaussian likelihood formula for its relative entropy and the chain rule yield \[ -\frac{\,\mathrm dH}{\,\mathrm d\log p} =p^2Q_{\rm obs}[|\delta m_h|^2] . \tag{93}\] This identity holds almost everywhere and in integrated form across activation points. One direct justification applies Itô’s formula to \((u+\epsilon)\log(u+\epsilon)\) of the bounded likelihood, then lets \(\epsilon\downarrow0\); zero-rate directions are omitted. This is the observation entropy identity (OpenAI 2026b, Lemma 19.2, Equation (19.6)), applied here to a terminal observation tilt instead of a spin tilt.

The ordinary log moment generating function of \(S\), for \(v\in\mathcal H_p\), is exactly \[\log\mathsf T_{p,R}^h[e^{\langle v,S\rangle}] =\phi_p(h+v)-\phi_p(h)+\frac{|v|^2}{2(R^2-p^2)} .\] Suppose that the Hessian of \(\phi_p\) in this subspace is at most \(C_{\rm curv}p^{-2}I\) on all shifts of length at most \(p\sqrt{2l_p(h)/C_{\rm curv}}\) from \(h\). The entropy variational inequality, tested against \(\langle v,S\rangle\), then gives \[ p^2|\delta m_h|^2 \le2\left(C_{\rm curv}+\frac{p^2}{R^2-p^2}\right)l_p(h). \tag{94}\] To check the finite shift radius, the centered log moment generating function is at most \(\frac12(C_{\rm curv}p^{-2}+(R^2-p^2)^{-1})|v|^2\). Testing at length \(\sqrt{2l_p/(C_{\rm curv}p^{-2}+(R^2-p^2)^{-1})}\) in the mean-shift direction proves the inequality by contradiction; this length is no larger than the assumed radius.

We next account for observations where that curvature bound is not available. The entropy bound \(l_p\le-\log A_p\), together with \(A\ge e^{-L}/2\), implies \[ Q_{\rm obs}(l_p>L+\log2+z)\le e^{-z},\qquad z\ge0. \tag{95}\] Under the ordinary joint law, \[S=P_pX+(R^2-p^2)^{-1/2}G_p,\] with an independent standard Gaussian \(G_p\) on \(\mathcal H_p\). For \(p\le\gamma R\), the projection tails of Proposition 3 and Gaussian concentration give, for each prescribed \(a_0>0\), a fixed \(V\) such that \[ \begin{split} Q_\mu\!\left[\frac{|S|^2}{M_p^2} \mathbf 1_{\{|S|>VM_p\}}\right]&\le Ce^{-a_0k_p},\\ Q_\mu\{\|h_p\|>Vp^2M_p\}&\le Ce^{-a_0k_p}. \end{split} \tag{96}\] To see the common scale, write the observation noise using a standard Gaussian \(G'_p\) on \(\mathcal H_p\). Since \(B_p\preceq2p^2P_p\), \[\frac{\|h_p\|}{p^2M_p} \le2\frac{\|P_pX\|}{M_p}+\sqrt2\frac{\|G'_p\|}{\sqrt{k_p}}, \qquad \frac{|S|}{M_p} \le\frac{\|P_pX\|}{M_p} +\frac p{\sqrt{R^2-p^2}}\frac{\|G_p\|}{\sqrt{k_p}}.\] The last coefficient is bounded on this range and \(\dim\mathcal H_p\le Ck_p\). Integration of the same tails allows any fixed polynomial moment as well. For a prefix event \(E\), conditional Jensen and the full and prefix likelihood bounds at most \(2e^L\) give \[ \begin{split} p^2Q_{\rm obs}[|\delta m_h|^2\mathbf 1_E] \le Ck_p\bigg( V^2Q_{\rm obs}(E) +e^L Q_\mu\!\left[ \frac{|S|^2}{M_p^2}\mathbf 1_{\{|S|>VM_p\}}\right]\bigg). \end{split} \tag{97}\] Indeed Jensen bounds \(|\delta m_h|^2\) by twice the sum of the second moments under the two conditional laws. For their parts \(|S|\le VM_p\) use the prefix probability of \(E\); for the remaining parts use respectively the full likelihood \(F_R^2/A\) and the prefix likelihood \(A_p/A\). Finally \(p^2M_p^2=k_p\).

Choose \(p_2\) by \(k_{p_2}=C_2k_{p_1}\), where \(C_2\) will be fixed below. Once it is fixed, \(p_2<\gamma R\) for \(K_0\) sufficiently large. Fix \(A_0>0\), choose a tail rate \(a_0>c_0^{-1}\) with a fixed margin, and then choose \(V\) in (96). On \([p_1,p_2]\) retain \(\|h_p\|\le Vp^2M_p\) and \(l_p\le L+\log2+A_0k_p\). The trial shifts in (94), in units \(p^2M_p\), then have length at most \(\sqrt{2(c_0^{-1}+A_0+1)}\), after enlarging the ball-curvature constant to be at least one. Choose a fixed ball containing the retained fields and these shifts before requesting its curvature constant from Proposition 3. Equations (95)–(97) show that discarded observations contribute at most \(Ck_pe^{-c k_p}\). Thus (93)–(94) imply \[\frac{\,\mathrm dH}{\,\mathrm d\log p}\ge-C_1H-Ck_pe^{-c k_p} \qquad(p_1\le p\le p_2).\] The logarithmic length of this interval is fixed.

For \([p_2,\gamma R]\), make a fresh auxiliary cap and extension request with the same \(\gamma\). Choose \(\eta\ll\gamma^2\), then \(\epsilon_{\rm aux}\le\epsilon\) sufficiently small compared with \(\eta\). Reapply the construction of Lemma 13 with these choices, and write \(s_0\) for its auxiliary width. Let \(\mathcal G_{\rm ext}=\mathcal G_n^{\rm aux,ext} \cap\mathcal G_n^{\rm FUA}\) be the resulting event for the required static tolerances, radii, threshold, and scale buffers. Lemma 10 gives \(\mathcal G_n^{\rm aux,ext}\) with arbitrarily high limiting ordinary probability, and \(\mathcal G_n^{\rm FUA}\) costs \(o(1)\). Assign a further fraction of the failure probability and intersect with the retained variance event; its exponent \(a\) and constants are unchanged.

On the full annulus \(|\alpha_p(h)|\le s_0p^2\), this request gives \(K_p(h)\preceq(1+\epsilon_{\rm aux})p^{-2}I \preceq(1+\epsilon)p^{-2}I\). The same annulus specifies admissible child fields for the extension. Choose \(0<\zeta<s_0\) and a fixed \(b_{\rm rad}>0\) so that shifts of length at most \(b_{\rm rad}p^2M_p\) from \(|\alpha_p(h)|\le\zeta p^2\) stay in the full annulus. After \(\zeta\) is fixed, impose its ordinary annulus estimate from Lemma 11 as one further fixed request and intersection. Thus the ordinary failure probability is \(Ce^{-a_\zeta k_p}\), with a fixed \(a_\zeta>0\); only this probability bound is used from the further request. The trial shifts fit whenever \(l_p\le b k_p\), where \(b=(1+\epsilon)b_{\rm rad}^2/2\). Choose \(C_2\) so large that \[\frac{L}{k_p}\le\frac1{C_2c_0} <\frac14\min\{b,a_\zeta\}\qquad(p\ge p_2).\] The likelihood bound and (95) give exponentially small tilted probabilities for both discarded events. Using (97) with a further fixed \(S\)-tail threshold proves \[\frac{\,\mathrm dH}{\,\mathrm d\log p} \ge-\left(2(1+\epsilon)+\frac{2p^2}{R^2-p^2}\right)H -Ck_pe^{-c k_p}.\] The extra coefficient has bounded integral, \[\int_{p_2}^{\gamma R}\frac{2p^2}{R^2-p^2}\,\,\mathrm d\log p \le-\log(1-\gamma^2).\] In both intervals the integrating factor charges a fixed power of \(p/p_1\), whereas \(k_p=np^3\) grows as its cube. Consequently the weighted errors are bounded by a polynomial in \(k_{p_1}\) times \((e^{-c k_{p_1}}+e^{-c k_{p_2}})\). They are \(o(L)\) after increasing \(K_0\). The first interval costs a fixed multiplicative factor. We obtain \[ H(\gamma R)\ge cL(p_1/R)^{2+2\epsilon}. \tag{98}\]

It remains to bound this entropy by the energy of \(f\). For a child field satisfying \(|\alpha_{\gamma R}(h)|\le s_0(\gamma R)^2\), the fresh request to Lemma 13 supplies an extension of the parent potential with upper Hessian \((1+\eta)R^{-2}I\). Put \[\mathcal A_R=\{|\alpha_R|\le s_0R^2/2\},\qquad \mathcal A_R^+=\{|\alpha_R|<3s_0R^2/4\}.\] The extension agrees with the true potential on \(\mathcal A_R^+\). The distance from \(\mathcal A_R\) to \((\mathcal A_R^+)^c\) is at least \(c_{\rm buf}R^2M_R\). Both transition laws assign \(\mathcal A_R^c\) probability \(Ce^{-c k_R}\), also with fixed polynomial moments of \(\|h_R\|/(R^2M_R)\). The precision increment satisfies \(\Delta\preceq(1-\gamma^2)R^2I\). On its range the extended transition, denoted by \(\overline{\mathsf T}\), has negative log-density Hessian at least \[\Delta^{-1}-(1+\eta)R^{-2}I\succeq c_\gamma R^{-2}I .\] The Bakry–Émery logarithmic Sobolev criterion (Bakry et al. 2014, Corollary 5.7.2) gives \[\mathop{\mathrm{Ent}}_{\overline{\mathsf T}}(g^2) \le C_\gamma R^2\overline{\mathsf T}[|\nabla g|^2].\] It applies to the semiconcave extension by mollification.

Choose a smooth cutoff \(\chi\) equal to one on \(\mathcal A_R\) and supported in \(\mathcal A_R^+\). The fixed buffer and the radial saddle derivative in Proposition 3 give \(|\nabla\chi|\le C/(R^2M_R)\). Apply the preceding inequality to \(\chi F_R\). Its cutoff-gradient term is at most \(Ce^{-c k_R}\), because \(R^2|\nabla\chi|^2\le C/k_R\) and the transition tail has that exponential rate. On agreement the ratio of the two transition densities is constant; their normalization and their exponentially small tails make that constant \(1+O(e^{-c k_R})\). To compare the entropies, use \(\mathop{\mathrm{Ent}}_{\mathsf T}(g)=\mathsf T[g\log g]-\mathsf T[g]\log\mathsf T[g]\): \(x\log x\) is bounded and Hölder continuous on \([0,1]\), and \(0\le F_R,\chi\le1\). The comparison, with a possibly smaller positive exponent, proves \[ \mathop{\mathrm{Ent}}_{\mathsf T_{\,\gamma R,R}^{h}}(F_R^2) \le C_\gamma R^2\mathsf T_{\,\gamma R,R}^{h} [|\nabla_{\mathcal H_R}F_R|^2]+Ce^{-c k_R}. \tag{99}\] Failure of the child condition is contained in the ordinary \(\zeta\)-annulus failure, since \(\zeta<s_0\). Its probability is \(Ce^{-c k_R}\) after changing \(c\), since \(k_{\gamma R}=\gamma^3k_R\). On it the conditional entropy of \(F_R^2\le1\) is at most \(1/e\). Average (99) under the ordinary child law and use (92). The tower property, \(A\ge N/2\), and the two inequalities of (28) at \(R\) give \[ \begin{split} H(\gamma R) &\le\frac{C_\gamma R^2}{A} Q_\mu\!\left[\|\nabla_{\mathcal H_R}F_R(h_R)\|^2\right] +\frac{Ce^{-c k_R}}A\\ &\le C\frac{V_R}{N}+C\exp\{L-c k_R^a\}\\ &\le C R^{-2}\frac{\mathcal E(f)}N+C\exp\{L-c k_R^a\}. \end{split} \tag{100}\] Here \(0<a\le1\) allows the transition error to be included in the displayed remainder, and \(k_R=L^M\) makes it \(C e^{L-cL^{Ma}}\). The error is negligible compared with (98), even if the latter is a negative power of \(L\). Comparing the two bounds and using \(T_np_1^2=(c_0L)^{2/3}\) gives \[T_n\,\frac{\mathcal E(f)}N \ge c\,T_nR^2L(p_1/R)^{2+2\epsilon} =c' L^{5/3-2(M-1)\epsilon/3}.\] The choice \(2(M-1)\epsilon/3<\kappa\) contradicts the assumed upper bound for \(K_0\) large. All ball radii, tail rates, annulus tolerances and factor buffers have been fixed before this last increase of \(K_0\) and before \(n\) tends to infinity. ◻

Proof. The first inequality is the product calculation of (OpenAI 2026b, Equation (19.18)), applied at each chosen endpoint. Its precision is \(t'I+W\) for the corresponding fixed \(t'>3\). The posterior interaction is then a scalar diagonal, hence its spin law is a product with fields \(h_i\). Tensorizing the two-point log-Sobolev bound gives \[D(\nu_h\Vert\mu_h) \le C(1+\|h\|_\infty)\frac{\mathcal E_{\mu_h}(f)}{N_h}.\] The field tail of (OpenAI 2026b, Equation (19.10)), on the common event for the chosen endpoints, restricts \(\|h\|_\infty\le C\sqrt{\log n+L}\). Its ordinary complement has probability at most \(e^{-2L}n^{-10}\), with polynomial moments; after the likelihood bound \(N_h/N\le e^L\) its entropy contribution is \(O(n^{-3})\). Equation (103) gives the displayed bound.

For the second inequality, entropy tensorization at a product posterior and Jensen’s inequality for the logarithm give, for positive \(z\), \[\mathop{\mathrm{Ent}}_{\mu_h}(z) \le\sum_i\mu_h[\mathop{\mathrm{Ent}}_{\mu_h(\cdot\mid X_{-i})}(z)] \le\sum_i\mu_h[\mathop{\mathrm{Cov}}_{\mu_h(\cdot\mid X_{-i})}(z,\log z)] =\mathcal I_{\mu_h}(z).\] In fact \(\mathop{\mathrm{Ent}}_\pi(z)\le\mathop{\mathrm{Cov}}_\pi(z,\log z)\) follows from \(\pi[\log z]\le\log\pi[z]\). The posterior entropy formula is \[H_{\rm end}=\frac1N Q_\mu[\mathop{\mathrm{Ent}}_{\mu_h}(f^2)].\] It remains to compare the averaged modified form to its initial value. For a fixed site and fixed \(x_{-i}\), let \[a_h=\mu_h(X_{-i}=x_{-i},X_i=+1),\qquad b_h=\mu_h(X_{-i}=x_{-i},X_i=-1)\] be the two joint posterior masses. Their contribution to \(\mathcal I_{\mu_h}(z)\) is \[\frac{a_hb_h}{a_h+b_h} (z_+-z_-)(\log z_+-\log z_-).\] The second factor is nonnegative and independent of the observation. The function \(c(a,b)=ab/(a+b)\), extended by zero on the coordinate axes, is concave: in the positive quadrant its Hessian is \(-2(b,-a)(b,-a)^{\mathsf T}/(a+b)^3\). Since \(Q_\mu[a_h]\) and \(Q_\mu[b_h]\) are the corresponding initial masses, Jensen’s inequality proves \(Q_\mu[\mathcal I_{\mu_h}(z)]\le\mathcal I_\mu(z)\). Take \(z=f^2+\epsilon\) and let \(\epsilon\downarrow0\) to include zeros. This proves the second bound. ◻

Proof. We prove the stronger square-density residual estimate that replaces the linear energy dependence in (OpenAI 2026a, Lemma 6.5). Use the zero-diagonal interaction \(J_*\), so the cube coordinates in this proof are spins, not the scaled edge coordinates. For a spin flip \(\sigma^{(i)}\), put \[d_i g(\sigma)=\frac{g(\sigma)-g(\sigma^{(i)})}{2\sigma_i}.\] The primary iterates of (OpenAI 2026a, sec. 6) are \[\begin{gathered} m^0=\sigma,\qquad h^1=h+J_*\sigma,\qquad m^l=\tanh h^l,\qquad b_l=1-\|m^l\|^2/n,\\ h^{l+1}=h+J_*m^l-jb_lm^{l-1},\qquad R_l=m^{l-1}-m^l . \end{gathered}\] For a fixed ordinary recipe admissible from level \(l\), let \(U\) be a terminal vector in that recipe. We claim that, whenever \(\sup_\sigma(|H_0|+\|dH_0\|)\le C_0\), \[ \left|\mu_h[f^2H_0R_l^{\mathsf T}U]\right| \le C\left(N_h+\sqrt{N_h\mathcal D_h(f)}\right). \tag{107}\] The constant has the dependence on the fixed recipe, depth, coefficient bounds and finite word diagnostics allowed in the source lemma, and is independent of \(f\). The residual claim itself does not require root stability.

Here are the base case and the induction, including the point where the improvement occurs. The pointwise estimate (OpenAI 2026a, Lemma 6.4, Equation (6.20)) gives \[\|U\|+\sum_i|d_iU_i|\le C .\] It also gives the same bound for \(\widetilde U=H_0U\): indeed \[d_i(H_0U_i)=H_0\,d_iU_i+U_i(\sigma^{(i)})d_iH_0, \qquad \|(U_i(\sigma^{(i)}))_i\| \le\|U\|+2\|\mathop{\mathrm{diag}}\nabla U\|_2\le C,\] and Cauchy–Schwarz bounds the sum of the added diagonal derivatives. Write \(C_U\) for a bound on \(\|\widetilde U\|+\sum_i|d_i\widetilde U_i|\).

Let \(p_i^\pm=\mu_h(\sigma_i=\pm1\mid\sigma_{-i})\). Then \(m_i^1=p_i^+-p_i^-\) and the conditional variance is \(v_i^1=4p_i^+p_i^-\), in particular \(v_i^1\le4p_i^\pm\) for each sign. Since a function of one spin is affine with slope \(d_if\), the form is \[\mathcal D_h(f)=\mu_h\sum_i v_i^1(d_if)^2 .\] The exact one-site integration by parts (OpenAI 2026a, Equation (6.31)) gives \[\begin{align*} \mu_h[f^2R_1^{\mathsf T}\widetilde U] &=\mu_h\sum_i v_i^1d_i(f^2\widetilde U_i)\\ &=\mu_h\sum_i v_i^1 f^2d_i\widetilde U_i +\mu_h\sum_i v_i^1(f_{i,+}+f_{i,-})d_if\, \widetilde U_i(\sigma^{(i)}). \end{align*}\] Here \(f_{i,\pm}=f(\sigma^{i,\pm})\), and \(d_i(f^2)=(f_{i,+}+f_{i,-})d_if\), also for signed \(f\). The first term is at most \(C_UN_h\) in absolute value. For the second set \[M_i(\sigma_{-i})=\max_\pm|\widetilde U_i(\sigma^{i,\pm})|.\] At each configuration, \[\left(\sum_iM_i(\sigma_{-i})^2\right)^{1/2} \le\|\widetilde U(\sigma)\| +2\|\mathop{\mathrm{diag}}\nabla\widetilde U(\sigma)\|_2\le 2C_U.\] Weighted Cauchy–Schwarz bounds that second term by \[\mathcal D_h(f)^{1/2} \left(\mu_h\sum_i v_i^1(f_{i,+}+f_{i,-})^2M_i^2\right)^{1/2}.\] The two separate bounds \(v_i^1\le4p_i^\pm\) imply \[v_i^1(f_{i,+}+f_{i,-})^2 \le8(p_i^+f_{i,+}^2+p_i^-f_{i,-}^2).\] Since \(M_i\) is independent of \(\sigma_i\), conditioning term by term gives \[\mu_h\sum_i v_i^1(f_{i,+}+f_{i,-})^2M_i^2 \le8\mu_h\!\left[f^2\sum_iM_i^2\right]\le32C_U^2N_h .\] This proves (107) for \(l=1\).

The finite induction in the source preserves this new right side. For clarity, let \(U\) be admissible from level \(l+1\), put \(a_l=j(b_l-b_{l-1})\), and define \[U_i'=\Phi(h_i^l;h_i^{l+1},a_l)U_i,\qquad \Phi(z;v,a)=\int_0^1 e^{a(1-u^2)/2}\frac{\cosh(u(z-v))}{\cosh z\cosh v}\,\,\mathrm du .\] Here \(|a_l|\le j\). The integral formula bounds \(\Phi\) and every needed fixed-order derivative uniformly in its two real field arguments: the hyperbolic derivatives introduce bounded hyperbolic-tangent factors, and the exponential factor and its derivatives are bounded for \(|a|\le j\). Together with the explicit level indices, these bounds make \(U'\) admissible from level \(l\). The scalar identity in (OpenAI 2026a, sec. 6) then gives the residual expansion below. Adjoin the auxiliary field \[y'=J_*U'-j\sum_q\theta_qm^{q-1} -j\sum_b\omega_bw^b,\qquad \theta_q=\frac1n\sum_i\partial_qU'_i,\quad \omega_b=\frac1n\sum_i\partial_{y^b}U'_i .\] Here \(w^b,y^b\) are the finitely many earlier auxiliary sources and fields of the recipe; the partials differentiate their explicitly displayed local arguments, holding parameters fixed, as in (OpenAI 2026a, Definition 6.2). Equation (6.33) of that source is \[\begin{align*} R_{l+1}^{\mathsf T}U ={}&R_l^{\mathsf T}y'+a_lR_l^{\mathsf T}U' -jb_{l-1}R_{l-1}^{\mathsf T}U' +j\sum_b\omega_bR_l^{\mathsf T}w^b\\ &+j\sum_{q\ge l}\theta_qR_l^{\mathsf T}m^{q-1} -a_ln\theta_l , \end{align*}\] with the \(R_{l-1}\) term absent when \(l=1\). All first-line terms are earlier residuals with ordinary admissible sources. Their scalar multipliers and their discrete-gradient norms are bounded by the fixed-recipe bounds. For \(q>l\) in the second line, use the source \(m^{q-1}/\sqrt n\) and multiplier \(j\sqrt n\,\theta_q\); (OpenAI 2026a, Equation (6.21)) gives the same boundedness for that multiplier. Finally the exact identity \[R_l^{\mathsf T}m^{l-1} =n(b_l-b_{l-1})-R_l^{\mathsf T}m^l\] cancels the displayed \(-a_ln\theta_l\). The remainder again uses an earlier residual, the source \(m^l/\sqrt n\), and a bounded multiplier. A bounded multiplier \(\zeta\) with bounded \(\|d\zeta\|\) can always be absorbed into \(H_0\), because \[d_i(\zeta H_0)=\zeta(\sigma^{(i)})d_iH_0+H_0(\sigma)d_i\zeta .\] Every induction call therefore retains the original \(f,N_h,\mathcal D_h(f)\). There are finitely many calls with a smaller residual index. This proves (107) at every fixed depth.

We spell out its consequences for selection and the mean. Put \(S_l=R_l^{\mathsf T}m^l/\sqrt n\). The primary derivative bounds in (OpenAI 2026a, Lemma 7.1) give \(|S_l|\le2\sqrt n\) and \(\|dS_l\|\le C_l\). Apply (107) with \(U=m^l/\sqrt n\) and \(H_0=S_l/\sqrt n\). Division by \(N_h\sqrt n\) yields the strengthened form of (OpenAI 2026a, Equation (7.1)): \[ \nu_h\!\left[\left(\frac{S_l}{\sqrt n}\right)^2\right] \le\frac{C_l}{\sqrt n} \left(1+\sqrt{\frac{\mathcal D_h(f)}{N_h}}\right). \tag{108}\] For a small fixed \(\rho>0\), choose a fixed depth \(d\) with \(\lfloor d/2\rfloor\rho^2/4>2\), and then \(e_\rho>0\) with \(2de_\rho<1\). The telescoping identity \[\frac{\|R_l\|^2}{n}=b_l-b_{l-1}-\frac{2S_l}{\sqrt n}\] shows that \(\max_{l\le d}|S_l|/\sqrt n\le e_\rho\) forces a consecutive pair \(R_{k-1},R_k\) to have both norms at most \(\rho\sqrt n/2\). Otherwise the disjoint pairs would make the sum of squared norms exceed \(2\), whereas telescoping bounds it by \(1+2de_\rho<2\).

As in that source lemma, choose smooth \(C_k\in[0,1]\), equal to one on this smaller pair region and supported where both norms are at most \(\rho\sqrt n\). The primary derivative bounds give \(\|dC_k\|\le C_\rho/\sqrt n\). Put \[\widehat C_k=C_k\prod_{i=2}^{k-1}(1-C_i),\qquad D_{\rm sel}=1-\sum_{k=2}^d\widehat C_k=\prod_{k=2}^d(1-C_k).\] Positive deficit implies \(\max_{l\le d}|S_l|/\sqrt n>e_\rho\). Chebyshev’s inequality and (108) give \[ \nu_h(D_{\rm sel}>0) \le\frac{C_\rho}{\sqrt n} \left(1+\sqrt{\frac{\mathcal D_h(f)}{N_h}}\right). \tag{109}\] The cutoffs are admissible bounded multipliers in (107).

Choose \(\rho\le\epsilon\) with \((\|J_*\|+4j)\rho<2\rho_0\). On the support of \(C_k\), the recursion gives \[F_h(h^k)=J_*R_k-jb_{k-1}(R_{k-1}+R_k) +j(b_k-b_{k-1})m^k,\] so \(\|F_h(h^k)\|\le(\|J_*\|+4j)\rho\sqrt n\). The root estimate of (OpenAI 2026a, Lemma 5.7), \[\|y-y_*\|\le \left(1+\frac{\|J_*\|+3j}{b}\right)\|F_h(y)\| \quad\hbox{when }\|F_h(y)\|<2\rho_0\sqrt n,\] and the Lipschitz property of \(\tanh\) therefore give \(\|m^k-m_*\|\le C_{\rm lead}\rho\sqrt n\), with a constant independent of the depth. For a deterministic unit vector \(e\), sum (107) for \(R_1,\ldots,R_k\), with source \(e\) and multiplier \(\widehat C_k\). Since \(\sigma-m^k=\sum_{l\le k}R_l\), this bounds its reweighted mean against \(e\) by \(C_\rho(1+\sqrt{\mathcal D_h(f)/N_h})\). The weighted root errors together cost at most \(C_{\rm lead}\epsilon\sqrt n\), because the cutoffs sum to at most one. The remaining deficit costs at most \(2\sqrt n\,\nu_h(D_{\rm sel}>0)\), controlled by (109). Sum over the fixed number of cutoffs, divide by \(\sqrt n\), and take the supremum over \(e\). This proves (106). ◻

Proof. We use the path and the field tests in (OpenAI 2026b, sec. 19.6); we verify that its entropy argument remains valid under (110). On source disorder events independent of the initial spin law, the calculation around Equation (19.20) there, using its Lemma 17.4, Equation (17.12), and Proposition 17.5, gives the following early field-test bound. On a fixed interval \([t_*,t_b]\) of small positive scalar precision, the diagnostic and stability failures have integrated probability \(o(1)\), uniformly under every initial spin law. This follows there by conditioning on the spin: the centered Gaussian-noise diagnostics have integrated failure \(o(1)\), the additional spin shift has normalized norm \(O(t)\), and the Gaussian low-band small-ball estimate is uniform in its center. No factor \(e^L\) is used for this interval. From \(t_b\) until the interaction coefficient reaches a fixed globally controlled range, the buffered stability test fails anywhere on the path with probability at most \(e^{-a n}\) under the ordinary law \(Q_\mu\), for a fixed \(a>0\). This is the later field bound in (OpenAI 2026b, Lemma 17.3, Section 19.4, and Section 19.6 after Equation (19.21)). The word diagnostics and stability constants on the retained regions are those of Lemma 19.

Here is the order of the fixed choices in that construction. Choose first a large scalar endpoint \(t_{\rm end}\) and a small interaction coefficient below which covariance is globally bounded. Choose \(t_b>0\) for the small-positive-field estimate, and obtain the buffer and rate \(a\) for the later test. The norm bound and the fixed total duration give constants \(C_{\rm jump}\) and \(C_{\rm aft}\) for the jump loss divided by \(t_*n\) and the loss after failure divided by \(n\), uniformly when \(1-\beta'\le t_*/3\). Choose a target drift budget \(\eta_1>0\) so that \[3\eta_1(1+C_{\rm aft}/a)<\lambda/4.\] Next choose \(0<t_*\ll t_b\) with \(C_{\rm jump}t_*\le\eta_1\), and finally \(1-\beta'>0\) sufficiently small relative to these choices, including \(1-\beta'\le t_*/3\). Start with precision \[B_*=t_*I+(1-\beta')W,\] which is positive when \(1-\beta'\) is small enough. Increase scalar precision to \(t_{\rm end}I+(1-\beta')W\), and then traverse \[B_v=(t_{\rm end}+4v)I+(1-\beta'+v)W,\qquad 0\le v\le\beta'.\] The latter rate \(4I+W\) is positive on the norm event. The endpoint is \((t_{\rm end}+4\beta')I+W\), so its posterior is a product. Before the globally controlled portion, the posterior interaction coefficient is either \(\beta'<1\) or in a fixed compact subinterval of \((0,1)\). Reduce the available diagonal enlargement in (105) to \(0<\eta<(\beta')^{-1}-1\). Once these stability margins and \(C_{\rm lead}\) are fixed, choose the mean accuracy \(\epsilon\) in Lemma 19 so that its leading error, integrated over the bounded path and converted to a quadratic mean bound, is less than \(\eta_1\). Then fix its residual tolerance, selection depth, finite word diagnostics and constant \(C_\epsilon\), and finally take \(n\) large.

On a valid posterior, apply (106) to \(f\) and to \(1\). With \(a_u=\|\nu_u[X]-\mu_u[X]\|/\sqrt n\), this gives \[a_u\le2C_{\rm lead}\epsilon+ \frac{C_\epsilon}{\sqrt n} \left(2+\sqrt{\frac{\mathcal E_{\mu_u}(f)}{N_u}}\right).\] By (103) and Cauchy–Schwarz, the \(Q_\nu\)-mean of the square root, also with any validity indicator, is at most \(\sqrt{\mathcal E(f)/N}\). Under (110) the resulting error is \[\frac{C_\epsilon}{\sqrt n} \left(2+\sqrt{\frac{\mathcal E(f)}N}\right) =O_\epsilon(n^{-1/2}+n^{-\omega/2})=o(1).\] This bound is applied at deterministic observation times; inserting \(\mathbf 1_{\{u<\tau\}}\) for a stopping time only decreases its Cauchy–Schwarz bound. Since \(a_u\le2\) always, \(a_u^2\le2a_u\). The early invalid paths cost at most four times their integrated probability, hence \(o(1)\). Thus the normalized quadratic mean-shift integral, on valid paths and before any later stopping, is at most \(\eta_1+o(1)\) by the preceding choice of \(\epsilon\).

Let \(u_g\) begin the globally controlled portion and let \(\tau\) be the first failure of the later buffered test in \([t_b,u_g]\), with the immediate and no-failure conventions of the source. The initial jump leaks at most \(C_{\rm jump}t_*n\) entropy: compare to pure noise as above and use \(\|B_*\|\le Ct_*\) and \(\|X\|^2=n\). The stopped form of (102), with the preceding integrated estimate, gives, with \(Q^\tau\) denoting the law of the stopped observations, excluding the spin, \[ D(Q_\nu^\tau\Vert Q_\mu^\tau) \le(C_{\rm jump}t_*+\eta_1+o(1))n . \tag{111}\] For the event \(E\) of a failure before \(u_g\), the ordinary probability is at most \(e^{-an}\). Data processing for its indicator then gives \[D(Q_\nu^\tau\Vert Q_\mu^\tau) \ge Q_\nu(E)\log\frac1{Q_\mu(E)}-\log2,\qquad Q_\nu(E)\le\frac{C_{\rm jump}t_*+\eta_1+o(1)}a .\] These are exactly the deductions in (OpenAI 2026b, Equations (19.21)–(19.22)), now with the enlarged energy range justified by Lemma 19. After a failure the remaining precision rate and duration are bounded, and both spin means have norm at most \(\sqrt n\); the remaining loss per failed path is therefore at most \(C_{\rm aft}n\). The total loss before \(u_g\) is bounded by \[\left(C_{\rm jump}t_*+\eta_1+o(1)\right) (1+C_{\rm aft}/a)n .\] The fixed choices above make this less than \(\lambda n/4\). The initial entropy is at least \(L/2\ge\lambda n/2\), hence \(H(u_g)\ge\lambda n/4\). In the remaining portion the spin covariance is bounded for every field. The linear entropy test then gives \(-\dot H\le CH\); over its bounded duration it retains a fixed fraction of \(H(u_g)\). This proves the lemma. ◻

Proof of Proposition 16. Fix \(C_{\mathrm{lev}},\xi\) and a desired disorder confidence. Apply Lemma 17 with \(\kappa<\min\{\xi,\,5/3-1.662\}\). After fixing its \(M\), choose its \(\delta>0\) as in that proof. For \(L\le n^\delta\), (87) and (83) imply both claimed inequalities.

For \(n^\delta<L\le\lambda n\), use the proportional branch of (OpenAI 2026b, Equation (19.16) and Section 19.4). It gives, at the product endpoint, \[ H_{\rm end}\ge cL(L/n)^{(2+2\epsilon)/3}. \tag{112}\] Here \(\epsilon>0\) is any fixed sufficiently small tolerance, and \(\lambda>0\) is chosen afterward. We justify the exact test class being used. The starting dimension in that source is \(\max\{k_{\min},c_0L\}\), with \(k_{\min}\asymp\sqrt{\log n}\). It equals \(c_0L\) throughout the present range for large \(n\). The proportional part of its initial jump estimate is the pure-noise argument reproduced in Lemma 17; it requires only the entropy at least \(L/2\) and density at most \(e^L\). The source’s support condition is used in its other, minimum-dimension branch. Its subsequent propagation uses the likelihood and conditional-entropy tails in (OpenAI 2026b, Equation (19.8)) and the ordinary path tests. These also require only the same density and entropy bounds. The continuation through its finite remaining scales uses those likelihood tails and conditional Pinsker bounds, then globally bounded covariance, and so preserves this test class. Thus (112) applies to (84) without a support restriction.

Since \(L\ge n^\delta\gg\log n\), comparison with the first bound of (104) gives \[T_n\,\frac{\mathcal E(f)}N \ge c n^{-2\epsilon/3}L^{7/6+2\epsilon/3} \ge c L^{7/6-(2\epsilon/3)(1/\delta-1)}.\] The additive \(O(n^{-3})\) is smaller than (112). The second product bound similarly gives \[T_n\,\frac{\mathcal I(f^2)}N \ge c n^{-2\epsilon/3}L^{5/3+2\epsilon/3} \ge c L^{5/3-(2\epsilon/3)(1/\delta-1)}.\] Choose \(\epsilon\) after \(\delta\) so that \[\frac{2\epsilon}{3}(1/\delta-1) <\min\{\xi,\,7/6-1.162\}.\] Then choose \(\lambda\) below the finitely many fixed rates and shift buffers in that source argument. This proves the profiles in the moderate range.

Finally let \(\lambda n\le L\le C_{\mathrm{lev}}n\). If the classical bound were false, then \(\mathcal E(f)/N\le Cn^{1.162-2/3}\). This meets (110) for a fixed positive \(\omega\), whereas the first product bound would give \[H_{\rm end}\le Cn^{1/2+1.162-2/3}+O(n^{-3})=o(n),\] contrary to Lemma 20. If the modified bound were false, (83) would instead give \(\mathcal E(f)/N\le Cn^{1-\xi}\), so that lemma applies with \(\omega=\xi\). The second product bound would give \(H_{\rm end}\le Cn^{1-\xi}=o(n)\), again a contradiction.

All the choices are uniform in \(f\). More explicitly, the variance exponent and \(M\) are fixed first, then the small \(\delta\), then the sharp cap and moderate-entropy tolerances, then \(\lambda\) and the fixed parameters of the order-\(n\) path. The finite word length in that last path is fixed after its mean accuracy. The final threshold \(K_0\) and \(n\) are chosen last. Intersecting this finite collection of disorder events preserves arbitrarily high limiting probability. This proves the proposition. ◻

Proof. In logarithmic coordinates \(l=\log z\), the logarithm of \(\mathcal U_d(e^l)\) is \[2l-0.8d\quad(l\le0.8d),\qquad l\quad(0.8d\le l\le1.1d),\qquad 1.1d\quad(l\ge1.1d).\] Consequently \(u\le z\), and \(u=z\) on the indicated level. This proves \(B\le\mu[u]\le1\) and \(L=1.1d-\log\mu[u]\), with the stated upper bound from the lower bound on \(B\).

Let \(E=\{\log z\ge0.7d\}\). Since \(u\le e^{-0.1d}z\) on its complement, \[\frac{\mu[u\mathbf 1_{E^c}]}{\mu[u]}\le\frac{e^{-0.1d}}B, \qquad \mu(E)\le e^{-0.7d}.\] Set \(e_d=e^{-0.1d}/B\). Data processing for the indicator of \(E\) gives \[D\!\left(\frac{u\mu}{\mu[u]}\middle\Vert\mu\right) \ge (1-e_d)\log\frac1{\mu(E)}-\log2 \ge0.7d(1-e_d)-\log2 .\] Uniformly for \(d\ge v/2\), the hypothesis on \(B\) makes \(e_d\to0\) and \(1+\log_+(d/v)=o(d)\) as \(v\to\infty\). The last lower bound therefore exceeds \(L/2\) for large \(v\). The level is nonempty, so \(d\le C_an\) by (113); the bound on \(L\), with \(C_{\mathrm{lev}}>2C_a+1\), puts it below \(C_{\mathrm{lev}}n\) for large \(n\). Its lower bound puts it above \(K_0\) for large \(v\).

Finally \(\mathcal U_d\) is nondecreasing and \(2\)-Lipschitz as a function of \(z\), while \(\log\mathcal U_d(e^l)\) is nondecreasing and \(2\)-Lipschitz as a function of \(l\). The product of their increments on an edge is at most four times the product of the increments of \(z\) and \(\log z\). Summing proves (115). ◻

Proof. Put \(H=\mathop{\mathrm{Ent}}_\mu(z)\) and \(d_j=(H/2)(1.1)^j\). The part of \(\mu[z\log z]\) with \(\log z<H/2\) is at most \(H/2\). Thus the levels \([d_j,1.1d_j]\) account for at least \(H/2\). If their tilted masses \(B_j\) were all less than \(c(H/d_j)^{1.1}\), their contribution would be at most \[1.1\sum_{j\ge0}d_jB_j <1.1cH^{1.1}\sum_{j\ge0}d_j^{-0.1}\le CcH .\] Choose \(c\) small. One level therefore has \(B\ge c(H/d)^{1.1}\). Apply Lemma 21 with \(v=H\) and \(r=1.1\), followed by (85) with any fixed \(\xi<5/3-1.662\). Homogeneity of \(\mathcal I\) gives \[T_n\mathcal I(z)\ge\tfrac14T_n\mathcal I(u) \ge c\,\mu[u]L^{1.662} \ge cB d^{1.662} \ge c'H^{1.1}d^{0.562}\ge c''H^{1.662}.\] The constants and the required lower threshold on \(H\) are uniform. ◻

Proof. First let \(0\le g\le1\) equal one on a nonempty set of mass \(a=e^{-L_a}\), and let its support have mass at most \(e^{-L_{\rm supp}}\). Take \[K=\left\lceil\log_2(1+L_a/L_{\rm supp})\right\rceil,\qquad g_i=K\bigl((g-i/K)_+\wedge K^{-1}\bigr),\quad 0\le i<K .\] Let \(S_i=\mu(g>i/K)\), \(T_i=\mu(g\ge i/K)\). Some \(i\) satisfies \(T_{i+1}\ge S_i^2\); otherwise induction gives \(a\le T_K<e^{-2^KL_{\rm supp}}<a\). For this \(i\), with \(N_i=\mu[g_i^2]\), \[a\le T_{i+1}\le N_i\le S_i,\qquad D\!\left(\frac{g_i^2\mu}{N_i}\middle\Vert\mu\right) \ge\log(1/S_i)\ge\tfrac12\log(1/N_i).\] Its logarithmic inverse mass \(L_i\) lies in \([L_{\rm supp},L_a]\), and \(L_a\le C_an\) by (113). For \(L\ge K_0\) large enough, the function \[e^{-L} \begin{cases}L^{1.662},&L\le n^\delta,\\ L^{1.162},&L>n^\delta \end{cases}\] is decreasing within each range and has a downward jump at their boundary. Since \(\mathcal E(g_i)\le K^2\mathcal E(g)\), (86) implies \[T_n\mathcal E(g)\ge \frac{ca}{(1+\log L_a)^2} \begin{cases}L_a^{1.662},&L_a\le n^\delta,\\ L_a^{1.162},&L_a>n^\delta . \end{cases}\] Absorbing the logarithmic denominator by lowering each exponent by \(0.008\) gives \[ T_n\mathcal E(g)\ge ca \begin{cases}L_a^{1.654},&L_a\le n^\delta,\\ L_a^{1.154},&L_a>n^\delta . \end{cases} \tag{118}\] Indeed \(L^{0.008}/(1+\log L)^2\) has a positive infimum for \(L\ge K_0\).

For arbitrary nonnegative \(g\) with the stated support, use \(\varphi_j=(g-2^j)_+\wedge2^j\), \(j\in\mathbb Z\), and let \(a_{j+1}=\mu(g\ge2^{j+1})\). Omit empty upper level sets. The normalized layer \(2^{-j}\varphi_j\) is one on that set and has the original support restriction. Its \(L_a=\log(1/a_{j+1})\) is at least \(L_{\rm supp}\). The right side of (118), divided by \(a\), is at least \(\Phi_n(L_{\rm supp})\): before the boundary it is at least \(L_{\rm supp}^{1.654}\), and after the boundary it is at least \(n^{1.154\delta}\), or at least \(L_{\rm supp}^{1.154}\) when \(L_{\rm supp}>n^\delta\). For each oriented edge the nonnegative increments of the \(\varphi_j\)’s sum to the increment of \(g\), so the sum of their squares is at most its square. Therefore \[T_n\mathcal E(g)\ge\sum_jT_n\mathcal E(\varphi_j) \ge c\Phi_n(L_{\rm supp})\sum_j4^ja_{j+1}.\] Pointwise, \(\sum_j4^j\mathbf 1_{\{g\ge2^{j+1}\}}\ge g^2/12\). Integration proves (117). ◻

Proof of Theorem 15. Fix the parameters in the theorem and intersect the finitely many events needed for the preceding results. The finite heat-bath chain has a strictly positive kernel at every positive time. Hence \(w_s^\rho>0\) for \(s>0\), for every initial law, and symmetry of the generator gives \[\frac{\,\mathrm d}{\,\mathrm ds}\mathop{\mathrm{Ent}}_\mu(w_s^\rho) =-T_n\mathcal I(w_s^\rho).\] When the entropy exceeds the threshold of Lemma 22, its derivative is at most \(-c\mathop{\mathrm{Ent}}_\mu(w_s^\rho)^{1.662}\). Integrating this differential inequality from any earlier positive time, and then letting that time decrease to zero, yields uniformly in the initial law \[\mathop{\mathrm{Ent}}_\mu(w_s^\rho) \le C\left(1+s^{-1/0.662}\right) \le C'\left(1+s^{-1.54}\right).\] This proves (80).

For the kernel, put \(Q_s=\mu[(w_s^\rho)^2]=e^{L_s}\). The atom bound gives \(Q_s\le e^{C_an}\), so its logarithm is within the range where the support profile was proved. Whenever \(L_s\) is large, set \(g=(w_s^\rho-Q_s/4)_+\). Its support has mass at most \(4e^{-L_s}\) by \(\mu[w_s^\rho]=1\), and \[\mu[g^2]\ge Q_s-2(Q_s/4)\mu[w_s^\rho]=Q_s/2,\qquad \mathcal E(g)\le\mathcal E(w_s^\rho).\] Its actual support logarithm is \[L_{\rm supp}=\log\frac1{\mu\{w_s^\rho>Q_s/4\}} \ge L_s-\log4.\] The support is nonempty by the preceding square-mass bound, so (113) also gives \(L_{\rm supp}\le C_an\). Since \(\dot Q_s=-2T_n\mathcal E(w_s^\rho)\), Lemma 23 and monotonicity of \(\Phi_n\) give \[-\dot L_s\ge c\,\Phi_n(L_s-\log4)\] above a fixed threshold. A shift of the argument by \(\log4\) only changes constants there. For any large \(K'\), the time allowed by this differential inequality above \(K'\) is at most \[ \int_{K'}^\infty\frac{C\,\,\mathrm dL}{\Phi_n(L)} \le C(K')^{-0.654}+C n^{-0.154\delta}. \tag{119}\] For completeness, below \(n^{\delta(1.154/1.654)}\) integrate \(L^{-1.654}\). The following constant portion of \(\Phi_n\), up to \(n^\delta\), costs at most \(n^\delta/n^{1.154\delta}\), and the integral of \(L^{-1.154}\) above \(n^\delta\) has the same order. Inverting (119) gives \[L_s\le C(1+s^{-1/0.654}) \qquad\hbox{for }s\ge Cn^{-0.154\delta},\] uniformly even for point initial laws. Reversibility and two half steps show that \[p_{sT_n}(\sigma,\tau) =\mu[p_{sT_n/2}(\sigma,\cdot)p_{sT_n/2}(\tau,\cdot)] \le\bigl(\mu[p_{sT_n/2}(\sigma,\cdot)^2]\, \mu[p_{sT_n/2}(\tau,\cdot)^2]\bigr)^{1/2}.\] The preceding bound, with \(1/0.654<1.54\), proves (81) after enlarging the vanishing threshold.

We finish by proving (82). For the fixed integer \(m\ge1\), let \(\Psi_m\) be the convex function on \((0,\infty)\) determined by \[\Psi_m(1)=\Psi_m'(1)=0,\qquad \Psi_m''(z)=\frac{(1+|\log z|)^{m-1}}{z}.\] It extends continuously to zero: in its twice-integrated formula the possible singularity is bounded by \(\int_0^1(1+|\log z|)^{m-1}\,\mathrm dz<\infty\). For a density \(z\), put \[V_m(z)=\mu[\Psi_m(z)],\qquad \mathsf M_m(z)=\left(1+\mu[z(\log_+z)^m]\right)^{1/m}.\] There are constants depending only on \(m\) such that \[ c_m\mathsf M_m(z)^m-C_m \le V_m(z)\le C_m\mathsf M_m(z)^m. \tag{120}\] Indeed for \(z\ge1\), \(\Psi_m'(z)=((1+\log z)^m-1)/m\). Integrating this on \([1,z]\) gives upper and lower bounds of order \(z(\log z)^m\) when \(\log z\) is large; the remaining range contributes at most a constant times \(1+z\). On \([0,1]\) \(\Psi_m\) is bounded.

For a strictly positive density \(z\), define the corresponding dissipation \(\mathcal D_m(z)=\mathcal E(z,\Psi_m'(z))\). We claim that, for \(\mathsf M_m(z)\) above a fixed threshold, \[ T_n\mathcal D_m(z) \ge c_{m,\xi}\mathsf M_m(z)^{m+2/3-\xi}. \tag{121}\] Put \(V=\mathsf M_m(z)\) and use levels \(d_j=(V/2)(1.1)^j\). If \(V\) is large, the part of \(\mu[z(\log_+z)^m]\) below \(V/2\) is at most \(2^{-m}V^m\). Geometric summation as in Lemma 22 gives a level \(d\ge V/2\) with \[B=\mu[z\mathbf 1_{\{d\le\log z\le1.1d\}}] \ge c_m(V/d)^{m+0.1}.\] Indeed otherwise \((1.1)^m\sum_jd_j^mB_j\le C_m c_m V^m\) would be too small. The test \(u=\mathcal U_d(z)\) from Lemma 21 therefore satisfies (84), and (85) gives \[ T_n\mathcal I(u)\ge c\,\mu[u]L^{5/3-\xi} \ge cB d^{5/3-\xi}. \tag{122}\]

We check that the new dissipation controls this layer with its additional weight \(d^{m-1}\). On an edge write \(a\ge b>0\), \(x=\log a\), \(y=\log b\). Then \[\Psi_m'(a)-\Psi_m'(b) =\int_y^x(1+|t|)^{m-1}\,\mathrm dt.\] If \(x\ge0.4d\), the last integral is at least \(c_m d^{m-1}(x-y)\). To verify this uniformly in \(y\), use the top half of \([0,x]\) when \(-x\le y\le x/2\), the entire interval when \(y\ge x/2\), and the negative half nearest \(y\) when \(y<-x\); in each case a fixed fraction of the interval has \(|t|\) at least a fixed fraction of \(x\). The two Lipschitz bounds in the proof of Lemma 21 then apply. If \(x<0.4d\), both endpoints are in the lower branch of \(\mathcal U_d\). On \([b,a]\) its derivative is at most \(2e^{-0.4d}\), while its logarithm has slope two in log coordinates. Since \(d^{m-1}e^{-0.4d}\le C_m\), the same conclusion follows using \((1+|t|)^{m-1}\ge1\). Thus in both cases \[d^{m-1}\bigl(\mathcal U_d(a)-\mathcal U_d(b)\bigr) \bigl(\log\mathcal U_d(a)-\log\mathcal U_d(b)\bigr) \le C_m(a-b)\bigl(\Psi_m'(a)-\Psi_m'(b)\bigr).\] Summation gives \(\mathcal D_m(z)\ge c_m d^{m-1}\mathcal I(u)\). Combining this with (122) and the lower bound on \(B\), and using \(2/3-\xi>0.1\), gives \[T_n\mathcal D_m(z) \ge cB d^{m+2/3-\xi} \ge c_m V^{m+0.1}d^{2/3-\xi-0.1} \ge c_m' V^{m+2/3-\xi}.\] This proves (121).

Along the finite semigroup at positive times, \[\frac{\,\mathrm d}{\,\mathrm ds}V_m(w_s^\rho) =-T_n\mathcal D_m(w_s^\rho).\] Equations (120)–(121) give, above a fixed threshold, \[\frac{\,\mathrm d}{\,\mathrm ds}V_m(w_s^\rho) \le-c V_m(w_s^\rho)^{1+(2/3-\xi)/m}.\] Integrating from an earlier positive time and letting that time decrease to zero yields \(V_m(w_s^\rho)\le C(1+s^{-m/(2/3-\xi)})\). The comparison (120) proves (82). This argument allows initial zeros, because \(\Psi_m\) is continuous at zero and the kernel is strictly positive afterward.

The three conclusions (80)–(82) depend only on the off-diagonal couplings. Let \(\mathcal G_n^{\rm aug}\) be the final event in the augmented Gaussian space, and put \[\mathcal G_n^{\rm off} =\left\{W_{\rm off}: \mathbb P(\mathcal G_n^{\rm aug}\mid W_{\rm off})>0\right\}.\] Then \(\mathbb P(\mathcal G_n^{\rm off})\ge \mathbb P(\mathcal G_n^{\rm aug})\). For fixed off-diagonal couplings the three inequalities are deterministic, so their validity on an augmented realization in \(\mathcal G_n^{\rm aug}\) descends to \(\mathcal G_n^{\rm off}\). Taking this as the theorem event proves the stated disorder quantifiers. ◻

Proof. Choose \(\xi>0\) so small that \[\frac{\xi}{2/3-\xi}<\frac{\beta}{2},\] then choose a fixed integer \(m>2A/\beta\). Apply (82) to the point initial law at \(\sigma\) and time \(s=d/T_n\). Set \(v=T_n/d\). Uniformly for \(1\le d\le C_\star T_n\), so \(C_\star^{-1}\le v\le n^{2/3}\), \[\frac{1+v^{1/(2/3-\xi)}}{n^\beta v^{3/2}} \le C_{\xi,C_\star} n^{-\beta+\xi/(2/3-\xi)} \le C_{\xi,C_\star}n^{-\beta/2}.\] Markov’s inequality under the transition law \(P_d(\sigma,\tau)=\mu(\tau)p_d(\sigma,\tau)\) now bounds the left side of (123) by \[\left( \frac{2\{1+\mu[p_d(\sigma,\cdot) (\log_+p_d(\sigma,\cdot))^m]\}^{1/m}} {n^\beta(T_n/d)^{3/2}}\right)^m \le C_m n^{-m\beta/2}\le C_m n^{-A}.\] The choices of \(\xi,m\) are fixed in that order before \(n\). The statement for \(\sigma\sim\rho\) follows by averaging these row-specific probabilities. ◻

Proof. The closed form of \(H\) is \(\mathcal E_g(F,G)=\Pi[\nabla F\cdot\nabla G]\). Its domain is the closure of the smooth compact cylinder functions, contains the constants and all coordinates, and satisfies \(\nabla x_i=e_i\) (OpenAI 2026d, Proposition 5.1). Smooth contractions on the core extend by closure, so this is a Dirichlet form. The same approximation, with smooth clipping to preserve a common bound, gives the chain and bounded product rules. In particular the gradient vanishes almost everywhere on a set on which the function is constant: apply the chain rule to smooth truncations in successively smaller neighborhoods of the constant, and use closedness. These rules show strong locality.

Here are the topological hypotheses for the diffusion theorem. The state space \(\mathbb R^\mathbb N\) is Polish in the product topology. The smooth compact cylinder core consists of continuous functions and contains a countable family separating its points. To obtain a compact nest, for each coordinate choose a smooth function \(v_j\), equal to one outside a sufficiently large coordinate interval and zero on a smaller interval, with \(\|v_j\|_{\mathcal E_g,1}\le\epsilon2^{-j}\). Such a choice is possible: coordinate tails have vanishing \(\Pi\)-mass, while a transition of increasing width makes the gradient norm vanish. The sum \(\sum_jv_j\) converges in the form norm and is at least one on the complement of a compact product box. The definition of \(1\)-capacity therefore bounds that complement’s capacity by \(\epsilon^2\). Taking \(\epsilon\downarrow0\) and enlarging successive boxes gives a compact nest. The form is thus quasi-regular. Since \(1\) has zero energy, its semigroup fixes \(1\); the associated process has infinite lifetime under \(\Pi\), and hence under \(w\Pi\).

The diffusion theorem and locality criterion for quasi-regular symmetric Dirichlet forms now give a continuous associated process (Ma and Röckner 1992, Theorems IV.3.5 and V.1.11). Its Fukushima decomposition is supplied by (Ma and Röckner 1992, Theorem VI.2.5). In particular, the coordinate martingales \(M^i\) are continuous and have brackets \[\langle M^i,M^j\rangle_t =2\int_0^t \nabla x_i\cdot\nabla x_j(X(v))\,dv =2\delta_{ij}t.\] The bracket identity is the energy-measure identity for the form, given by (Fukushima et al. 2011, Theorem 5.2.3 and Equation (5.2.34)) after the quasi-regular regularization of (Ma and Röckner 1992, VI). To apply its bounded formula, first take a bounded smooth clip \(u_N=c_N(x_i)\). For every bounded form-domain test \(f\), the chain and product rules give \[2\mathcal E_g(u_Nf,u_N)-\mathcal E_g(u_N^2,f) =2\Pi[f|c_N'(x_i)|^2].\] Choose the clips with \(|c_N'|\le1\) and \(c_N(x_i)\to x_i\) in form norm. Passing to the limit in the energy measures and polarizing gives the displayed coordinate brackets. The multidimensional Lévy characterization, for every finite set of indices, gives \(M^i=\sqrt2W_i\) with the asserted countable family of Brownian motions. Brownian motion here is relative to the filtration of the process, so its future increments are independent of the initial sigma field. The usual form exceptional set has \(\Pi\)-measure zero; it can be discarded for \(w\Pi\). These facts also give continuous coordinate paths.

We verify the generator used for \(\theta_i\). The finite-coordinate density formula for \(\Pi\), on any finite set \(J\), is \[ \frac{q_{J^c}\!\left(D-\sum_{j\in J}(x_j^2-r_j^{-1})\right)} {q(D)} \prod_{j\in J}\left(\frac{r_j}{2\pi}\right)^{1/2} e^{-r_jx_j^2/2}. \tag{164}\] Here \(q\) is the density of the full centered Gaussian square sum, and \(q_{J^c}\) is the density after the coordinates in \(J\) are deleted (OpenAI 2026d, Proposition 2.3). Both densities exist, \(q(D)>0\), and \(q_{J^c}\) has bounded derivatives of any prescribed finite order. For the last assertion, leave sufficiently many fixed indices outside \(J\); the product of their characteristic-function moduli is bounded by \(\prod (1+4t^2r_j^{-2})^{-1/4}\), which is integrable even after multiplication by the prescribed power of \(|t|\). Fourier inversion then gives the assertion. In particular (164) is bounded by a constant times the product Gaussian density.

The set \(R=0\) is polar. Indeed a smooth cutoff equal to one on \(\{R\le\epsilon\}\), zero on \(\{R\ge2\epsilon\}\), and with derivative \(O(\epsilon^{-1})\), has squared form norm \(O(\epsilon^m+\epsilon^{m-2})\), by (164). Letting \(\epsilon\downarrow0\) proves zero capacity. A diffusion starting outside its exceptional set never hits a polar set, which proves the stated pathwise positivity of \(R\).

For a cylinder test whose index set \(J\) contains \(\{1,\ldots,m,i\}\), integrate by parts in (164). The derivative of the factor \(q_{J^c}\) is radial in the \(J\) coordinates. Its pairing with the gradient of \(\theta_i\) vanishes, because \(\sum_{j\in J}x_j\partial_j\theta_i=0\). Consequently the nonpositive generator on this function is \[\mathcal L\theta_i=\Delta\theta_i-\sum_{j\in J}r_jx_j\partial_j\theta_i.\] This integration by parts is legitimate across \(R=0\). More explicitly, cut off at \(R=\epsilon\) and at large finite-coordinate radius. The function and its first two derivatives near zero are bounded by \(C_i(1+|x_i|)(1+R^{-3})\). The products with derivatives of the scale-\(\epsilon\) cutoff satisfy the same bound on \(R\asymp\epsilon\). Their squared integrals near zero vanish for \(m=12>6\). Terms with a derivative on \(q_{J^c}\) are bounded by the same Gaussian integrals with polynomial factors, since its first derivative is bounded. Gaussian tails handle the outer cutoff. Thus the cutoffs converge in the form norm, proving \(\theta_i\) belongs to the form domain, and the weak generator pairing passes to the limit. The displayed \(\mathcal L\theta_i\) is in \(L^2(\Pi)\) by the same bounds. Extending the pairing from the core shows \(\theta_i\in\operatorname{Dom}(H)\) and \(H\theta_i=-\mathcal L\theta_i\).

A direct differentiation gives, with the head vector extended by zeros, \[\partial_j\theta_i=\frac{\delta_{ij}-\theta_i\theta_j\mathbf 1_{\{j\le m\}}}{R}, \qquad \mathcal L\theta_i =\theta_i\left(\frac{3-m-2\mathbf 1_{\{i\le m\}}}{R^2} -r_i+\overline r\right).\] The Fukushima drift is the time integral of this \(L^2\) generator. Its martingale has brackets with the coordinate martingales equal to \(2\int_0^t\partial_j\theta_i(X(v))\,dv\), and self-bracket \(2\int_0^t|\nabla\theta_i(X(v))|^2\,dv\). It therefore equals \(\sqrt2\sum_j\int_0^t\partial_j\theta_i(X(v))\,dW_j(v)\): the difference has zero bracket. Only the head and index \(i\) occur in this finite sum. Integrability follows, for instance, from \[\mathbb E\int_0^T|\nabla\theta_i(X(v))|^2\,dv \le T\|w\|_\infty\mathcal E_g(\theta_i)<\infty,\] because \(\|\widetilde S_vw\|_\infty\le\|w\|_\infty\). This proves (163). Countability permits a common null set for all indices. ◻

Proof. The local integrability follows directly from \[\int_0^T L(t)\,dt =\frac12\sum_i r_i^{-2}(1-e^{-2r_iT})<\infty.\] For \(L_N=\sum_{i\le N}r_i^{-1}E_i\), put \[F_N(z)=D+z\widehat L_N(z) =D+\sum_{i\le N}\frac{z}{r_i(z+2r_i)}.\] The rational function \(F_N\) is strictly increasing between its negative poles, since \(F_N'(z)=\sum_{i\le N}2(z+2r_i)^{-2}>0\). Its zeros interlace those poles and are strictly negative, using \(D>0\). Its reciprocal is therefore a positive constant plus a sum of positive multiples of \((z+a)^{-1}\), \(a>0\): the residue at a zero is \(1/F_N'(-a)>0\). Repeated rates can be grouped before this argument.

Write these sums as Stieltjes transforms of positive measures. Their measures weighted by \((1+a)^{-1}\), with the constant as mass at the point at infinity, have total mass \(F_N(1)^{-1}\le D^{-1}\). Weak compactness on \([0,\infty]\) and monotone convergence of \(F_N(z)\) give \[\frac1{D+z\widehat L(z)} =c+\int_{[0,\infty)}\frac{\nu(da)}{z+a}, \qquad \int\frac{\nu(da)}{1+a}<\infty.\] By (158), splitting the sum at \(r_i=z\) gives \[ D+z\widehat L(z)\asymp_g \sqrt z\qquad(z\ge1). \tag{178}\] For sufficiently large \(z\), the part with \(r_i\le z\) is comparable to \(\sum_{r_i\le z}r_i^{-1}\asymp\sqrt z\), and the other part to \(z\sum_{r_i>z}r_i^{-2}\asymp\sqrt z\). Positivity and continuity adjust the constants on the remaining compact interval \(z\ge1\). Hence the transform tends to zero at infinity and \(c=0\). It is bounded as \(z\downarrow0\), so \(\nu\) has no atom at zero. The function \[K(t)=\int_{(0,\infty)}e^{-at}\nu(da)\] has the required Laplace transform by Tonelli’s theorem. The weighted integrability of \(\nu\) implies smoothness for \(t>0\), and the representation makes \(K\) positive and decreasing.

The upper bound in (171) follows from \[e^{-1}tK(t)\le\int_0^t e^{-v/t}K(v)\,dv \le\widehat K(1/t)\le C_g\sqrt t.\] For the lower bound choose a fixed small \(\eta>0\). When \(t\le\eta\), (178) bounds \(\widehat K(\eta/t)\) below by \(c_g\sqrt{t/\eta}\), whereas the upper bound just proved bounds its integral over \([0,t]\) by \(C_g\sqrt t\). Choose \(\eta\) so that the first constant is larger than twice the second. The integral over \([t,\infty)\) is at most \(\eta^{-1}t e^{-\eta}K(t)\), proving the lower bound for \(t\le\eta\). Positivity and monotonicity adjust the constant on \([\eta,1]\). Finally \[|K'(t)|=\int a e^{-at}\nu(da) \le \frac{C}{t}K(t/2)\le C_g t^{-3/2}.\]

For \(r_it\le1\), integration of \(K(v)\le C_gv^{-1/2}\) gives the first bound in (172). For \(r_it>1\), write \(K_i(t)=\int_0^t K(t-v)e^{-2r_iv}\,dv\) and split at \(t/2\). The first half is bounded by \(C_g/(r_i\sqrt t)\); the second by \(C_g\sqrt t\,e^{-r_it}\), which has the same bound. For the derivative use \(K_i'=K-2r_iK_i\). When \(r_it\le1\) the preceding bounds suffice. When \(r_it>1\), write \[K_i'(t)=K(t)e^{-2r_it} -2r_i\int_0^t e^{-2r_iv}\bigl(K(t-v)-K(t)\bigr)\,dv.\] On \(v\le t/2\) the derivative bound on \(K\) makes the integrand difference at most \(C_g v t^{-3/2}\); on \(v>t/2\) the exponential factor pays the integral of \(K\). This gives \(|K_i'(t)|\le C_g/(r_it^{3/2})\). It also proves (173), uniformly away from zero.

The remaining estimates follow by summing (172). For example (158) gives \[\sum_i\frac{(1+r_i)^{2\xi}}{(1+r_it)^2} \le C_{g,\xi}t^{-3/2-2\xi}\quad(0\le\xi<1/4).\] The restriction on \(\xi\) is exactly integrability at infinity after the change of variable \(v=r_it\): the comparison integral there is \(\int_0^\infty v^{1/2+2\xi}(1+v)^{-2}\,dv\). This proves (175). If \(h<t\), integrate the bound on \(K_i'\) between \(t\) and \(t+h\), and then use the same sum with \(\xi=0\); this gives (176) and (174). If \(h\ge t\), bound the two values separately. Finally the locally integrable function \[D\int_0^tK(v)\,dv+(K*L)(t)\] has Laplace transform \(1/z\), and hence equals \(1\) for almost every \(t>0\). Its causal distributional derivative gives (177). ◻

Proof. The bracket estimate and the Burkholder–Davis–Gundy inequality give \[\|Z_i^q(t)\|_{L^j}^2 \le C_{C_0,j}\int_0^t e^{-2r_i(t-v)}v^{-1/2}\,dv \le C_{C_0,j}\frac{\sqrt t}{1+r_it}.\] For the last inequality split at \(t/2\); on the second half use \(v^{-1/2}\le C t^{-1/2}\), and on the first use \(e^{-2r_i(t-v)}\le e^{-r_it}\).

Write \(p_i=y_i^2/(2r_i)\). The bracket of the first series, at a fixed time \(t\), is bounded pathwise by \[C_{C_0}\int_0^t v^{-1/2} \sum_i p_i r_iK_i(t-v)^2\,dv \le C_{g,C_0}\sqrt t,\] by (174) and \(\sum_i p_i=1\). This proves its bound in (181). In its increment, the old portion of the bracket is at most \[C_{g,C_0}\int_0^t v^{-1/2} \min\{1,h^2/(t-v)^2\}\,dv \le C_{g,C_0}\sqrt h ,\] and the new portion is bounded by \(C\int_t^{t+h}v^{-1/2}\,dv\le C\sqrt h\). For the displayed integral, if \(t\le2h\) discard the minimum; if \(t>2h\), split at \(t-h\) and at \(t/2\). These bounds prove the first increment estimate.

For the second series, the martingale moment inequality followed by Minkowski’s inequality bounds its \(L^j\) norm squared by \[C_j\int_0^t\sum_i K_i(t-v)^2 \|q(v)Z_i^q(v)\|_{L^j}^2\,dv \le C_{g,C_0,j}\int_0^t (t-v)^{-1/2}\,dv \le C_{g,C_0,j}\sqrt t.\] We used (180), the deterministic bound on \(q\), and (175) with \(\xi=0\). For the increment use (176) on the old portion: its squared norm is bounded by \[C_{g,C_0,j}\int_0^t (t-v)^{-1/2} \min\{1,h^2/(t-v)^2\}\,dv \le C_{g,C_0,j}\sqrt h .\] The new portion has the same bound by integrating \((t+h-v)^{-1/2}\) from \(t\) to \(t+h\).

All these estimates hold for partial sums. At each fixed time their tails tend to zero in \(L^j\): for the first series use \(\sum_{i>N}p_i r_iK_i(t-v)^2\to0\) and its constant bound inside the bracket; for the second use the summable bound just displayed. Dominated convergence applies in both cases. Taking \(j>4\), (182) gives uniformly small moduli of continuity, by a dyadic chaining estimate (equivalently the Kolmogorov continuity criterion). Fixed-time convergence on a finite grid then proves convergence in probability in \(C([0,1])\). The same chaining has a convergent series \(\sum_{k\ge0}2^{k/j}2^{-k/4}\), and bounds the \(L^j\) norm of the supremum for \(j>4\). Taking a higher order gives the asserted bound for every finite order. ◻

Proof. We now justify the infinite passage in (169). The constraint says \(R_N(x)=\sum_{i>N}(x_i^2-r_i^{-1})\) \(\Pi\)-almost surely. The conditioned edge law describes the tail, after any sufficiently large fixed head, by a bounded density relative to its product Gaussian law (OpenAI 2026d, Proposition 2.3). Consequently, for \(N\) beyond that head, \[ \Pi[|R_N|]\le C_g\left(\sum_{i>N}r_i^{-2}\right)^{1/2} \longrightarrow0. \tag{184}\] Indeed the Gaussian tail is centered and has variance \(2\sum_{i>N}r_i^{-2}\).

The density of \(X(t)\) is bounded by \(\|w\|_\infty\) for all elapsed \(t\ge0\). The constraint therefore holds for almost every elapsed time almost surely, and (184) gives \[\mathbb E\int_0^T|R_N(X(v))|\,dv \le T\|w\|_\infty\Pi[|R_N|]\longrightarrow0 .\] The continuous process \(f\) is bounded on \([0,T]\) almost surely. First restrict to a bound for that supremum, and then let the bound increase. It follows that \(fR_N(X)\to0\) in probability in \(L^1(0,T)\). Also \[ \|L-L_N\|_{L^1(0,T)} \le\frac12\sum_{i>N}r_i^{-2}\longrightarrow0 . \tag{185}\] Convolving (169) with \(K\), its left side therefore tends to \(f\) in local distributions by (177). One can see this directly by writing the difference as the derivative of \(K*(L_N-L)*f\); its undifferentiated factor tends to zero in \(L^1_{\mathrm{loc}}\) by (185). The convolved residual also tends to zero in probability in \(L^1_{\mathrm{loc}}\), since \(K\) is locally integrable.

The convolved deterministic terms are \(\sum_{i\le N}K_i(t)y_i^2\), which converge uniformly on compact subintervals of \((0,1]\) to \(A_y(t)\) by its definition and bound above.

For a finite stochastic sum, stochastic Fubini gives \[K*\bigl(E_i*(QY_i\,dW_i)\bigr)(t) =\int_0^t K_i(t-v)Q(v)Y_i(v)\,dW_i(v).\] It can first be used with \(QY_i\) stopped at a finite bound. For such a bound a sufficient stochastic Fubini integral is at most \[C\int_0^t K(t-s) \left(\int_0^s E_i(s-v)^2\,dv\right)^{1/2}ds \le C\int_0^t K(t-s)\sqrt s\,ds<\infty.\] Removing that bound by localization gives the displayed identity as local distributions.

To specify the limit without a formal exchange of infinite stochastic sums, for \(C_0>0\) set \[q_{C_0}(v)=Q(v)\wedge(C_0v^{-1/4}),\qquad \zeta_{C_0}=\inf\{v>0:Q(v)>C_0v^{-1/4}\}.\] By Lemma 30, the partial sums of \(\mathsf G_{q_{C_0}}^y+\mathsf N_{q_{C_0}}\) converge in probability in \(C([0,1])\) on the whole probability space. Before \(\zeta_{C_0}\), (167) gives \(Y_i(v)=e^{-r_iv}y_i+Z_i^{q_{C_0}}(v)\), so these clipped partial sums agree there with the convolved finite stochastic sums.

Fix a deterministic \(C_0>0\) and a rational \(a\in(0,1]\). On the whole probability space, the clipped partial sums converge in probability in \(C([0,a])\), and the convolved residual \(K*(fR_N(X))\) converges in probability to zero in \(L^1(0,a)\) as shown above. We may therefore choose a deterministic subsequence \(N_j\uparrow\infty\) along which both convergences hold almost surely. Only then restrict to the event \(\{a<\zeta_{C_0}\}\). On this event, passage in (169) proves (183) as a distribution on \((0,a)\). Take \(C_0\) through the positive integers and \(a\) through the rationals in \((0,1]\), and intersect the corresponding whole-space probability-one convergence events. The continuous versions then give equality for \(0<t<\min\{\zeta_{C_0},1\}\), and by continuity at a positive hitting time \(\zeta_{C_0}\le1\) and at \(t=1\) when \(\zeta_{C_0}>1\). The positive continuous \(Q\) is finite at zero, and \(\sup_{v\le1}v^{1/4}Q(v)<\infty\) almost surely. A sufficiently large integer \(C_0\) therefore covers the whole interval. This proves the claimed identity and its precise interpretation. ◻

Proof. By (173), \(2r_iK_i\to K\) uniformly on a fixed compact positive interval, and these functions are uniformly bounded there. In the convex combination \(A_y=\sum_i p_i\,2r_iK_i\), the contribution of a fixed finite set tends to zero in probability and the contribution of its complement is uniformly close to \(K\). This proves (186).

For the stochastic assertion, its squared norm is at most \[C_{C_0}\int_0^t v^{-1/2}\, \mathbb E\!\left[\sum_i p_i r_i K_i(t-v)^2\right]\,dv .\] For every \(v<t\), (172) implies \(r_iK_i(t-v)^2\to0\), whereas (174) bounds it uniformly. Escape of \(p_i\) makes the expression in brackets tend to zero in probability, and its uniform bound gives convergence of its expectation. Dominated convergence against \(v^{-1/2}\) proves the assertion. ◻

Proof. Choose a fixed smooth increasing function \(f\) that is zero on \((-\infty,1]\) and one on \([2,\infty)\). For each nonempty \(A_j(d)\) use the factor \(f(-\log\mu^{\otimes j}(A_j(d))/a_j(d))\), and omit empty events. Multiply these factors and a smooth function of \(n^{-6}\sum_e\theta_e^2\) that is one below \(1\) and zero above \(2\). This gives (196) and a plateau equal to one when each probability there is at most \(e^{-2a_j(d)}\) and the size statistic is at most one.

To check derivatives, the \(k\)th derivative of the logarithm of a nonempty replica probability in \(\theta_e\) is the difference of two cumulants of \(\sum_{a=1}^jX_i^aX_j^a\), under the restricted and unrestricted replica probabilities. The variable is bounded by \(j\le4\), so every fixed-order cumulant is bounded by a constant depending only on its order. Since \(a_2(d)\ge c_0 n^{3\beta}\) and \(a_4(d)\ge c_0 n^{8\beta}\) on \(\mathcal{D}_n\), every derivative of positive order of a replica factor gains at least \(n^{-3\beta}\). Derivatives of the size factor gain at least \(n^{-3}\): where a derivative is nonzero, \(|\theta_e|\le\sqrt2 n^3\). There are \(O(\log n)\) factors. The product rule now proves the stated bound, for example with any sufficiently small \(b<\min(3\beta,3)\).

We prove the Gaussian assertion with the diagonal and eigenvector sign randomization fixed as in Section 3. On spectral events of limiting probability as close to one as desired, Lemma 6 gives, for the uncapped zero-field spherical law and a sufficiently small fixed \(\rho_0>0\), \[\mathbb P_{\rm sph}^{\otimes2}\{ |\rho_2|>w\} \le C e^{-c_0nw^3},\qquad C_0p\le w\le\rho_0,\] where one can take any permitted reference scale with \(np^3\) above the tight threshold. The smallest pair threshold \(r_2(d)\) in (195) is \(\gg n^{-1/3}\), so such a \(p\) is available below all the nonvacuous shrinking pair thresholds. In a band \(z<|\rho_2|\le2z\), truncated at \(\rho_0\), we apply the weighted two-replica overlap-cell consequence of Lemma 8. On the retained event \(\|W\|\le3\), the logarithm of the positive zero-field angular weight has overlap derivative \(O(n)\), uniformly over the orthonormal two-frame. It therefore oscillates by at most a fixed constant on each cell of width \(2/n\). Summing the weighted cell inequality over the band enlarges its spherical endpoints by at most \(1/n\) and costs at most \(\exp(Cnz^4+C\log n)\) after increasing \(C\). For the pair thresholds \(w\) at issue, \(z\ge w\gg n^{-1/3}\), so the enlargement is absorbed by decreasing the tail constant \(c_0\). Shrink \(\rho_0\) so that this cost is bounded by \(\exp(c_0nz^3/2+C\log n)\). Summing the dyadic bands from \(w\) to \(\rho_0\) therefore bounds the Haar mean of the restricted cube partition function, divided by the squared spherical partition function, by \(\exp(-c_1nw^3)\) for all the thresholds in question. Here \(nw^3\ge c n^{6\beta}\) absorbs \(O(\log n)\). The likelihood ratio \(L_\circ=Z_{\rm cube}/Z_{\rm sp}\) is at least \(1/2\) with probability tending to one on these restrictions, by Equation (19). Markov’s inequality consequently proves the pair plateau, because \[\frac{nr_2(d)^3}{a_2(d)}=n^{3\beta}\longrightarrow\infty.\] For overlaps bounded away from zero, the fixed-overlap estimate in (OpenAI 2026d, Lemma 8.2) gives an \(e^{-c n}\) tail with Gaussian probability tending to one. This pays \(2a_2(d)=2n^{1-3\beta}r_2(d)^3=o(n)\) for any such nonvacuous threshold.

Here is the corresponding four-replica calculation. If \(r_4(d)<1\), then \(r_2(d)\le n^{-14\beta/3}=o(1)\) and \(r_2(d)^2/r_4(d)\le n^{-6\beta}\). For one row of four independent uniform signs, let \(Y\in\{-1,1\}^6\) be its six pair products and let \(H\) be its four-product. For \(n\) independent rows write \(\bar Y=n^{-1}\sum_iY_i\) and \(\bar H=n^{-1}\sum_iH_i\). Put \(\Lambda(v)=\log\mathbb Ee^{v\cdot Y}\), and let \(\mathbb P_v\) be the row probability with density \(e^{v\cdot Y-\Lambda(v)}\), with expectation \(\mathbb E_v\). For several rows the same notation denotes its product probability. Around the origin, \[\Lambda(v)=\tfrac12|v|^2+O(|v|^3),\qquad \mathbb E_v H=O(|v|^2).\] Indeed the six coordinates of \(Y\) are centered and have identity covariance, while \(\mathbb EH=\mathbb E(HY)=0\); Taylor’s theorem on this finite space gives both uniform remainders. If the empirical pair vector is the lattice value \(u\) with \(|u|\le\sqrt6 r_2(d)\), put \(E_d=\{|\bar H|>r_4(d)\}\). The change of measure gives \[\mathbb P_{\rm cube}^{\otimes4}\{\bar Y=u,\ E_d\} =e^{-n|u|^2+n\Lambda(u)} \mathbb P_u\{\bar Y=u,\ E_d\} \le e^{-n|u|^2+n\Lambda(u)}\mathbb P_u(E_d).\] Under \(\mathbb P_u\) the rows remain independent. Since \(|\mathbb E_uH|\le r_4(d)/2\) for large \(n\), Hoeffding’s inequality (Hoeffding 1963) bounds the last probability by \(2e^{-nr_4(d)^2/8}\). Together with the expansion of \(\Lambda\), this gives \[ \mathbb P_{\rm cube}^{\otimes4}\{\bar Y=u,\ |\bar H|>r_4(d)\} \le 2\exp\{-\tfrac n2|u|^2+Cn|u|^3-cn r_4(d)^2\}. \tag{197}\]

For completeness, the reverse Gram cost is also explicit. Put \(G_{aa}=1\) and \(G_{ab}=u_{ab}\). The Gram density of four independent uniform points of the sphere of radius \(\sqrt n\) is a polynomial normalization times \(\det(G)^{(n-5)/2}\) on positive correlation matrices. For \(|u|=o(1)\), \(\log\det G=-|u|^2+O(|u|^3)\). A cell of width \(c/n\) about \(G\) thus has probability at least \[\exp\{-\tfrac n2|u|^2-Cn|u|^3-C\log n\}.\] At a fixed spectrum with bounded norm, let \(F\in\mathbb R^{n\times4}\) be a uniform orthonormal frame and put \[\mathcal A_\lambda(G)=Z_{\rm sp}^{-4}\mathbb E_F \exp\{(n/2)\mathop{\mathrm{Tr}}(F^{\mathsf T}\mathop{\mathrm{diag}}(\lambda)F G)\}.\] The exponent has Lipschitz constant \(O(n)\) in \(G\), so the integrand changes by at most a factor \(e^{O(1)}\) across the cell. The integral of \(\mathcal A_\lambda\) against the spherical Gram probability equals one. Restricting it to the cell therefore bounds the Haar mean at Gram \(G\) by \[\exp\{\tfrac n2|u|^2+Cn|u|^3+C\log n\}.\] Multiply this by (197) and sum the at most \((n+1)^6\) possible Grams. The remaining error \(Cnr_2(d)^3+C\log n\) is negligible relative to \(nr_4(d)^2\), since \[nr_2(d)^3=n^{6\beta}nd^{-3/2},\qquad nr_4(d)^2=n^{20\beta}nd^{-3/2}.\] On \(L_\circ\ge1/2\), division by the cube partition function changes the four-replica numerator by at most \(16\). Markov’s inequality proves its plateau \(e^{-2a_4(d)}\), since \(nr_4(d)^2/a_4(d)=n^{12\beta}\). A union over \(O(\log n)\) scales is harmless. Finally, the Gaussian mean of \(n^{-6}\sum_e\theta_e^2\) is \(O(n^{-5})\). The size plateau follows from Markov’s inequality. Enlarging the spectral restrictions completes the proof that \(\chi_n=1\) with probability tending to one. ◻

Proof. Let \(\psi\) be smooth and nondecreasing, zero on \((-\infty,2]\), with \(0\le\psi'\le1\) and \(\psi'=1\) on \([3,\infty)\). Put \[a_N(z)=(\log n)^2(z-1)+3N\psi(z),\qquad W_N(l)=\bigl(1+e^{a_N(l/N)}\bigr)^{-1}.\] For \(l\le3N\) the first inequality in (199) is immediate. For \(z=l/N\ge3\), \(\psi(z)\ge z-3\), so \(l-a_N(z)\le3N\); the same inequality follows from \(W_N\le e^{-a_N}\). If \(l\le N/2\), then \(a_N(l/N)\le-(\log n)^2/2\), proving the plateau.

All \(l\)-derivatives of \(a_N(l/N)\) of positive order at most \(K\) are bounded: the first is \((\log n)^2/N+3\psi'(l/N)\) and the others are \(3N^{1-k}\psi^{(k)}(l/N)\). Derivatives of the logistic function have a factor \(W_N(1-W_N)\) times a bounded polynomial in \(W_N\), which proves the first part of (200). If \(W_N\ge n^{-L}\), then \(a_N\le L\log n+O(1)<(\log n)^2\); hence \(l/N<2\) for large \(n\). In this region \(a_N\) is affine, and its slope in \(l\) is \((\log n)^2/N\). The same derivative formula proves (201). It also proves the second part of (200) for \(l/N\le2\). In the remaining region \(W_N\le e^{-(\log n)^2}\), so the first global bound with its logistic factor gives the second bound there as well, since \(N\) is polynomial in \(n\). Finally the two logarithms are bounded by \(C(1+|a_N(l/N)|)\). Since \(\psi\) is Lipschitz and \((\log n)^2/N\le1\) for large \(n\), this proves (202). ◻

Proof. Positivity and domination by \(p_d\) follow by induction and the semigroup property of \(P_d\). The cap is (199). At a split the two children have equal lengths and use the same rule, so their kernels are identical. The convolution of this symmetric kernel with itself is symmetric; multiplication by the endpoint function \(W_N(\log\bar k_I)\) preserves symmetry. This starts from detailed balance of \(p_d\).

Every subinterval of length \(h\) larger than a fixed constant contains a tree node \(J\) of length at least \(c_1h\). One may require \(J\) to be a proper descendant at the cost of reducing \(c_1\). Keep the checks in \(J\) and discard the other weights, all of which lie in \([0,1]\). The remaining transition is an ordinary kernel, followed by \(k_J\), followed by an ordinary kernel. Since ordinary kernels preserve \(\mu\) and constants, its density is at most \(e^{CN(|J|)}\le e^{C'N(h)}\). A discarded failure or top factor also lies in \([0,1]\). ◻

Proof. The log-moment assertion of Theorem 15 says that, for each fixed integer \(p\ge1\) and sufficiently small fixed \(\xi>0\), on disorder restrictions of arbitrarily high limiting probability, \[\sup_x\left(1+\sum_yP_d(x,y)\log_+^p p_d(x,y)\right)^{1/p} \le C_{p,\xi}\left(1+(T/d)^{1/(2/3-\xi)}\right),\quad 1\le d\le CT.\] Choose \(\xi\) so that \(\xi/(2/3-\xi)<\beta/2\). Division by \(N(d)\) shows that the right side is at most \(C_{p,\xi}n^{-\beta/2}N(d)\) uniformly in this range: the largest possible extra factor is \(T^{1/(2/3-\xi)-3/2}=n^{\xi/(2/3-\xi)}\). Markov’s inequality therefore gives \[\sup_xP_d(x,\{y:\log p_d(x,y)>N(d)/2\}) \le C_{p,\xi}n^{-p\beta/2}.\] By (204), \(\bar k_I\le p_{|I|}\). At each check, its failure probability under an ordinary path is thus bounded by the last display plus \(e^{-(\log n)^2/2}\). The probability of a first failure is at most the sum of these probabilities, because all preceding weights are at most one. There are \(O(T)\) nodes. Choose \(p\) large after \(A\) and use Lemma 33 for the plateau. All estimates were uniform in the starting state and in \(d\), which proves the asserted uniformity. ◻

Proof. The tree is laminar: every check whose interior intersects the interior of \(I\) is a descendant or an ancestor of \(I\). Disjoint siblings may share an endpoint, but their factors are fixed by the exterior and boundary data. The ancestor factors, the top factor, and the external test are fixed by the same data. The remaining factors are exactly \(V_I\). The Markov property gives (205) as the conditional law, and then gives (206) and the conditional independence. This proves the resampling assertion as well. It applies to tree nodes; an arbitrary subinterval can have crossing checks.

At a fixed start \(x\), \[\sum_y\mu(y)\bar k_I(x,y)\mathbf 1_{\{\bar k_I(x,y)<\delta\}}\le\delta.\] Integrate the node first in the tested history and discard all other weights. The bound on \(\Phi\) and the lower bound on its mass show \(\mathsf Q_\Phi\{\bar k_I(X_a,X_b)<\delta\}\le n^{M_++M_-}\delta\). This proves the endpoint assertion by choosing \(\delta=n^{-L}\).

Let \(\mathcal F_t^\to\) contain just the marked prefix of the node from \(a\) to \(t\). For \(t-a\le2d/3\), its conditional density is exactly \[\frac{d(\mathsf C_I^{x,y}|_{\mathcal F_t^\to})} {d(\mathsf P_x^I|_{\mathcal F_t^\to})} =\frac{\mathsf E_x[V_I\mathbf 1_{\{X_b=y\}}\mid\mathcal F_t^\to]} {\mu(y)\bar k_I(x,y)}.\] The interval remaining after \(t\) contains a proper node of length at least \(c_1d\). Keep its checks and discard all others in the numerator. By Lemma 35, that numerator is at most \(\mu(y)e^{C'N(d)}\). This proves (207). Detailed balance for marked paths includes self-resampling rings, and the proper-descendant weights are invariant under reversal because equal-length children use identical symmetric kernels. Reversal therefore sends \(\mathsf C_I^{x,y}\) to \(\mathsf C_I^{y,x}\) on the reflected node, proving the last assertion. No reversal of the entire stopped forest is needed. ◻

Proof. First use deterministic times with \(m\) larger than a fixed constant. Inside each gap choose a contained proper checked node of length comparable to \(m\). They are disjoint. Dropping all other weights and applying Lemma 35 shows that the joint law of the observed states, given any initial state, is bounded by their product \(\mu\) law times \(\exp(C_2N(m))\). This estimate counts the retained weights without conditioning on survival. Apply (196) at a dyadic scale comparable to \(m\), first to all relevant pairs, and then to the quartet on the event that its pairs obey the same threshold. The ratios \(a_2(m)/N(m)=n^{2\beta}\) and \(a_4(m)/N(m)=n^{7\beta}\) tend to infinity. Their exponents are at least a positive power of \(n\), so they pay the cap and any polynomial test density with any prescribed inverse-polynomial precision. A vacuous threshold or bounded \(m\) gives the bound directly from \(|\rho_j|\le1\).

Use a time mesh of sufficiently small inverse-polynomial spacing. The probability that any mesh cell has two rings is arbitrarily small after paying the polynomial test density: under ordinary paths the total clock rate is \(n\), and the union bound is \(O(n^2T)\) times the spacing. A single ring changes either overlap by at most \(2/n\) per affected time. This is smaller than the displayed thresholds, whose smallest quartet value has order \(n^{-1/2+10\beta}\). A union over the polynomially many grid tuples proves the simultaneous assertion. A resampled interior has the same tested marginal by Lemma 38; a union proves the last claim for any fixed number of copies. ◻

Proof. The size support gives \(\sum_e|\theta_e|\le Cn^4\), hence \(\min_x\mu(x)\ge e^{-Cn^4}\). By Cauchy–Schwarz, each absolute one-site field is at most \(Cn^{7/2}\). On a leaf, prescribe one ring of each site in successive subintervals of length \(1/n\), with its sign set to the desired endpoint, and no other rings. The clock event has probability \(e^{-n|I|}n^{-n}\), and each prescribed mark has probability at least \(\tfrac12e^{-Cn^{7/2}}\). Thus every leaf transition probability is at least \(e^{-Cn^5}\), uniformly in its endpoints. Its logarithmic density has absolute value at most \(Cn^5\). The last assertion of Lemma 34 and the kernel recursion propagate this property up the tree: composing two positive kernels costs a fixed multiple of the previous logarithmic bound plus a polynomial, and applying \(W_N\) has the same property. The depth is \(O(\log n)\), so all these absolute logarithms remain polynomial, with a fixed exponent. In particular \(\mu(y)\bar k_I(x,y)\ge e^{-n^{C_0}}\) for a fixed \(C_0\).

The total number of ordinary rings on a node is Poisson with parameter \(n|I|\le CnT\). Its tail above any sufficiently large polynomial is smaller than \(e^{-2n^{C_0}}\) times any inverse power of \(n\). Dividing by the last lower bound and using \(V_I\le1\) proves polynomial conditional count moments for every boundary. Each fixed Taylor coefficient of the ring product is a polynomial of fixed degree in its number of rings, since each single-ring coefficient in (210) is bounded. This proves the claimed raw moments, both conditionally and under the polynomially tilted tested laws. The copy statement follows from their preserved marginals.

At a leaf, (211) has an empty product; the preceding count bound and the bounded coefficients of \((1+u\mu q_e)/(1+u q_e(y))\) give the coarse bound for its pre-density. Suppose normalized coefficients of order \(l\le K\) at one tree level are bounded by \(D^l\). For a composition, division by its base value writes its ratio as a conditional average of the product of the two child ratios and \(\mu_u(z)/\mu(z)\). Its order-\(l\) coefficient is therefore at most \((C_K(D+1))^l\). Taylor’s chain rule for \(W_N(\log\bar k_I^u)\) uses the global ratios (200); coefficients of the logarithm are polynomials in nonconstant pre-ratio coefficients, each product having total order \(l\). The same bound applies to either survival or failure ratios and to the post-density. Starting with a polynomial \(D\) at the leaves and iterating \(D\mapsto C_K(D+1)\) for \(O(\log n)\) levels proves these coefficient bounds. Formula (211) itself follows by inserting the path likelihood and then dividing by \(\mu_u(y)/\mu(y)\) in the definition of the pre-density.

To check the last forest assertion, fix its designated external endpoints. The interiors of its maximal roots are the corresponding conditional laws (205), and their top survival or failure factors are fixed. Expand a coefficient of total order at most \(K\) by the product rule. A differentiated node contributes one of the normalized ratios just bounded; the raw ring factors have the conditional polynomial moments proved above. There are polynomially many nodes and at most \(K\) positive orders, so the number and magnitude of the resulting terms are bounded by \(n^{C_K}\). Stationary initialization, if present, contributes only the bounded coefficients in (209). Integrating this conditional bound against the nonnegative endpoint test and its base endpoint measure proves the asserted factor of the base mass. ◻

Proof. Include a possible initial density \(z_0=d\mathsf Q|_{\mathcal F_0}/ d\mathsf P|_{\mathcal F_0}\). With bounded integrated rates, the marked point-process likelihood formula (Jacod 1975, Theorem 5.1) gives \[\sqrt Z\,e^{\mathcal J_{[0,\tau]}/2} =\sqrt{z_0}\prod_{s\le\tau\,\mathrm{ring}}\sqrt{b_{i_s,\zeta_s}(s)} \exp\left\{-\int_0^\tau\sum_{i,\zeta}r_{i,\zeta}(\sqrt{b_{i,\zeta}}-1)\,dt \right\}.\] The product allows zero factors; a mark with \(b=0\) has no mass under \(\mathsf Q\). The product-exponential following \(\sqrt{z_0}\) is the stochastic exponential for intensities \(r\sqrt b\) relative to \(r\). Its expectation conditional on the initial data is at most one, as follows by integrating the successive conditional waiting and mark densities, also when a mark factor is zero; this is the nonnegative likelihood supermartingale of (Jacod 1975, Proposition 4.3). Cauchy–Schwarz gives \(\mathsf E_{\mathsf P}\sqrt{z_0}\le1\). Thus the right side has \(\mathsf P\) expectation at most one. For general ratios, stop only when the integrated reference, changed, and square-root rates reach a finite level. Let \(Y=\sqrt Z e^{\mathcal J/2}\) on \(\{Z>0\}\), with extended values, and set \(Y=0\) on \(\{Z=0\}\). The localized likelihood expressions converge to \(Y\) on \(\{Z>0\}\); assigning zero on \(\{Z=0\}\) can only decrease their lower limit. Fatou’s lemma gives \(\mathsf E_{\mathsf P}Y\le1\). No positivity restriction is imposed on \(b\) in this localization. Since \(\mathsf Q\{Z=0\}=0\), this expectation bound also makes \(\mathcal J<\infty\) \(\mathsf Q\)-almost surely. On the event in (214), \(\sqrt Z e^{-\mathcal J/2}\le e^{(B-A)/2}\). Multiplying this by \(Y\) and integrating under \(\mathsf P\) proves the inequality.

For a tested forward law, \(\log Z\le(M_++M_-)\log n\) by Definition 37. For a conditional portion, (207) gives \(\log Z\le CN(|I|)+L\log n\) outside an arbitrarily small boundary event. Since \(N(|I|)\ge c n^\beta\), applying (214) with these bounds proves (215) after increasing \(C\).

Let \(C_{i,H}\) be the number of rings at site \(i\) in a unit block \(H\), and \(\Lambda_{i,H}=\int_H\sum_\zeta r_{i,\zeta}b_{i,\zeta}\,dt\). The exponential count supermartingale (Jacod 1975, Proposition 4.3) gives \[\mathsf E_{\mathsf Q}\exp\{-C_{i,H}+(1-e^{-1})\Lambda_{i,H}\}\le1,\] with the conditional version when the boundary is fixed. Ordinary unit counts have every fixed moment bounded; paying a polynomial test density and using the preserved copy marginals shows \(C_{i,H}\le n^{\eta/2}\) with arbitrarily high inverse-polynomial probability, simultaneously on any polynomial collection of blocks. The last exponential bound then gives \(\Lambda_{i,H}\le n^\eta\) with the same precision. Its tail inequality also bounds every moment of \(\Lambda_{i,H}\) by a constant times a higher moment of \(1+C_{i,H}\). The conditional assertions here are first averaged over the boundary law. Finally, \((\sqrt b-1)^2\le b+1\) bounds \(\mathcal J\) by the sum of the new and ordinary integrated intensities. The moment assertions follow, using the polynomial count moments from Lemma 40. ◻

Proof. The order-\(l\) prefix coefficient is a martingale starting from zero. At a ring its jump is \(\sum_{h=1}^l\gamma_{h,e}\ell_{l-h,e}([\inf I,t))\); the prefix factors are predictable and every \(\gamma_h\) is centered. The martingale square-function inequality (Burkholder et al. 1972, Theorem 1.1), applied at successive ring times, and Hölder’s inequality bound its \(L^p\) supremum by a constant times \[\sum_{h=1}^l\|C_{e,I}^{1/2}\|_{L^{2p}} \left\|\sup_t|\ell_{l-h,e}([\inf I,t])|\right\|_{L^{2p}},\] where \(C_{e,I}\) is the number of rings at the two sites. It is Poisson with mean \(2d\), whose fixed moments are \(O(d^p)\) for \(d\ge1\). Induction in \(l\), with the same assertion at the higher moment order \(2p\), proves (217).

Detailed balance for marked paths states that the forward path measure multiplied by its initial stationary mass equals the reversed measure multiplied by its final stationary mass. Apply this once at the base matrix and once using (209), and divide. This gives (218), including at self-resampling rings. The Taylor coefficients of the two endpoint factors in that identity are bounded through order \(K\). Thus the reversed bounds follow from the forward ones. Choose \(p\) sufficiently large in (217) to pay the polynomial test density, the number of entries and intervals, and the desired inverse-polynomial precision. This proves (219); resampling preserves the marginal. On a unit block a fixed-order coefficient is a fixed-degree polynomial in the two-site count, which gives the last assertion in the same way. ◻

Proof. For each site write \(\alpha_i\) for the signed measure in the first integral with \(X_j\) omitted. Split it according to \(b\le2\) and \(b>2\). On the first set \(|b-1|\le C|\sqrt b-1|\), so Cauchy–Schwarz and \(\sum_\zeta r_{i,\zeta}=1\) bound the square integral of its absolute density by \(C\mathcal J_{H,i}\). On the second set \(|b-1|\le C(\sqrt b-1)^2\), so its total variation is at most \(C\mathcal J_{H,i}\). On each unit block this variation is also bounded by a constant times the new and ordinary integrated site intensities, hence by \(n^\eta\) on the event of Lemma 41. If \(a_{ik}\) is the sum of the two variations on the \(k\)th unit block, these facts imply \[ \sum_k a_{ik}^2\le Cn^\eta\mathcal J_{H,i}. \tag{222}\] In particular the large ratios have been treated by total mass, not by a pointwise square-integrability assertion.

First suppose all times in \(H\) have age at least a fixed multiple of \(d\) from the forward start of its checked root. Squaring the contribution of a fixed site \(i\) and averaging its other endpoint gives the kernel \(n^{-1}\sum_jX_j(t)X_j(s)=\rho_2(X(t),X(s))\). By (208), on unit block indices this is bounded by \(n^{O(\beta)}(1+|k-k'|)^{-1/2}\). Adjacent blocks use the bound one. The \(\ell^1\) sum of this kernel over one index is \(O(d^{1/2})\), so \[\frac1n\sum_j\left|\int_H X_j\,d\alpha_i\right|^2 \le n^{O(\beta)}d^{1/2}\sum_k a_{ik}^2.\] Omitting \(j=i\) adds at most \(n^{-1}(\sum_k a_{ik})^2\), which is absorbed by the right side since \(d\le CT\). Summing over the contributing site and dividing by \(\binom n2\) gives, using (222), \[|D(H)|_2^2\le \frac{n^{O(\beta)}d^{1/2}}n \sum_i\sum_k a_{ik}^2 \le n^{O(\beta)}d^{1/2}\mathcal J_H/n.\]

For the fourth power use the kernel \(n^{-1}\sum_j\prod_{h=1}^4X_j(t_h)=\rho_4(X(t_1),\ldots,X(t_4))\). The age assumption and (208) bound its unit-block kernel by a constant times \[n^{O(\beta)}\sum_{1\le h<h'\le4}(1+|k_h-k_{h'}|)^{-3/4}.\] For each chosen pair the convolution is at most \(Cd^{1/4}\sum_k a_{ik}^2\); the other two \(\ell^1\) sums contribute at most \(d\sum_k a_{ik}^2\). Thus the fourth-power integral is at most \(n^{O(\beta)}d^{5/4}(\sum_k a_{ik}^2)^2\). The omitted-diagonal term is \(n^{-1}(\sum a_{ik})^4\) and is again absorbed, since \(d^{3/4}/n\le Cn^{-1/2}\). Now sum over \(i\) and divide by \(\binom n2\). Since \(\sum_i\mathcal J_{H,i}^2\le\mathcal J_H^2\), this gives \[|D(H)|_4^4\le\frac{n^{O(\beta)}d^{5/4}}n \sum_i\left(\sum_k a_{ik}^2\right)^2 \le n^{O(\beta)}d^{5/4}\mathcal J_H^2/n.\] The fourth root is the second bound.

For a general \(H\), partition it into unit blocks near the forward root start and pieces whose lengths grow geometrically and are at most a fixed multiple of their ages. Apply the preceding estimates on each piece. For \(\gamma=1/4\) or \(5/16\), Cauchy–Schwarz gives \(\sum_jd_j^\gamma\sqrt{\mathcal J_{H_j}} \le C d^\gamma\sqrt{\mathcal J_H}\), because \(\sum_jd_j^{2\gamma}=O(d^{2\gamma})\). This partition uses the forward age even when the compensation orientation is reversed; it is a deterministic division of the integrals and changes no filtration. Finally, replacing both overlap kernels by one gives the exponent \(1/2\) for either norm. That proof uses only a common upper bound on the absolute block masses, so bounded \(e\)-dependent predictable weights and higher insertion orders are permitted in this last assertion. ◻

Proof. For \(l=1\), (220) is the representation, and Lemma 43 gives both bounds. For \(l\ge2\), expand a raw coefficient by the smallest dyadic node containing all its positive-order ring insertions. If all insertions lie in a leaf, keep that leaf term. Otherwise, at that unique node \(J\) the term is \(\ell_{j,e}(J_{\mathrm{early}})\ell_{l-j,e}(J_{\mathrm{late}})\) for some \(1\le j<l\). This is an exact partition of the ring expansion, including rings carrying Taylor order larger than one.

On a leaf, expand according to the last ring and its assigned order as in (220). The remaining prefix coefficients are predictable and are bounded by a power allowance on this unit block, by Lemma 42. The unit-block conclusion of Lemma 43 applies. At a split of length \(y\), the earlier coefficient is known before the later child and is at most \(n^\eta y^{j/2}\) by (219). Apply the inductive representation to the later child, then include this earlier factor by its supremum norm. At any one scale the later children are disjoint and number at most \(Cd/y\). Consequently \[\sum_{J:\,|J|\asymp y}\sqrt{\mathcal J_{J_{\mathrm{late}}}} \le C\sqrt{(d/y)\mathcal J_H}.\] For the \(L^2\) or \(L^4\) bound the power before this summation is \(y^{(l-1)/2+\gamma}\), with \(\gamma=1/4\) or \(5/16\). Relative to the claimed right side, this scale thus has the factor \[(y/d)^{(l-2)/2+\gamma}.\] It is geometrically summable for \(l\ge2\). The leaf terms cost at most \(C\sqrt{d\mathcal J_H}\) with the respective factor \(n^{-1/2}\) or \(n^{-1/4}\), and are also bounded by the claimed expression. All compensations used the expectation identity with the earlier coefficient still predictable. After absolute values, each representation vector is bounded by a fixed polynomial in raw clock counts and prefix coefficients, and in \(\int_H\sum_{i,\zeta}r_{i,\zeta}|b_{i,\zeta}-1| \le\int_H\sum_{i,\zeta}r_{i,\zeta}b_{i,\zeta}+n|H|\). For the sharper representations below, an earlier endpoint projection adds a conditional coefficient with the uniform coarse bound of Lemma 40. That lemma, the intensity moments in Lemma 41, and Hölder’s inequality therefore give polynomial fixed moments for the representation norms, including a fixed number of copies. The conditional exceptional-set rule then applies to each of the polynomially many terms and proves the stated probability formulation. ◻

Proof. All conditional estimates in this proof are first established under the tested boundary law followed by its interior sampling. We then use the conditional exceptional-set rule following Lemma 40. The polynomial coefficient and representation moments proved in Lemmas 40 and 44 permit any required stronger working precision. This yields the stated high-probability bounds on conditional expectations, without asserting them for every possible boundary.

Proof. The additive precision in the lemma is useful for very small tested masses. To prove it, first suppose each \(z_\alpha\ge n^{-M}\) for a fixed \(M\). Normalize each integral by \(z_\alpha\) and use the tested probability law. The estimates below then have the factor \(\prod_\alpha z_\alpha\). All exceptional contributions may be made smaller than \(n^{-A'}\) for an arbitrary fixed \(A'\) by the precision convention in Definition 37 and Proposition 45. If some \(z_\alpha<n^{-M}\), Lemma 40 and the polynomial path moments bound its unnormalized coefficients by \(n^{C(K,B)}z_\alpha\). Choosing \(M\) after \(A\) discards all such terms at cost \(n^{-A}\). We shall use this reduction throughout the proof.

Proof. On the support of a term containing a differentiated history, the static support conditions hold and \(\sum_q\ell_q\le2n^{-10}\). This follows from the supports of \(\Gamma\) and its derivatives, and of \(\chi\) and its derivatives. Split each \(\zeta_q\) into its positive and negative parts. The base masses of these two checked integrals are at most \(n^{O(\beta)}\). Indeed the post-check kernel caps at the ends of the one or two legs dominate the joint law of the two observation spins by \(\exp(C'N(cT))\mu^{\otimes2}\). For a stationary query the first spin already has law \(\mu\); for a uniform-start query the first leg cap holds from each initial spin. Choose the first dyadic \(d_*\ge cT\). Then \(d_*\le2cT\), \(r_2(d_*)\le r_2(cT)\), and \(a_2(d_*)\ge c_1n^{3\beta}\). The pair cutoff therefore gives \[\mathbb E_q\left[|\zeta_q|\prod_Iw_I\right] \le n^{1/3}r_2(cT) +n^{1/3}\exp(C'N(cT)-c n^{3\beta}) \le C n^{2\beta}+n^{-A'}\] for arbitrary fixed \(A'\). We used \(N(cT)=O(n^\beta)\).

Apply the ordinary chain rule to \(\chi(n^{10}\sum\ell_q)\) and to \(F(\widetilde Y)\). A term has at most \(K\) positive-order history coefficients, with total order at most the differentiating order; repeated histories are represented by independent copies. The coefficients from derivatives of \(F\) are evaluated at the base \(\widetilde Y(J)\), hence are common across \(e\), and their total absolute sums are bounded by (258). For loss factors, expand (255). The mass factors in Lemma 47, summed over all choices of queries and first-failure nodes and multiplied by their \(n^{10}\) factors, give at most \[\left(n^{10}\sum_{q,I}\ell_{q,I}\right)^r\le2^r\] when there are \(r\) loss factors. Trace mass factors cost at most \(n^{O_K(\beta)}\). The additive precisions in that lemma are chosen after the polynomial number of nodes and queries and after the factors \(n^{10K}\), so their total is negligible.

If a static cutoff is also differentiated, its pointwise derivative bound from Lemma 33 multiplies these estimates. If only the static cutoff is differentiated, that bound is \(O(n^{-b})\) for some \(b=b(\beta)>0\), and its remaining outer factor is bounded. Consequently the fourth averaged derivative is at most \(n^{-a+C'_K\beta}(\log n)^{C'}+O(n^{-b})\), higher averaged derivatives are at most \(n^{C'_K\beta}(\log n)^{C'}T^{(k-4)/2}\), and the supremum bound is at most \(n^{C'_K\beta}(\log n)^{C'}T^{j/2}\). These estimates hold everywhere: outside the indicated supports the corresponding product-rule terms vanish. Choose \(\beta\) after \(K\) so that \(C'_K\beta<a/2\) and \(C'_K\beta<0.005\), while satisfying the preceding sections’ smallness requirements. Absorb logarithms by a still smaller fixed power. This gives (259)– (261) for some \(a_0>0\) and \(\delta<0.01\). ◻

Proof. We use the Taylor replacement principle (Chatterjee 2006), in the finite mixture form of the stationary comparison (OpenAI 2026d, sec. 8). Put \(\nu_\lambda=(1-\lambda)\mathsf N(0,1) +\lambda(\delta_{-1}+\delta_1)/2\) for \(0\le\lambda\le1\). Differentiating its finite product measure gives exactly \[ \frac d{d\lambda}\mathbb E_{\nu_\lambda^{\otimes\binom n2}}\Phi =\sum_e\mathbb E_{\lambda,-e} \left[\mathbb E_\xi\Phi(J_{-e},\xi) -\mathbb E_Z\Phi(J_{-e},Z)\right], \tag{262}\] where \(\xi\) is a symmetric sign, \(Z\) is standard normal, and \(\mathbb E_{\lambda,-e}\) averages the other entries.

Independently of \(J_{-e},\xi,Z\), sample \(V\) from \(\nu_\lambda\) and set \(J^0=(J_{-e},V)\). Taylor-expand at this completed matrix: \[ \Phi(J_{-e},x)= \sum_{k=0}^{K-1} \frac{\partial_e^k\Phi(J^0)}{k!\,n^{k/2}}(x-V)^k +\mathcal R_e(x,V), \qquad |\mathcal R_e(x,V)| \le\frac{|x-V|^K}{K!\,n^{K/2}}\,n^\delta T^{K/2}. \tag{263}\] The remainder bound is global by (261), including every point of the segment. Conditional on \(J_{-e},V\), the orders \(k\le3\) cancel exactly between the two proposals. In fact \[ \Delta_k(V):=\mathbb E_\xi(\xi-V)^k-\mathbb E_Z(Z-V)^k =\sum_{j=4}^k\binom kj(-V)^{k-j} \bigl(\mathbb E\xi^j-\mathbb EZ^j\bigr), \tag{264}\] where the sum is empty for \(k\le3\). Independence of the completed entry from both proposals is what permits this conditional cancellation.

For \(4\le k<K\), on \(|V|\le\log n\) the absolute value of (264) is at most \(C_K(\log n)^K\). The complementary mixture tail, even with any fixed moment of \(V\), is smaller than every inverse power of \(n\). Its contribution is negligible by the global polynomial bounds (261). When \(e\) is uniform, \(J^0\) has the full iid mixture law independently of \(e\). Therefore (262)–(263) yield, uniformly in \(\lambda\), \[\begin{align*} \left|\frac d{d\lambda}\mathbb E_{\nu_\lambda^{\otimes\binom n2}}\Phi\right| &\le C_K(\log n)^K \sum_{k=4}^{K-1}n^{2-k/2} \mathbb E_\lambda\!\left[ \frac1{\binom n2}\sum_e|\partial_e^k\Phi(J)|\right]\\ &\hspace{2cm}+C_K n^{2-K/2+\delta}T^{K/2}+o(1). \tag{265}\end{align*}\] The remainder uses the bounded \(K\)th moments of \(\xi,Z,V\). By (259), the \(k=4\) term tends to zero. For \(k\ge5\), (260) and \(T=n^{2/3}\) give \[n^{2-k/2+\delta}T^{(k-4)/2} =n^{-(k-4)/6+\delta}=o(1).\] The remainder is \(n^{2-K/6+\delta}=n^{-2+\delta}\) because \(K=24\). Thus (265) tends to zero uniformly in \(\lambda\). Integration over \([0,1]\) proves the lemma. ◻