Functional universality of critical SK autocorrelations

Proof. We use the eigenvalue-only spectral estimate of [22, 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

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,

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 (2.2) 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

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 2.1 then shows that the expressions in braces in (2.3) 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 \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\R. Thus the first m coordinates of the required conditional law have density

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

Given the tail, use polar disintegration of the first m Gaussian coordinates at the radius (2.5). This produces (2.6), 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 (2.4) 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 (2.6), 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 (2.4). 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 [21], 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 (2.8).□

Source statements and the radial consequence. The spectral, saddle, and density assertions are the conclusions of [21], 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 [21]. 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)|_{\cH_p}. At an intermediate evaluation ratio x\in[\gamma^2,\gamma], Lemma 16.2, “Evaluation with the induction extension,” in [21] 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

after decreasing s_0. This proves the second part of (2.11) 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

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\cH_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 \cH_p\subset\cH_q and

For h\in\cH_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\cH_p and both signs,

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 2.3. Before this conditioning, write

where the z_i are independent standard normals. On an active coordinate B_{p,i}=m-D_i', and direct expansion gives

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

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 2.3. 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 2.5, 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 [21, Section 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 [21, Section 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 (3.3) and its pair-tail assertion. The uncapped argument uses the corresponding uncapped spherical tail from [21, Section 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

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

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

and \|A\|\le C bound the diagonal terms by C|\rho|. It remains to show

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 [21, 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 (3.4), 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

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,

and \|m_r^{\mathrm{sp}}\|\le CM_r proves the second part of (3.4).□

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 2.3, 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 3.1 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 3.2. 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 (3.5). The spherical part follows by the same argument.

For the first estimate in (3.6), split according to L_r\ge n^{-0.12}. On this event (3.4) 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 (3.5) give CM_r^2n^{-0.06}. The field exceptions are negligible. Finally

Interpolation of the decaying second moment with an arbitrarily high fixed moment from (3.5) 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

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, (3.3) gives

Choose C large and use Lemma 3.1 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 (3.7). 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 3.1, 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}\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 (3.8).

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

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

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

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 3.5, and Lemma 3.1 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 (2.9), 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

Apply (3.7) in expectation and interpolate with the higher moments in (3.5) and the coordinate moments following Lemma 3.1. Hölder’s inequality on probability times logarithmic scale gives (3.9), 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 [21, Proposition 16.1]. On |\alpha_p|\le3s_0p^2/4, take the infimum of the quadratics

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}\Id, its saddle gradient, and a bounded density error. The static value and gradient errors from Lemma 2.7 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

The frozen Hessian is at most (1+Cs_0)p^{-2}\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

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

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

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

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)\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^+\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 (4.4).□

Proof. Condition on W_p, the physical field h_p, and its spectral coordinates, as in Lemma 3.7. 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

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 (4.6).□

Proof of Proposition 4.2. Let \widehat s_p be s_p^+ on the full coarse annulus and zero outside it. We now restrict to \mathcal G_n. Average (4.5) over c/(2\gamma)\le p\le c/\gamma with normalized logarithmic measure, and condition on W and the observation path through c. Combining (4.4) and (4.6), 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 (3.8). We obtain

where b>0, \mathcal F_c is the observation filtration, and \operatorname{err}_c\ge0 satisfies

uniformly at deterministic scales. To verify the exceptional terms in this assertion, first apply (3.8) to the parent scale integral before taking its conditional expectation. Its failures and those of (4.6) 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 (4.7), 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

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 3.1 removes the truncation. All bounds were uniform at deterministic scales, so Fubini’s theorem gives their integrated version.□

Proof. The continuation in [21, Section 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

This is [21, 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 [21, 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,

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

Apply these inequalities to the polynomially weighted ordinary exceptions in the terminal approximation estimates of [21, Section 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

The observation entropy identity [21, Lemma 19.2], with bounded precision rates on these intervals, now proves (4.8). 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

Every path likelihood and every posterior density ratio is bounded by fixed constants. Taylor’s formula for relative entropy gives, uniformly at every posterior,

Thus H(s) is comparable to \eta^2V_s(f). Integrating (4.8) 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

Rescaling f proves the lemma.□

Proof of Proposition 4.1. 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\Cov(X,f\mid h_p). Consequently

Integrate the exceptional probabilities in Proposition 4.2 in logarithmic scale, then apply Markov’s inequality and Lemma 2.5. 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\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

because

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 4.6 therefore gives (4.1). 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. Expand f and a in the normalized product Hermite basis. We index the basis by finitely supported multi-indices \beta and write the coefficients as f_\beta,a_\beta. Gaussian integration by parts gives

The formula is valid for weak derivatives: one first uses finite-coordinate Hermite tests with cutoffs, then removes the cutoffs by Cauchy–Schwarz.

Reduction to one chaos grade. Group the unknowns as pairs (f^{(d)},a^{(d-2)}), where the superscript denotes Hermite chaos order and negative orders are omitted. The derivative of f^{(d)} and the creation part of 2y_i a^{(d-2)} both have order d-1 and must be kept together. The remaining, annihilation part of 2y_i a^{(d)} has summed norm at most

Here \sup_i l_i<\infty follows from \sum_i l_i^2<\infty. It follows from (5.3) that each fixed-grade pair has a square-summable compensated derivative.

These pairs also approximate the full triple in its graph norm. To see this without separating divergent creation terms, multiply the grade-d pair by e^{-sd}, with s>0. Relative to damping the output of (5.3) by e^{-sd} in order d-1, the only discrepancy is the annihilation term, with multiplier e^{-s(d+2)}-e^{-sd}. Its squared norm is bounded by

which tends to zero. For fixed s, truncating the damped grades then converges in all three norms. It remains to approximate one grade.

The array represented by a fixed grade. Fix d and the finite set O of coordinates with odd multiplicity. The relevant coefficients of f have indices \mathbf1_O+2j, where j is a count vector with |j|=h=(d-|O|)/2; those of a use |j|=h-1. Different parity sets are orthogonal in the norms of the common derivative–creation terms, so finitely many such sets suffice for approximation. If h=0, only a finite-coordinate monomial is involved. Suppose henceforth that h\ge1.

Set

View u and v as symmetric arrays on h and h-1 ordered places. Give each place mass \mu_i=l_i^2. The coefficient norms are equivalent to the resulting product L^2 norms: the ordered-place multiplicities and the displayed factorials have ratios bounded above and below for fixed h and O. This comparison includes repeated indices.

For i\notin O, the identity

shows that the common term in (5.3) is exactly a multiple of u(j+e_i)-v(j). Its norm is therefore equivalent, again with fixed-grade constants, to the sum over places of

The finitely many indices in O contribute bounded operators in the coefficient norms, as does the annihilation term at this fixed grade. We have thus reduced the approximation problem to symmetric arrays with a common one-place trace v.

The sum of the one-place forms. For one place, take the base Hilbert space \mathfrak H=\ell^2(\mu). We include the scalar trace in the form norm by defining

The trace v is unique because \sum_i l_i=\infty. This is a densely defined closed form. Indeed, finite-support sequences belong to D(q); and a Cauchy sequence in the form norm has convergent traces and convergent deviations, whose coordinatewise limits identify the trace and deviation of its limit in \mathfrak H. Replacing the tail of u by its trace proves that eventually constant sequences are a form core.

On \mathfrak H^{\otimes h} take the sum of the h coordinate forms. By the slice description of each coordinate form, its norm on symmetric arrays is precisely, up to fixed factors, the norm of u, the norm of v, and the integrated deviations just obtained. In particular, it requires v to belong to the base (h-1)-place Hilbert space; it requires no further form regularity of v.

For completeness, let A_q be the nonnegative selfadjoint operator associated with q. The commuting projections \mathbf1_{[0,M]}(A_q)^{\otimes h} converge to the identity in the norm of the sum form. On their ranges the form is bounded by hM times the squared Hilbert norm. Algebraic tensors are dense there, and each factor can in turn be approximated in its one-place form norm by eventually constant sequences. This proves that such tensors are a core for the sum form. Symmetrizing preserves convergence and identifies the slice traces. We may consequently approximate (u,v) by arrays constant in each place outside a finite set. This argument does not require traces on intersections of two or more limiting faces.

Return to polynomials. The correspondence is already visible with one place and even parity. If u_i=v outside a finite set E, the polynomial

has P(S_y,y)=\sum_i u_i(y_i^2-l_i) and \partial_sP=v. Its compensated derivatives vanish outside E. Thus the constant tail of a coefficient array is represented by the centered square sum. For several places, its powers give the corresponding tensor coefficients, as the following calculation shows.

Choose a finite set E containing O and all exceptional indices of one of these approximating arrays. A symmetric array constant in each place outside E is determined by its value U_j for every count vector j=(j_i)_{i\in E} with |j|\le h: the remaining r=h-|j| places can have arbitrary indices outside E, including coincident ones. Define

The degree-d Hermite coefficients of M_j(y)S_{E^c}(S_y,y)^r have the prescribed counts j in E. For outside counts k with |k|=r, their coefficient is

This follows directly from the multinomial expansion, and so includes repeated outside indices. Multiplying this polynomial by U_j c(j)/(r!C_E(j)), and summing over the finitely many j, realizes exactly the desired top-grade array u. Here c(j) is restricted to E. Differentiation in s replaces S_{E^c}^r by rS_{E^c}^{r-1}. Its top coefficients have the same limiting one-place trace, since r(r-1)!=r!. They therefore realize v, also by trace uniqueness.

Every such polynomial pair has a square-summable compensated derivative, since differentiation after subtracting 2y_i\partial_s P leaves only the finitely many individual-coordinate derivatives. All its moments are finite; the centered sum S_y has exponential moments in a neighborhood of zero. Induction on the grade now removes the lower Hermite grades of these polynomial pairs. More explicitly, their lower grades satisfy the same summability identities and, by induction, already belong to the polynomial closure. Subtracting their approximants shows that the desired top-grade pair lies in that closure. The fixed-grade approximations and the preceding damping argument prove the lemma.□

Completion of the proof of Proposition 5.1. We can now use Lemma 5.2. Work in the uniform-angle times product Gaussian norm, equivalent to the localized conditioned norm. First project onto finitely many spherical harmonics in \vartheta; the projections converge in the first angular Sobolev norm and commute with the tail identities. Apply the lemma to each resulting coefficient. This yields polynomials in S_y and finitely many y_i, with smooth angular coefficients, converging in f, a, the compensated derivatives, and the angular derivative.

Multiply these approximants by a smooth cutoff of S_y supported in a slightly larger radius annulus and equal to one on the original support. Under multiplication by \chi(S_y) the scalar component becomes \chi a+\chi'f, while the compensated derivative becomes \chi G. This is continuous in all the norms under consideration. Finally substitute S_y=C-R^2. The approximants are now genuine cylinder functions in x, smooth because their first-block radii stay away from zero. Their Euclidean gradients have exactly the components in (5.2). Ordinary cutoffs in their finitely many other coordinates make them compactly supported, with convergence in the same norms. Removing the localizations proves equality of the cylinder closure and the angular graph.

It remains to identify the kernel. If G=0, the restriction of F to almost every finite initial-block sphere is constant, since the sphere is connected. Thus F is measurable, modulo null sets, with respect to the complement of every sufficiently large initial block. This completed tail field is trivial. To check this directly under \Pi, eliminate one fixed initial block. Its tail law has an integrable density with respect to product Gaussian measure. Approximating this density and its product with any bounded observable by cylinder functions shows that events beyond increasing indices are asymptotically independent of that observable, uniformly over the events. An event in every completed tail field is consequently independent of itself and has probability zero or one. Hence F is constant. Conversely constants satisfy the angular identities with G=0 and are in the closed domain by what we have proved.

The same characterization applies to x_a: it is square integrable by Proposition 2.3, and its angular weak gradient is e_a. This proves the last assertion.□

Proof. On the active cap, \nabla_{h_p}F_p=P_p\Cov(X,f\mid h_p). Conditional Cauchy–Schwarz therefore gives

Combining the covariance bound (2.11), the annulus estimate (3.1), and the variance estimate (4.1), and multiplying by the squared change-of-variable factor p^4n^{2/3}, proves (5.4). Conditional Jensen and \|B_p\|\le Cp^2 also bound J_p^f. The first line of (5.5) 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 ith 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

by (4.1). The degree-j contraction into the vector indexed by i has operator norm at most

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 (5.5) in the unweighted vector norm.□

Proof. Pass first to a subsequence on which the energy converges to its liminf. Equation (5.4) bounds the weak spectral vector limits in L^2(\Pi;\ell^2). To identify them, fix A and \psi as in (5.1). We will compare the angular pairing of the slopes at fixed cap scale to an integration by parts at the smallest scale r.

Define

where the rotation acts on spectral coordinates. By (3.2) and (5.4),

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 (3.2); compact support of the test, or its fourth-moment version, controls the coordinate factors.

Transport between two scales. Let r\le c<p\le2c, with all indices of A active at scale c. The increment covariance B_p-B_c is scalar on those indices. Rotation therefore commutes with the Gaussian convolution expressing the observation bridge. Differentiating its numerator and denominator gives the exact identity

In the first term, replacement of \psi(u_c) by \psi(u_p) costs O((cn^{1/3})^{-1})+o(1), using (3.2) and (5.4). The other term is bounded in absolute value by

Here \mathbb E[(F_p-F_c)^2]\le V_c(f) follows from the posterior martingale and conditional variance decomposition.

Starting at each p\in[q,2q], descend by factors of two, using r as the final scale if necessary. The coordinate errors sum to O((rn^{1/3})^{-1})=O(n^{-\delta}). For the score errors, (4.1) bounds (5.8), up to a fixed tight factor, by the root mean square of \mathcal R_A\phi_p divided by n^{1/3}p. Average over the starting p. In every resulting parent window, (3.9) and Cauchy–Schwarz apply, including a partial final window by enlarging it to its fixed-ratio buffer. Under the biased law the expected contribution of a window of scale Q is at most

The comparison through L_\circ\ge1/2 transfers this estimate to ordinary disorder integrals. Its sum is bounded by C_A(q^{\tau'/2}+k_r^{-\tau'/2}). Markov’s inequality therefore shows that the averaged T_p^A and T_r^A differ by a quantity tending to zero in probability as n\to\infty and then q\downarrow0. The estimate is independent of the choice of f within the stipulated size and energy bounds.

Integration by parts at the smallest scale. The observation density in the active h_r coordinates is proportional to

In u_r coordinates the quadratic coefficient in direction i is n^{2/3}r^4/B_{r,i}. Its pairwise differences are

for fixed leading indices. The common divergent part of the coefficient cancels in an angular derivative. Also \phi_r^{\rm sp} is radial on the cap, so that \mathcal R_A\phi_r=\mathcal R_A\log L_r. Its contribution to the integration-by-parts identity tends to zero by (3.6), the field-coordinate moments, and the same biased-to-quenched comparison. Cauchy–Schwarz applies because F_r is bounded in L^2 and the relevant score has vanishing L^2 norm after multiplication by the compact cylinder test.

For every function b(h_r), Q_\mu[F_r(h_r)b(h_r)]=Q_\mu[f(X)b(h_r)]. We may therefore use (3.2) in the remaining terms of the integration-by-parts identity and then the weak cylinder convergence of f. Equations (5.9) and (5.1) have the same limiting quadratic coefficient, since adding the scalar b has no angular effect. We obtain

Together with the preceding scale comparison, this proves that each weak limit G of the averaged slopes satisfies (5.1). Proposition 5.1 identifies it with \nabla F, and weak lower semicontinuity in (5.4) proves (5.6).

Finally, these arguments can be realized simultaneously for the countably many scales and tests needed in the sequel. Represent the bounds by tight random constants on the high-probability disorder sets, and include the error variables in the subsequential realization of Remark 2.2. The errors tend to zero along a sufficiently rapidly decreasing outer sequence of scales. On the resulting limit samples, weak Hilbert-space extraction applies to any of the particular sequences f_n under consideration. This establishes the stated quenched form of the proposition.□

Proof. Separate all coincidences among the site labels; the V remaining site vertices have distinct labels. Condition on the image under U of the chosen orthonormal basis of B. The residual map between the orthogonal complements is Haar. The Gram matrix of the first two displayed vectors tends to \left(\begin{smallmatrix}1&1\\1&2\end{smallmatrix}\right), while the marked axes are asymptotically orthonormal and orthogonal to these two vectors. Thus the basis can be chosen with converging coefficients. For bounds, an arbitrary orthonormal basis suffices.

At any fixed number of distinct rows, its image entries multiplied by \sqrt n converge jointly with all moments: the first column equals 1, and the remaining entries tend to independent standard normals. This follows by representing finitely many columns by independent Gaussian vectors projected off \mathbf1 and then orthonormalizing. At fixed graph and moment orders, products of vertex factors and row polynomials have a common polynomial growth bound. The bounds on higher row moments make these products uniformly integrable, so their expectations converge also when the continuous vertex functions are not polynomials. The vertex functions can therefore be retained in the calculation of residual Haar moments and integrated afterward.

We give the counting argument, using the orthogonal Haar integration formula [10, Section 3.2]. Suppose that splitting the edge endpoints between B and B^\perp leaves e edges and 2b residual endpoints. The finite-block endpoints contribute n^{-(e-b)} times a polynomial in the scaled row variables. Odd residual endpoint counts have zero expectation. For 2b endpoints the integration formula is a sum over a physical pairing \pi and a spectral pairing \sigma, whose coefficient has order

Here d(\pi,\sigma) is b minus the number of loops in the paired union. This order also follows directly by inverting the pairing Gram matrix: after removing the diagonal power of n, the off-diagonal entries have orders given by the pairing distances. The leading inverse terms are paths of minimum total distance.

We spell out the spectral wiring represented by a pairing. Split each matrix edge into its two endpoints, declaring each endpoint either finite-block or residual. The 2b residual endpoints are paired by \sigma; keep the original matrix edges as well. Each connected component of this endpoint diagram is then either a closed alternating cycle, or a path ending at two finite-block endpoints. Summing its spectral indices gives a trace or path of (6.8). Define t(\sigma) to be the number of closed components of this diagram. The notation t(\pi) means the number obtained from exactly this construction with \pi in place of \sigma.

Writing \mathsf P=\Id-UP_BU^t, a term of the conditional expectation therefore has the form

times its vertex factors and a polynomial in the scaled rows. Here v_\alpha is the physical vertex incident to endpoint \alpha, and M_c,M_p abbreviate the compressed products just defined. There are 2e-2b finite-block endpoints, each supplying a row entry n^{-1/2}\xi_{i,r}; this explains the factor n^{-(e-b)}.

The physical covariance \mathsf P is the projection off the image of B. Each pairing of unequal site vertices thus costs another factor n^{-1}, with polynomially bounded factors in the scaled rows. Let D be the number of such pairs. Each trace costs at most Cn; each path costs at most a constant. If the original graph has a components, then

Indeed, add the D unequal-site pairing connections to the graph, and let c\le a be the resulting number of components. Its cycle rank is e+D-V+c. The residual trace circuits are edge-disjoint circuits in this enlarged graph, so t(\pi)\le e+D-V+c\le e+D-V+a. Open paths ending in B add no circuits. A pairing switch creates at most one new circuit, which proves the second inequality.

Before summing site labels, the resulting bound is

It compensates for the site sum and the a normalizing factors. Every surviving term has a limit: the normalized trace and path factors converge by hypothesis, the row expectations converge by Gaussian orthogonalization, and the rescaled leading Haar coefficients converge by the Gram-matrix expansion.

The same count proves concentration. In a product of normalized sums, identifying vertices in two original components saves a power of n, as does a physical pairing across components. If only the spectral pairing crosses components, consider the first cross-component switch along a shortest pairing path. Before it, each residual circuit or open path lies inside one original component. Joining two circuits loses a circuit; joining paths, or a path and a circuit, cannot create a circuit. This switch therefore gains at most zero circuits, instead of the one allowed by (6.9), and again saves a power of n. Surviving terms remain inside the original components. Their row limits and leading Haar coefficients factor; for the latter, shortest pairing paths cannot join distinct paired components. Applying this factorization to even powers of centered sums proves convergence in every fixed L^p. The argument also gives bounds uniform over bounded spectral data.

We verify the graph representation for the heat-bath expressions. Repeatedly use the exact rule

A difference of an explicit spin identifies an existing site; a difference of a field function adds a factor W_{ij} joining the differentiating site to that field vertex. The induction invariant is that each summand obtained from one copy of Y_a has a connected set of site labels and exactly one marked factor. Initially its summand is v_{ja}X_j on one site. In an application of H, a newly summed site must differentiate an existing factor: it either becomes an existing site or joins one by a field edge. The factor X_i-t_i remains at that same site. Flip evaluations from the product rule introduce only field edges to sites already present. The final difference D_i has the same property, with i retained as the distinguished slope site. A term in which a new difference strikes no factor vanishes, so there is no unattached summation site. The slope inner product identifies the distinguished sites of two such pieces; a projected quadratic contraction joins them by a projection edge instead. Both are connected. Each piece carries one marked factor v_{ja}=n^{-1/2}(\sqrt n\,v_{ja}), so the two marks together supply the normalization 1/n.

The shifts of fields caused by flips may be Taylor expanded to a sufficiently large fixed order. Under conditional Haar, the off-diagonal entries and centered diagonal entries are at most n^{-0.49} outside a probability exponentially small in a positive power of n. Haar concentration on the complement of the prescribed spin direction has bounded Lipschitz constants, and the entry means are O(n^{-1}); a union bound gives the assertion. Each fixed expression has only polynomially many summands. A large enough fixed Taylor order therefore makes all remainders vanish in any prescribed fixed moment. Polynomial bounds absorb the exceptional event. For the diagonal in h_i=(U\Lambda y)_i-W_{ii}, Taylor-expand about \Tr\Lambda/n and retain the powers of W_{ii}-\Tr\Lambda/n as loop factors in the graph. Their edge matrix is \Lambda-(\Tr\Lambda/n)\Id; only the high-order Taylor remainder is discarded by the entrywise bound. This proves the required representation.

For these applications the edge matrices are powers of \Lambda, fixed spectral cutoffs, marked projections, and the scalar shifts just described. All hypotheses concerning (6.8) follow from the stated spectral inputs. Indeed, finite-rank removal changes a normalized trace by O(n^{-1}). Expanding each R=\Id-\sum_rb_rb_r^t in a path reduces it to finitely many products of ordinary mixed spectral inner products between y/\sqrt n, \Lambda y/\sqrt n, and the marked axes. Their limits are given by the two empirical spectral laws, the leading eigenvalue limits, and y_a/\sqrt n\to0. At fixed q, the cutoff endpoint 2-q^2 is a continuity point of the limiting laws.

Finally set R_q=P_q-P_{\rm mark} in the extra edge. Normalized traces containing R_q tend to zero as q\downarrow0, because the semicircle mass of [2-q^2,2] vanishes. Finite paths also vanish: R_q kills the marked axes and, for every fixed k,

Paths containing finite-block compressions reduce to these statements by expanding those projections. There are only finitely many contractions at each fixed order. The limit order is n\to\infty at fixed q, followed by q\downarrow0.□

Proof. We first separate the differences that strike a conditional mean from those that remain on f. The exact product rule is

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

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 (6.12) in the norm obtained by summing \mu[w_S|\cdot|^2] over S. For positive derivative orders,

For example, Cauchy’s formula on a fixed circle of radius less than \pi/2, applied to \sech^2z, proves this inequality: on that circle |\sech^2z|\le C\sech^2u uniformly for real u. The difference integral for D_A t_i consequently gives

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\sech^2u-\log\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

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:

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

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 ath 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 (6.15), 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

Its contribution is at most C_m\sqrt n\epsilon_n^2N_l(f)=o_{n;m}(1)N_l(f). We have proved

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

The constants depend only on K, once n is sufficiently large for the fixed m.

The conditional field under \nu is exactly

Since the first two derivatives of \log\sech^2u are bounded, the nonnegative deterministic quantities

satisfy |\delta_i|\le a_i and |D_j\delta_i|\le L_{ij} for i\ne j. They obey

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 (6.17) follows from |\delta_i|\le a_i=o(1) uniformly in i. Taylor’s formula for \tanh and \sech^2, now at the tilted fields, also gives

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

Use the version of (6.10) for \nu to write

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,

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 (6.18) and the same row square bounds. Consequently

Moreover |t_i-t_i'|\le Ca_iw_i, so pointwise Cauchy–Schwarz and \|a\|_2\le Cm show

These two bounds prove (6.17).

Apply (6.17) 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

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 (6.11). 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 (6.10) give

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, (6.19) implies

Define the symmetric square-root lift \rho by

Its 4jth absolute moment is bounded by the right side of (6.20). 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 6.1. For the Gaussian matrices in this section, removing the diagonal preserves a bounded operator norm and the entry bound used in Lemma 6.3, on events whose probability tends to one. The fixed-order contractions give moment convergence there. Corollary 6.4 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 (6.3) gives precisely those same moments. Finally the contraction and projection arguments preceding this subsection give (6.4) and (6.5) 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,

is bounded for fixed Q: the quotient extends to a polynomial at zero. Thus the correction in (6.22) 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

uniformly in the spin. The discrete product rule then writes the scaled slope of (6.22) as \sum_aF_{,a}(x)A_a^Q plus this remainder and terms (D_iF_{,a})(x)V_a(X^{(i)}). Their coefficients satisfy

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

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

Equation (6.4), 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 5.4 to f_{n,Q_j}. Its limiting gradient is \nabla F by Proposition 5.1, and the constant in the coarse bound (5.6) is independent of the fixed polynomial index. Lemma 6.5 therefore yields

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

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

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 5.4 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 (6.4) gives

We identify its pairing with n^{1/3}Df_n.

Fix q>0 small and use the logarithmic averages over [q,2q] in Lemma 5.3. By (5.5), r_{n,q}=n^{1/3}(f_n-\overline{S f_n}_q) has L^2 norm at most C/q. Lemma 6.7 bounds the limiting absolute value of its slope pairing against \psi_aA_a^{Q_j} by

This vanishes when j\to\infty at fixed q, after n\to\infty.

For the smoothed part, the second estimate of (5.5) 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, (6.5) and weak convergence of the averaged slopes therefore give

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:

It follows from 0\le w_i\le1 and (6.4). For fixed J, its right side tends as j\to\infty to \|\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

and these iterated estimates to obtain

Finite vectors of smooth compact cylinders are dense in L^2(\Pi;\ell^2). Optimizing proves (6.23). The remaining claims follow from Lemmas 6.5 and 6.6.□

Proof. Use the quenched subsequential realization from Section 2. For a fixed coordinate the source has bounded L^2 norm on the represented sequence, by the static moment convergence. Its resolvent optimizer therefore has bounded L^2 norm and bounded scaled energy. Markov contraction also gives a polynomial supremum bound: \|x_a\|_\infty\leq n^{1/6} and

The lower form bound (6.23), together with weak lower semicontinuity of the mass term, bounds the upper limit of the variational expression. Source pairings pass to the limit by first cutting off x_a, using static cylinder convergence, and then removing the cutoff with the fixed-coordinate moment bounds.

Conversely, the correctors (6.22) give the variational lower bound for every smooth compact cylinder test: first take n\to\infty at a fixed recovery polynomial Q_j, and then take j\to\infty. The polynomial supremum exponent may depend on j, which is harmless in this order of limits. The cylinder core in Proposition 5.1 extends it to the full form domain. The two variational bounds identify the limiting resolvent. The same argument uses one subsequence for the finitely many sources and parameters under consideration, proving the joint assertion.□

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

Here (4.1) 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 (3.5), 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 (4.1) holds with a fixed constant. Average the weighted sum over the biased rotation law, use (3.5), 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 (4.1) 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 7.1. Let B_{n,>\ell}(z) denote the tail trace in the proposition. If it is at least \epsilon, its normalized optimizer is

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 7.3 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,

Letting \ell\to\infty and then \eta\downarrow0 proves the asserted tail estimate.

Combine it with Lemma 7.2. 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 (4.1), 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

The pair-tail estimate following (3.3), the annulus estimate (3.1), and the biased-to-quenched transfer used for (3.5) give

For the first inequality, split posterior overlaps at C'r, and on the complementary event use the likelihood-threshold argument from (3.5). The second follows by L_\circ comparison and Markov’s inequality. The n^2 matrix components cause no loss in (4.1), 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 (7.4).□

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

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

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

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 (8.3) is at most C(n+1). On A=\{|\rho|\ge\epsilon\} it is at most

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

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,

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,

Use (8.4) 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 (8.5) with \eta=\kappa_\epsilon/8, division by Z^2 proves (8.2) with a_\epsilon=\kappa_\epsilon/2.

For the last assertion, an exact identity is

On the event supplied by (8.2), 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

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,

Write E_kg=\mu[g\mid X_{-k}]. Testing the equation MY=b_0 against qg gives the exact identity

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 (8.12) is -st_k. The remaining sum is sE_kg. Cauchy–Schwarz and conditional Jensen now give

which proves (8.9).

By (8.13), 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

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 (8.10). Finally, (8.9) and (8.8) 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 (8.15) are bounded by constants. For the second of them introduce the nonnegative function

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

Lemma 8.2, 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 (8.8), Lemma 8.3, and Cauchy–Schwarz:

Combining these estimates proves (8.16) and (8.17). 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

The coefficient of u in (A(u)-B)/d^2 is

whose absolute value is at most Ch_e. For higher coefficients, (8.13) gives

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

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 (8.10) and (8.1), 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

Since

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 8.1. First use only second-feature traces in (8.21). For finitely many traces, \mathfrak h_e+\mathfrak h_e^4\le C_m\sum_a(h_{e,a}+h_{e,a}^4). Equation (8.16) 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 (7.4) and the bound B_n^{[2]}(z)\le1/z imply

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 (8.17) makes their right side in (8.21) 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 7.1, the Gaussian trace vector converges to the finite vector in (7.1); 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

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

This countable sum defines a nonnegative measure. By Proposition 7.1, its division by 1+\lambda has finite mass B_g(1) almost surely. It follows in particular that the series in (2.7) 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 5.1. Thus the resolvent pairings, the divided spectral measure determined by them, and \mathcal A_g are measurable.

Apply Lemma 9.1 to the finite-dimensional trace convergence in Proposition 7.1 and Theorem 8.1. For either entry law,

The common coefficient is the deterministic c_* of Theorem 6.8; 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

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,

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

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,

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 (2.3) shows

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

The two renormalized constraints therefore coincide. Comparing the first Gaussian coordinate in their conditioning densities gives

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 9.3, 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 (9.1) proves the same assertion for \mathcal A_g(s) at every fixed positive s.□

Completion of the proof of Theorem 1.1. Proposition 9.2 proves convergence of the laws in C_{\rm loc}((0,\infty)) for both entry distributions and identifies their common limit by (2.7). 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 (9.1) excludes the zero function, and Proposition 9.4 establishes sample-to-sample randomness.□

Proof. Let K=\supp Q_{\rm crit} denote the closed topological support. Since C_{\rm loc}((0,\infty)) is separable, Q_{\rm crit}(K)=1. Consider all shifts of finitely many edge points by rational amounts, restricted to the event that the shifted points remain strictly ordered. There are countably many such shifts. By Lemma 9.3, outside a single null set, every valid shift g' of g gives a configuration for which the conditioned measure, the form identities, and (2.7) are defined, and for which \mathcal A_{g'} belongs to K. We fix such a configuration g, also satisfying the spectral bounds of Section 2. Choose and fix b with g_1+b>0, and put

In particular, l_a is strictly decreasing, \sum_a l_a=\infty, and l_a\le C a^{-2/3} for all sufficiently large a. All constants below may depend on this fixed configuration and on b.

Choosing the modified variances. We leave g_1 fixed and modify g_2,\ldots,g_m, along a sequence of integers m\to\infty. Let J=60 and define

Then A_m\to\infty and A_m=O(m^{1/3}). There are arbitrarily large m such that

Indeed, if the reverse strict inequality held eventually, the identity A_{m+1}=A_m+l_{m+1} would imply A_{m+1}>A_m(1+1/(2m)), and hence A_m\ge c m^{1/2}, contrary to the upper bound. Restrict henceforth to integers satisfying (A.1), and set M=A_m/(m-1). For all sufficiently large such integers,

Choose strictly decreasing positive numbers on indices 2,\ldots,m whose sum is A_m, with the first J-1 numbers close to 2M and the others close to M. This is possible by adjusting the latter group by a relative amount O(J/m) and then making small distinct perturbations of zero total sum. For large m these numbers are less than l_1 and greater than l_{m+1}, by (A.1). They therefore define an ordered modification of the edge points through g_i'=1/l_i'-b. Approximate each of the finitely many shifts g_i'-g_i by a rational number, sufficiently closely to retain strict order and all the strict comparisons just imposed. We continue to write g^{(m)} and l_a^{(m)} for the resulting configuration and variances. The approximation can also be chosen so that the following bounds hold, with constants independent of m:

Here D^{(m)} is the value of (2.3) for g^{(m)}. To check the first conclusion in (A.5), the finite-change identity from that equation gives

Before rational approximation our choice of sum makes V_m=1 exactly; afterwards we can require |V_m-1|\le m^{-1}. The square-sum bound follows from mM^2\le C m^{-1/3} and \sum_{a>m}l_a^2\le C m^{-1/3}. By the choice of the original configuration, each \mathcal A_{g^{(m)}} belongs to K.

Concentration after conditioning. Let \gamma_m be the product law of centered normals on indices a>1 with variances l_a^{(m)}, and let

This series converges in L^2(\gamma_m), and (A.5) implies S_m\to0 in probability under \gamma_m. Write \Pi_m=\Pi_{g^{(m)}}. Eliminating x_1 in the conditioning prescription shows that the a>1 marginal of \Pi_m has density

relative to \gamma_m. Under the full law, x_1^2=V_m-S_m, and the sign of x_1 is independent and symmetric.

We give bounds near the singularity in (A.6), because concentration under the product law alone would not suffice. Put t_m=m^{1/10}. For either sign, the centered Gaussian square-sum formula and (A.5) give

since t_m\max_{a>1}l_a^{(m)}\to0. Under either exponential tilt the coordinates remain independent normals, with variances l_a^{(m)}/(1\mp2t_m l_a^{(m)}), comparable to the original variances. The sum of the squares of any fixed four coordinates among 2,\ldots,J has density bounded by C/M: its characteristic function has absolute value at most C(1+M^2u^2)^{-1}, whose integral is O(M^{-1}). Convolution with the remaining coordinates does not increase this bound. The same assertion holds for the infinite sum by its L^2 convergence. Undoing the tilt and using (A.2)–(A.7), we obtain a density p_m of S_m satisfying

The factors h_m(v) are uniformly bounded on the complement of a fixed neighborhood of v=V_m. Near that point, which stays bounded away from zero, (A.8) supplies an exponentially small factor, while the singularity (V_m-v)^{-p/2} is integrable for every p<2. Consequently, concentration of S_m at zero gives

The same estimate, integrated over |S_m|>\varepsilon, shows that for each fixed \varepsilon>0 there are positive constants C_\varepsilon,c_\varepsilon such that

for all sufficiently large m. Integrating the negative tail of S_m also gives convergence of x_1^2 to 1 in every fixed moment. Finally, Hölder’s inequality applied to (A.9) yields, for each fixed p<\infty,

The first coordinate and the trace tail. Write H_m=H_{\Pi_m} and \mathcal E_m(F)=\int|\nabla F|^2\,d\Pi_m. Choose an odd smooth compactly supported function \phi on \R which equals 1 near 1 and -1 near -1. The preceding concentration and moment estimates imply

For 0\le s\le S, the spectral theorem gives

Semigroup contraction therefore shows that the first coordinate’s contribution to \mathcal A_{g^{(m)}}(s) tends to 1, uniformly for s in bounded intervals. Coordinates 2,\ldots,J contribute at most \sum_{a=2}^J\Pi_m[x_a^2]=O(M), which tends to zero. It remains to control the infinitely many coordinates beyond J. We use the angular integration-by-parts estimate of Section 7. Here the angular identity (5.1) is exact, so there is no likelihood-score error. The remaining task is to make the coefficient and inverse-moment bounds uniform in m.

Let R_0=\sum_{j=1}^Jx_j^2. We first establish the uniform inverse moments needed for angular integration by parts. By (A.9), Hölder’s inequality, and R_0\ge\sum_{j=2}^Jx_j^2,

The Gaussian inverse moment is finite because J-1=59>48. On \{x_1^2\ge1/2\} the quantity R_0^{-4} is bounded by 16; on the complement, Cauchy–Schwarz and (A.10) absorb the preceding polynomial bound. Thus

For a>J put

By (A.4), \delta_{aj}\ge(1-q)/l_a^{(m)}. The calculation in (7.3), now in the limiting coordinates, gives

Together with (A.11) and (A.12), these imply

Each rotation x_j\partial_a-x_a\partial_j preserves the square-sum constraint. Its derivative of the Gaussian log density is -\delta_{aj}x_ax_j, so the corresponding derivative along \mathcal V_a is exactly -x_a. Applying (5.1) with the coefficients defining \mathcal V_a therefore gives

for smooth cylinder tests. To justify the singular coefficients, multiply them first by \chi(R_0/\varepsilon), where \chi is smooth, zero below 1, and one above 2. Direct calculation gives

The last expression tends to zero in L^2(\Pi_m) by (A.11) and (A.12). At infinity we use a radial cutoff in a finite coordinate set containing 1,\ldots,J,a and all test coordinates. Every rotation in \mathcal V_a preserves this radius, so differentiating that cutoff produces no error. This proves (A.14) on the cylinder core, which then extends the identity to the form domain.

Fix z>0 and a finite set I\subset\{J+1,J+2,\ldots\}. Applying (A.14) to a vector of tests (f_a)_{a\in I} and using Cauchy–Schwarz and (A.13) yields

By the resolvent variational formula, followed by monotone convergence over finite I,

For every s_0>0 the scalar inequality e^{-s\lambda}\le C_{s_0,z}(z+\lambda)^{-1} holds for s\ge s_0 and \lambda\ge0. Hence (A.15) also makes the semigroup trace beyond J tend to zero uniformly on compact subsets of (0,\infty).

We have proved \mathcal A_{g^{(m)}}\to1 in C_{\rm loc}((0,\infty)). Each approximating function belongs to K, so closedness of K proves the proposition.□