Proof. We use the eigenvalue-only spectral estimate of (OpenAI 2026b, Proposition 3.1). For every deterministic sequence \(L_n\to\infty\) with \(L_n\le\log n\), that proposition controls both resolvent traces for \(s\in[L_nn^{-2/3},5]\), with positive denominators. At \(s=L_nn^{-2/3}\), multiplying its first-trace error by \(n^{1/3}\) gives \[O\!\left(L_n^{-1}+L_n^{-1/8} +L_n^{-1/4}\sqrt{\log(2+L_n)}\right)=o(1).\] The semicircle resolvent equals \((2+s-\sqrt{s(4+s)})/2\), whose expansion supplies the main term \(\sqrt{L_n}\) and an additional error \(O(L_nn^{-1/3})=o(1)\).

The second-trace estimate yields, for the same interval of \(u\), \[\#\{a:d_a\le u\} \le 4u^2\sum_a(d_a+u)^{-2} \le Cnu^{3/2}.\] Indeed \(0<d_a+u\le2u\) for every counted eigenvalue. It follows that \(g_{a,n}\ge c a^{2/3}\) once \(a>CL_n^{3/2}\). The uniform norm bound is part of the same proposition.

These conclusions hold for every arbitrarily slowly divergent \(L_n\). If (5) failed, one could select \(L_j\to\infty\) and \(n_j\to\infty\), with \(\log n_j\ge L_j\), along which the failure probability stayed positive, and embed the values \(L_{n_j}=L_j\) in a sufficiently slow divergent sequence. This contradicts the preceding estimate. The identical criterion, applied to the least index after which the counting consequence holds, makes that index tight. This proves the second assertion as well. ◻

Proof. For fixed admissible \(L\), the identity \[D_n(b_n)=n^{1/3}-\sum_a\frac1{g_{a,n}+L} -\sum_a\left(\frac1{g_{a,n}+b_n}-\frac1{g_{a,n}+L}\right)\] separates the divergent trace from an absolutely convergent difference. The tail of the second sum is bounded by a constant times \(|L-b_n|\sum_{a>m}a^{-4/3}\) on each set of bounded spectral data, so that sum passes to the limit. Lemma 2 then shows that the expressions in braces in (6) are Cauchy in probability, and that \(D_n(b_n)\) has the asserted joint limit. If \(\sum_a l_a\) were finite, those expressions would tend to \(+\infty\) as \(L\to\infty\); this contradicts the finite limit. Changing \(b\) changes \(D(b)\) by the corresponding absolutely convergent resolvent difference.

We give the density argument explicitly. Let \(q\) be the density of \(S\), and let \(q_m\) be the density of \(\sum_{a>m}(Z_a^2-l_a)\). After any fixed coordinates are omitted, a sufficiently large fixed group of remaining coordinates supplies an integrable bound for the characteristic function: the modulus of the factor for coordinate \(a\) is \((1+4t^2l_a^2)^{-1/4}\). Fourier inversion therefore gives continuous, bounded densities. The same bound applies to the finite sums and proves uniform convergence of their densities when the variances converge and their squared tails vanish.

The centered sum after any finite deletion has support all of \(\mathbb R\). Its unbounded right support comes from one square. To reach an arbitrarily negative interval, make a sufficiently long finite block of squares small, using \(\sum l_a=\infty\), and keep the remaining centered sum in a bounded interval of positive probability. Convolution with a finite block whose square-sum density is positive in the interior of its support now gives \(q(s)>0\) and \(q_m(s)>0\) for every \(s\in\mathbb R\). Thus the first \(m\) coordinates of the required conditional law have density \[ \frac{q_m\!\left(D(b)-\sum_{a\le m}(x_a^2-l_a)\right)}{q(D(b))} \prod_{a\le m}\frac{e^{-x_a^2/(2l_a)}}{\sqrt{2\pi l_a}}. \tag{9}\] Convolution shows that these densities are consistent. They depend measurably on the spectral data, so their extension is a random probability measure \(\Pi_g\).

For completeness, they admit a disintegration on the constraint itself. Let \(q^{\rm head}_m\) be the density of \(\sum_{a\le m}(Z_a^2-l_a)\). Give the tail product Gaussian law the weight \[\frac{q^{\rm head}_m\!\left(D(b)-\sum_{a>m}(x_a^2-l_a)\right)}{q(D(b))}.\] Given the tail, use polar disintegration of the first \(m\) Gaussian coordinates at the radius (8). This produces (9), has an absolutely continuous tail law, and enforces the constraint almost surely.

In finite dimension, conditioning independent normals of variances \(l_{a,n}\) on \(\sum_a x_a^2=n^{1/3}\) gives exactly the spherical Gibbs measure in the stated coordinates: the term involving \(b_n\) in the Gaussian exponent is constant on that sphere. The density convergence above and \(q(D(b))>0\) imply convergence of each cylinder marginal. Their densities are bounded by a constant times the corresponding product Gaussian density. To retain an arbitrary single coordinate uniformly, choose the integrable Fourier bound from a fixed group of leading coordinates other than that coordinate. This proves (7) and uniform integrability of all fixed polynomial moments.

The argument applies to every admissible \(b\) on sets where its distance from the pole is bounded below and its magnitude is bounded above; these sets exhaust the admissible choices. The finite conditioned spherical law is independent of \(b_n\), so its limiting cylinder marginals are also independent of \(b\). The conclusion holds simultaneously for all admissible \(b\). First take a common probability-one event for rational parameters. On every compact admissible interval, the resolvent-difference series converges uniformly, so \(D(b)\) is continuous. The squared variance tails and the Fourier majorants above are also uniform on that interval. Thus the conditional cylinder densities in (9), and their integrals against bounded continuous cylinder tests, depend continuously on \(b\). Equality for rational parameters therefore extends to the whole interval. ◻

Proof. First test against a bounded continuous function supported away from \(2\). Before conditioning, the Gaussian variance of its integral is \(O(n^{-1})\), and its mean converges to the stated integral by the semicircle law. Integrate out a fixed number of leading Gaussian coordinates when imposing the norm constraint. Their square-sum density gives a bounded density ratio, so the concentration persists after conditioning. Each of those leading coordinates has mass \(n^{-1/3}x_a^2=o(1)\) in probability by (7). Finally both the finite measure and its proposed limit have mass one, since \(\int(2-\lambda)^{-1}d\rho_{\rm sc}(\lambda)=1\). This extends convergence across the edge. The same argument permits bounded tests with cuts at continuity points in the bulk. ◻

Proof. In the joint weighted law, \(Z_{\rm cube}\) cancels the Gibbs denominator. For each fixed cube vertex, Haar rotation sends that vertex to uniform surface measure on the sphere of radius \(\sqrt n\). The remaining energy weight is \(\exp(y^{\mathsf T}\Lambda y/2)\), proving the first assertion. The same invariance gives the conditional Haar law on the stated coset. Finally, the uncapped zero-field calculation in the proof of the static cap comparison of (OpenAI 2026a), Section 15, paragraph “Zero field and overlaps bounded away from zero,” gives \(E_UZ_{\rm cube}^2/(Z_{\rm sp})^2=1+o(1)\). Together with \(E_UL_\circ=1\), this is (11). ◻

Source statements and the radial consequence. The spectral, saddle, and density assertions are the conclusions of (OpenAI 2026a), Sections 15.2–15.3, “Spectral estimates at the bottom scale” and “Spherical saddles, norm densities, and tails.” The static comparison and its projection tails are Proposition 15.1 in that section. Its order of choices is the one stated here: fix the radius, errors, and geometric ratios, then decrease the upper cutoff. The overlap estimate is the spherical estimate preceding that proposition.

The upper covariance bounds are Theorem 18.6, “Uniform curvature of the true cap potentials,” in (OpenAI 2026a). Its lower endpoint is \((\sqrt{\log n}/n)^{1/3}\), with fixed-factor changes permitted, so it covers our scale \(r\) and its fixed buffers. We also use a consequence retained in that theorem’s proof. Let \(\gamma\) be the fixed geometric ratio chosen in that proof, and set \(R_{p,\alpha}^H=(P_pR_{p,\alpha}P_p)|_{\mathcal H_p}\). At an intermediate evaluation ratio \(x\in[\gamma^2,\gamma]\), Lemma 16.2, “Evaluation with the induction extension,” in (OpenAI 2026a) bounds the radial entry of \(p^2(K_p-R_{p,\alpha}^H)\) above by \(-0.26\). The theorem’s proof chooses \(s_0\) so that every target saddle lies inside the parent agreement annulus with fixed slack, and applies exactly this deterministic evaluation. Therefore \[p^2(K_p)_{ee}\le\frac1{1+\alpha/p^2}-0.26 \le\frac1{1-s_0}-0.26\le0.9\] after decreasing \(s_0\). This proves the second part of (14) with a fixed tolerance. ◻

Proof. First use an ordinary fixed-ratio list. Monotonicity of \(P_p\) and a fixed-factor change in \(M_p\) extend the uncapped projection tail to every \(p\). Removing \(B_p\) from the interaction changes its normalizing factor by \[\mu\exp(-X^{\mathsf T}B_pX/2)\ge \mu\{\|P_pX\|\le VM_p\}\, \exp(-\|B_p\|V^2M_p^2/2)\ge e^{-Ck_p},\] where \(V\) is fixed and large. We used \(\|B_p\|=O(p^2)\) on the spectral event. Thus the projection tail persists at cap zero field. A field \(\|h\|\le Rp^2M_p\) adds at most \(Rk_pv\) on the shell \(\|P_pX\|=vM_p\); this is dominated by \(ak_pv^6\) for large \(v\). The zero-field measure is even, so its linear-tilt normalizer is at least one. This gives the same tail on the field ball, with an adjusted starting threshold. The argument also bounds \(\|P_qX\|/M_q\) under the \(p\)-cap posterior whenever \(q\ge p\) and \(q/p\) is bounded: use the uncapped \(P_q\) tail, the same denominator lower bound, and \(P_ph=h\in\mathcal H_q\). This is the tail needed to compare two nearby caps. The spherical estimates give the identical conclusion.

Now choose a fixed geometric list with consecutive ratios close to one, and let \(q\) be its first scale above \(p\). Then \(\mathcal H_p\subset\mathcal H_q\) and \[\|W_p-W_q\|\le C(1-p/q)q^2, \qquad \operatorname{ran}(W_p-W_q)\subset\mathcal H_q.\] For \(h\in\mathcal H_p\) in the prescribed ball, restrict first to \(\|P_qX\|\le VM_q\). The change in log weight there is at most \(C(1-p/q)V^2k_q\). On its complement, the projection tail controls the exponential of the quadratic change: the exponent \(C(1-p/q)k_qv^2\) is dominated by the negative \(ak_qv^6\) tail. Choose \(V\) first and the list ratio second. The resulting log-partition change is as small as desired in units \(k_q\), for both cube and sphere. Their value comparison at \(q\) therefore implies the value comparison at \(p\), since \(k_p/k_q\) is arbitrarily close to one.

For the mean, use the value comparison on a slightly larger field ball. The frozen Gaussian saddle identity and its norm-density bound give, for any unit \(v\in\mathcal H_p\) and both signs, \[\log\mu_{p,h}e^{\pm t v^{\mathsf T}(X-R_{p,\alpha}h)} \le 2\epsilon_v k_p+\tfrac12t^2v^{\mathsf T}R_{p,\alpha}v+C.\] Jensen’s inequality, with \(t=\theta p^2M_p\), bounds the projected mean discrepancy divided by \(M_p\) by \(C\theta+(2\epsilon_v+C/k_p)/\theta\). Choose \(\theta\) from the required mean tolerance and then \(\epsilon_v\) sufficiently small. This proves the claim uniformly at every intermediate scale. ◻

Proof. Under \(\mathbf P\), the field in spectral coordinates has exactly its spherical distribution. Generate the spherical spin from independent centered Gaussian coordinates \(y_i\) of precision \(D_i'=d_i+b_n n^{-2/3}\), conditioned on \(\sum_i y_i^2=n\); the choice of \(b_n\) is that of Proposition 4. Before this conditioning, write \[h_{p,i}=B_{p,i}y_i+B_{p,i}^{1/2}z_i, \qquad m=p^2+b_n n^{-2/3},\] where the \(z_i\) are independent standard normals. On an active coordinate \(B_{p,i}=m-D_i'\), and direct expansion gives \[\mathbf E_0\bigl(\|h_p\|^2-m^2\|y\|^2\bigr) =-m^2\mathop{\mathrm{Tr}}R_{p,b_n n^{-2/3}}.\] Here \(\mathbf E_0\) denotes the unconditioned Gaussian expectation. The subtraction of \(m^2\|y\|^2\) removes the large contribution of the leading coordinates: its coefficient on the standardized \(y_i^2\) term is \((B_{p,i}^2-m^2)/D_i'=D_i'-2m=O(p^2)\).

As a quadratic form in the standardized \((y,z)\) variables, the centered quantity has squared Frobenius norm at most \(Cp^4k_p\) and operator norm at most \(Cp^2k_p^{1/3}\). Indeed the cross coefficients are \(O(p^3/\sqrt{D_i'})\), while the spectral estimates give \[\min_iD_i'\ge c n^{-2/3},\qquad \sum_{d_i\le p^2}(D_i')^{-1}\le Cnp.\] The inactive coordinates are bounded by the capped second resolvent trace. The Gaussian quadratic-form exponential-moment formula therefore bounds deviations larger than \(p^2k_p^{9/10}\) by \(C\exp(-k_p^{a_0})\) for some \(a_0>0\).

This estimate survives conditioning on \(\|y\|^2=n\). To see this directly, use the same exponential tilt in the Markov bound, with its parameter at most one eighth of the inverse operator norm. The tilted joint Gaussian covariance, and hence its \(y\)-marginal covariance, remain comparable to the original ones. A fixed number of leading covariance eigenvalues bounds the density of \(n^{-2/3}\|y\|^2\) from above. Its original density at \(n^{1/3}\) is bounded below on the chosen spectral set, by Proposition 4. The ratio of norm densities therefore costs only a constant.

After conditioning, the deterministic target for \(\|h_p\|^2\) is \(\ell_p(b_n n^{-2/3})^2\asymp p^2k_p\). The deviation just proved is a relative \(O(k_p^{-1/10})\) error, so it first places the inferred saddle inside any fixed surrounding annulus for all sufficiently large \(n\). On that annulus, the derivative of \(\ell_p(\alpha)^2\) is comparable to \(k_p\). Consequently the preceding deviation bound implies \(|\alpha_p-b_n n^{-2/3}|\le Cp^2k_p^{-1/10}\). Since \(b_n n^{-2/3}/p^2=O(k_p^{-2/3})\), this proves the fixed-scale assertion.

For the whole path take a sufficiently fine inverse-polynomial mesh in \(p^2\). The functions \(B_p\), the saddle targets, and their relevant derivatives have polynomial bounds; the positive-part operation defining \(B_p\) is continuous at coordinate activations. Conditional on the spin, each coordinate of the noise has independent Gaussian increments. A maximal Gaussian estimate in every mesh interval, followed by a union bound over coordinates and intervals, makes their oscillation smaller than the polynomial tolerance just used. The fixed-scale exponential estimate absorbs this union bound because \(k_r=n^{3\delta}\). Finally, Lemma 6, Markov’s inequality, and \(L_\circ\ge1/2\) outside a vanishing disorder exception transfer the result to \(Q_\mu\). ◻

Proof. Given the overlap \(\rho=n^{-1}X\cdot X'\), the Haar angular integral of a cube pair equals that of a spherical pair. The difference is entirely in their overlap distributions. The replica Gram comparison of (OpenAI 2026a, sec. 15.4, Equation (15.10)) bounds the ratio on overlap cells of width \(2/n\) by \(\exp(C+Cn\rho^4)\). Above \(C_1p\) and below a small fixed cutoff, the spherical pair tail costs \(\exp(-a n\rho^3)\); choosing the cutoff small pays for the comparison on all dyadic bands. At overlaps bounded away from zero, the normalized zero-field comparison has a strictly negative exponential rate, by (OpenAI 2026a, sec. 15.4, p. 118). A bounded field changes logarithmic weights by at most \(Cnp^{5/2}\), which preserves that rate when the upper scale is small. Below \(C_1p\) the comparison factor is \(\exp(C+Cnp^4)\). This proves (17) and its pair-tail assertion. The uncapped argument uses the corresponding uncapped spherical tail from (OpenAI 2026a, sec. 15.3).

We give the finer cell estimate, now taking \(p=r\) throughout the remaining comparison. On \(|\rho|\le C_1p\), Stirling’s formula and the spherical overlap density show that the binomial mass of a cell and its spherical mass differ by a relative error \[O(np^4+p+n^{-1}).\] It remains to compare the angular integrands within one cell. Write a spherical pair on an orthonormal sum-and-difference frame \((q,q')\). For \(A=W_p\) their tilt exponent is \[\frac n2\bigl((1+\rho)q^{\mathsf T}Aq +(1-\rho){q'}^{\mathsf T}Aq'\bigr) +\sqrt{2n(1+\rho)}\,h^{\mathsf T}q.\] Its oscillation on a cell is bounded by \(C(|q^{\mathsf T}Aq-{q'}^{\mathsf T}Aq'|+p^{5/2})\). This is uniformly bounded, and its average against the normalized tilted spherical pair law on these cells is \(O(p)\). For the latter assertion, the exact identity \[q^{\mathsf T}Aq-{q'}^{\mathsf T}Aq' =\frac{2Y^{\mathsf T}AY' -\rho(Y^{\mathsf T}AY+{Y'}^{\mathsf T}AY')} {n(1-\rho^2)}\] and \(\|A\|\le C\) bound the diagonal terms by \(C|\rho|\). It remains to show \[E_{\mathrm{sp}}|Y^{\mathsf T}AY'|\le Cnp.\] At a bounded cap field, apply exponential Markov with parameter \(cp^2\) to either sign of \(Y^{\mathsf T}AY'\). The saddle means have squared norm \(O(np)\), the covariance trace-square is \(O(n/p)\), and a sufficiently small \(c\) leaves a fixed precision margin. The Gaussian logarithmic moment generating function is therefore at most \(Ck_p\). Conditioning both norms costs a constant by the joint norm-density bound in (OpenAI 2026a, Equation (15.6)). Integrating the resulting tail, beginning at a sufficiently large multiple of \(np\), proves the displayed estimate. For uncapped zero field, freeze a Gaussian parameter equal to a small fixed multiple of \(p^2\). The same moment generating function bound applies, and the deficit-filling lower density estimate in that subsection costs at most \(\exp(Ck_p)\); this gives the same integrated bound.

For completeness, the passage from this averaged oscillation to comparison of angular integrals does not require a pointwise \(O(p)\) derivative bound. The oscillation of the exponent on a cell is bounded by a fixed constant, so the mean value theorem bounds the difference of its exponentials by a constant times its integrated absolute derivative. The spherical overlap density varies by a bounded factor on a cell. Summing over cells is thus bounded by the preceding \(O(p)\) integral. Slightly enlarging the overlap cutoff pays for boundary cells by the pair tail.

At \(p=r\), \(np^4+p+n^{-1}=O(n^{-1/3+4\delta})=O(n^{-1/4})\). The normalized pair integral consequently equals \(1+O(n^{-1/4})\), which proves the first part of (18), since \(E_U L_r=1\). Apply the same comparison with the test \(n\rho\). Its absolute value on the retained cells is at most \(Cnp=CM_p^2\), and its change within one cell is bounded by \(2\). Hence \[E_U\|L_rm_r\|^2 =\|m_r^{\mathrm{sp}}\|^2+O(M_r^2n^{-1/4}).\] The first-moment identity \(E_U[L_rm_r]=m_r^{\mathrm{sp}}\) turns this into an \(L^2\) bound on \(L_rm_r-m_r^{\mathrm{sp}}\). Finally, \[L_r(m_r-m_r^{\mathrm{sp}}) =(L_rm_r-m_r^{\mathrm{sp}})-(L_r-1)m_r^{\mathrm{sp}},\] and \(\|m_r^{\mathrm{sp}}\|\le CM_r\) proves the second part of (18). ◻

Proof. In the uncapped comparison, insert a bounded Lipschitz cylinder function of the scaled spectral coordinates in each replica numerator. Changing the overlap within a cell moves each scaled coordinate by \(O(n^{-5/6})\); the preceding weight comparison therefore remains valid. The variance over Haar frames of the numerator normalized by \(Z_{\mathrm{sp}}\) tends to zero. Together with \(L_\circ\to1\) and Proposition 4, this proves convergence of the quenched cube cylinder integrals to \(\Pi\). Polynomially growing continuous tests follow by the higher moment bounds and truncation. ◻

Proof. Lemma 9 confines the random field to a bounded range except with power-exponential probability. At a fixed such field, the Haar-averaged pair-tail numerator divided by the squared spherical partition function is at most \(Ce^{-ak_p}\), by Lemma 10. Under the biased frame law \(L_p\,dU\), the set \(L_p<e^{-ak_p/4}\) has mass at most \(e^{-ak_p/4}\). On its complement, the expectation of the posterior pair-tail probability is at most \(e^{ak_p/4}\) times that pair-tail integral. A further Markov bound gives an exponentially small posterior pair tail outside an exponentially small biased exception. Since \(\|m_p\|^2=\mu_{p,h}^{\otimes2}[X\cdot X']\), this implies \(\|m_p\|^2\le Cnp\) there. The deterministic bound \(\|m_p\|\le\sqrt n\) pays for all exceptions and proves the cube part of (19). The spherical part follows by the same argument.

For the first estimate in (20), split according to \(L_r\ge n^{-0.12}\). On this event (18) bounds the biased second moment by \(CM_r^2n^{-0.13}\). The complementary biased probability is at most \(n^{-0.12}\); Cauchy–Schwarz and the fourth moment in (19) give \(CM_r^2n^{-0.06}\). The field exceptions are negligible. Finally \[\nabla_{u_r}\log L_r =n^{1/3}r^2 P_r(m_r-m_r^{\mathrm{sp}}),\qquad n^{1/3}r^2M_r=n^{(5/2)\delta}.\] Interpolation of the decaying second moment with an arbitrarily high fixed moment from (19) proves the fourth-moment assertion. ◻

Proof. Use \(t=p^2\) as the channel parameter and take the filtration generated by the eigenframe and observations through time \(t\). In particular, every disorder event is measurable at time zero. Between coordinate activations, \(\dot B_t=P_p\), and the Gaussian likelihood identity gives \[d\log L_p=a_p^{\mathrm{err}}\cdot d\mathcal B_t +\tfrac12\|a_p^{\mathrm{err}}\|^2dt,\] where \(\mathcal B\) is the innovation Brownian motion in the active subspace. The bracket of the martingale is the score integral. There is no singular term at an activation because the added precision starts at zero. These statements also follow by Itô differentiation of the posterior partition function, including the compensating interaction change \(-dB_t\).

Put \(H_q=Ck_q(q^\tau+k_q^{-\tau})\), with \(\tau\) small and fixed. At the first endpoint, the likelihood-density identity gives \(\mathbf P(\log L_p<-H_q)\le e^{-H_q}\). At the last endpoint, (17) gives \[\mathbf P(\log L_p>H_q,\ h_p\text{ in the bounded range}) \le \exp(-H_q+C+Ck_q q).\] Choose \(C\) large and use Lemma 9 outside the range. An exponential martingale inequality now bounds the event that the bracket exceeds a sufficiently large multiple of \(H_q\) while these endpoint inequalities hold. This proves (21). The logarithmic number of intervals is absorbed by the power-exponential bound. Both means have norm at most \(\sqrt n\), so the integral has a polynomial deterministic bound; integrating the exception proves the expectation assertion. ◻

Proof. On the shrinking annulus of Lemma 9, the spherical covariance has norm \(O(p^{-2})\). By rotation invariance within the flat cap its restriction to \(e^\perp\), \(e=h_p/\|h_p\|\), is scalar. The frozen Gaussian linear-tilt bound gives an \(O(p^{-1})\) error between the spherical radial mean and the saddle mean. Dividing by \(\|h_p\|\asymp p^2M_p\) shows that the transverse covariance differs from \((p^2+\alpha_p)^{-1}\mathop{\mathrm{Id}}\) by \(O(p^{-2}k_p^{-1/2})\). The remaining radial covariance is rank one. Thus its contribution, and the shrinking saddle parameter, are already bounded by the right side of (22).

It remains to compare \(K_p\) and \(K_p^{\mathrm{sp}}\). The spherical posterior, evaluated along the cube observation path, has innovation drift \(K_p^{\mathrm{sp}}a_p^{\mathrm{err}}dt\). Subtracting the two posterior mean equations gives \[da_p^{\mathrm{err}} =(K_p-K_p^{\mathrm{sp}})\,d\mathcal B_t -K_p^{\mathrm{sp}}a_p^{\mathrm{err}}dt\] between activations. Multiply \(p^2\|a_p^{\mathrm{err}}\|^2\) by a smooth nonnegative cutoff supported on an enlarged scale interval and equal to one on \([q,2q]\), with bounded derivative in \(\log p\). Itô’s formula and integration by parts give \[\int_q^{2q}\|p^2(K_p-K_p^{\mathrm{sp}})\|_{\mathrm F}^2\,d\log p \le C\int_{\mathrm{enlargement}}\|a_p^{\mathrm{err}}\|^2\,d(p^2) +C|\mathcal M|.\] Activations add squared new mean coordinates with nonnegative coefficients, so they may be discarded on this side of the inequality. The cutoff is zero at both endpoints; the displayed energy identity is applied over the entire interval, not at a stopping time. Its martingale is \[\mathcal M=2\int\chi(t)t \langle a_{\sqrt t}^{\mathrm{err}}, (K_{\sqrt t}-K_{\sqrt t}^{\mathrm{sp}})\,d\mathcal B_t\rangle.\] For concentration, stop this stochastic integral alone at the first failure of the annular covariance bounds, and set its integrand to zero on \(\mathcal G_n^c\), a time-zero event. Its bracket is then bounded by \(C\int\|a_p^{\mathrm{err}}\|^2d(p^2)\). The stopped integral agrees with \(\mathcal M\) on \(\mathcal G_n\) and the whole-path annulus event. The martingale inequality, Lemma 13, and Lemma 9 therefore prove the assertion. Division by \(k_q\asymp k_p\) gives the stated normalization. ◻

Proof. The coordinates on which \(A\) acts belong to the cap for all sufficiently large \(n\). By (12), the joint density of the frame and spectral observation relative to independent Haar and the spherical observation law is \(L_p\). Condition on the physical capped matrix \(W_p\), the physical vector \(h_p\), and the spectral coordinates of that vector. The spherical observation density is now fixed, and \(L_p\) depends only on the fixed physical data and spectrum. The remaining rotation of the spectral frame in \(e^\perp\cap\mathcal H_p\) is therefore exactly Haar. In physical coordinates, the differentiation direction is \(UA\widetilde h_p\), which is orthogonal to \(h_p\) and has length \(|A\widetilde h_p|\). The spherical mean in the cap is radial. Haar averaging consequently gives \[\frac{\mathbf E|\mathcal R\widetilde\phi_p(\widetilde h_p)|^2} {n^{2/3}p^2} \le C\mathbf E\left[ |Au_p|^2\frac{p^2\|a_p^{\mathrm{err}}\|^2}{k_p}\right].\] Apply (21) in expectation and interpolate with the higher moments in (19) and the coordinate moments following Lemma 9. Hölder’s inequality on probability times logarithmic scale gives (23), with a smaller positive exponent. No event depending on the unused spectral refinement was inserted before the Haar average. ◻

Proof. We give the construction and its tail comparison, following the cap extension method of (OpenAI 2026a, Proposition 16.1). On \(|\alpha_p|\le3s_0p^2/4\), take the infimum of the quadratics \[\phi_p(z)+\nabla\phi_p(z)\cdot(y-z) +\frac{1+\eta}{2p^2}\|y-z\|^2.\] Each of these supports lies above \(\phi_p\) on that smaller annulus. Choose \(s_0\) small compared with \(\eta\) and \(\gamma^2\). The radial buffer between the smaller and larger annuli then determines a fixed \(\beta_{\mathrm{buf}}>0\) such that every segment starting in the smaller annulus and of length at most \(\beta_{\mathrm{buf}}p^2M_p\) remains in the larger one. For such pairs \(y,z\), the supporting inequality follows by integrating the coarse Hessian bound along their segment. For \(d=y-z\) with \(\|d\|\ge\beta_{\mathrm{buf}}p^2M_p\), freeze the spherical saddle at \(z\). The frozen Gaussian bound bounds the spherical logarithmic increment by the quadratic with Hessian \((p^2+\alpha_p(z))^{-1}\mathop{\mathrm{Id}}\), its saddle gradient, and a bounded density error. The static value and gradient errors from Lemma 8 are at most \(\epsilon_v k_p\) and \(\epsilon_gM_p\). Relative to \(\|d\|^2/p^2\), their combined contribution, including the density error, is at most \[\frac{2\epsilon_v+C/k_p}{\beta_{\mathrm{buf}}^2} +\frac{\epsilon_g}{\beta_{\mathrm{buf}}}.\] The frozen Hessian is at most \((1+Cs_0)p^{-2}\mathop{\mathrm{Id}}\). Thus, after choosing \(s_0\), then \(\beta_{\mathrm{buf}}\), then the static errors, the excess quadratic term dominates this residual for all sufficiently large \(n\). This proves the supporting property, and the infimum is a semiconcave extension agreeing on the smaller annulus.

For the transition tails freeze instead the child saddle \(\alpha=\alpha_c(h_c)\). As reference use the Gaussian transition tilted by the parent quadratic \(\|y\|^2/(2(p^2+\alpha))\) with no linear term. Writing \(R_u=R_{u,\alpha}\), its law is characterized by \[R_ph_p\sim N(R_ch_c,R_c-R_p), \qquad R_c^{-1}=R_p^{-1}-\Delta.\] Since the child saddle satisfies its norm constraint, the expected squared norm of \(h_p\) is \(\ell_p(\alpha)^2\). Its covariance norm is at most \(C\gamma^{-2}p^2\). Therefore deviations in radius by \(vp^2M_p\) cost \[C\exp(-a_0\gamma^2k_pv^2), \qquad v\ge C(\gamma\sqrt{k_p})^{-1}.\] The radius \(\ell_p(\alpha)\) lies strictly inside the parent agreement annulus, because \(|\alpha|\le s_0c^2\).

Calibrate the reference quadratic to the spherical potential at this radius. On a bounded range of scaled radii the true potential is at most the reference quadratic plus \(\epsilon k_p+O(1)\), with arbitrarily small fixed \(\epsilon>0\), by the frozen norm-density bound and static comparison. Outside a sufficiently large bounded range, 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 integrating the tail \(\exp(-ak_pv^6)\) against the linear field tilt. Its growth is absorbed by the reference quadratic. For the extension, use the support centered at the point of radius \(\ell_p(\alpha)\) in the direction of \(y\). At radial distance \(vp^2M_p\) its excess above the reference is at most \[\epsilon k_p+\epsilon k_pv+C(\eta+s_0)k_pv^2+O(1).\] Choose \(\eta+s_0\) small compared with \(\gamma^2\), and choose \(\epsilon\) after the fixed gap to the boundary of agreement. These errors are strictly smaller than the Gaussian deviation cost outside agreement.

The normalizing integrals relative to the reference are bounded below by \(\exp(-\epsilon'k_p)\) with any needed fixed \(\epsilon'>0\). Indeed integrate over a sufficiently thin fixed relative shell around \(\ell_p(\alpha)\). Both potentials agree there, their values at the central radius are given by static comparison, and their gradients are bounded in units of \(M_p\). Gaussian concentration gives a positive lower bound for the reference mass of this shell for large \(n\). This proves the asserted transition tails; polynomial moments are absorbed by the same Gaussian deviation estimate and the subquadratic bound above. ◻

Proof. The coarse radial deficit gives \(D_{ee}\le-\varepsilon_1<0\). For each transverse vector, maximize the quadratic form of \(D\) over its radial coordinate. The resulting form is \(D_{\perp\perp}+zz^{\mathsf T}/|D_{ee}|\), proving \(D\preceq E_p\). Since \(K_p\succeq0\) and \(p^2K_p\preceq(1+\varepsilon_0)\mathop{\mathrm{Id}}\), both \(D\) and the maximizing radial coordinate for a unit transverse vector are bounded by constants. Comparing that vector with \(D\preceq s_p^+\mathop{\mathrm{Id}}\) proves the norm estimate. The trace of the rank-one correction is bounded, while the trace of the positive part of \(D_{\perp\perp}\) is at most \(\sqrt{\dim\mathcal H_p}\,\|D\|_{\mathrm F}\). This proves (27). ◻

Proof. Condition on \(W_p\), the physical field \(h_p\), and its spectral coordinates, as in Lemma 15. The remaining frame in \(e^\perp\cap\mathcal H_p\) is Haar, and \(E_p\) is fixed under this conditioning. For a uniformly rotating unit vector \(v\) in that space, the Gaussian representation of the uniform sphere gives \[\mathbb P\left(v^{\mathsf T}E_pv> \frac{C(\mathop{\mathrm{Tr}}E_p+t\|E_p\|)}{k_p}\right) \le C(e^{-t}+e^{-c_0k_p}).\] The same bound applies to a fixed vector of norm at most one. Choose a constant-resolution net of the child unit sphere and project it onto \(e^\perp\). Its cardinality is at most \(\exp(C\dim\mathcal H_c)\le\exp(Cx^3k_p)\). With \(t=C_2x^3k_p\), a sufficiently large \(C_2\) pays for the net; taking \(\gamma\) sufficiently small also pays for the second exponential term. The net approximation for a positive quadratic form proves (29). ◻

Proof of Proposition 17. Let \(\widehat s_p\) be \(s_p^+\) on the full coarse annulus and zero outside it. We now restrict to \(\mathcal G_n\). Average (28) over \(c/(2\gamma)\le p\le c/\gamma\) with normalized logarithmic measure, and condition on \(W\) and the observation path through \(c\). Combining (27) and (29), the operator-norm term has coefficient at most \(Cx^{-2}x^3\le C\gamma\). Choose \(\gamma\) so that this coefficient, denoted by \(\theta\), is less than \(1/2\). The trace term is controlled by Cauchy–Schwarz and the integrated Frobenius estimate (22). We obtain \[ \widehat s_c\le \theta\operatorname{avg}_{c/(2\gamma)\le p\le c/\gamma} \mathbf E[\widehat s_p\mid W,\mathcal F_c] +C_\gamma(c^{b}+k_c^{-b})+\operatorname{err}_c, \tag{30}\] where \(b>0\), \(\mathcal F_c\) is the observation filtration, and \(\operatorname{err}_c\ge0\) satisfies \[\mathbf E[\mathbf1_{\mathcal G_n}\operatorname{err}_c] \le Ce^{-n^\xi}\] uniformly at deterministic scales. To verify the exceptional terms in this assertion, first apply (22) to the parent scale integral before taking its conditional expectation. Its failures and those of (29) have exponential probability and at most polynomial cost. Since \(W\) is included in the conditioning, the indicator of \(\mathcal G_n\) commutes with that conditional expectation. No assertion of Haar independence conditional on \(\mathcal G_n\) is used. If the child field is outside its annulus, the inequality holds by truncation.

Choose \(a\le b\) sufficiently small that \(\theta\gamma^{-a}<1\). For a sufficiently large \(C'\), the function \(b_0(c)=C'(c^a+k_c^{-a})\) dominates its parent average multiplied by \(\theta\), plus \(C_\gamma(c^b+k_c^{-b})\), with a fixed margin. It also dominates the coarse constant bound at the upper scales where the parent interval no longer fits below \(p_*\). Take positive parts above \(b_0\) in (30), use conditional Jensen, and average over \(\mathbf P\) with \(\mathbf1_{\mathcal G_n}\). At each iteration the scale increases by at least \((2\gamma)^{-1}>1\), so there are only \(O(\log n)\) generations; their errors sum to at most \(Ce^{-n^\xi}\sum_{j\ge0}\theta^j\). Thus \[\mathbf E[\mathbf1_{\mathcal G_n} (\widehat s_c-b_0(c))_+]\le Ce^{-n^\xi}.\] Enlarging \(C'\) and applying Markov’s inequality proves the asserted probability bound, since \(b_0(c)\) is bounded below by an inverse polynomial on the scale range. Lemma 9 removes the truncation. All bounds were uniform at deterministic scales, so Fubini’s theorem gives their integrated version. ◻

Proof. The continuation in (OpenAI 2026a, sec. 19.4) increases the observation precision from \(B_{p_{\mathrm{top}}}\) through finitely many cap intervals, a bridge to a classical field, and a bounded-duration continuation to a diagonal-interaction endpoint. For an initial change of spin law from \(\mu\) to \(\nu\), let \(\nu_s,\mu_s\) be the two posteriors, let \(l_s=D(\nu_s\Vert\mu_s)\), and put \(H(s)=E_{Q_\nu}l_s\). On this continuation, the estimate used is \[ \dot H(s)\ge-CH(s)-e^{-a n}. \tag{31}\] This is (OpenAI 2026a, Equation (19.17)). Its proof uses upper bounds on observation likelihoods and posterior entropy tails, rather than a support restriction on the posterior. We explain its application when the initial density lies between \(1/2\) and \(2\). Every observation likelihood then has the same bounds, every posterior density ratio lies between \(1/4\) and \(4\), and \(l_s\le\log4\).

At the classical part, the ordinary posterior covariance is bounded on a fixed normalized field neighborhood outside an event of probability \(e^{-an}\); farther along the continuation it is bounded for every field. The linear entropy inequality in that neighborhood gives \(\|m_\nu-m_\mu\|^2\le Cl_s\) whenever \(l_s\) is bounded: a trial field shift of size proportional to \(\sqrt{l_s}\) fits the neighborhood. The bounded density ratios pay for the ordinary exceptional paths.

To propagate this mean estimate across a preceding cap or bridge interval, use the approximate gradient \(g_s\) furnished by the extended potential there. The transport identity (OpenAI 2026a, Lemma 18.3) holds under either initial law. If \(U\) is its linear transport from the current time \(s\) to the end of the interval, subtraction gives, for a vector \(v\) in the current rate subspace, \[\begin{align*} \langle v,m_{\nu,s}-m_{\mu,s}\rangle ={}&E_{Q_{\nu,h}^{\mathrm{cont}}} \langle Uv,m_{\nu,\mathrm{end}}-m_{\mu,\mathrm{end}}\rangle\\ &+(E_{Q_{\nu,h}^{\mathrm{cont}}} -E_{Q_{\mu,h}^{\mathrm{cont}}}) \langle Uv,m_{\mu,\mathrm{end}}-g_{\mathrm{end}}\rangle. \end{align*}\] Here the two continuation laws live on the same path space, and the same path function \(U\) is used in both expectations. Its operator norm is bounded by a fixed constant on these fixed intervals. At agreement, the terminal approximation error in the tested subspace has bounded norm. Truncate it there. Data processing gives \(D(Q_{\nu,h}^{\mathrm{cont}}\Vert Q_{\mu,h}^{\mathrm{cont}})\le l_s\), so Pinsker’s inequality bounds the second term by \(C\sqrt{l_s}\|v\|\).

The complements have exponentially small averaged squared cost. Indeed, for any nonnegative terminal cost \(W_1\), the bounded full-path and prefix likelihoods give both \[E_{Q_\nu}E_{Q_{\nu,h}^{\mathrm{cont}}}W_1 \le C E_{Q_\mu}W_1, \qquad E_{Q_\nu}E_{Q_{\mu,h}^{\mathrm{cont}}}W_1 \le C E_{Q_\mu}W_1.\] Apply these inequalities to the polynomially weighted ordinary exceptions in the terminal approximation estimates of (OpenAI 2026a, sec. 19.4). Conditional Cauchy–Schwarz controls the first transport term by the terminal squared mean difference. Taking the supremum over unit \(v\) costs no dimension factor because all the bounds are norm bounds uniform in \(v\). Since terminal expected posterior entropy is at most \(H(s)\), backward induction through the finite list yields \[E_{Q_\nu}\|P_{\mathrm{rate}} (m_{\nu,s}-m_{\mu,s})\|^2 \le CH(s)+Ce^{-a n}.\] The observation entropy identity (OpenAI 2026a, Lemma 19.2), with bounded precision rates on these intervals, now proves (31). This also proves that its constants apply uniformly to the bounded-density perturbations below.

Normalize \(\|f\|_\infty\le1\), set \(\eta=e^{-n^{1/3}}\), and take \[\frac{d\nu}{d\mu}=\frac{1+\eta f}{1+\eta\mu f}.\] Every path likelihood and every posterior density ratio is bounded by fixed constants. Taylor’s formula for relative entropy gives, uniformly at every posterior, \[c\eta^2\mathop{\mathrm{Var}}_{\mu_s}(f) \le D(\nu_s\Vert\mu_s) \le C\eta^2\mathop{\mathrm{Var}}_{\mu_s}(f).\] Thus \(H(s)\) is comparable to \(\eta^2V_s(f)\). Integrating (31) over a bounded duration and dividing by \(\eta^2\) gives \(V_{p_{\mathrm{top}}}(f)\le CV_{\mathrm{end}}(f)+Ce^{-a'n}\). At the endpoint the posterior is a product measure, so variance tensorization and conditional-variance contraction give \[V_{\mathrm{end}}(f) \le E\sum_i\mathop{\mathrm{Var}}(f\mid X_{-i},\text{endpoint observations}) \le\sum_i\mu[\mathop{\mathrm{Var}}(f\mid X_{-i})]=\mathcal E(f).\] Rescaling \(f\) proves the lemma. ◻

Proof of Proposition 16. For fixed \(W\), the posterior mean of \(f\) is a martingale in the observation filtration. Its diffusion coefficient, with parameter \(t=p^2\), is \(P_p\mathop{\mathrm{Cov}}(X,f\mid h_p)\). Consequently \[-\frac{d}{d(p^2)}V_p(f) =Q_\mu\|P_p\mathop{\mathrm{Cov}}(X,f\mid h_p)\|^2 \le Q_\mu\bigl[\|K_p\|\mathop{\mathrm{Var}}(f\mid h_p)\bigr].\] Integrate the exceptional probabilities in Proposition 17 in logarithmic scale, then apply Markov’s inequality and Lemma 6. On disorder events of probability arbitrarily close to one, intersected with \(\mathcal G_n\), their total ordinary path mass is at most \(Ce^{-n^\xi}\). The same statement holds for the annulus exceptions used with the coarse bound between \(p_*\) and \(p_{\mathrm{top}}\). These events are chosen before, and independently of, \(f\).

On the exceptional paths the elementary estimate \(\|P_p\mathop{\mathrm{Cov}}(X,f\mid h_p)\|^2\le Cn\|f\|_\infty^2\) suffices. On the remaining paths, put \(\epsilon_p=C(p^a+k_p^{-a})\) below \(p_*\) and use the fixed coarse tolerance above it. Backward integration from \(p_{\mathrm{top}}\) gives an amplification bounded by \[\exp\left(\int_{p^2}^{p_{\mathrm{top}}^2} \frac{1+\epsilon_{\sqrt t}}{t}\,dt\right) \le Cp^{-2},\] because \[\int_r^{p_*}(p^a+k_p^{-a})\,d\log p\le C,\] and the remaining scales form a fixed interval bounded away from zero. The amplification and the elementary exceptional-path costs are polynomial in \(n\); they are absorbed by decreasing \(\xi\). Lemma 21 therefore gives (24). The same integrated exceptional event works for every lower endpoint \(p\), so the assertion is simultaneous in scale as well as in the test. ◻

Proof. On the active cap, \(\nabla_{h_p}F_p=P_p\mathop{\mathrm{Cov}}(X,f\mid h_p)\). Conditional Cauchy–Schwarz therefore gives \[|\nabla_{h_p}F_p|^2 \le\|K_p\|\mathop{\mathrm{Var}}(f\mid h_p).\] Combining the covariance bound (14), the annulus estimate (15), and the variance estimate (24), and multiplying by the squared change-of-variable factor \(p^4n^{2/3}\), proves (35). Conditional Jensen and \(\|B_p\|\le Cp^2\) also bound \(J_p^f\). The first line of (36) follows from the same Jensen inequality applied to \(f-F_p(h_p)\).

For the second line, condition on \(X\) and write the channel as \(h_p=B_pX+B_p^{1/2}Z\), with \(Z\) standard Gaussian on the active space. Flipping spin \(i\) translates \(Z\) by \(-2X_iB_p^{1/2}e_i\). Expand the Gaussian translation in Hermite tensors. The degree-one term in the half-difference is the \(i\)th component of \(J_p^f\). For the higher degrees, subtract the constant \(f(X)\) from the translated function. The total squared norm of the Hermite coefficients of \(n^{1/3}(F_p(h_p)-f(X))\), averaged over \(X\), is at most \[C p^{-2}T\mathcal E(f)+o(1)\] by (24). The degree-\(j\) contraction into the vector indexed by \(i\) has operator norm at most \[\frac{2^j}{\sqrt{j!}}\|B_p\|^{1/2} \left(\max_i(B_p)_{ii}\right)^{(j-1)/2}.\] Indeed its Gram matrix, apart from the displayed scalar and spin signs, is a Schur power of \(B_p\); the Schur product bound gives the estimate. Since \(\max_i(B_p)_{ii}\le\|B_p\|\le Cp^2\), summing degrees \(j\ge2\) costs \(O(p^2)\) times the coefficient norm. This proves the second line of (36) in the unweighted vector norm. ◻

Proof. Pass first to a subsequence on which the energy converges to its liminf. Equation (35) bounds the weak spectral vector limits in \(L^2(\Pi;\ell^2)\). To identify them, fix \(A\) and \(\psi\) as in (32). We will compare the angular pairing of the slopes at fixed cap scale to an integration by parts at the smallest scale \(r\).

Define \[T_p^A=Q_\mu[\psi(u_p)\mathcal R_AF_p(h_p)],\] where the rotation acts on spectral coordinates. By (16) and (35), \[\mu[\psi(x)(Ax)\cdot U^tJ_p^f]-T_p^A =O\bigl((pn^{1/3})^{-1}\bigr)+o(1).\] For the finitely many coordinates involved, \(B_p/p^2=I+O((pn^{1/3})^{-2})\). Replacing \(x\) by \(u_p\) in the test then uses Cauchy–Schwarz and (16); compact support of the test, or its fourth-moment version, controls the coordinate factors.

Proof. We first separate the differences that strike a conditional mean from those that remain on \(f\). The exact product rule is \[D_j(FG)=F D_jG+(D_jF)G(X^{(j)}), \qquad G(X^{(j)})=G(X)-2X_jD_jG(X).\] If \(i\in S\), applying \(D_i\) first to \((X_i-t_i)D_if\) gives \(D_if\); these terms therefore sum to \(mD_Sf\). For \(i\notin S\), the terms in which no difference strikes \(t_i\) give \(HD_Sf\). Every remaining term, with its overall minus sign, is \[ (D_A t_i)\prod_{j\in B}(-2X_j)\, D_{\{i\}\cup(S\setminus A)\cup B}f, \qquad i\notin S,\quad \varnothing\ne A\subset S,\quad B\subset A. \tag{52}\] Indeed, the iterated product rule first evaluates \(D_{\{i\}\cup(S\setminus A)}f\) at \(X^A\), and expanding the flips at the sites of \(A\) gives the displayed sum over \(B\).

Fix the positions of \(A\) and \(B\), and write \(a=|A|\), \(b=|B|\) and \(l=1+m-a+b\). We estimate the sum over \(i\) of (52) in the norm obtained by summing \(\mu[w_S|\cdot|^2]\) over \(S\). For positive derivative orders, \[|\tanh^{(a)}u|\le C^a a!\mathop{\mathrm{sech}}^2u.\] For example, Cauchy’s formula on a fixed circle of radius less than \(\pi/2\), applied to \(\mathop{\mathrm{sech}}^2z\), proves this inequality: on that circle \(|\mathop{\mathrm{sech}}^2z|\le C\mathop{\mathrm{sech}}^2u\) uniformly for real \(u\). The difference integral for \(D_A t_i\) consequently gives \[ |D_A t_i|\le w_i C^a a!\prod_{j\in A}|W_{ij}|. \tag{53}\] The constant is uniform for each fixed \(m\) and all sufficiently large \(n\), since all field shifts in that integral have absolute value at most \(2m\epsilon_n\) and \(|\log\mathop{\mathrm{sech}}^2u-\log\mathop{\mathrm{sech}}^2v|\le2|u-v|\).

Suppose first that \(b\ge1\), and select \(j_0\in B\). Apply Cauchy–Schwarz to the sum over \(i\), using \(\sum_iW_{ij_0}^2\le K^2\) for its first factor. The second factor contains \(w_i^2\), the squared derivative of \(f\), and \(\prod_{j\in A\setminus\{j_0\}}W_{ij}^2\). Put \(R=S\setminus A\) and \(A_0=A\setminus B\). The weights \(w_Rw_B\) occur in both input and output norms and are retained. Bound the additional output weights \(w_{A_0}\) by one. Summing the sites of \(A_0\) now costs at most \(K^{2(a-b)}\), by row square sums; the remaining \(b-1\) factors at sites of \(B\) cost at most \(\epsilon_n^{2(b-1)}\). Finally, \(w_i^2\le w_i\) supplies the input weight at site \(i\). Here only the distinctness restrictions involving the summed-out sites of \(A_0\) are removed; the retained tuple \((i,R,B)\) still consists of distinct sites. After summing that tuple and integrating, the resulting norm is at most \[ C^{a+1}a!\epsilon_n^{b-1}N_l(f),\qquad b\ge1. \tag{54}\] Thus all terms with \(b\ge2\) have coefficients \(o_{n;m}(1)\).

When \(b=0\), taking absolute values in the sum over \(i\) would lose the operator norm bound on \(W\). Instead expand the difference integral once more: \[\begin{align*} D_A t_i={}&\Bigl(\prod_{j\in A}W_{ij}\Bigr) \left\{\tanh^{(a)}(h_i) -\tanh^{(a+1)}(h_i)\sum_{j\in A}W_{ij}X_j+R_{i,A}\right\}, \tag{55}\\ |R_{i,A}|\le{}&C_m w_i\epsilon_n^2. \end{align*}\] The first correction has the indicated sign because the field in the integral is \(h_i-2\sum_{j\in A}s_jW_{ij}X_j\), with each \(s_j\) uniform on \([0,1]\).

Here is the operator estimate for the first two terms. Regard \[(T_a u)_{j_1,\ldots,j_a} =\sum_i\prod_{r=1}^a W_{ij_r}\,u_i\] as a map from \(\ell^2\) of sites to \(\ell^2\) of ordered site tuples, initially allowing repetitions. Its Gram matrix is \((WW^t)^{\circ a}\), the entrywise \(a\)th power, so \(\|T_a\|\le K^a\). To see the norm bound directly, the entrywise product of two matrices is the compression of their tensor product to the span of the vectors \(e_i\otimes e_i\); iterate this observation for \(WW^t\). In each first-correction term, one factor \(W_{ij_r}\) is replaced by \(W_{ij_r}^2\). The matrix \((W_{ij}^2)_{i,j}\) has row and column sums at most \(K^2\), hence operator norm at most \(K^2\). The same tensor-product bound therefore controls this corrected map by \(K^{a+1}\). Multiplication by \(X_{j_r}\) on its output is an isometry.

Fix \(R=S\setminus A\) and apply these operator estimates pointwise in \(X\), before integrating against \(w_R\mu\). Restrictions that the sites of \(A\) be distinct and avoid \(R\) are coordinate projections on the output of these maps. The restriction \(i\notin R\) is a coordinate projection on the input, and \(i\notin A\) is automatic from \(W_{ii}=0\). The output weights \(w_A\) may again be discarded. The common weight \(w_R\) remains, and the factors \(\tanh^{(a)}(h_i)\) and \(\tanh^{(a+1)}(h_i)\) are bounded multiples of \(w_i\). Their multiplication on an input with squared norm \(\sum_iw_i|D_{\{i\}\cup R}f|^2\) thus costs no inverse powers of \(w_i\). Summing \(R\) proves the required bound \(C^{a+1}a!N_l(f)\) for the first two terms of (55), after increasing \(C\). For the remainder, signs may be discarded: the matrix with entries \(\prod_{r=1}^a|W_{ij_r}|\) has squared Frobenius norm \[\sum_i\left(\sum_jW_{ij}^2\right)^a\le nK^{2a}.\] Its contribution is at most \(C_m\sqrt n\epsilon_n^2N_l(f)=o_{n;m}(1)N_l(f)\). We have proved \[ \text{norm of the fixed-position terms with }b=0 \ \le\bigl(C^{a+1}a!+o_{n;m}(1)\bigr)N_{1+m-a}(f). \tag{56}\]

It remains to control \(HD_Sf\). This term is measured with \(w_S\mu\), so we require an operator estimate in precisely that measure. Fix \(S\) and write \(\nu=w_S\mu\), without normalizing it. Denote its conditional means and variances by \(t_i'\) and \(w_i'\), its heat-bath operator by \(H'=\sum_i(X_i-t_i')D_i\), and its first two difference seminorms by \(N_1'\) and \(N_2'\), using the measure \(\nu\) and weights \(w_i'\). We claim that \[ \|HF\|_{L^2(\nu)} \le C\{N_2'(F)+(1+m)N_1'(F)\}, \qquad C^{-1}w_i\le w_i'\le Cw_i. \tag{57}\] The constants depend only on \(K\), once \(n\) is sufficiently large for the fixed \(m\).

The conditional field under \(\nu\) is exactly \[h_i'=h_i+\delta_i,\qquad \delta_i=D_i\log w_S.\] Since the first two derivatives of \(\log\mathop{\mathrm{sech}}^2u\) are bounded, the nonnegative deterministic quantities \[a_i=2\sum_{k\in S}|W_{ik}|, \qquad L_{ij}=C\sum_{k\in S}|W_{ik}W_{jk}|\] satisfy \(|\delta_i|\le a_i\) and \(|D_j\delta_i|\le L_{ij}\) for \(i\ne j\). They obey \[\|a\|_2\le 2mK,\quad \|a\|_\infty\le2m\epsilon_n, \quad \|L\|\le CmK^2, \quad \Bigl(\sum_jL_{ij}^2\Bigr)^{1/2}\le CK a_i.\] These statements follow respectively from the triangle inequality for columns of \(W\) and the expression of \(L\) as a sum of nonnegative rank-one matrices. The variance comparison in (57) follows from \(|\delta_i|\le a_i=o(1)\) uniformly in \(i\). Taylor’s formula for \(\tanh\) and \(\mathop{\mathrm{sech}}^2\), now at the tilted fields, also gives \[\begin{align*} D_i t_j'&=w_j'(W_{ij}+E_{ij}), &|E_{ij}|&\le C(L_{ij}+W_{ij}^2),\tag{58}\\ |D_iw_j'|&\le Cw_j' r_{ij}, &r_{ij}&=|W_{ij}|+L_{ij},\qquad i\ne j. \end{align*}\] In deriving the first bound, the square of \(L_{ij}\) is absorbed into \(L_{ij}\) since its maximum tends to zero. The absolute error matrix is bounded entrywise by a nonnegative matrix of norm at most \(C(1+m)\); indeed \((W_{ij}^2)\) has norm at most \(K^2\). Also \(\sup_i\sum_jr_{ij}^2\le C\) and \(\max_{i,j}r_{ij}=o(1)\), for \(n\) large at fixed \(m\).

For completeness, these coefficient bounds give the operator estimate without an independence assumption on the slopes. Put \(F_i=D_iF\) and \(F_{ij}=D_iD_jF\). The product rule gives the exact identity \[D_iH'F=H'F_i+F_i -\sum_{j\ne i}(D_it_j')(F_j-2X_iF_{ij}).\] Use the version of (50) for \(\nu\) to write \[\begin{align*} \|H'F\|_{L^2(\nu)}^2 ={}&N_1'(F)^2 +\sum_{i\ne j}\nu[w_j'D_j(w_i'F_i)F_{ij}]\\ &-\sum_{i\ne j}\nu[w_i'F_i(D_it_j') (F_j-2X_iF_{ij})]. \end{align*}\] In the last sum, the \(F_j\) term is bounded by \(C(1+m)N_1'(F)^2\): apply the signed norm bound to \(W\) with input \((w_i'F_i)_i\), and the absolute matrix bound to \((E_{ij})\). The \(F_{ij}\) term is bounded by \(CN_1'(F)N_2'(F)\), by Cauchy–Schwarz and the row square bounds for \(r\). Finally, \[D_j(w_i'F_i)=w_i'F_{ij}+(D_jw_i')(F_i-2X_jF_{ij}).\] The first summand contributes \(N_2'(F)^2\). The other two summands cost, respectively, \(CN_1'(F)N_2'(F)\) and \(CN_2'(F)^2\), using (58) and the same row square bounds. Consequently \[\|H'F\|_{L^2(\nu)}^2 \le C\{N_2'(F)^2+N_1'(F)N_2'(F)+(1+m)N_1'(F)^2\}.\] Moreover \(|t_i-t_i'|\le Ca_iw_i\), so pointwise Cauchy–Schwarz and \(\|a\|_2\le Cm\) show \[\|(H-H')F\|_{L^2(\nu)}\le CmN_1'(F).\] These two bounds prove (57).

Apply (57) to \(F=D_Sf\) and sum squares over the ordered tuples \(S\). A difference at a site already in \(S\) vanishes. For all other sites, the variance comparison replaces \(w_i'\) by \(w_i\) with a fixed factor. Appending one or two ordered sites to \(S\) enumerates exactly the ordered tuples in \(N_{m+1}\) or \(N_{m+2}\); there is no further factorial. Minkowski’s inequality therefore yields \[\left(\sum_S\mu[w_S|HD_Sf|^2]\right)^{1/2} \le C\{N_{m+2}(f)+(1+m)N_{m+1}(f)\}.\] Together with the term \(mD_Sf\), this controls every contribution except the positional sums already bounded above.

There are \(\binom ma\) choices of the positions of \(A\), and \(\binom ab\) choices for \(B\). The terms with \(b=0\) have \(l=m-a+1\) and coefficients bounded by \(C^{a+1}a!\binom ma\); those with \(b=1\) have \(l=m-a+2\) and coefficients bounded by \(C^{a+1}a!a\binom ma\). Use \(a!\binom ma\le m^a\) and absorb the factor \(a\le2^a\) by increasing \(C\). Both bounds, as well as the \(HD_Sf\) and \(mD_Sf\) contributions, fit the coefficient \(C^{m-l+3}(m+1)^{m-l+2}\) in (51). The finitely many \(b\ge2\) terms and Taylor remainders combine into \(o_{n;m}(1)\). ◻

Proof. The masses are at most one, and the spectral theorem and (50) give \[\int\lambda^{2j}\,d\omega_{n,a}(\lambda)=N_1(H^jY_a)^2.\] The \(j=1\) bound gives tightness. Bounds at higher even orders give uniform integrability for each fixed power of \(\lambda\), so every weak subsequential limit has the prescribed moments. In particular, (59) implies \[ \int\lambda^{2j}\,d\omega(\lambda) \le C^{2j}(2j+2)^{4j}. \tag{60}\]

Define the symmetric square-root lift \(\rho\) by \[\int g(t)\,d\rho(t) =\frac12\int\{g(\sqrt\lambda)+g(-\sqrt\lambda)\}\,d\omega(\lambda).\] Its \(4j\)th absolute moment is bounded by the right side of (60). For an arbitrary integer \(k\ge1\), choose \(4j\) between \(k\) and \(k+3\) and use Hölder’s inequality. Since the mass is at most one, this gives \(\int|t|^k\,d\rho(t)\le(Dk)^k\) with a constant \(D=D(K)\). The exponential series and \(k!\ge(k/e)^k\) therefore show that \(\int e^{c|t|}\,d\rho(t)<\infty\) for small enough \(c>0\). This is the claimed square-root exponential moment.

If another measure on \([0,\infty)\) has the same moments, its symmetric square-root lift has the same integer moments as \(\rho\), and the same argument gives an exponential moment. The Fourier transforms of the two lifts extend analytically to a strip about the real axis. Their Taylor series at zero agree, so the identity theorem and uniqueness of Fourier transforms of finite measures identify the lifts, and hence the original measures. Every subsequential limit is therefore \(\omega\), proving weak convergence of the full sequence.

For density, let \(g\in L^2(\rho)\) be orthogonal to all polynomials. Cauchy–Schwarz implies that the finite signed measure \(g\rho\) has an exponential moment with a smaller positive exponent. All derivatives at zero of its Fourier transform vanish. The same analytic argument shows that this transform, and hence \(g\rho\), is zero. Thus polynomials are dense in \(L^2(\rho)\). Applying the bounded projection \(p(t)\mapsto(p(t)+p(-t))/2\) shows that even polynomials are dense in its even subspace. The isometry \(f(\lambda)\mapsto f(t^2)\) identifies this subspace with \(L^2(\omega)\), proving the first density assertion. Finally, the symmetric square-root lift of \((1+\lambda)\omega+\delta_0\) is \((1+t^2)\rho+\delta_0\). It also has an exponential moment after decreasing \(c\), and the same proof establishes the second density assertion. ◻

Completion of the proof of Proposition 26. For the Gaussian matrices in this section, removing the diagonal preserves a bounded operator norm and the entry bound used in Lemma 28, on events whose probability tends to one. The fixed-order contractions give moment convergence there. Corollary 29 therefore constructs the unique positive measure with those moments and proves the exponential bound and polynomial density. The moments are deterministic and the same for every fixed leading index. Applying the conditional Haar calculation to the deterministic spectral data of (43) gives precisely those same moments. Finally the contraction and projection arguments preceding this subsection give (44) and (45) with this measure. This proves all assertions of the proposition. ◻

Proof. Set \(V_a=(Q(H)-1)Y_a\). Since \(Q-1\) vanishes at zero, \[\|V_a\|_2^2 =\int\frac{(Q(\lambda)-1)^2}{\lambda} \,d\omega_{a,n}(\lambda)\] is bounded for fixed \(Q\): the quotient extends to a polynomial at zero. Thus the correction in (62) tends to zero in \(L^2\). Static convergence of cylinders gives the claimed convergence of values and norms. The bounds \(\|H\|_{\infty\to\infty}\le2n\) and \(\|Y_a\|_\infty\le\sqrt n\) give the polynomial supremum bound.

Taylor’s formula in the finitely many coordinates of \(F\) gives \[n^{1/3}D_iF(x)=\sum_aF_{,a}(x)v_{ia}+r_i, \qquad \sum_i|r_i|^2=o(1),\] uniformly in the spin. The discrete product rule then writes the scaled slope of (62) as \(\sum_aF_{,a}(x)A_a^Q\) plus this remainder and terms \((D_iF_{,a})(x)V_a(X^{(i)})\). Their coefficients satisfy \[|D_iF_{,a}(x)|^2\le Cn^{-2/3}\sum_bv_{ib}^2.\] The microscopic potentials need not be independent of these coefficients. Conditionally on all spins except \(i\), put \(p_i(\xi)=(1+\xi t_i)/2\). The identity \[\frac{w_ip_i(\xi)}{p_i(-\xi)}=(1+\xi t_i)^2\le4\] implies \(\mu[w_i|V_a(X^{(i)})|^2]\le4\mu[|V_a(X)|^2]\). Summing over \(i\) and using the unit norms of the finitely many \(v_b\) gives \[\left\|n^{1/3}Df_{n,Q}-\sum_aF_{,a}(x)A_a^Q\right\|_{w,n}\to0.\] Equation (44), with multiplier \(F_{,a}F_{,a'}\), now proves the energy limit. ◻

Proof. Choose a smooth compact cylinder \(F\) with \(\Pi[\|\nabla F\|^2]>0\), possible because finite-coordinate marginals have full support. For each fixed \(j\), apply Proposition 25 to \(f_{n,Q_j}\). Its limiting gradient is \(\nabla F\) by Proposition 22, and the constant in the coarse bound (37) is independent of the fixed polynomial index. Lemma 30 therefore yields \[\Pi[\|\nabla F\|^2] \le C\left(\int Q_j^2\,d\omega\right)\Pi[\|\nabla F\|^2].\] Let \(j\to\infty\) to obtain \(1\le Cc_*\). Only one fixed-degree sequence is used at a time. There is no need to impose a common polynomial supremum exponent on a sequence whose degree increases with \(n\). ◻

Proof. Conditional single-site integration gives the exact formula \[D_w^*(\psi A_a^Q) =\sum_i(X_i-t_i)(E_i\psi)(A_a^Q)_i.\] Replacing \(E_i\psi\) by \(\psi\) gives \(\psi HQ(H)Y_a\). The cylinder Lipschitz estimate is \(\sum_i|E_i\psi-\psi|^2\le Cn^{-2/3}\) pointwise. Since \((A_a^Q)_i\) is independent of \(X_i\), Cauchy–Schwarz gives \[\begin{split} &\left\|\sum_i(X_i-t_i)(E_i\psi-\psi)(A_a^Q)_i\right\|_2^2\\ &\quad\le Cn^{-2/3} \mu\!\left[\sum_i(X_i-t_i)^2|(A_a^Q)_i|^2\right] =Cn^{-2/3}\|A_a^Q\|_{w,n}^2=o(1). \end{split}\] Finally, \(\|HQ(H)Y_a\|_2^2=\int\lambda Q^2\,d\omega_{a,n}\). ◻

Proof. Only the lower bound remains. Pass to a subsequence attaining the lower limit of the scaled energies. Proposition 25 gives the weak gradient \(G=\nabla F\) and its coarse bound. For finitely many smooth compact cylinders \(\psi_a\), put \(Z_{n,j}=\sum_a\psi_a(x)A_a^{Q_j}\). Equation (44) gives \[\lim_{j\to\infty}\lim_{n\to\infty}\|Z_{n,j}\|_{w,n}^2 =c_*\sum_a\Pi[\psi_a^2].\] We identify its pairing with \(n^{1/3}Df_n\).

Fix \(q>0\) small and use the logarithmic averages over \([q,2q]\) in Lemma 24. By (36), \(r_{n,q}=n^{1/3}(f_n-\overline{S f_n}_q)\) has \(L^2\) norm at most \(C/q\). Lemma 32 bounds the limiting absolute value of its slope pairing against \(\psi_aA_a^{Q_j}\) by \[\frac{C\|\psi_a\|_\infty}{q} \left(\int\lambda Q_j^2\,d\omega\right)^{1/2}.\] This vanishes when \(j\to\infty\) at fixed \(q\), after \(n\to\infty\).

For the smoothed part, the second estimate of (36) replaces \(n^{1/3}D\overline{S f_n}_q\) by \(\overline{J^{f_n}}_q\), with pairing error at most \(Cq\limsup_n\|Z_{n,j}\|_{w,n}\). This vector belongs to \(\mathcal H_{2q}\) and has bounded unweighted \(L^2\) norm. For any fixed polynomial index \(J\), (45) and weak convergence of the averaged slopes therefore give \[\mu\!\left[\psi_a(x)\overline{J^{f_n}}_q\cdot wA_a^{Q_J}\right] \longrightarrow \left(\int Q_J\,d\omega\right)\Pi[\psi_aG_a]\] as \(n\to\infty\) and then \(q\downarrow0\) along the gradient extraction.

The last step fixes the polynomial before taking \(q\downarrow0\), whereas the residual estimate takes \(j\to\infty\) first. The following bound reconciles these orders without uniform graph estimates in the degree: \[\limsup_{n\to\infty} \|w(A_a^{Q_j}-A_a^{Q_J})\|_{L^2(\mu;\mathbb R^n)} \le\left(\int(Q_j-Q_J)^2\,d\omega\right)^{1/2}.\] It follows from \(0\le w_i\le1\) and (44). For fixed \(J\), its right side tends as \(j\to\infty\) to \(\|\mathbf 1_{\{0\}}-Q_J\|_{L^2(\omega)}\), and then tends to zero as \(J\to\infty\). Boundedness of \(\overline{J^{f_n}}_q\) makes this estimate uniform in the cap scale. All these bounds may be expressed with upper and lower limits, so no prior convergence of the raw slope pairings is required. We first take the finite-size limit, then send \(j\to\infty\) at fixed \(q\) to remove the residual, approximate the remaining test by a large fixed \(J\), send \(q\downarrow0\), and finally send \(J\to\infty\). The limiting cross pairing is \(c_*\sum_a\Pi[\psi_aG_a]\).

Apply the elementary inequality \[\|n^{1/3}Df_n\|_{w,n}^2 \ge2\langle n^{1/3}Df_n,Z_{n,j}\rangle_{w,n} -\|Z_{n,j}\|_{w,n}^2\] and these iterated estimates to obtain \[\liminf_nT\mathcal E(f_n) \ge2c_*\sum_a\Pi[\psi_aG_a]-c_*\sum_a\Pi[\psi_a^2].\] Finite vectors of smooth compact cylinders are dense in \(L^2(\Pi;\ell^2)\). Optimizing proves (63). The remaining claims follow from Lemmas 30 and 31. ◻

Proof. The posterior means of \(f\) and \(X\) are martingales as the observation precision increases. If \(c<p\leq2c\), orthogonality of martingale increments bounds the change in their pairing by \[n^{-1/3} \big(Q_\mu|F_p-F_c|^2\big)^{1/2} \big(Q_\mu|m_p-m_c|^2\big)^{1/2} \leq Cn^{-2/3}c^{-1} \big(Q_\mu|m_p|^2\big)^{1/2}+o(1).\] Here (24) bounds the first increment by \(Cn^{-1/3}c^{-1}\); its exponentially small remainder can be summed over at most \(n\) components with polynomially bounded supremum. The moment estimate (19), with \(M_p=(np)^{1/2}\), makes the last bound of order \(n^{-1/6}p^{-1/2}\), apart from tight constants. On a geometric list beginning at \(r\), these deterministic weights sum to \(O(n^{-\delta/2})\).

For completeness, no simultaneous bound on the maximum of all these random constants is required. Restrict first to the common disorder events on which (24) holds with a fixed constant. Average the weighted sum over the biased rotation law, use (19), and then use \(L_\circ\geq1/2\) and Markov’s inequality to return to ordinary disorder probability. The probability of the discarded disorder events can be made arbitrarily small.

At the fixed upper endpoint \(p_*\), the difference from the original pairing is bounded directly using (24) and \(|X|=\sqrt n\). It is \(O(n^{-1/6})\). Combining this endpoint estimate with the geometric sum proves the lemma. All inequalities use vector norms, so their constants do not grow with the vector dimension. ◻

Proof of Proposition 34. Let \(B_{n,>\ell}(z)\) denote the tail trace in the proposition. If it is at least \(\epsilon\), its normalized optimizer is \[f=\frac{(z+TH)^{-1}(x_a)_{a>\ell}}{\sqrt{B_{n,>\ell}(z)}}.\] It has \(\|f\|_{z,n}=1\) and \(\|f\|_\infty\leq n^{1/6}/(z\sqrt\epsilon)\), since \(|(x_a)_{a>\ell}|\leq n^{1/6}\) pointwise. It is therefore an admissible test for the uniform estimates above. Lemma 36 and the angular calculation imply that its pairing with the source vanishes in probability in the double limit \(n\to\infty\), then \(\ell\to\infty\). That pairing equals \(\sqrt{B_{n,>\ell}(z)}\). In the localized form just described, Markov’s inequality gives, for each fixed \(\epsilon>0\) and every \(\ell\geq L_\eta\), \[\limsup_{n\to\infty} \mathbb P\{B_{n,>\ell}(z)>\epsilon\} \leq\eta+C_\eta\epsilon^{-1}s_\ell.\] Letting \(\ell\to\infty\) and then \(\eta\downarrow0\) proves the asserted tail estimate.

Combine it with Lemma 35. The tail estimate and tightness of one sufficiently long fixed-coordinate sum first give tightness of the full \(B_n(z)\). Each limiting finite partial sum is therefore bounded by the same tightness bounds, uniformly in its length: apply convergence of that partial sum and its pointwise bound by \(B_n(z)\). Monotonicity now shows that the limiting series is finite almost surely. The tail estimate then proves full trace convergence, jointly at finitely many parameters. Finiteness at all positive \(z\) follows first at positive rationals and then everywhere by resolvent monotonicity. ◻

Proof. Test the feature against a vector \(f\) with \(\|f\|_{z,n}\leq1\) and a polynomial supremum bound. By (24), replacing \(f\) by \(F_r\) in this pairing costs \(O(n^{-\delta})+o(1)\): the feature has norm one and \(Q_\mu|f-F_r|^2\leq Cr^{-2}n^{-2/3}+o(1)\). The remaining pairing uses the conditional mean \(M_r^{[2]}=\mathbb E[XX^t/n\mid h_r]\). If \(X,X'\) are independent posterior replicas, then \[|M_r^{[2]}|_{\rm HS}^2 =\mathbb E\left[\left.\left(\frac{X\cdot X'}n\right)^2 \right|h_r\right].\] The pair-tail estimate following (17), the annulus estimate (15), and the biased-to-quenched transfer used for (19) give \[\mathbf E|M_r^{[2]}|_{\rm HS}^2 \leq Cr^2+C\exp(-n^\xi), \qquad Q_\mu|M_r^{[2]}|_{\rm HS}^2=o_{\mathbb P}(1).\] For the first inequality, split posterior overlaps at \(C'r\), and on the complementary event use the likelihood-threshold argument from (19). The second follows by \(L_\circ\) comparison and Markov’s inequality. The \(n^2\) matrix components cause no loss in (24), because their exponentially small remainders remain negligible after summation. The posterior Cauchy–Schwarz bound now makes the pairing vanish uniformly. As in the preceding proof, whenever the trace exceeds a fixed \(\epsilon\), the normalized resolvent optimizer has a fixed polynomial supremum bound; in this case even a constant bound suffices. The variational formula proves (67). ◻

Proof. Let \(Z\) denote the partition function with uniform cube probability as its reference measure. The moment generating function \(m_t(v)=\mathbb E_t e^{vJ_{12}}\) satisfies \[m_t(v)\le e^{v^2/2},\qquad \log m_t(n^{-1/2})\ge\frac1{2n}-\frac C{n^2},\] with constants independent of \(t\). If \(Z_A^{(2)}\) is the two-replica partition function restricted to an event \(A\) depending on \(\rho\), independence of the entries therefore gives \[ \frac{\mathbb E_t Z_A^{(2)}}{(\mathbb E_tZ)^2} \le C\,\mathbb E_{\mathrm{cube}}\!\left[ e^{n\rho^2/2}\mathbf1_A\right]. \tag{70}\] Here the expectation on the right is over two independent uniform cube spins. Indeed the pair exponent uses \((X_iX_j+X_i'X_j')^2=2+2X_iX_i'X_jX_j'\); summing over \(i<j\) produces \(n\rho^2/2\) after division by the squared first moment, with only a bounded factor remaining.

Put \[I(r)=\frac{(1+r)\log(1+r)+(1-r)\log(1-r)}2, \qquad -1\le r\le1,\] with \(0\log0=0\). The binomial bound \(\mathbb P_{\mathrm{cube}}(\rho=r)\le e^{-nI(r)}\) and the fact that there are \(n+1\) possible overlaps show that the right side of (70) is at most \(C(n+1)\). On \(A=\{|\rho|\ge\epsilon\}\) it is at most \[ C(n+1)e^{-n\kappa_\epsilon},\qquad \kappa_\epsilon=I(\epsilon)-\epsilon^2/2>0. \tag{71}\] For the strict inequality, \(I'(r)-r=\operatorname{arctanh}(r)-r>0\) for \(0<r<1\), and continuity handles \(r=1\).

We next obtain a lower bound for the quenched partition function. The unrestricted second-moment estimate and Paley–Zygmund imply \[\mathbb P_t\{Z\ge\tfrac12\mathbb E_t Z\}\ge\frac c{n+1}.\] Clip every entry to \([-\log n,\log n]\), and let \(Z_L\) be the resulting partition function. The probability that any entry changes is at most \(Cn^2e^{-(\log n)^2/2}\), uniformly in \(t\). Changing one clipped entry changes \(\log Z_L\) by at most \(2\log n/\sqrt n\). Bounded differences hence gives, for each \(\eta>0\), \[\mathbb P_t\left\{ |\log Z_L-\mathbb E_t\log Z_L|>\eta n \right\} \le2\exp\!\left(-\frac{c\eta^2n}{(\log n)^2}\right).\] This upper bound is smaller than \(c/(2(n+1))\) for large \(n\). The preceding Paley–Zygmund event thus forces \(\mathbb E_t\log Z_L\ge\log\mathbb E_tZ-\eta n\) for all sufficiently large \(n\), after adjusting \(\eta\) by a fixed factor. Concentration once more shows that, for every fixed \(\eta>0\), \[ \sup_t\mathbb P_t\{ Z<e^{-\eta n}\mathbb E_tZ\}\longrightarrow0. \tag{72}\] Use (71) and Markov’s inequality to bound \(Z_A^{(2)}\le e^{-3\kappa_\epsilon n/4}(\mathbb E_tZ)^2\) with probability tending to one. On the event in (72) with \(\eta=\kappa_\epsilon/8\), division by \(Z^2\) proves (69) with \(a_\epsilon=\kappa_\epsilon/2\).

For the last assertion, an exact identity is \[ \frac1{\binom n2}\sum_{i<j}\mu[pX_iX_j]^2 =\frac{n\mu^{\otimes2}[p(X)p(X')\rho^2]-(\mu p)^2}{n-1}. \tag{73}\] On the event supplied by (69), the replica integral on the right is at most \(\epsilon^2+C^2n^{2K}e^{-a_\epsilon n}\). First let \(n\to\infty\) and then \(\epsilon\downarrow0\). Cauchy–Schwarz gives the asserted average of absolute correlations. This average is bounded by \(\mu p\le1\), which also gives the uniform convergence in expectation. ◻

Proof. For a site \(k\notin e\), the energy multiplier in \(M(u)\) is \(1+uq\). At an endpoint \(k\in e\), let \(s\) be the other spin in the pair. Its conditional mean changes from \(t_k\) to \((t_k+us)/(1+ust_k)\), while the marginal measure on \(X_{-k}\) acquires the factor \(1+ust_k\). Thus its energy term has the multiplier \[ \frac{1-u^2}{1+ust_k} \tag{79}\] relative to \(\mu[w_kD_kf\cdot D_kg]\). Since \(|t_k|\le1\), these formulas show that \(\|M_a\|_{M\to M^*}\le C^a\) and, for \(a\ge2\), that \(M_a\) uses only the two endpoint energies. In particular, \[ \begin{aligned} \|M_aY\|_*&\le C^a d\sqrt{\Gamma_e},& |M_a(Y,Y)|&\le C^a d^2\Gamma_e\quad(a\ge2),\\ \|M_1Y\|_*&\le C\sqrt B. \end{aligned} \tag{80}\]

Write \(E_kg=\mu[g\mid X_{-k}]\). Testing the equation \(MY=b_0\) against \(qg\) gives the exact identity \[ (b_1-M_1Y)(g) =T\sum_{k\in e}\mu[w_kD_kY\cdot sE_kg]. \tag{81}\] To verify it, the mass terms and all sites outside \(e\) cancel. At an endpoint, \(D_k(qg)=s(g(X_k=1)+g(X_k=-1))/2\), whereas the first coefficient in (79) is \(-st_k\). The remaining sum is \(sE_kg\). Cauchy–Schwarz and conditional Jensen now give \[|(b_1-M_1Y)(g)|^2 \le d^2\Gamma_e T\sum_{k\in e}\mu[w_k|E_kg|^2] \le\frac{2T}{z}d^2\Gamma_e\|g\|_M^2,\] which proves (76).

By (80), \(R_e^2\le C_z(1+\|b_1\|_*^2)\). For the second feature, \(\|qS^{[2]}\|_{L^2(\mu)}=1\), so \(\|b_1\|_*^2\le1/z\). For the first feature, use the stationary heat-bath process at the fixed disorder to write \[\begin{align*} \|b_1\|_*^2 &=n^{1/3}\int_0^\infty e^{-zs} \mathbb E_\mu\!\left[ q(X_0)q(X_{Ts})\rho(X_0,X_{Ts})\right]\,ds\\ &\le n^{1/3}\left(\frac1z \int_0^\infty e^{-zs} \mathbb E_\mu[\rho(X_0,X_{Ts})^2]\,ds\right)^{1/2} =n^{1/3}\sqrt{B_n^{[2]}(z)/z}. \end{align*}\] The last equality is the Hilbert–Schmidt feature identity \(\langle XX^{\mathsf t}/n,X'X'^{\mathsf t}/n\rangle=\rho(X,X')^2\). This proves (77). Finally, (76) and (75) give the second-power average. Multiplying it by the respective pointwise bound on \(R_e^2\) proves both fourth-power estimates. ◻

Proof. The first two terms in (82) are bounded by constants. For the second of them introduce the nonnegative function \[p_Y=\frac{z|Y|^2+T\sum_k w_k|D_kY|^2}{d^2}.\] Its mass is \(B/(1+B)\le1\). The Markov resolvent representation yields \(\|Y\|_\infty\le\|S\|_\infty/z\), with vector norm intended, and hence \(\|p_Y\|_\infty\le C_z n^2\) for the first feature and \(\|p_Y\|_\infty\le C_z n^{5/3}\) for the second. The endpoint formula gives \[\left|d^{-2}M_1(Y,Y)-\mu[qp_Y]\right|\le2\Gamma_e.\] Lemma 39, first with \(p=1\) and then with \(p=p_Y\), therefore shows that these first two terms have vanishing pair averages in expectation, uniformly in \(t\). Their fourth powers have the same property because the terms are bounded. The remaining terms are controlled by (75), Lemma 40, and Cauchy–Schwarz: \[\operatorname{avg}_e R_e\le C_z n^{-1/6},\qquad \operatorname{avg}_e\sqrt{\Gamma_e}\le\sqrt{2/n}.\] Combining these estimates proves (83) and (84). The deterministic bound follows from the same estimates and \(B_n^{[2]}(z)\le1/z\). ◻

Proof. We first consider one trace. Complexify the finite-dimensional form space, extending the forms and dual pairings complex bilinearly. The coefficient bounds for \(M(u)\) imply that \(M^{-1/2}M(u)M^{-1/2}\) differs from the identity in norm by \(O(|u|)\) on a fixed disk. Its inverse is therefore analytic and uniformly bounded on a smaller fixed disk. Put \(r(u)=b(u)-M(u)Y\) and \(A(u)=b(u)M(u)^{-1}b(u)\). Completing the quadratic expression around \(Y\) gives the algebraic identity \[A(u)=2b(u)Y-M(u)(Y,Y)+r(u)M(u)^{-1}r(u).\] The coefficient of \(u\) in \((A(u)-B)/d^2\) is \[\frac{M_1(Y,Y)+2(b_1-M_1Y)(Y)}{d^2},\] whose absolute value is at most \(Ch_e\). For higher coefficients, (80) gives \[\frac{\|r_1\|_*}{d}=R_e,\qquad \frac{\|r_a\|_*}{d}\le C^a\sqrt{\Gamma_e}\quad(a\ge2).\] The direct term \(-M_a(Y,Y)/d^2\) is at most \(C^a\Gamma_e\). Coefficients of the residual quadratic term are bounded, using the analytic inverse, by \(C^a(R_e^2+R_e\sqrt{\Gamma_e}+\Gamma_e)\le C^ah_e^2\). Thus every coefficient of order \(a\ge2\) in \((A(u)-B)/d^2\) is bounded by \(C^ah_e^2\).

The actual trace divides \(A(u)\) by the total mass \(1+u\mu q\). Thus, with \[\delta(u)=\frac{B(u)-B}{d^2},\] its linear coefficient is \(O(h_e)\) and its coefficients of order \(a\ge2\) are bounded by \(C^a(h_e+h_e^2)\). From (77) and (68), \(h_e\le C_z(1+n^{1/6})\) for the first feature; for the second it is bounded. Consequently \(|u|h_e=o(1)\) uniformly for \(|u|\le2\log n/\sqrt n\). Summing the coefficient bounds on this interval gives \[\sup|\delta(u)|=o(1),\qquad |\delta'(u)|\le Ch_e,\qquad |\delta^{(a)}(u)|\le C(h_e+h_e^2),\quad 2\le a\le4.\] Since \[\frac1{1+B(u)}=\frac1{d^2}\frac1{1+\delta(u)},\] the chain rule bounds its fourth derivative, and that of any fixed smooth test of it, by \(C(h_e+h_e^4)\). The derivatives of \(u=\tanh\theta\) through order four are bounded on the interval in question, so the same estimate holds for interaction displacement \(\theta\).

For several traces apply the multivariable chain rule and bound every mixed product using \(\mathfrak h_e+\mathfrak h_e^4\). Finally, at any real base point the coefficient argument applies at \(\theta=0\), where the denominator of the compactification is at least one. The bound \(h_e\le C(1+n^{1/6})\) supplies the claimed polynomial bounds, uniformly in that base point. ◻

Proof of Theorem 38. First use only second-feature traces in (88). For finitely many traces, \(\mathfrak h_e+\mathfrak h_e^4\le C_m\sum_a(h_{e,a}+h_{e,a}^4)\). Equation (83) therefore makes the mixture derivative tend to zero uniformly in \(t\). In particular, for one \(z\), choose the test \(F(v)=1-v\). The Gaussian convergence (67) and the bound \(B_n^{[2]}(z)\le1/z\) imply \[\sup_t\mathbb E_t\frac{B_n^{[2]}(z)}{1+B_n^{[2]}(z)}\longrightarrow0.\] Since \(b\le(1+1/z)b/(1+b)\) for \(0\le b\le1/z\), this proves the first assertion of the theorem and, by Jensen, \(\sup_t\mathbb E_t\sqrt{B_n^{[2]}(z)}\to0\).

Now use first-feature traces. Equation (84) makes their right side in (88) tend to zero uniformly in \(t\). Integrating from \(t=0\) to \(t=1\) shows equality of the limiting expectations of every smooth test of finitely many compactified traces. Such tests determine probability laws on \([0,1]^m\). By Proposition 34, the Gaussian trace vector converges to the finite vector in (64); its compactification therefore has all coordinates strictly positive almost surely. The continuous mapping theorem for the inverse map \(v\mapsto v^{-1}-1\) recovers the same joint limit for the sign traces. ◻

Proof. Set \(d\eta_n=(1+\lambda)^{-1}d\nu_n\) and regard \(\eta_n\) as a finite measure on the compact space \(K=[0,\infty]\), giving no mass to infinity. Its total mass is the Stieltjes transform at \(z=1\), and is therefore tight. Finite measures on \(K\) with bounded total mass form a compact set, so the laws of \(\eta_n\) are tight.

For \(z>0\) the function \[k_z(\lambda)=\frac{1+\lambda}{z+\lambda},\qquad k_z(\infty)=1,\] is continuous on \(K\). A countable dense family of these functions, including \(k_1=1\), determines a finite measure even when that measure has mass at infinity. Indeed their closed linear span contains \(1\) and \(r_z(\lambda)=(z+\lambda)^{-1}\), with \(r_z(\infty)=0\). The partial-fraction identity \(r_zr_w=(r_z-r_w)/(w-z)\), followed by uniform limits when \(w=z\), shows that this span is an algebra. It separates the points of \(K\), so it is dense in \(C(K)\). Consequently every subsequential law is the law of \(d\eta=(1+\lambda)^{-1}d\nu\), with no mass at infinity. Here joint convergence on the countable determining family identifies the law of the random measure, rather than only its mean.

For \(s>0\), define \(h_s(\lambda)=(1+\lambda)e^{-s\lambda}\) and \(h_s(\infty)=0\). On each compact interval \(a\leq s\leq b\), \(a>0\), the map \(s\mapsto h_s\) is continuous in the uniform norm on \(K\). Thus weak convergence of finite measures on \(K\) gives convergence of their \(h_s\) integrals uniformly on \([a,b]\): a finite uniform-norm net of the functions \(h_s\) reduces the assertion to finitely many continuous tests, while weak convergence bounds the total masses. The resulting map from measures to \(C_{\rm loc}((0,\infty))\) is continuous. The continuous mapping theorem proves the claim. ◻

Proof. For a fixed edge configuration \(g\), let \(E_g\) be the spectral resolution of \(c_*H_\Pi\) and put \[\nu_g(I)=\sum_{a\geq1}\langle x_a,E_g(I)x_a\rangle_\Pi.\] This countable sum defines a nonnegative measure. By Proposition 34, its division by \(1+\lambda\) has finite mass \(B_g(1)\) almost surely. It follows in particular that the series in (10) defines a continuous function \(\mathcal A_g\) on \((0,\infty)\).

These constructions are measurable in \(g\). The density prescription for \(\Pi\) makes each cylinder integral measurable. Resolvent pairings can then be computed by the variational supremum over a countable family of smooth compact cylinders. The family can be chosen independently of \(g\) by ordinary approximation of a function and its first derivatives on finite-dimensional compact sets; it is a form core by Proposition 22. Thus the resolvent pairings, the divided spectral measure determined by them, and \(\mathcal A_g\) are measurable.

Apply Lemma 43 to the finite-dimensional trace convergence in Proposition 34 and Theorem 38. For either entry law, \[\operatorname{Law}(A_{n,J})\Longrightarrow Q_{\rm crit}:=\operatorname{Law}(\mathcal A_g) \quad\text{in }C_{\rm loc}((0,\infty)).\] The common coefficient is the deterministic \(c_*\) of Theorem 33; no further change of time is involved. ◻

Proof. We prove the claim by finite-dimensional GOE density ratios. Let \(\mathsf P_n\) denote the ordered GOE edge law in dimension \(n\). When a sequence-valued argument is needed, append the values \(g_{n,n}+j\), \(j\geq1\), after its final coordinate. This convention does not affect any fixed coordinate for large \(n\), so \(\mathsf P_n\Longrightarrow\mathsf P\) in the product topology; it also commutes with \(T_h\) whenever \(n>m\). In the variables \(g_{a,n}=n^{2/3}(2-\lambda_a)\), the ordered density is proportional to \[\exp\left\{-\frac n4\sum_{a=1}^n (2-n^{-2/3}g_{a,n})^2\right\} \prod_{i<j}(g_{j,n}-g_{i,n}).\] Translation of the first \(m\) variables has Jacobian one. Its Gaussian log-density ratio is \(n^{1/3}\sum_{k\leq m}h_k+O(1)\) on sets where the first \(m\) coordinates are bounded. For a fixed \(k\leq m\), the spectral estimates of Section 2 give, tightly, \[\sum_{m<i\leq n}\frac1{g_{i,n}-g_{k,n}}=n^{1/3}+O(1), \qquad \sum_{m<i\leq n}\frac1{(g_{i,n}-g_{k,n})^2}=O(1).\] To see the first assertion, compare the reciprocals with \((g_{i,n}+b_n)^{-1}\), where \(g_{1,n}+b_n\geq1\). Their difference is summable by the square-tail bound, while \(n^{1/3}-\sum_i(g_{i,n}+b_n)^{-1}\) is tight. Tight separation of the first finitely many eigenvalues controls the remaining denominators.

The logarithm of the Vandermonde ratio involving indices \(k\leq m<i\) contains \[\sum_{m<i\leq n}\log\left(1-\frac{h_k}{g_{i,n}-g_{k,n}}\right).\] Beyond a sufficiently large fixed index, the fractions have absolute value at most \(1/2\). Taylor’s inequality and the preceding two bounds show that this sum is \(-h_kn^{1/3}+O(1)\). The finitely many remaining factors, including pairs within the first \(m\) coordinates, are bounded away from zero after imposing positive margins on the old and shifted gaps. The linear terms cancel those of the Gaussian factor. Consequently the new-to-old density ratio has a positive lower bound on sets of probability arbitrarily close to one, after any fixed localization inside the strict-order event.

Here is the measure-theoretic direction of the resulting implication. Let \(\chi\) be a continuous function of finitely many leading coordinates, with \(0\leq\chi\leq1\), supported where the shifted order has a positive margin and the leading coordinates are bounded. For every \(\epsilon>0\), the density-ratio bound supplies \(c_\epsilon>0\) such that, for every bounded nonnegative continuous function \(f\) of the edge sequence, \[\mathsf P_n(f)\geq c_\epsilon \left(\mathsf P_n[\chi\,f\circ T_h] -\epsilon\|f\|_\infty\right).\] The common spectral exceptional sets have probability at most \(\epsilon\) for all sufficiently large \(n\). Pass to the edge limit. For a compact subset of a \(\mathsf P\)-null set, take continuous tests between zero and one, equal to one on that compact subset and supported in open sets of arbitrarily small \(\mathsf P\)-mass. The inequality bounds its \((T_h)_*(\chi\mathsf P)\)-mass by \(\epsilon\). Inner regularity gives the same bound for every \(\mathsf P\)-null set. Letting \(\epsilon\downarrow0\) proves absolute continuity for this localized measure. Finally exhaust \(\mathcal O_h\) by such continuous localizations. This proves the lemma without requiring a uniform bound on the density ratio over the entire edge law. ◻

Proof. Lower \(g_1\) by \(t>0\), leaving the other points fixed, and denote the resulting sequence by \(g^{(t)}\). Choose the auxiliary parameter \(b\) large enough for both Gaussian prescriptions. Only \(l_1\) changes, and (6) shows \[D_{g^{(t)}}(b)-D_g(b) =-\big(l_1^{(t)}-l_1\big).\] At a common coordinate sequence \(x\), the finite centered-square sums in the two prescriptions differ by \(-(l_1^{(t)}-l_1)\). Thus, wherever these sums converge, \(S_{g^{(t)}}(x)=S_g(x)-(l_1^{(t)}-l_1)\). Consequently \[S_{g^{(t)}}(x)-D_{g^{(t)}}(b)=S_g(x)-D_g(b).\] The two renormalized constraints therefore coincide. Comparing the first Gaussian coordinate in their conditioning densities gives \[\frac{d\Pi_{g^{(t)}}}{d\Pi_g}(x) =\frac{\exp(t x_1^2/2)}{\Pi_g[\exp(t x_1^2/2)]}.\] The denominator is finite: increase \(b\) until \(g_1+b>t\) and use the bounded conditional first-coordinate density relative to the centered Gaussian of variance \((g_1+b)^{-1}\). The independence of the conditioned law from \(b\) then gives the displayed identity for \(\Pi_g\).

For a nonnegative random variable with unbounded support, exponential tilts with parameter tending to infinity have means tending to infinity. Indeed, for any \(M\), the mass below \(M\) after tilting is at most \(e^{-t/2}\Pi_g[x_1^2\leq M]/\Pi_g[x_1^2\geq M+1]\). It follows that \(v(g^{(t)})\to\infty\) as \(t\to\infty\).

Suppose that \(v(g)\) had an essential finite upper bound. By Lemma 45, this same bound would hold for \(v(g^{(t)})\) almost surely for every positive integer \(t\). Intersecting these countably many full-measure events contradicts the preceding divergence. Therefore \(v(g)\) is essentially unbounded. The lower bound (89) proves the same assertion for \(\mathcal A_g(s)\) at every fixed positive \(s\). ◻

Completion of the proof of Theorem 1. Proposition 44 proves convergence of the laws in \(C_{\rm loc}((0,\infty))\) for both entry distributions and identifies their common limit by (10). The same deterministic \(c_*>0\) occurs in both models. The spectral representation and the trivial form kernel prove nonnegativity, monotonicity, and almost-sure decay. Equation (89) excludes the zero function, and Proposition 46 establishes sample-to-sample randomness. ◻