Critical mixing in the Sherrington–Kirkpatrick model

Proof. The Gaussian–chi representation is the classical tridiagonal construction; see [18, Theorem 2.1 and Section 2.3] for the general beta-ensemble model. In the present normalization, successive orthogonal Gaussian elimination gives a tridiagonal matrix with the same spectrum as J. Its independent diagonal entries are N(0,2/n) and its adjacent entries are \chi_{n-i}/\sqrt n, 1\le i<n. To see the independence, rotate the first column below the diagonal onto the second coordinate. Orthogonal invariance leaves a fresh independent GOE matrix on the remaining coordinates, and the procedure can then be iterated.

Let A be the deterministic Jacobi matrix with zero diagonal and adjacent entries \sqrt{1-i/n}, and write the tridiagonal matrix as A+\Delta. Centered entries of \Delta are sub-Gaussian at scale n^{-1/2}. Gaussian norm concentration and the second moment of a chi variable give adjacent-entry means O((n(n-i))^{-1/2}). For

the positive walk expansion gives

Indeed the positive series

is a sum over returning walks. Their count divided by 2^j is O((j+1)^{-1/2}). A walk staying above i/2 gains e^{-cij/n} from the adjacent weights, so its contribution is bounded by

By reflection and the binomial tail bound, the relative count of walks that visit below i/2 is at most Ce^{-ci^2/(j+1)}. Their sum is at most C_p d^{1-2p}e^{-cdi}. Since nd^3\ge1, di\ge i/(nd^2); exponential decay in this last quantity bounds the sum by C_p d^{1-2p}(1+i/(nd^2))^{(1-2p)/2}, which is again C_pa_i^{1-2p}.

The perturbation is small in the resolvent metric already at d=n^{-1/3}\sqrt\ell:

The block-energy estimate below follows the partial-sum and summation-by-parts method used in [24, Proposition 7 and Lemmas 8–9]; see also [30, Lemma 5.6 and Section 6]. The resolvent precision needed here is verified explicitly. With b_i=\sqrt{1-i/n} and b_0=b_n=0, the Jacobi energy is

It controls

The identity 2\Delta_{i,i+1}u_iu_{i+1} =\Delta_{i,i+1}(u_i^2+u_{i+1}^2) -\Delta_{i,i+1}(u_{i+1}-u_i)^2 separates each noisy adjacent term into a diagonal contribution and a squared difference.

The maximum entry of \Delta is o(1), so the difference terms and the terms on the bulk half are harmless. Divide the first half into consecutive blocks of length comparable to a_i^{-1}, merging a short last block. The values of a_i are comparable within each block. Sub-Gaussian maximal bounds, applied separately to each shifted entry sequence, show that centered diagonal partial sums from the beginning of a block are at most \epsilon a_i, except with probability Ce^{-c\epsilon^2na_i^3} per block.

For a block B of length comparable to a_B^{-1}, summation by parts against a centered diagonal sequence \zeta_i with these partial sums gives

To see the scale, average the endpoint value of u^2 over the block and bound the variation of u^2 by Cauchy–Schwarz. Multiplication by the partial-sum bound \epsilon a_B then gives the two terms displayed. The first-half means are O(n^{-1}), a relative O((na_B^2)^{-1})=o(1) contribution to the potential term.

A dyadic shell contains O(na_B^3) blocks. At d=n^{-1/3}\sqrt\ell, the smallest value of na_B^3 is \ell^{3/2}. Choose \epsilon\downarrow0 with \epsilon^2\ell^{3/2}\gg\log\ell. The block counts and the exponents then grow geometrically on successive shells, so the exceptions are summable. Summing the block inequalities proves (5.3). Positivity of R^{-1}-\Delta at this value of d also places the upper edge below 2+n^{-2/3}\ell=2+o(d_0^2).

The relative perturbation estimate has placed the spectrum below the resolvent pole. The trace estimate requires more precision: its error must be small on the unnormalized trace scale. We first estimate the linear and quadratic perturbation terms, then identify the deterministic Jacobi trace through its Hermite equation.

Expanding the resolvent and using (5.3) gives

The quadratic remainder is nonnegative, so its expectation controls it by Markov’s inequality. Its centered-noise expectation is O(d^{-2}), by (5.2), Cauchy–Schwarz for neighboring entries, and n^{-1}\sum_i a_i^{-4}=O(d^{-2}). For the mean contribution, first use R\preceq Cd^{-2}I. On the bulk half the adjacent-entry means are O(n^{-1/2}) and the diagonals of R^2 are bounded; on the other half the means are O(n^{-1}) and the trace bound applies.

The linear term has standard deviation O(d^{-2}) and mean O(d^{-1}+1). These estimates extend to each fixed-ratio interval in d. In an eigenbasis of A, with \rho_i=(2+d^2-\lambda_i(A))^{-1}, the quadratic term is

It is nonnegative and decreases with d, regardless of the signs in \Delta. For the linear term L(d)=\Tr R^2\Delta,

The diagonal bounds give this logarithmic derivative the same O(d^{-2}) fluctuation scale as L(d) on a fixed-ratio interval. Integrating it controls the supremum there. Markov’s inequality for Q and the corresponding second-moment bound for L, at threshold (nd^3)^{1/4}d^{-2}, have summable failures over the dyadic intervals because nd^3 starts at \ell^3 and grows geometrically. Finally

which gives the stated uniform trace error.

It remains to identify the deterministic trace. Reversing the Jacobi indices in its principal-determinant recurrence gives monic Hermite polynomials. The characteristic polynomial P_n satisfies

Therefore b_A=P_n'/(nP_n) obeys

The right side is O((nd)^{-1}) by (5.2). Continuation from large z selects the smaller branch and gives b_A(2+d^2)=b_{\mathrm{sc}}(2+d^2)+O((nd^2)^{-1}). Together with the perturbation estimate this proves (5.1).

For a fixed integer p\ge2, take the (p-1)st finite difference at spacing hd^2, with \Delta_a f(u)=f(u+a)-f(u). The scalar identity is

For an eigenvalue v of J, take t=2-v. The upper-edge bound already proved in (5.1) gives t+d^2\ge d^2-o(d_0^2)\ge d^2/2 for all sufficiently large n, uniformly for d\ge d_0. Hence

The same comparison holds for semicircle depths t\ge0. For fixed h>0, (5.1) gives the semicircle limit for the finite difference with a relative error tending to zero. Letting h decrease after n tends to infinity now squeezes the pth power to its semicircle value.

For the edge counts at a shrinking scale, put t_i=(2-\lambda_i(J))/d^2 and consider the finite measure

To identify its limit, use the coordinate x=(1+t)^{-1}. The upper-edge bound places the pushed-forward measures on [0,1+o(1)], and for every fixed integer j\ge0 their moments are

The trace-power estimates include the total mass, corresponding to j=0. Polynomial approximation on a fixed compact interval therefore determines the weak limit of the pushed-forward finite measures. For d\to0, the semicircle change of variables identifies its pullback as

The interval [1,2] has positive limiting mass and boundary of mass zero. Since its weight (1+t)^{-2} is bounded above and below, it contains order nd^3 eigenvalues. This conclusion is uniform over shrinking scales: any violating sequence would have the same moment limits and hence the same interval-mass limit. The second-power trace bound gives at most Cnd^3 eigenvalues in any fixed multiple of the d^2 upper-edge window. For d in a fixed interval bounded away from zero, ordinary semicircle convergence gives the same conclusions with constants depending on that interval. Applying the same argument to -J gives the lower-edge bound.

For the interlacing assertion, the two eigenvalue counting functions differ by at most a fixed integer. Write z for the evaluation point and let (z-cd^2,z+cd^2) be free of both spectra. The function (z-v)^{-p}, defined outside this gap, has an extension across the gap of total variation O_p(d^{-2p}). Integration by parts against the difference of the counting functions therefore bounds the raw trace difference by O_p(d^{-2p}). This argument also applies when z is interior to the full spectral range. For p\ge2 the error is relative O((nd^3)^{-1}); for p=1 it is o(\sqrt{n/d}), the unnormalized error scale in (5.1).□

Proof. After diagonalization and normalization it is enough to consider

Argue by subsequences. If an a_i stays bounded below, write the sum as a_i(g_i^2-1)+2b_ig_i+W. Cantelli’s inequality gives a positive lower bound for \Pprob(W\le a_i/2), and Chebyshev’s inequality removes W<-C after a fixed large C is chosen. On the remaining event the one-coordinate target lies in [-a_i/2,C], a compact interval strictly above the minimum -a_i-b_i^2/a_i of the quadratic. Its density there has a positive lower bound. This proves the claim along the subsequence.

If \max_i a_i\to0 but \|b\| stays bounded below, write g=z\,b/\|b\|+g_\perp. The coefficient of z^2 tends to zero, and the random correction to its linear coefficient has variance O((\max_i a_i)^2). Thus that linear coefficient stays bounded away from zero with high probability. The remaining term has bounded second moment. Restrict it to a fixed compact interval and the cross coefficient to a small interval; the change of variables in z then has a bounded Jacobian and a preimage in a fixed compact interval, where the Gaussian density is bounded below. In the remaining case Fourier inversion gives a local Gaussian limit. The quadratic-product formula supplies the integrable bound

The same proof permits an o(1) displacement of the target in standard-deviation units.□

Proof. The Gaussian shell formula gives

The last term is O(\log n) in the direction needed for the lower bound. Replacing \lambda^0 by 2+Cd_0^2 in the first two terms increases them by only O(nd_0^3), by convexity and (5.1). At this new argument, concavity of log determinant gives the upper bound

The linear correction has mean O(1) and standard deviation O(\sqrt{\log n}), by (5.2).

Integrating the deterministic Hermite transform estimate shows that the deterministic log determinant differs from the semicircle value by O(\log n). Beyond a fixed large z, the same Hermite equation and \|A\|\le2 give normalized-transform error O(n^{-1}(z-2)^{-3}), so the integral to infinity is harmless. Finally,

has value 1/4 at z=2, as follows by integrating b_{\mathrm{sc}}, and is increasing thereafter. This proves the lower bound. The upper bound follows from the same Gaussian representation at a fixed positive shift, using the length-density upper bound and then sending the shift to zero.□

Proof. Test (5.4) at \lambda=2+\gamma s. Before conditioning, X\sim N(0,D^0) and h_s=rg+sX with an independent white Gaussian g. Put \delta=\lambda-\lambda^0. Since k\gg\log n, we have r/d_0\to\infty and \delta=\gamma r^2(1+o_n(1)) for fixed \gamma>0. The partial fraction identity

together with (5.1) and \Tr D^0=n gives

The leading expressions agree at \gamma=1; their errors are small in the stated order of upper scale and dimension.

The signal square is s^2X^{\mathsf T}D^2X. Subtract (s^2/\delta^2)(\|X\|^2-n), which has mean zero and vanishes on the exact sphere. In a white eigen-coordinate with t=\lambda^0-\lambda_i(J)>0, the remaining quadratic coefficient is

The pole in D^0 has disappeared. The spectral trace bounds give operator norm O_\gamma(r^{-2}) and squared Hilbert–Schmidt norm O_\gamma(n/r) for this quadratic matrix. The noise square has the same bounds. Gaussian exponentiation at a deviation of order nr therefore gives exponent \min\{(nr)^2/(n/r),(nr)/r^{-2}\}\asymp k. Conditional on X, the cross term is Gaussian with variance O(n/r) when the signal square is O(nr). The radial conditioning comparison preserves these rates after polynomial loss. Testing on either side of \gamma=1 proves the sharp assertions.

For a prescribed rate, first choose a large fixed L. At a lower trial point 2+\gamma r^2, the L^3k modes at depth comparable to L^2r^2 contribute noise-square mean comparable to nr/L, whereas the trace deficit is comparable to n\sqrt\gamma r. Choose \sqrt\gamma smaller than a fixed multiple of L^{-1}. A fixed-fraction Gaussian small-ball bound on these modes then has failure e^{-cL^3k}, uniformly in X because translation decreases the probability of a centered ball.

For the upper width, test at 2+L^2r^2. The trace deficit is of order nLr, while the noise and signal means are O(nr/L) and O(nr/L^3). The noise quadratic matrix has norm O(L^{-4}r^{-2}) and squared Hilbert–Schmidt norm O(n/(L^5r)); the canceled signal matrix has the corresponding bounds O(L^{-6}r^{-2}) and O(n/(L^9r)). Their Gaussian tail exponents at the deficit scale are at least cL^3k. For an actual time t\le s, write h_t=\sqrt t\,g+tX and use this same trial point. Both noise and signal prefactors decrease, so the upper-width estimate retains its nominal k-scale rate. Choose L from the requested rate, then the lower trial constant \gamma, then the upper scale so that these spectral windows are available. Finally choose the logarithmic threshold to absorb the polynomial shell cost. On the exact sphere \|X\|=\sqrt n; Gaussian norm concentration applied to h_t=\sqrt t\,g+tX gives the stated field bound uniformly over the deterministic choices 0\le t\le s.□

Completion of Proposition 6.1. Normalize the replica integral bounds from Section 6.1 using (6.1). To handle large overlaps between replica groups after normalization, put q_\eta(U)=\mu_h^{\otimes2}(|R_{12}|>\eta). The global pair bound gives

On the retained lower-partition event F\ge-\epsilon k, Markov’s inequality shows that q_\eta\le e^{-c_\eta n/2} outside Haar mass e^{-c_\eta n/2+2\epsilon k}. A union bound applies to any fixed number of replica pairs. Their fixed tilts cost at most C_pnr^2, since \|X^a\|=\sqrt n and \|m_{\mathrm o}\|\le\sqrt n. Choosing the upper scale after \eta therefore makes the tilted exterior exponentially small in n as well.

On the remaining replica configurations the truncated integral estimates apply. Markov’s inequality with repeated tilted replica groups gives (6.2), choosing the denominator accuracy arbitrarily small first. The pair concentration estimate transfers in the same way, taking the denominator error below its strict rate. The polynomial Gram-box and shell-density factors cost C_*\log n in these logarithmic estimates and in the exponents of the raw failure bounds. Choosing K after the rate and accuracy absorbs them when k>K\log n, as stated.

Finally, for an event with an arbitrarily prescribed raw rate,

gives any desired weighted rate by starting with more than twice that rate. For a bounded posterior failure probability 0\le q(U)\le1 having only a fixed strict rate, split instead at F=\delta k:

The second term has any fixed rate by the strong upper tail, and the first loses only \delta from its rate. Taking \delta small proves the stated weighted conclusions.□

Proof. For two independent prior vectors, the sphere–sphere and cube–sphere overlap distributions agree exactly. The cube–cube overlap has lattice spacing 2/n. Stirling’s formula and the spherical density c_n(1-R_{12}^2)^{(n-3)/2} show that its mass, divided by the lattice spacing and the spherical density, equals

Choose C' sufficiently large. Outside this window the second integrals, including any fixed bounded-degree polynomial insertions, are negligible.

Indeed the negative-square-tilt argument of Section 6.1 applies at depths much larger than (\lambda-2)_+^{1/2}+d_0; combine it with (6.3), then with the global pair estimate. A cube-only first moment averages exactly to the spherical first moment.

The following cancellation is needed to turn (7.1) into an operator estimate. Before length conditioning, write independent Gaussian replicas as

with white \eta_i, and use Fourier inversion for

The reciprocals of the two length densities at zero, in these units, are at most \log^C n.

The Fourier inversions, including any fixed number of derivatives, cost only powers of \log n apart from the insertion size. Choose an edge width d_*=n^{-1/3}\log^M n larger than the polylogarithmic saddle width. The spectral count supplies N\asymp\log^{3M}n modes on which D_i/L\ge\log^{-C}n. A frequency vector for the three statistics is a symmetric 2\times2 matrix T. At least one eigenvalue of T has size comparable to \|T\|, so the quadratic-product formula on those modes gives

The noncentral factor does not increase the modulus, because the real part of the inverse complex precision is positive. This bound is integrable with any prescribed fixed frequency polynomial, with a polylogarithmic integral. It remains so after deleting finitely many modes, as required below.

We now assemble the signed comparison before estimating it. Let \pi_{\mathrm c} and \pi_{\mathrm o} be the uniform cube and sphere priors, put E_U(x)=x^{\mathsf T}Jx/2+h^{\mathsf T}x, and write H_U(x)=(x-m_{\mathrm o})(x-m_{\mathrm o})^{\mathsf T}-D. Set Z_{\mathrm c}=Z. For i\in\{\mathrm c,\mathrm o\} define

All matrix data rotate with U. For a symmetric A commuting with D, expansion of the square gives

where

Conditional on the prior overlap, Haar integration gives the same angular integral in all four terms. The three pair laws containing a sphere use the same continuous overlap density; only the cube pair uses the lattice and its multiplier in (7.1).

After whitening, the common signed angular insertion is

Under the complex Fourier tilt its covariance differs from identity by

Its Hilbert–Schmidt norm, and the tilted mean using \|D^{1/2}m_{\mathrm o}\|/L\le\log^C n, are bounded by frequency polynomials times powers of \log n. Centering in (7.3) cancels the baseline and the terms belonging to only one replica. With zero tilted mean the remaining Wick terms have the forms

The first two are bounded using \|K\|^2 and the Hilbert–Schmidt bounds on \Delta_T. For the last, |\operatorname{Tr}K\Delta_{12}|\le C\|T\|\operatorname{Tr}(|K|D)/L. Terms containing the tilted mean obey the same envelope. Fourier inversion therefore bounds the signed density of the insertion and its overlap derivative by a power of \log n times

The frequency decay above absorbs all fixed derivative polynomials.

The spectral estimates give

The last two follow from A_0D=I-(\lambda-2)D and \|A_0\|\le5. Thus (7.4) is at most L^2\log^C n for A=I and n^{2/3}\log^C n for A=A_0. Insert a smooth cutoff in the central overlap window. The lattice spacing in the last statistic of (7.2) is 2/L, so its Riemann-sum error is the same envelope times L^{-1}\log^C n. The leading continuous integrals cancel in the four-term sum, and (7.1) leaves a squared error bounded by n^{-1/3} times the envelope, up to logarithms. The mass calculation has envelope one.

Consequently the root-mean-square errors are

Here n^{-1/3}L^2=n and n^{-1/3}n^{2/3}=n^{1/3} are the two squared matrix scales. Markov’s inequality gives the stated bounds outside Haar mass o(1), with any fixed exponent slack.

Finally let Q_i be the normalized raw second moment about m_{\mathrm o}. Then

so the common D cancels and

The shell density-ratio bound controls \widehat M_{\mathrm o} in operator norm by L\log^C n, and its A_0 sandwich by \log^C n. The scalar mass error therefore costs at most n^{1/2+o(1)} in the unweighted comparison and less in the weighted one. The spherical weighted raw moment itself is polylogarithmically bounded. This proves all claims.□

Proof. In the conditioning above, y is fixed and m=\tanh y is its deterministic profile. The root map is proper, since \Phi_J(y)-y is bounded for fixed J,n. Its critical target values have Lebesgue measure zero, and the Gaussian field has an everywhere positive smooth density. The area formula therefore expresses the expected root sum as the integral of the observation density at \Phi_J(y) times |\det D\Phi_J(y)|; exhaustion by compact balls and monotone convergence handle arbitrary nonnegative tests.

The derivative is

The conditioned observation has covariance \nu I+mm^T/n, determinant \nu^n(1+j_0), and inverse \nu^{-1}I-mm^T/[n\nu^2(1+j_0)]. After dividing its density by \prod_i\varphi_{d,\nu}(y_i), the log ratio per site, apart from the determinant prefactor, is

This proves the formula, including the conditional matrix law and all nonnegative restrictions.□

Proof. The identity \det(I+AB)=\det(I+BA) writes the determinant in \mathcal W as the symmetric determinant \det(I+\sqrt V(bI-J-2mm^{\mathsf T}/n)\sqrt V). First compare the conditional matrix with an unconditioned GOE matrix W_0. For q>0, put u=m/\sqrt{nq} and E=u^\perp. The conditional law leaves P_EWP_E as a fresh GOE block. Its remaining row and scalar are Gaussian; with e=P_E\mathbf1/\sqrt n they have the representation

where g_E,\xi are independent standard Gaussians, also independent of the E block. Share that block with W_0. The row change, the plant ee^{\mathsf T}, and the TAP spike are a perturbation of fixed rank. Its norm is polynomial in n,\|y\| and the fresh Gaussian norms: |M|/\sqrt q\le1, a^2\le q\langle(y-d)^2\rangle, and \nu,2q+s\ge s with s^{-1} polynomial on the present range. When q=0, m=0 and the conditioning is independent of W, so the same conclusion is immediate.

Here is a useful bound for this reduction. For symmetric H and rank-j symmetric N,

It follows from singular-value interlacing after factoring H+\mathrm i\varepsilon I: the factor H(H+\mathrm i\varepsilon I)^{-1} is a contraction and the remaining change has rank j. Take \varepsilon=n^{-5}. Markov’s polynomial derivative inequality on [-1/n,1/n], followed by integration on a small interval around a maximizer, gives for every real polynomial P of degree at most n

Indeed \|P'\|_\infty\le Cn^3\|P\|_\infty on this interval; iterating this bound controls the imaginary displacement, and an interval of length cn^{-3} controls the maximum by the integral. Truncate enormously large fresh entries at negligible cost by crude determinant bounds.

It remains to bound the unconditioned square. For real z, put x_i=1+z+bv_i. The exact identity is

where H_1,H_2 are independent real normals of variance 1/n and h_1,h_2 are independent circular normals with \mathbb E|h_j|^2=1/n. For completeness, represent each determinant as the top coefficient of the exponential in one set of exterior variables \bar\psi,\psi. Averaging W_0 adds -[X_{11}^2+X_{22}^2+2X_{12}X_{21}+2PQ]/(2n) to the exponent, where X_{ab}=\sum_i v_i\bar\psi_i^a\psi_i^b, P=\sum_i v_i\bar\psi_i^1\bar\psi_i^2, and Q=\sum_i v_i\psi_i^1\psi_i^2. Decoupling these four quadratic terms with the displayed Gaussian variables and extracting the site coefficients gives the identity.

Move each H_j contour to H_j+\mathrm i b. The factors become 1+z+\mathrm i v_iH_j, while the two Gaussian densities cost e^{nb^2} in modulus. This exactly cancels the factor e^{-nb^2} in \mathcal W^2. The contour move is valid because the integrand is an entire polynomial times a Gaussian. If e_1,e_2 are the eigenvalues of the two-by-two Hermitian matrix with diagonals H_1,H_2 and squared off-diagonal modulus |h_1|^2+|h_2|^2, the remaining modulus is at most \prod_{j=1}^2[(1+z)^2+e_j^2]^{n/2}, since 0\le v_i\le1.

More explicitly, write R_H^2=H_1^2+H_2^2+2(|h_1|^2+|h_2|^2)=e_1^2+e_2^2. The Gaussian exponent is -nR_H^2/2. For R_H\le1, the inequality \log(1+x)\le x-cx^2 on a fixed bounded interval bounds the remaining integrand by C\exp(-cnR_H^4); for R_H>1, split off a fixed compact annulus and then use \log(1+x)\le x-cx for x sufficiently large, obtaining an integrable Ce^{-cn\min(R_H^4,R_H^2)} bound. There are six real Gaussian variables, so their normalized integral is at most Cn^3\int_{\mathbb R^6}e^{-cn\min(|u|^4,|u|^2)}du \le Cn^{3/2}. The bounded-rank and real-shift reductions multiply this by another fixed polynomial. This proves the second-moment bound.□

Proof.

Scalar comparison and exclusion of remote parameters. First restrict to a fixed polynomial ball in y. The bound a^2\le q\langle(y-d)^2\rangle and j_0/(1+j_0)\le1/2 give a Gaussian tail envelope outside it. On the ball, the logarithmic likelihood and its parameter derivatives are polynomially bounded, since \nu\ge s and s^{-1} is polynomial. Divide q,M,a into a polynomial number of bins with sufficiently fine inverse-polynomial mesh. Replacing their empirical values by a bin representative changes the log likelihood by O(1). In the transport and variance comparisons through the retained-range bound below, the symbols q,M,a,b,\nu,d,C_a,j_0,\alpha denote these frozen representatives; y and m=\tanh y remain the variables being integrated. In the variance comparison, j_0^{-1} is used only where C_a>1; Cauchy–Schwarz and the polynomial ball then also bound q^{-1} polynomially.

Suppose \alpha<1 stays a fixed distance from one. The map z'_i=y_i-\alpha\tanh y_i has Jacobian \prod_i(1-\alpha v_i); here \alpha is the frozen bin value. Direct expansion of the two Gaussian exponents gives

where

The inequality uses \log(1-uv)\ge v\log(1-u) for 0\le v\le1. In particular H(u)\ge qu^2/6 for u\le q, with a cubic or stronger penalty on the negative side when q is small.

The remaining branch is C_a>C_0:=1+j_0q, equivalently \alpha>q. Compare there with the product Gaussian on y having variance \nu C_a^2/j_0. Cauchy–Schwarz on a gives the remaining rate f(C_a)/2, where

The bounds f(C_0)\ge-2q^3/3, f'(C_0)\ge2q/(1+q), and f''\ge1 exclude \alpha>q+C_AR_*^{3/2} at budget AnR_*^3, where R_*:=\max(r,q). The H comparison excludes \alpha<-C_AR_*. On the retained range

where \delta can be made arbitrarily small by decreasing the upper scale.

The frozen-symbol convention for those two comparisons ends here. We also need a fixed negative rate away from q=0 as s\downarrow0. Fix q_0>0. For a probability law P with finite second moment and \E_P\tanh^2Y\ge q_0, define

Put \psi_s(P)=(1+j_0)\alpha^2/2 and

The endpoint scalar inequality of Lemma 13.3, proved independently below, gives \mathscr R_s\le0, with equality only for the indicated Gaussian. At s=0, a consistent equality law with q>0 would be N(M,q). Its mean equation M=\E_{N(M,q)}\tanh Y forces M=0, since the difference of the two sides is strictly increasing and vanishes at zero. But then \E_{N(0,q)}\tanh^2Y<\E Y^2=q, a contradiction.

For completeness, this pointwise exclusion has the uniformity needed by the integral. Fix a bound on \E_PY^2. The resulting set of laws with q\ge q_0 is weakly compact. Its first moments are uniformly integrable, so the average of y\tanh y and hence \psi_s are continuous there; \nu\ge q_0 keeps the denominators separated from zero. Relative entropy is lower semicontinuous, making \mathscr R_s upper semicontinuous. Its maximum at s=0,q\ge q_0 is therefore strictly negative, and remains so for sufficiently small s. The second-moment bound also makes replacement of nearby Gaussian parameters uniformly e^{o(n)} in the product density. A finite cover by neighborhoods on which one bounded continuous entropy test has this slack bounds the scalar integral by e^{-cn}. This is the compact-set integral bound in the proof of Lemma 13.3, with its smoothing variance set to zero and \nu=q+s\ge q_0; that proof records the finite-cover argument of [28, Lemma 5.4]. The Gaussian envelope G(y)\le e^{Cn-c\sum_i y_i^2} on q\ge q_0 removes the second-moment restriction. This proves the fixed-q_0 exclusion; the cubic rate for shrinking q is derived next.

Empirical moments at the cubic scale. The transport comparison handles the negative and moderate scalar parameters, and variance inflation excludes the remaining positive range. On the retained range we can now impose product-Gaussian moment diagnostics and use root consistency to locate the overlap scale.

Return to the root integral. For the shifted-coordinate reference of the bin containing y, write \alpha_{\rm bin},d_{\rm bin},\nu_{\rm bin} for its frozen values and put

The rate parameters in this product-reference estimate also use the frozen bin values. For any fixed A'>0, exponential Markov bounds give the following outside product probability e^{-A'nR_*^3}: the second moment on |Z_i|>R_*^{-\eta} is at most C A'R_*^3, and clipped moments through any prescribed fixed degree differ from their Gaussian values by O(\sqrt{A'}R_*^{3/2}). Choose \eta sufficiently small for that degree. These estimates use a fixed exponential tilt for the tail square and a tilt of order \sqrt{A'}R_*^{3/2} for each clipped polynomial; their constants stay bounded as A'\downarrow0.

The scalar change of measure costs at most e^{\delta nR_*^3} on a retained bin. Hence a reference exception of size e^{-A'nR_*^3} contributes at most e^{-(A'-\delta)nR_*^3} to the scalar integral, up to the polynomial bin factor. For a small target budget choose A' proportional to that budget and then make \delta<A'/2 by shrinking the upper scale. This preserves the small O(\sqrt{A'}) moment errors needed below.

For the consistency and residual conclusions below, the symbols q,M,a,b,\nu,d,C_a,j_0,\alpha again denote the exact empirical quantities defined at the start of the section. Expand the exact identity m=\tanh(z'_{\rm bin}+\alpha_{\rm bin}m) on the clipped region and use the tail second moment on its complement. Replacing the bin parameters in the resulting expressions by the empirical quantities is included in the displayed errors, using the initially chosen sufficiently fine mesh. The first consistency estimate is

It implies s+d^2\le C_AR_*q. If q\le r, the inequality r^2=s\le C_Arq forces q\ge c_Ar; otherwise this is automatic. Thus R_*=O_A(q) and |d|=O_A(q). On this narrower range the retained H is nonnegative. Keeping the next terms gives

Apart from an o(q^2) error, the last constant tends to zero with the small budget. The H and f penalties then force |\alpha|\ll q. Since d=M+s, division of the two consistency equations by q^2 gives

The first ratio is nonnegative. These two relations therefore force s/q^2 and d/q close to one, proving the small-budget location. In particular, bins with q\gg r have a fixed negative rate cnq^3. This tail also pays a prescribed budget Ak in the large-budget calculation: choose q_{\min}/r so that c(q_{\min}/r)^3>A, with slack for the polynomial bins.

Finally the total scalar integral is polynomial. The exceptional bins just considered have strict negative rates. On the retained bins the reference product integral is one and H\ge0; indeed H(q)\ge q^3/6 and this sign persists up to q+O_A(q^{3/2}). Summing over the polynomial grid proves the claim.

Fourth and sixth moments and the cubic residual. The scale comparison is not yet enough for the matrix estimates: they also use the first correction to the fourth moment. We therefore keep the cubic-order terms in the same expansion. The cubic residual will require a separate cancellation at small exclusion budget.

The same expansion has linear coefficient 1+\alpha+O(q^2) and cubic coefficient -1/3+O(q). The consistency equation for q gives

whereas direct expansion of the fourth moment gives

Since \nu=q+O(q^2), their difference is -8q^3+o(q^3). Likewise \langle m^6\rangle=15\nu^3+o(q^3) and \langle m^3\rangle=3\nu d+O_A(q^3)=3qM+O_A(q^3). The clipped empirical errors are o(q^3) at this scale even for a fixed large budget. On the complement, boundedness of \tanh and the original-scale tail second moment O_A(q^4) control the remainders.

For the residual, use the identity

Expand it in z'_{\rm bin} through degree six, including the occurrences of M and q in this expression. Replacing empirical moments by their Gaussian values costs O(\sqrt{A'}q^3)+o(q^3). The order-q^2 terms cancel: the cubic Gaussian moment starts with 3\nu d and the fourth with 3\nu^2. After division by q^3, the remaining leading coefficient is continuous in the bounded ratios \nu/q,d/q,\alpha/q, with \nu/q bounded below.

At the small-budget limit these ratios approach 1,1,0. The leading coefficient vanishes there by an exact reflection identity. If Y\sim N(\nu,\nu), its density obeys \varphi(-y)=e^{-2y}\varphi(y), so pairing y and -y gives

With F(y)=y and F(y)=\tanh y, this gives \E Y=\E[Y\tanh Y] and \E\tanh Y=\E\tanh^2Y. The integrated residual is therefore exactly zero at \alpha=0,d=\nu. Continuity and the empirical error prove the stated small leading constant.

Nonlattice and spanning diagnostics. On bounded standardized inputs, the normalized profiles m/\sqrt q and (y-lm)/q^{3/2} converge to the nonsingular pair z and z^3/3-z. Their Gram matrix on a fixed bounded input set is consequently bounded above and below. Put w=y-lm. Then \langle wm\rangle=0, \langle w^2\rangle\asymp q^3, and 1-M^2/q=1-O_A(q) is bounded below. The normalized squared norm of the projection of w onto \operatorname{span}(m,\mathbf1) is \langle w\rangle^2/(1-M^2/q)=O_A(q^6). Removing both directions therefore leaves squared norm comparable to nq^3, as asserted.

For phase dispersion we need derivative control of the actual profiles, because the frequencies may grow with n. In a frozen reference bin, write Y(z) for the inverse of

The retained parameters satisfy d=O_A(q), \nu=q+O_A(q^2), \alpha=O_A(q), and l=(a+dM)/q=1+q+O_A(q^2). The last expansion follows by expanding ym=m^2+m^4/3+m^6/5+\cdots on the clipped region and using (8.2) and the tail-square bound. On every fixed bounded z interval, the functions

therefore satisfy f_q(z)=z+O_A(\sqrt q) and g_q(z)=z^3/3-z+O_A(\sqrt q) in C^1. For example,

which converges uniformly to z^2-1 on that interval.

Put R=\|(t_1,t_3)\| and v=(t_1,t_3)/R. Each polynomial v_1z+v_3(z^3/3-z), with \|v\|=1, has an interval of fixed positive length on which its derivative is bounded above and away from zero in absolute value. Compactness in v permits a finite choice of such intervals and uniform constants. The C^1 bounds give the same derivative bounds for the actual function v_1f_q+v_3g_q when the upper scale is small. On its chosen interval this function is monotone. For all sufficiently large R, counting periods of R(v_1f_q+v_3g_q)+\theta shows that a fixed positive fraction of the interval stays a fixed distance from \pi\mathbb Z, uniformly in \theta. For \tau\le R in a fixed bounded range, compactness in R,v,\theta\bmod\pi gives the same conclusion with possibly smaller constants. Gaussian density is bounded below on the bounded intervals used. No approximation error is multiplied by the growing frequency in this argument.

Use a smooth periodic test supported a little inside this separated phase set and a smooth cutoff inside the bounded input interval. For each fixed choice, product concentration gives failure e^{-cn}. All parameter derivatives on the retained bins are polynomially bounded, since q^{-1} is polynomially bounded. The parameter l is determined by the existing bins through (a+dM)/q. Refine their mesh, and mesh frequencies and phase, at inverse-polynomial accuracy chosen after D, so that every test changes by less than the fixed separation slack even for R\le n^D. There are only polynomially many tests. Their union retains failure e^{-c'n}, and the scalar change of measure preserves that rate after the upper scale is decreased. The same bounded-input concentration gives the Gram estimates used above.□

Proof.

Conditional law and Haar concentration. We use Haar concentration in the Frobenius metric on proper rotations: an L-Lipschitz function on SO(N) has tail 2e^{-cNz^2/L^2} about its mean. See [26, Sections 5.2–5.3, Proposition 5.13 and Theorem 5.16]. A constant-mesh net of size e^{O(j)} then gives an operator bound on a j-dimensional space, using polarization for bilinear forms. All such tests below condition first on deterministic eigenvalues, so they also apply to resolvents defined separately on spectral bins.

In \mathsf Q_y, the scalar data and the vectors determined by y are fixed. The regression in Lemma 8.1 gives

where W_E is a centered GOE block and w\sim N(0,I_E/n) is independent of it. The mean p_0 is the projected regression mean of Wm+\sqrt s\,g, plus the plant column. We may next condition on the eigenvalues of W_E and retain its Haar eigenframe. On the intersection with \mathcal G_{\rm spec}, interlacing of W_E, W_E+ee^{\mathsf T}=J|_E, and J supplies the required counts and \lambda_{\max}(W_E)\le\lambda_{\max}(J). The concentration probabilities are estimated before that intersection is imposed.

Diagonal truncation and the compensating moments. Compare with D=(b+b^{-1})I-J. In K and K_T, replace the positive diagonal entry v_i^{-1}-1 by \min\{v_i^{-1}-1,q^{1-\eta}\}. This gives a lower precision. Let B be the difference of either truncated precision, or of K_o, from D, and set B_E=P_EB|_E and f=P_EBu.

The scalar moments give the quantities needed by the Schur complement. With \mu_B=n^{-1}\operatorname{Tr}B_E and \nu_B=n^{-1}\operatorname{Tr}B_E^2,

The error in the vector relation is in Euclidean norm.

For the diagonal precisions, the truncated entry expands as m^2+m^4+\cdots on |y|\le q^{1/2-\eta_0}, with 0<\eta_0<\eta/2. The missed bounded powers on the complement are O(m^2) and have average O_A(q^4) by the tail-square diagnostic. The bulk mean is therefore 3q^2-q^2=2q^2 to leading order, equal to the centered variance \langle(m^2-q)^2\rangle=2q^2+o(q^2). In the root direction,

Subtracting the scalar b^{-1}-1=q+q^2+O(q^3) leaves at least 2q^2+o(q^2). Also

This is the leading value of \|f\|^2.

The root identity y=m+m^3/3+O(m^5) on the clipped region gives

The tail contributes O_A(q^4/q) to its squared error. In the formula for p_0, the extra e coefficient is (M-(q/\nu)d)/\sqrt q=O_A(q^{3/2}), giving p_0=f/3+O_A(q^{3/2}). The bounds involving e follow similarly from \langle m^3\rangle=3qM+O_A(q^3).

For K_o, put L_B=(P_{\mathcal P}(\operatorname{diag}(m^2)-qI)P_{\mathcal P})|_{\mathcal P}. Its norm is O_A(q): in the cubic direction the numerator with an extra factor m^2 is O(q^4) and the normalization is of order q^3. Since V_{\mathcal P}=bI_{\mathcal P}-L_B,

Here (L_B)_{uu}=2q-8q^2+o(q^2) and (L_B^2)_{uu}=10q^2+o(q^2). Projection onto \mathcal P in the second identity loses only O_A(q^3), by the preceding approximation of the cubic column. Substitution gives the same root coefficient and column estimates; the e entries again use the third moment.

Finally, the diagonal restored in K and K_T is positive and is supported where v_i^{-1}-1>q^{1-\eta}. The tail-square bound limits this support to O(nq^{3+\eta'}) sites for some fixed \eta'>0. Thus restoring it is a positive perturbation of rank o(nq^3).

Resolvents away from the edge. Augment by one scalar \beta for the negative spike -ee^{\mathsf T}, so the augmented form contains \beta^2-2\beta e^{\mathsf T}x_E. Put \Delta_{\mathrm{sp}}=q^{8/5} and define

Then \dim N=O(n\Delta_{\mathrm{sp}}^{3/2}). The high-block resolvent for the unperturbed comparison, also after subtracting (1-\epsilon)q^2, is R=(2+O(q^2)-W_E)^{-1}_H.

Its traces divided by n for the first and second powers are 1+o(1),\,\asymp \Delta_{\mathrm{sp}}^{-1/2}. On the conditional Haar orientation with the spectra satisfying the interlacings, except at arbitrarily high rate,

Indeed dyadic blocks at depths t_1,t_2\ge\Delta_{\mathrm{sp}} give centered bilinear deviations at most Cq^{1-\eta}\max(t_1,t_2)^{3/4} by Haar concentration and nets. After division by \sqrt{t_1t_2}, the dyadic sum is bounded by a logarithmic factor times q^{1-\eta}\Delta_{\mathrm{sp}}^{-1/2} =q^{1/5-\eta}. The mean costs O(q^2/\Delta_{\mathrm{sp}})=O(q^{2/5}). Both vanish for small \eta; the sum includes a constant-width bulk bin.

Uniform tests on the remaining coordinates. For remaining variables \gamma,x_N,\beta, normalized by \gamma^2+\|x_N\|^2+\beta^2/\sqrt{\Delta_{\mathrm{sp}}}\le1, write x=x_N+R(p\gamma+e\beta). The needed simultaneous Haar estimates are

Here little-oh can mean any specified fixed leading error.

For a fixed symmetric weight T, normalized Gaussian quadratic tails and polarization give error

for a Haar quadratic test on fixed orthonormal axes, at failure e^{-Bk}. The same bound holds for an independent Gaussian column with covariance I_E/n. A subspace test instead pays its dimension through a net. Since q\asymp_A r, fixed changes of the constants allow nq^3 in place of k.

Fix, before sampling W_E and w, a fixed-dimensional space S containing e,c_3,p_0,f. On S, compression of R differs from its trace mean by O_A(q^{3/2}\Delta_{\mathrm{sp}}^{-1/4}+q^3/\Delta_{\mathrm{sp}}); compression of R^2 has relatively vanishing error. Test w separately, with zero mixed means, so S stays deterministic. Also \|P_N|_S\|=O(\Delta_{\mathrm{sp}}^{3/4}). These estimates give \|x\|=O_A(1) and

The terms f^{\mathsf T}x_N and \beta f^{\mathsf T}Re are o(q^2) on the normalized variable ball. This proves the f^{\mathsf T}x line of (9.2). The same tests give

The fixed-axis estimates have identified the projection of the optimizing vector. We next expose the low frame and the two resolved columns. Rotations preserving these data control the remaining components and hence the final weighted quadratic test.

Now expose N,Rp,Re and put Y=S+N+\operatorname{span}(Rp,Re). Before this exposure, the part of Y beyond S is invariant under rotations fixing S, hence is a uniform subspace of S^\perp. Its compressions of B_E^j, j=1,2, have deviations from their scalar means of size O_A(q^{j(1-\eta)}\Delta_{\mathrm{sp}}^{3/4}); mixed compressions with S have the same bound and mean zero. Together with the estimates on B_Ee, e^{\mathsf T}B_Ee, and P_Sx, this gives

as well as \|P_SB_Ex\|=o(q) and \|P_{Y\cap S^\perp}B_Ex\|=O_A(q^2).

After the exposure, only rotations fixing Y pointwise remain. Split R=R_{\rm near}+R_{\rm far} at a small fixed depth \zeta. The conditional mean of the far restriction on Y^\perp is its trace on that complement divided by its dimension. Since \dim Y=O(n\Delta_{\mathrm{sp}}^{3/2}), \|R_{\rm far}\|\le C/\zeta, and n^{-1}\Tr R_{\rm near}=O(\sqrt\zeta), this scalar is

Residual Haar concentration and the low-dimensional nets make this the quadratic coefficient on the needed vectors in Y^\perp. The component of B_Ex in Y contributes o(q^2) to the far quadratic and mixed terms, by its two projection bounds and the fixed far norm. The mixed term with Rf has zero leading conditional mean; f^{\mathsf T}Rf is already covered by the fixed-axis test.

For the near part, \|R_{\rm near}\|\le C/\Delta_{\mathrm{sp}} and its compression to S is small by the fixed-axis tests. Its action on the O_A(q^2) component in Y\cap S^\perp is negligible at scale q^2. On the remaining vectors, the conditional square-root norm is bounded by a constant times

This is the Haar Lipschitz bound plus its net cost. First choose \zeta small, then the upper q scale. These estimates prove the last line of (9.2).

All these failures were estimated under the Gaussian/Haar conditional law before imposing \mathcal G_{\rm spec}. On its intersection, interlacing supplies the spectra of W_E used in the trace and count bounds. Integration in y gives the stated exclusion.

Schur complements, counts, and projections. Substitution of the simultaneous tests into the high-block Schur complements gives the perturbed-minus-unperturbed difference

By (9.2) and u^{\mathsf T}Bu\ge2q^2+o(q^2), the displayed difference is bounded below on the normalized variable ball by

This is the one-sided root-direction cancellation and the bulk cancellation.

To see the remaining positive margin, let x_{\rm phys}=\gamma u+x_E. The shifted augmented base form is exactly

The original spectral bound makes its first term at least a fixed multiple of \epsilon q^2\|x_{\rm phys}\|^2. At the high optimum, x_H=R(p\gamma+e\beta), and

After absorption, the base complement therefore controls c\epsilon q^2(\gamma^2+\|x_N\|^2+ \beta^2/\sqrt{\Delta_{\mathrm{sp}}}). The o(q^2) difference preserves this margin and proves the lower precision bound.

The one-sided estimate proves positivity. The trace and shell-density arguments also need enough eigenvalues near the critical edge. Removing the fixed-rank directions makes the bulk cancellation two-sided, so the same calculation can now be repeated at dyadic thresholds.

After removing the fixed-rank u,e interactions, the unaugmented N-Schur complement for a truncated precision differs from the base one by o(q^2) on both sides, because \mu_B-\nu_B=o(q^2). At a shifted threshold a\le\Delta_{\mathrm{sp}}/2 the high block remains positive. Schur inertia therefore identifies the number of full eigenvalues below a with the negative-eigenvalue count of the low complement, up to the removed fixed ranks. The two-sided comparison bounds this count by the W_E edge counts at thresholds displaced by o(q^2).

The same estimates hold on a finite dyadic grid Cq^2\le a\le\Delta_{\mathrm{sp}}/2: the shift is at most half the high gap, so the high resolvents remain comparable and their normalized traces change by O(\sqrt{\Delta_{\mathrm{sp}}}). This gives the upper count Cna^{3/2} and at least cnq^3 eigenvalues below Cq^2 for a sufficiently large fixed C. Above a fixed multiple of \Delta_{\mathrm{sp}}, the weighted dyadic form bound gives the same upper count. Restoring the positive discarded diagonal preserves upper counts and removes at most its rank O(nq^{3+\eta'})=o(nq^3) from the lower spread. Dyadic integration of these counts gives all trace-square bounds.

For K_o, the correction to D has fixed rank. At b+b^{-1}, the semicircle transform is exactly b; (5.1) and the fixed-rank trace error O(q^{-2}) =o(\sqrt{n/q}) give its stated inverse trace. For a truncated precision, write A_0 as that precision minus an O(q^{1-\eta}) term and use the lower bound at q^2 to obtain \|A_0\mathsf K^{-1/2}\|\le Cq^{-\eta}. Restoring the positive diagonal preserves this bound by inverse order. The fixed-rank structure gives the stated \eta=0 bound for K_o.

It remains to control variances in the distinguished directions. For projection bounds at e,c_3 first use D. At R_1=(b+b^{-1}-W_E)^{-1}, deterministic fixed-axis compressions have means 1+O(q) and errors O_A(q), \|R_1 e\|^2\asymp_A 1/q (lower via modes at sufficiently enlarged width to get any desired rate). Thus 1-e^T R_1 e\gtrsim_A q by the principal-block top eigenvalue bound at 2+q^2/2 and shifting. Also e^T R_1 p=O_A(q^{3/2}), c_3^TR_1 p=O_A(q). Sherman-Morrison for the spike followed by inclusion of u (whose inverse variance bound after inclusion is the global operator bound) gives the orders stated.

For a truncated precision or K_o, the reduced Schur form controls the base reduced form up to a factor two. View a projection variance as the square of the dual norm of its physical source in the augmented positive form. The high precision stays comparable to R^{-1}, and its minimizing vector changes by -(R^{-1}+B|_H)^{-1}P_H(B_Ex+f\gamma). For a source at e or c_3, (9.2) and its bounded R^{1/2} norm bound the change in testing this optimum by

The reduced energy controls q^2 times the square of this parenthesis, so the additional squared dual norm is O_A(1). The remaining high square also costs O_A(1). Adding these costs to the base variances preserves the orders q^{-1} for e and 1 for c_3; the global lower precision bound supplies the q^{-2} order for u. Inverse order again permits restoration of the diagonal.□

Proof. First remove the null set of critical target fields from Lemma 8.1. For the remaining fields every root is regular, and the probability of an excluded root is bounded by its expected count. On the complementary field event, V^{1/2}K_TV^{1/2} =I+V^{1/2}(bI-J-2mm^{\mathsf T}/n)V^{1/2} is positive. The Jacobian D\Phi_J is similar to this matrix and thus has positive determinant at every root.

Properness and the identity homotopy outside a sufficiently large ball give degree one. Hence the signed root count is one. With all signs positive there is exactly one root. More generally, for a regular target the numbers of positive and negative roots obey N_+-N_-=1, so the excess multiplicity is N_++N_--1=2N_-. This identity also bounds multiplicity errors by the excluded negative-root count.

On the retained event the unique root is a measurable function of (J,h); extend it by zero elsewhere when a globally defined profile is needed. Conditional on (J,h) in the planted experiment, the planted spin has law \mu_h. Thus an equivariant test involving this profile may replace the plant by a posterior replica, with the excluded event charged separately. This is a conditional posterior identity under planted disorder; the later likelihood comparison performs the transfer to ordinary disorder.□

Proof. Write m(\lambda)=\operatorname{Tr}(C^{-1}+\lambda I)^{-1} on \lambda>-\|C\|^{-1} and C_\lambda=(C^{-1}+\lambda I)^{-1}. The function m decreases continuously from infinity to zero, and

If |m(0)-x_0|\le L, there is nothing to prove. Otherwise set \varepsilon=\operatorname{sign}(m(0)-x_0) and x_1=x_0+\varepsilon L. There is a unique \lambda with m(\lambda)=x_1, and \varepsilon\lambda>0. Exponential tilting of the squared length gives its density f_\lambda under covariance C_\lambda and the identity

This identity treats the two signs with exactly the same inequality.

First we bound the tilt without assuming a local central limit theorem near the final pole. This coarse bound will place the covariance in a region comparable to its original value; sharper density estimates can then improve the logarithmic tilt bound to a constant. Since \operatorname{Tr}C_\lambda=x_1\asymp n, its standard deviation is at most \sqrt2x_1\le Cn. The mean-density bound of Lemma 5.2 gives f_\lambda(x_1)\ge c/n. If \lambda>0, the inequality m(\lambda)\le n/\lambda implies \lambda\le C; if \lambda<0, every eigenvalue increases.

In both cases at least c n eigenvalues of C_\lambda are bounded below by a fixed positive constant. The quadratic Gaussian product formula therefore gives

Consequently M\ge c n^{-1/2}e^{|\lambda|L/2}, so |\lambda|L\le C\log n. This is the promised control near a pole: no local central limit theorem is being assumed there, and the density loss is only polynomial.

Choose a>0 so small that aC_0<1/2. Because q^2L=\sqrt{nq^3}\ge\log^5 n, the preceding bound forces |\lambda|<a q^2. Thus C_\theta is comparable to C for every \theta between zero and \lambda, and \operatorname{Tr}C_\theta^2\asymp L^2. We can now use the spectral spread at the tilted covariance. For these tilts, the largest eigenvalue divided by the square root of the trace square is O((nq^3)^{-1/2})=o(1). To check the Fourier bound explicitly, if a_i are the eigenvalues of C_\lambda and \sigma^2=2\sum_i a_i^2, concavity of x\mapsto\log(1+ux) on [0,\max_i a_i^2] gives

Since a_{\max}^2/\sigma^2\le1/16 for large n, the right-hand side is bounded by (1+t^2/4)^{-2}, an integrable function independent of the parameters.

Hence its density has supremum at most C/L. Its density at the mean is at least c/L by the same mean-density lemma. Applying these bounds to f_\lambda in the ratio identity improves the preceding estimate to |\lambda|L\le C. Finally,

□

Proof.

Projection cuts and the three priors. The plant is gauged to \mathbf1 as above. Let \mathsf P be the n\times3 matrix whose columns are the three orthogonalized profiles spanning \mathcal P, normalized by n^{-1}\mathsf P^T\mathsf P=I_3. Its first column is m/\sqrt q, and its second is the normalized vector \mathbf1-(M/q)m; its third is the normalized residual of y after projection onto those first two columns. We write \mathsf P_j for column j, and empirical contractions of matrices always include the factor n^{-1}.

Put z=X-m,\ N=nq^3,\ L=\sqrt{n/q}, and use the statistics

Here \vartheta denotes the scaled projection coordinate. For a pair write t=\langle(z-\mathsf P\delta)(z'-\mathsf P\delta')\rangle,\ U=n t/L, so t=qU/\sqrt N. The three-column notation always uses empirical contractions over sites. The exact length constraint is \langle z^2\rangle=b-2\sqrt q\,\delta_1. We tilt by the common energy z^T(J-b)z/2; with p the product of independent spins of means m (translated by -m), this gives precisely the posterior by the root equation.

We cut p smoothly to |\vartheta|\le\epsilon_0 (one below \epsilon_0/2), \epsilon_0>0 an arbitrarily small fixed constant. Except at a small rate in the true counting law with good spectrum, the cut omits negligible normalized posterior mass (n^{-D}, D any fixed constant). Indeed a posterior spin can replace the plant in tests with the equivariant axes m,y-lm by uniqueness and (8.2), so |\langle m z\rangle|\ll q,\ |\langle(y-lm)z\rangle|\ll q^3. The overlaps of plant and replica concentrate about q to any fixed tolerance times q by (5.1), the spherical observation estimates, (6.1)–(6.3) and q/r\simeq1.

Thus |\langle z\rangle| (overlap with the gauged plant minus M) is bounded by arbitrarily small fixed times q, and the subtraction (M/q)\langle mz\rangle in the e axis is likewise small. Orthogonalizing the cubic axis uses |\langle y-lm\rangle|=O(q^3) and residual norm asymptotic, in empirical units up to constant factors, to q^{3/2}, by the clipped Gaussian moment diagnostics and the tail-square bound. Here multi-root fields do not affect counting claims: the count of positive-sign regular roots is at most one plus the number of negative-sign roots by degree, whose expectation on good spectrum is exponentially small. Null singular fields may be ignored. The orbit weight comparisons thus allow us to discard further orbits at a small integrated rate measured by w_0\,d\pi, so that the cut loss is negligible on most Haar orientations.

Use two continuous auxiliary priors, not necessarily of mass one. We fix their length-density units here. Put

For a Gaussian prior, the notation \delta_{\rm D}(\mathscr L(z)-nb)\,dz means the coarea density at the raw squared length nb. Since this shell is \|z+m\|^2=n, it is surface measure divided by 2\sqrt n. In particular, if T=(\mathscr L-nb)/L, then \delta_{\rm D}(\mathscr L-nb)=L^{-1}\delta_{\rm D}(T). The spin prior already lies on this shell.

With \varphi_C the density of N(0,C), define the uncut pre-energy measures

where

The constants \mathfrak c_{g_0},\mathfrak c_o are fixed by the zero-projection saddle normalization below. Multiplication by the common energy e^{z^T(J-bI)z/2} changes the underlying Gaussian precisions to K and K_o, respectively.

For Z\sim N(0,C_o), C_o=K_o^{-1}, put T_o(Z)=(\mathscr L(Z)-nb)/L. The reference mass used throughout is projection-uncut but still shell-constrained:

Thus its normalized physical law is exactly the density disintegration of N(0,C_o) conditional on T_o=0. Both factors in the last formula are invariant under rotations fixing \mathcal P, so Z_* is an orbit invariant. We cut g_0 by the same projection cutoff as p, and multiply it there by a projection-only factor to form g. All cuts below are separable across replicas.

The three priors serve different parts of the comparison. The reference prior has an orbit-invariant tilted mass; the variable-variance Gaussian has the precision needed for covariance control. The projection multiplier aligns its single-replica marginal with the spin marginal. What remains is a pair-density calculation in which the signed sum cancels common errors and errors additive in the replica types.

Single-prior projection saddles. We begin by defining the single-prior saddle functions. Write p_\ell=\mathsf P_{\ell\bullet}^T\in\mathbb R^3 for the profile row at site \ell. The independent site laws are

Their constraint vectors and per-site targets are

For a spin, z_\ell^2+2m_\ell z_\ell=v_\ell identically, so its shell constraint is automatic; adding a square coordinate would make the constraint covariance singular. For Gaussians the square coordinate and its target are equivalent to the translated length coordinate by a determinant-one triangular change.

Set

On the local interior branch let \nabla\Lambda_i(\theta_i)=c_i(\delta) and \mathsf H_i=\nabla^2\Lambda_i(\theta_i). The branch and its positive Hessian are established on the small projection cut below. With \mathfrak c_p=\mathfrak r_p=1, define

This is a smooth saddle function per ordinary d^3a, not an exact density for an atomic spin projection. The factor n^{3/2} converts the three raw projection sums \sqrt n\,a to a; the Gaussian raw length-density coordinate keeps its original units. Primitive lattice covolumes are excluded here and will belong to the mixed counting measures.

At zero projection the constraint Hessians are S_{\mathsf P}, \operatorname{diag}(S_{\mathsf P},2\kappa_2), and \operatorname{diag}(bI_3,2b^2). Hence the choices

make all three saddle functions agree at zero. These are saddle normalizations, not divisions by the exact shell masses. The additional multiplier defining g on its cut is f_p^a/f_{g_0}^a.

These constant normalizations cost only polynomial for the unconditioned product Gaussian density integrals. On the cut we have

also with the corresponding Taylor bounds after up to two scaled derivatives (bounded derivatives for the prefactor log, and O(N |\vartheta|^{3-j}) for the rate-part log).

We include details of the expansions and bounds; all constants may use the scalar diagnostics above. In particular S_{\mathsf P}=I-O(q); Q^{-1}S_{\mathsf P} Q and its inverse are bounded. At typical sites, meaning the clipped region of those diagnostics with cutoff exponent sufficiently tiny, the profiles \mathsf P have bounded empirical absolute moments through any prescribed fixed degree, and y such moments at scale \sqrt q. Outside that region \langle y^2{\bf1}_{\rm tail}\rangle=O(q^4); the scaled profiles \mathsf P Q are bounded there by C |y|.

These bounds follow directly from (8.2), l=1+O(q), and the residual cubic normalization just noted. Centered moments involving \mathsf P and exponentially damped site fluctuations of order through fixed j\ge2 thus have tail bounds for their profile factors up to C_j(1+q^{4-3j/2}).

For p solve the linear-projection saddle by tilting by \mathsf P\theta, \theta=O_Q(|\vartheta|), where O_Q denotes a bound in Q^{-1} units. Indeed on that parameter ball the response beyond linear v\mathsf P\theta has empirical L^2 size O(q^{3/2}|Q^{-1}\theta|^2). Its scaled derivatives of order j through fixed orders have the bound C_j q^{3/2}|Q^{-1}\theta|^{\max(2-j,0)}. This follows by the tanh derivatives (second bounded by a constant times the absolute small argument on typical sites, using the moment bounds), and on the tail the derivatives are exponentially damped since the new field differs from y by at most a small fraction of |y|.

Thus scaled smooth inversion applies at uniform small radii. Evaluating the Legendre exponent at \theta=S_{\mathsf P}^{-1}\delta instead of the minimizer costs O(q^3 |\vartheta|^4) per site, by the ordinary Hessian bounds. The density exponent per site, apart from the Gaussian projection term -\delta^T S_{\mathsf P}^{-1}\delta/2, is

Indeed the third log moment term is -\langle mv (\mathsf P\theta)^3\rangle/3, with 111 profile average \langle mv \mathsf P_1^3\rangle=3\sqrt q+O(q^{3/2}), 112 average O(q) by (8.2).

The fourth term is -\langle v(1-3m^2)(\mathsf P\theta)^4\rangle/12, with 1111 average 3+O(q), 1112 average O(\sqrt q). The fifth coefficient has a factor O(m), and the sixth remainder is bounded with tail damping. All other scaled monomials thus cost O(q^3) by typical moments and the tail bounds. In the two leading powers \theta_1 can be replaced by \delta_1. The estimates retain their Taylor derivative orders: the nonlinear Legendre correction does also by scaled smooth inversion and the stated derivative bounds (the unscaled shift in the minimizing coefficient has norm O(q^{3/2}|\vartheta|^2) with the corresponding derivatives).

For g_0, first optimize the projection coefficients in the formal saddle and then optimize the length coefficient. The Gaussian projection exponent is exact. At this canonical projection saddle, the length target differs from the mean by -2\sqrt q\,\delta_1-\langle(v\mathsf P S_{\mathsf P}^{-1}\delta)^2\rangle per site. This is a saddle mean, rather than the trace of an exactly projection-conditioned finite-dimensional Gaussian. The length curvature in the partial log moment is 2\langle v^2\rangle+O(q|\vartheta|^2). Thus the additional exponent is the negative square of that deficit divided by 4\langle v^2\rangle, to O(q^3|\vartheta|^3); its leading quadratic cancels the explicit rank factor.

Since \langle(v\mathsf P S_{\mathsf P}^{-1}\delta)^2\rangle=|\delta|^2+O(q^3|\vartheta|^2), this gives exactly (11.3). These expansions use only the analytic bounded-variance Gaussian length tilt, adjusting the projection tilt by the corresponding three-dimensional linear solve. Length coefficients are O(q |\vartheta|), and scaled Taylor derivative estimates are unchanged. The calculation for o replaces the variance profile by constant b, giving the same terms. The Gaussian central prefactors for all three priors have uniformly bounded log derivatives in \vartheta by the nonsingular saddle Hessians and the profile bounds. This proves (11.2).

The signed pair comparison. We next form the pair saddle functions on |\vartheta|,|\vartheta'|\le\epsilon_0 and |t|/q\le\epsilon_1, where \epsilon_1 is fixed small and \epsilon_0 may be much smaller. For types i,j\in\{p,g_0,o\}, put

so n^{-1}\sum_\ell h_\ell=t at the projection constraints. Define

Keep the targets and the \delta’s inside h_\ell fixed when differentiating. On the local interior branch let \nabla_\theta\Lambda_{ij}(\theta_\beta,\beta)=c^{ij}, and put

Thus F_{ij}(0)=0; it is the partial infimum over the single-constraint coefficients, minus its value at zero cross tilt. It is a formal optimized log moment, not the conditional log moment of an unsmoothed atomic event.

At the optimizing law write

These are averages of sitewise covariances. Implicit differentiation and the envelope identity give

In particular the last expression is the variance after regressing the cross statistic on the single constraints.

Let \beta_{ij}(t) solve F_{ij}'(\beta_{ij})=t, with t=LU/n. The formal joint function in ordinary d^3a\,d^3a'\,dU, retaining every raw Gaussian length density, is

The factor n^3L converts the six raw projection sums and the raw cross sum nt=LU. Dividing this formula by its two single saddle functions, and converting U back to t, gives

The determinant quotient uses each pair’s own constraint dimension: 6,7,8 for pp, mixed pairs, and two Gaussian replicas. The Gaussian shell units and all explicit rank factors cancel in this quotient. Define the dimensionless formal ratio

It is equally the ratio in U units, since both densities then acquire the factor L/n. The full formal pair ratio is

This is an algebraic quotient of saddle formulas. The local-limit argument below relates it to the actual mixed statistic measures. After the multiplier defining g, both effective single factors are f_p^a; the projection cut does not change the conditional calculation.

The signed comparison and its verification below use the four ordered pairs i,j\in\{p,g_0\}. With signs s_p=1,s_{g_0}=-1 and E_*=1+|W|+|W'|+|U|, the full signed pair-density comparison needed is

Here cuts can multiply both sides; the coefficients are bounded and independent of the statistics. The constant \zeta can be made as small as needed by reducing the projection cut and the upper r. Central-prefactor relative errors O(q/N) are allowed.

Here is a verification. Abbreviate F_{ij} by F for the pair under discussion. At \beta=0 the canonical laws are independent single-replica laws. Write their site means and variances as \lambda_i,\sigma_i, where \sigma_i denotes variance rather than its square root, and put d_i^0=\lambda_i-\mathsf P\delta, with the primed target for the second replica.

Uniformly with scaled \vartheta derivatives,

Indeed the common linear response and nonlinear bounds were just obtained (for Gaussian the shifted length coefficient changes the response only nonlinearly).

The variance assertions use tanh or Gaussian differentiation as above; the last identity uses the exact target, automatic also for spins. The first bounds hold in any needed fixed typical-site empirical moment; on the tail d_i^0 and its scaled derivatives are bounded by C |y|.

Consequently

In the conditional curvature the centered-fluctuation product is orthogonal to the constraints, and the two linear remainder terms are regressed on them, proving the second formula, also to differentiated orders. For the size bounds with scaled projection derivatives used here, \sigma_i=v at zero, and \sigma_i and its scaled derivatives after empirical mean subtraction have L^2 size O(q): the logistic first derivative of variance brings a factor of the magnetization, and higher Taylor terms in the site shift or the Gaussian length coefficient already pay order q (use the same parameter estimates as for the single saddles).

On the tail all changes of the site fields and their scaled derivatives cost O(1+|y|) besides bounded quadratic coefficients, against fluctuation moments damped exponentially. Constraint covariance matrices and their needed scaled derivatives at the independent saddles are bounded, with eigenvalues bounded below using just bounded-profile sites. Regression coefficients for the two linear remainders and their derivatives therefore use ordinary unscaled constraint units (profiles \mathsf P, squares for Gaussian), with size O(q^{3/2}) by (11.5).

Thus F'(0) to quadratic order, of size O(q^3(|\vartheta|+|\vartheta'|)^2), is common to all four pairs. The curvature discrepancy from B(\vartheta)B(\vartheta') and its derivatives are O(q^2); it is common at zero projections, and to linear order varies by an additive sum of single-type terms. The third derivative is O(q) at \beta=0, with O(q) projection derivatives. Indeed it is the third cumulant averaged over sites of the regression-adjusted cross observable.

At zero cross tilt the product of centered site fluctuations contributes the average product of individual third moments, O(q) with derivatives. The linear corrections have coefficients d_i^0,d_j^0, and regression coefficients on site constraints D=(\mathsf P z,\mathsf P z',\text{applicable squares}) of size O(q^{3/2}(|\vartheta|+|\vartheta'|)). Their cumulant contributions and derivatives satisfy the stated bound by typical moments in (11.5); on tails they use at most exponentially damped polynomials in |y|+1, of tail average O(q^3).

For clarity, fourth cross derivatives stay bounded up to |\beta|\le C\epsilon_1 q. The constraint-coefficient shifts have first derivatives of norm O(q^{3/2}(|\vartheta|+|\vartheta'|)+|\beta|), second derivatives bounded. Indeed bootstrap their first derivatives at O(q), so shifts are O(q^2), giving only small parameter changes even in Q^{-1} units. The constraint Hessian D_2 (averaged covariance of D) stays well-conditioned. Centered site fluctuations of z,z' remain exponentially damped in |y| on tails, with bounded means: for a coupled Gaussian integrate the small quadratic tilt exactly, and for spins the conditional odds or Gaussian-integrated odds still have fields y+O(1)+o(1)|y|.

Thus the regression-adjusted observable H=(z-\mathsf P\delta)(z'-\mathsf P\delta')+\theta'_\beta D, with \theta_\beta the optimizing single coefficients, has bounded empirical centered fourth moment, using the O(q) coefficient times the worst \mathsf P_3 profile. Implicit differentiation then bounds \theta''_\beta by C\|\langle\kappa(D,H,H)\rangle\|\le C', closing the bootstrap (or use continuation at a larger threshold and small \epsilon_1); \kappa denotes joint cumulant.

The fourth derivative of F uses only the fourth cumulant of H and the constraint regression of its square. Similarly \partial_\beta D_2 is bounded (using at most two constraint factors and one H), and \partial_\beta^2 D_2=O(1+q^{-1/2}): with two H’s the worst four-profile tail costs Cq^2 q^{4-6}, while the \theta_\beta'' term costs a third constraint moment. At zero cross tilt \partial_\beta D_2 varies with bounded scaled projection derivatives by the same estimates. Hessian lower bounds throughout use just typical sites; in particular F'' stays bounded below. All these statements apply to mixed constraint dimensions without dummy square variables for spins.

The pair conditional exponent is therefore

The preceding calculations give the cross mean, curvature, and third cumulant at the independent saddle, with their projection derivatives. We now convert them into a conditional density. Both the prefactor and the Legendre exponent must be retained, because each contributes to the odd term in the signed comparison.

For the unweighted reference, the exact conditional law uses two independent uniform directions in dimension n-3. Writing B=B(\vartheta), B'=B(\vartheta'), its density is

on |t|<\sqrt{BB'}. The formal \widehat h_{oo}^{\,t} has the same power and replaces c_{n-3} by \sqrt{n/(2\pi)}; their ratio is 1+O(1/n), a permitted relative error. The reference Legendre rate exponent is \frac n2\log(1-t^2/(BB'))=-nt^2/(2BB')+O(nt^4). The remaining factor (1-t^2/(BB'))^{-3} belongs to the reference saddle prefactor. The pair conditional prefactor is (F''(\beta)\det D_2(\beta)/\det D_2(0))^{-1/2} at the dual \beta, while the reference prefactor is (BB')^{-1/2} up to O(t^2+1/n) relative error. The curvature part at \beta=0 thus has a common constant, additive linear log terms of size O(q^2(|\vartheta|+|\vartheta'|)), and error O(q^2(|\vartheta|+|\vartheta'|)^2). The log change at \beta is a projection-independent linear term in t, up to O(qE_*^C/N), by the preceding derivative bounds and dual inversion.

We organize the exponent difference by its dependence on the pair types. Common terms and terms additive in one type cancel after the signed sum. The remaining leading terms are odd in the normalized overlap; quadratic products of the smaller terms fit the remainder.

In the exponent difference, the t^2 terms give a common c_2 q U^2 plus additive single-type terms O(q U^2 (|W|+|W'|)/\sqrt N) and error O(q E_*^C/N). The term linear in F'(0) is common up to that error, of size O(q E_*^3/\sqrt N); its square contribution to the rate is negligible. The cubic term at leading order is odd, of order qU^3/\sqrt N, with coefficient at zero projections. All fourth-order remainders cost Cq E_*^C/N. Indeed nt^2=q^{-1}U^2,\ nt^3=U^3/\sqrt N,\ nt^4=q U^4/N.

In F''(0)-B B' the terms through projection degree one are just from the empirical covariance of the two site-variance profiles (both v at zero). Hence the inverse-curvature discrepancy has a common constant and single-replica-summed linear terms, with remainder O(q^2 E_*^2/N). In ntF'(0), the response product has a common bilinear part, leaving error O(q E_*^C/N) after multiplication. The log-prefactor shift uses \beta=(t-F'(0))/F''(0)+O((t-F'(0))^2); even its O((1+q^{-1/2})\beta^2) Taylor remainder fits.

Products of two of the small terms cost O(q E_*^C/N). Exponentiating (with the small Gaussian-growth envelope stated, using small cutoffs), the common and additive terms cancel from the signed sum, leaving just the claimed odd terms. Finally (11.2) allows the outside projection factor to be replaced by one on those odd terms to the stated remainder. We check the remainder sizes term by term, including at the ends of the cut window. Writing \mathcal W=|W|+|W'|, the linear-mean term ntF'(0)/F''(0) has common leading polynomial of size C q |U|\mathcal W^2/\sqrt N and remainder C q |U|\mathcal W^3/N, while nF'(0)^2\le C q^3\mathcal W^4/N.

The inverse curvature difference from 1/(B(\vartheta)B(\vartheta')) has quadratic-and-higher Taylor error C q^2\mathcal W^2/N. The cubic coefficient discrepancy from its value at zero projections is Cq\mathcal W/\sqrt N, and replacing t-F'(0) by t inside that cube also fits the stated error. Hence besides the common c_2 q U^2 and projection-independent log-prefactor constant (common because both variances are v at zero), log contributions retained are of size C q E_*^3/\sqrt N, common, single-type summed, or odd as indicated.

In exponentiating, both the leading polynomials in the log and the full log deviations (using n|t-F'(0)|^4 for the Legendre remainder) have absolute upper bounds by constants plus Cq E_*^2 on these cuts, before any outside projection log which adds O(|\vartheta|+|\vartheta'|+(|\vartheta|+|\vartheta'|)E_*^2). This yields the stated Gaussian-growth envelope even at window ends, and the common and single-type-summed terms indeed drop out after multiplying by the common projection weights and summing both signs.

The zero-projection coefficients and the exact common/additive cancellation are also written explicitly in Lemma 11.5. This proves the smooth expansion (11.4). Individually the full ratios have at most the same Gaussian-growth envelope, and differ from one on bounded normalized statistics by O(q)+o(1).

Density inversion, tails, and the common angular kernel. The four pair laws have different statistic measures. We keep their lattice coordinates and translated lengths exact, and smooth only the other coordinates. Each remainder is estimated on its own measure. The smooth leading terms are transferred to common volume before we perform the signed cancellation.

• •

  On the central region just used, one can evaluate orbit integrals as lattice sums and smooth densities with the saddle formulas, each to relative remainder O(q/N). Use lattice variables \sum x for each sign replica and \sum xx' for a pair of signs, on the translated lattice generated by differences of sign patterns. One can use a_1,a_3 as the remaining continuous-approximation projection coordinates for sign replicas. Smooth all nonlattice variables at inverse arbitrarily high polynomial scale with compactly supported smooth kernels, except exact Gaussian length constraints \sum(z^2+2mz)=nb.

  Changing an angular test by this smoothing costs an arbitrarily small inverse polynomial absolute error times the corresponding positive partition integrals (on slightly enlarged regions). Indeed projections and residual Gram factor parametrize the complement rotations, with Gram nonsingular here, so energies, pre-factors (in log), smooth cutoffs and the bounded-degree polynomial tests have polynomial variation. Only the Gaussian length itself needs remain imposed on the original site integral; inverse site variances need not be bounded.

  Projection multipliers also have polynomial log derivatives. When lattice variables are present keep them exact in this procedure. For instance for two signs one may hold (\sum x,\sum x',\sum xx') and smooth the two nonconstant projection coordinates for each sign; a_2,a'_2,U are recovered using those constant-coordinate sums, the other a’s and known site data. These are nonsingular affine or triangular coordinate changes (equivalently use the three raw statistics in site units together with the nonconstant profiles).

  For a mixed pair the raw lattice variable is just \sum x for the sign, and one also smooths the nonlattice overlap. Deterministic statistic Jacobians are understood in saddle densities and in converting lattice mass units. The tilt optimization using projected deviations rather than raw cross products is an affine change at the fixed projection constraints, with the same partial Legendre rate.

For the density estimation, tilt to the canonical product saddle at the evaluation point; variation of tilting factors across the mollifier costs negligible relative error. Fourier inversion there gives the Gaussian prefactor, including lattice covolume, to relative O((1+q^{-2})/n). Indeed covariance in site-sum units (normalized by \sqrt n) is bounded positive definite. We can use the well-conditioned equivalent raw coordinates (including zz' or xx' rather than the projection-subtracted statistic).

Maximum centered site variance in any such direction is O(q^{-3}), average third absolute and fourth moments O(1+q^{-1/2}), O(1+q^{-2}), by the bounds above. In the central Fourier expansion up to small times \sqrt N in normalized frequency units, each site factor stays close to one. The cubic log term integrates to zero at first order by oddness, and the fourth remainder and squared cubic term, with Gaussian damping, give the indicated error. One can evaluate at the exact mean here even with smoothing (or keep its transform, which changes central frequencies negligibly).

For more detail about Fourier tails, bounded typical sites damp through small constant frequencies in unnormalized (single-site) units by nonsingular covariance on that subset. Up to arbitrarily high polynomial frequencies thereafter there is exponential-in-size damping off the central region of the reciprocal-lattice fundamental cell. For a sign pair, phase comparisons on the four patterns test the two nonconstant profile combinations plus constants by single-spin flips (with both signs for the other spin).

Thus if nonconstant coefficients for either replica are not tiny, the nonlattice diagnostics give damping; with tiny coefficients on the bounded typical sites, only the reciprocal lattice of the constant three-pattern statistics can escape the test. More explicitly these statistics are the coefficients of x,x',xx', and equality of the four phases characterizes exactly that dual lattice. Typical-site pattern probabilities are bounded below. The single sign test works identically. For Gaussian variables on typical sites condition on any site signs first.

Quadratic Gaussian phases separated from zero give contraction irrespective of the linear part. With tiny quadratic coefficients, Gaussian linear phases separated from zero on a fixed fraction also give contraction; for a mixed sign pair one tests both sign choices to detect the cross coefficient. The Gaussian profile columns are well-conditioned on the bounded typical sites. Their conditional real means are bounded there. Thus failure of contraction from these tests requires all Gaussian-related coefficients tiny, when on this subset they only perturb the sign phase test by a small amount.

This argument can take successively small fixed cutoffs, or pass to sequences where contraction vanishes (quadratic phases must tend to zero, then Gaussian linears for each sign choice, then the empirical sign-phase dispersions). Beyond polynomial frequencies the mollifiers suffice; in unsmoothed length directions the Gaussian quadratic factors on typical sites themselves give a bound (1+c|\omega_{\rm lengths}|^2)^{-c'n}, also in the joint inversion. Here length direction means the single-site quadratic coefficients; the linear parts do not affect this bound.

Thus one is approximating ordinary joint constraint densities in the Gaussian length dimensions, and possibly smoothed/lattice joint masses elsewhere. Factoring a result into single prefactors and a conditional saddle expression (11.4) just uses determinant Schur factorization and rate subtraction at the level of these formulas; no division by tiny unsmoothed atomic projection probabilities is required. These estimates justify inversions in the different constraint dimensions and also the denominator calculations for conditional densities.

• •

  Overlap tails outside |t|\le\epsilon_1 q, with the projection cuts, are negligible (e^{-cN} times polynomial) in each pair orbit integral relative to Z_*^2, even for polynomial tests. First take \epsilon_1 q<|t|<c_* with c_* small fixed. Conditional density ratios can here be estimated as upper bounds on polynomially small boxes. At independent single canonical laws use exponential Markov on (z-\mathsf P\delta)(z'-\mathsf P\delta') up to tilts C c_*, without having to keep its constraint regressions optimized.

  Its mean is O(q^3); unconditional curvature differs from B B' by O(q^2) by (11.5); third cumulant is O(q), fourth derivatives bounded. Indeed single saddle parameters remain fixed in this tilt and the damping on tails is preserved (no large new profile coefficients, |\mathsf P\delta|\le C\epsilon_0 |y| there, and site constants do not enter higher cumulants). Thus relative to the reference conditional overlap density the log cost, besides polynomial losses, is at most

Imposing exact Gaussian length densities costs at most polynomial under these tilts by bounded typical-site Gaussian smoothing as in the inversion bound.

Projection marginals relative to reference cost at most \exp(C\epsilon_0^3 N) besides constants by (11.2), including our modification. Boxes can straddle lattice coordinates without needing any matching lower bound there for the numerator. In the physically tilted reference C_o, the length deficit of one of (z\pm z')/\sqrt2 is at least n|t|/2 from trace, on taking \epsilon_0\ll\epsilon_1. This costs \exp(-c n |t|^3) by negative square tilt of strength small-times t^2 (and conditioning at polynomial cost).

Indeed the covariance loses normalized trace at most C\sqrt{\xi}+o(q) at added precision \xi\ge0 small by (5.1) and finite-rank comparison; lengths have central denominators and upper density bounds polynomial under the tests by Gaussian Fourier damping. Equivalently exact shells here can use buffered radial conditioning on the spheres translated by m. The absorption is uniform down to the moving endpoint |t|=\epsilon_1q, since

is the scalar cost divided by n|t|^3, while the projection cost has relative size at most C(\epsilon_0/\epsilon_1)^3. Hold \epsilon_1 fixed, choose c_* and then the upper bound on q small, and finally choose \epsilon_0/\epsilon_1 small. The reference penalty then leaves a fixed multiple of n|t|^3, which is at least a fixed multiple of N. Angular comparisons over these boxes have polynomially controlled variation.

For larger |t|, use directly the angular semicircle pair upper bound from (6.3) in the complement of \mathcal P, whose Gram eigenvalues here are 1\pm t+o_{\rm small}(1), and ignore projected components at o_{\rm small}(n) cost. The log gain besides independent baselines is at most nt^2/2+o_{\rm small}(n). Indeed \log Z_*/n\ge1/4-1/2-o_{\rm small}(1) from its C_o formula by central length density, the semicircle log determinant upper bound and (9.1); finite-rank and prior normalization changes do not increase the log determinant by a nonvanishing order per spin.

The sign-sign prior pays the strict Cramér slack already used, with biased signs and translations changing bounds by o_{\rm small}(n). For Gaussian pairs, simultaneous length/overlap Chernoff gives at least the rate -\frac12\log(1-t^2) in the small-q limit, strictly larger than t^2/2 throughout this range. For two Gaussian types this uses fixed tilts of the Gram matrix, with log moments tending to the iid unit-variance values (v\le1); for mixed types condition on the sign vector, of bounded entries after translation and near-unit empirical length, and tilt the one Gaussian length and its projection on that vector, with the same limiting rate.

This follows immediately from the normal moment formulas since \langle|v-1|\rangle=q; the mixed vector entries are bounded. Interior finite tilts suffice including by approaching the endpoints, using compactness and only requesting strict slack. Length targets differ from 1 negligibly, and their exact Gaussian densities again cost polynomial under the tilts. Projection/pre-rank factors cost o_{\rm small}(n). These estimates also handle singular pair Grams directly, by splitting the fixed overlap range into sufficiently small fixed bins. This proves the tail assertion.

• •

  Here is the accuracy of kernel integration and of converting lattice sums to ordinary integrals. Under the tilted reference, condition two C_o-normals independently on their length shells, whose density denominators in L units are bounded above and below as in (5.6). Conditional on projections a,a' before length conditioning, their white coordinates \eta_i have fixed projections described by standard-normal coordinates \alpha,\alpha' (exactly whitened a,a'), and fresh standard orthogonal complements.

  In particular |W|\le C|\alpha|, and raw projection covariance eigenvalues are bounded below since K_o has bounded norm. Lengths and U are linear-quadratic forms in the remaining normals. The quadratic parts in a Fourier direction of norm one have operator and Hilbert-Schmidt norms at most C/\sqrt N,C, with spread giving determinant decay (1+c |v_f|^2/N)^{-c'N} at frequency v_f. Indeed they use C_o/L tensored with a two-by-two symmetric matrix (whose norm in such a direction is bounded below), up to the finite-rank compressions/subtractions, and (9.1) applies.

  The length linear part uses \|C_o^{1/2}m\|/L\le C. Consequently central densities with any fixed derivatives in a,a',U, including the projection density, are bounded by

To see the U-tail factor before inversion, tilt in U by parameter small-times the target U; up to \epsilon_1\sqrt N the quadratic norm shift is small. Its centered determinant cost is \exp(C\,{\rm parameter}^2) (off-diagonal tilt), and the projected translations cost at most \exp(o_{\rm small}(1)(|\alpha|^2+|\alpha'|^2)). The tilted spread is unchanged up to constants.

For the weighted matrix test, subtract C_o times the mass before forming the Hilbert–Schmidt square. The resulting insertion must be averaged with its sign: its absolute value would lose the Gaussian cancellation at the scale required here. Its insertion is

The next kernel estimate controls this signed average and, after the projection variables are integrated out, its smaller odd part.

Lemma 11.4, applied with C=C_o, A=A_0 and E=\mathsf P/\sqrt n, proves the signed-density bound (11.6) times Cnq, including its fixed derivatives. We verify its hypotheses explicitly. The formula for K_o, S_{\mathsf P}=I-O(q), and \|J\|\le3 give \|K_o\|\le C; therefore C_o\succeq C^{-1}I. The lower precision bound, trace-square bound, and edge spread follow from (9.1). Its first projection estimate also gives

Thus

The lower variance bound uses the edge spread. The density estimate (5.6), with its Fourier upper bound, now puts both shell density denominators between fixed positive constants in L units. After integrating out a,a', the odd parts of both the mass and matrix densities gain N^{-1/2}.

The lemma proves this by comparing the cross-frequency signs before projection conditioning; hence it does not assume an artificial symmetry of the translated shells. Projection cuts multiplying the leading odd term can be removed at exponentially small cost, since (11.6) gives Gaussian tails in the whitened projection variables.

Lattice sums after the density approximation use steps bounded by C/\sqrt n in the raw a’s and C/L in full-overlap L units (fixing the other projection coordinates first, with affine shifts). Their covolumes are exactly the prefactor ones from Fourier inversion. Passing to U has bounded derivatives on the cut: besides subtracting a\cdot a'/L, converting a full x-overlap to a z-overlap subtracts the m-linears with derivative \sqrt{nq}/L=q. Converting constant-axis sums to a_2’s uses \mathsf P_2’s formula with bounded coefficients.

Thus the centered-cell integration error for smooth leading terms is O(1/n) times their respective polynomial-moment envelope scale. More explicitly use the individual expansions leading to (11.4) with second derivatives of the explicit constant/common/additive and odd terms, keeping the projection factors via (11.2); in raw a,a',U units these derivatives cost just polynomials times e^{\zeta E_*^2}. Remainders O(q E_*^C e^{\zeta E_*^2}/N) require no derivatives.

The type of error matters here. Density-ratio errors, including relative saddle errors, are scalar functions of the statistics and multiply the same angular kernel as the leading terms. We make the exact denominator in this assertion explicit. Use the chart \Phi, Jacobian J_\Phi, mixed measures dm_i, and exact unweighted reference density f_{oo}^{\rm unw} from Lemma 11.2. For the reference single projection, isotropy gives the exact formula

Its ratio to f_o^a is a dimension-only factor 1+O(1/n) by Stirling’s formula. Together with the exact conditional overlap density above, this proves, uniformly on the regular window,

No raw shell density, rank factor, or normalizing constant has been removed from either side.

For the numerator inversion, first smooth the uncut statistic laws only in their nonlattice coordinates on a slightly enlarged window, and then apply the smooth projection cuts and the multiplier defining g. Let \widehat f_i^{\rm mod}, for i=pp,pg,gp,gg, denote the corresponding formal pair function with these factors included. Uniform inversion gives at each pair’s own mixed point

The factors are applied after smoothing, so this identity makes no relative-error assertion at a zero of a convolved cutoff. Commuting those factors through a mollifier of radius \delta instead changes the tested integral by at most \delta n^C times the positive mass on the enlarged window. The multiplier’s logarithmic derivatives and the cutoff derivatives have polynomial bounds there. The positive-mass estimate below and a sufficiently high inverse-polynomial choice of \delta make this separate error negligible.

Write \widehat R_i=\widehat f_i^{\rm mod}/\widehat f_{oo} in this chart, and let L_i be the finite smooth individual expansion used above for quadrature. Thus \widehat R_i=L_i+\widehat e_i, where |\widehat e_i|\le C(q/N)E_*^C e^{\zeta E_*^2}. Consequently the exact ratios used by the angular disintegration are

Here 1/n\le q/N, and the individual formal ratios have the Gaussian-growth envelope already proved. Only the smooth L_i undergo quadrature. Their signed sum has the leading odd term in (11.4), with its stated envelope remainder. These are absolute remainder bounds on each own mixed measure, including cutoff boundaries; no regularity of e_i is needed. Bounds (11.6) integrate or sum them against the absolute value of the signed angular average, not against the angular average of the absolute insertion. Thus the envelope in Lemma 11.3 is \mathscr E=E_*^C e^{\zeta E_*^2}.

Holding the length constraints exact ensures that at those points both angular averages use the same residual norms and Gram. Thus a scalar saddle error against an angular kernel is integrated using that kernel’s reference signed density at the points, not by integrating absolute Wick polynomials. Mollification changes the angular test itself and therefore requires a separate estimate. For that change we use absolute bounds: comparison of positive angular integrands at neighboring regular Grams, with polynomial derivative costs relative to base exponential weights.

Such costs times an arbitrarily small inverse polynomial are negligible here. Indeed the positive partition integrals with smooth cut to this region are polynomially bounded in reference units already by the scalar upper saddle bounds against the mass version of (11.6), summing also on the meshes; neighboring unsmoothed mass needed on a smaller region obeys the same type of bound by the angular energy comparison itself and unit mollifier mass. One can slightly enlarge regular windows in this justification and use smooth boundaries, without moving length or lattice statistics.

Cut boundaries can all use smooth tapers, neglecting the t-boundary by the preceding tails or (11.6). Lemmas 11.2 and 11.3 below state this disintegration and quadrature step in a single explicit chart, including its Jacobian and primitive lattice covolumes. They justify (11.4) inside both mass and signed-kernel tests without identifying the distinct atomic and continuous statistic measures.

Transfer of moments and final parameter choices. We now finish (11.1). All four pair laws have been related to the same angular kernel, with their individual remainders controlled. We can therefore square differences of the single-replica integrals.

Under Haar averaging, the prior density ratios multiply the common angular integral at fixed statistics. When the reference prior density and the two energy factors in that integral are combined and divided by Z_*^2, they give the tilted reference shell kernel. This is the mixed-measure identity made explicit below. The mass of modified g, divided by Z_*, tends to one in probability: its first and second moments follow from (11.2), the individual pair comparisons, and the tails.

The Hilbert-Schmidt second moments for the difference of the cut p and g integrals, still with common denominator, are bounded as follows:

Each row follows by expressing the square as a pair integral and using (11.4), (11.6).

The insertion determines whether an odd-part gain is needed. For the mean the inner product kernel L U+a\cdot a' needs no odd gain. For the raw second moment its square differs from L^2U^2 by at most L^2 E_*^4 C/\sqrt N, so the leading odd term is harmless. For the last test use exactly the signed H bounds. All constants here may use a fixed small upper bound on \epsilon_1,\zeta,\epsilon_0 satisfying the preceding absorptions.

On orientations obeying (10.1), the uncut g_0 physical law has centered covariance at most (1+o(1))K^{-1} and raw second moment at most C K^{-1}, exactly by the shell argument of (5.6). Here displacement of the standardized length target is bounded rather than vanishing, so use centered conditional Gaussian variance in that argument (joint Gaussian limit with any white projection, variance can only decrease). Projection and length density bounds needed for uniform moments follow after any bounded codimension compression by (9.1).

The modification defining g changes these estimates by o(1) in covariance units. Indeed W has uniform sub-Gaussian tails, (11.2) gives a factor tending to one with arbitrarily small Gaussian growth, and the removed tail begins at order \sqrt N. Cauchy–Schwarz applies uniformly to each projection moment test.

We spell out the normalization and centering. Write

where both priors include their cuts and g includes its projection multiplier. Since L/\sqrt N=q^{-2}, the four root-mean-square errors in (11.7) are, in order,

Choose the failure probabilities in Markov’s inequality in advance, with their sum and the preceding orientation losses below 1/4. For small q and large N, the resulting orientations have M_p,M_g bounded away from zero and close to one. For any matrix or vector test F,

The normalized modified Gaussian g has raw second moment O(q^{-2}), so its mean is O(q^{-1}). For F=z and F=zz^T, the last display and the first three errors show that the normalized mean difference is O(q^{-1/2})+o(q^{-1}), and the normalized raw second-moment difference is O(q^{-3/2})+o(q^{-2}). Subtracting the two mean outer products costs at most O(q^{-3/2})+o(q^{-2}). Consequently the cut spin covariance is bounded above by (1+o(1))K^{-1}+o(q^{-2})I.

For the weighted estimates choose the loss in (9.1) separately, denoting it by \eta_{\rm mat}, with 2\eta_{\rm mat}<\gamma. The normalized modified Gaussian g has raw weighted second moment O(q^{-2\eta_{\rm mat}}). Use the last row of (11.7) in the normalization identity with F=A_0(zz^T-C_o)A_0. The subtraction is a constant under each normalized law, so the difference of raw weighted second moments is

Here \gamma>1/2 is exactly what makes the first error smaller than the target. The cut spin raw weighted second moment is therefore o(q^{-\gamma}). It bounds the weighted covariance; Jensen’s inequality bounds the squared weighted mean by the same quantity. In the unweighted case the raw second moment is O(q^{-2}), so the mean is O(q^{-1})=o(q^{-1-\eta}) for the independently prescribed \eta>0.

Finally restore the removed spin mass. If it is at most n^{-D}, then \|X-m\|\le2\sqrt n bounds the changes in normalized means and raw second moments by Cn^{1/2-D} and Cn^{1-D}, respectively; the same bounds, up to constants, hold after A_0 since \|A_0\|\le5. Choose D large. The sharp unweighted coefficient follows by choosing the precision, shell, transfer, and q/r tolerances so that

This proves (11.1) with all the prescribed small constants.

The same event also provides the mass anchor needed in the next section. We may require M_p\ge1/2, and positivity of the omitted integral gives

It holds simultaneously with the norm bounds on the retained orientation set. On any fixed compact observation-time interval bounded away from zero, contained in the established small-time regime, and on which r is bounded below, the translated posterior raw second moment \E[(X-m)(X-m)^T\mid J,h] is bounded by a fixed constant in operator norm on these orientations, using the covariance and mean estimates.

All choices in this static assertion begin with coarse scalar tolerances and matrix budgets giving fixed constants in (8.2)–(10.1), and with the orientation failure allocations just described. Choose the overlap cutoff and then the projection cutoff small, including for the Gaussian envelopes, and use a sufficiently small scalar and overlap rate to put q/r close to one and control the projection-cut loss.

Stricter diagnostics do not require increasing the coarse constants. Typical-site moment and tail cutoff exponents can be chosen for the finite derivative orders used. The upper bound on r is then sufficiently small in terms of these choices, which have not used dynamical probability/moment inputs.□

Proof. Two ordered residual pairs with the same positive Gram matrix are related by an orthogonal map. If its determinant is negative, compose with a reflection on the orthogonal complement of the target pair; this fixes the pair and corrects the determinant. Haar invariance and Fubini therefore give, for every finite pair measure \mu, atomic or continuous, and every integrable B,

where \mathcal A_B is its angular average. For signed insertions use absolute integrability. Moving the rotation from the interaction matrix to the pair vectors gives precisely this identity for orbit averages in the application; all matrix factors in the insertion move together. For the reference prior, o is invariant under the rotations: its isotropic Gaussian density, translated shell, and projection rank factor are all invariant. Applying this disintegration to e^{\mathcal E(z)+\mathcal E(z')}F(z,z')\,o(dz)o(dz'), and then using Z_*^{-1}e^{\mathcal E}o=\operatorname{Law}(Z\mid T_o=0), proves f_{oo}^{\rm unw}\mathcal A_F=Z_*^2k_F. For a signed insertion absolute integrability justifies Fubini; the insertion keeps its sign inside the angular average.

The chart formulas follow by expanding \sum x_i and x^Tx'. Its Jacobian is triangular, with the three new diagonal factors 1/(\sqrt n c),1/(\sqrt n c),1/L. For pp, the four sign-pattern counts are

Their integrality gives the stated lattice and determinant 16. The nonnegativity boundaries lie outside the central regular window. In particular the conditional R step is 4, so three independent step-2 lattices would give the wrong normalization.

The mixed lattices have only the one sign-sum coordinate. Finally the asserted integral formula follows by multiplying and dividing the smoothed scalar density by the positive unweighted reference density evaluated at the same point. This is an identity in the chosen mixed measures, not a Radon–Nikodym derivative of an atomic statistic law with respect to a continuous law.□

Proof. Bound the four errors separately on their own meshes, perform quadrature on the four smooth terms, and only then add their ordinary-volume integrals. No matching support or regularity of the residual functions is needed. The relevant absolute value is |\widetilde k|, the absolute value of the signed angular average; the insertion is not replaced by its absolute value.□

Proof. Both B=C/L and B_\Pi=C^{1/2}\Pi C^{1/2}/L have operator norm O(N^{-1/2}), Hilbert–Schmidt norm O(1), and nuclear norm O(\sqrt{nq}). Their difference has rank at most d. Conditioning on a_i amounts to writing

where \xi_i is standard Gaussian on S^\perp.

Take Fourier transforms in the two lengths and in U, allowing a real cross tilt u with |u|\le c\sqrt N. The precision then has real part bounded above and below by fixed positive constants. Let \Sigma,\mu denote the complex Gaussian covariance and mean. The resolvent identity, the preceding three norm bounds, and the fixed-rank projection correction give

where P is a polynomial in the frequencies, |u|, and 1+|\alpha_1|+|\alpha_2|. The length linear coefficient is C^{1/2}m/L; projection conditioning adds coefficients of norm at most C(|\alpha_1|+|\alpha_2|)/\sqrt N.

Write \Sigma_{11}=I+\Delta_1, \Sigma_{22}=I+\Delta_2, R=\Sigma_{12}, and s=\mu_1^TD\mu_2. Wick’s formula is exactly

All zero-mean terms except the trace square cost a polynomial by Hilbert–Schmidt bounds. The exceptional term satisfies

The mean terms cost C\sqrt{nq}P. In particular the terms \operatorname{Tr}D^2 and \operatorname{Tr}(D^2\Delta_i) have canceled identically. This is an estimate for the signed Gaussian expectation, not for the absolute insertion.

For real Fourier variable v\in\mathbb R^3, determinant damping is

Before fixed-rank changes, Q is the tensor product of C/L with the symmetric two-by-two frequency matrix. The latter has an eigenvalue of magnitude at least a constant times |v|. The spread assumption supplies cN singular values of size c|v|/\sqrt N; interlacing loses only a fixed number. A real cross tilt conjugates the Fourier quadratic matrix by bounded invertible operators, so the same bound holds with changed constants.

Every fixed frequency polynomial is integrable against this bound, uniformly for large N. Fourier inversion therefore proves the density bounds and their fixed derivatives. To obtain the Gaussian tail in U, shift the cross contour by u=c_1U. Its centered determinant cost is O(u^2), since the cross tilt has trace zero and bounded Hilbert–Schmidt norm. The projected mean cost is at most C\epsilon(|\alpha_1|^2+|\alpha_2|^2) on the stated window. Choose c_1 and then \epsilon small. The factor e^{-c_1U^2} and the projection Gaussian density absorb these costs. Division by the bounded shell denominators completes these bounds.

The parity estimate uses the unconditioned projection variables. Integrate them out before Fourier inversion, so that reversing the cross frequency can be compared at the level of the Gaussian formula. Reversing U reverses both the cross Fourier frequency and the real cross tilt. Conjugation by \operatorname{diag}(I,-I) preserves the determinant and all zero-mean contractions of H. The inverse cross block satisfies the sharper operator estimate

by the block inverse formula.

The length linear coefficients have bounded norms times |v|. Their Gaussian exponential factors therefore differ by at most N^{-1/2} times a frequency polynomial. This estimate keeps the determinant damping: the two exponents and the segment between them have nonpositive real part, because the linear coefficients are imaginary and the real precision is positive. For the mass this proves the extra N^{-1/2}. For H, its zero-mean Wick expression costs Cnq times this factor. The mean terms can be bounded directly by C\sqrt{nq}P, which suffices since \sqrt{nq}\le nq/\sqrt N for q\le1. Fourier inversion with the same real tilt proves the odd-part bounds and their derivatives.□

Proof. At zero constraints both canonical site laws have mean zero and variance v_i.

Their third centered moments are -2m_iv_i for signs and zero for Gaussians. The cross observable is orthogonal to all single-replica constraints. The first derivative of the optimizing coefficients thus vanishes, so the first three optimized cross derivatives are its ordinary first three cumulants. This proves the displayed identities.

Let D_2(\beta) be the constraint covariance. At zero cross tilt it is block diagonal. Each diagonal block of D_2'(0) vanishes: the corresponding third cumulant contains a centered factor from the other replica. Hence \left.\partial_\beta\log\det D_2(\beta)\right|_0=0. The conditional saddle prefactor is proportional to [F''(\beta)\det D_2(\beta)/\det D_2(0)]^{-1/2}, whose logarithmic t derivative is therefore -F'''(0)/(2\kappa_2^2). Its cubic Legendre term is nF'''(0)t^3/(6\kappa_2^3). Now use t=qU/\sqrt N, nt^2=U^2/q, and nt^3=U^3/\sqrt N. Reference finite-codimension corrections O(t^2+1/n) lie within O(q/N) on bounded normalized statistics.

Away from zero the Taylor estimates following (11.5) organize the logarithms as \log R_{ij}=L_0+C_*+A_i+A'_j+B_{ij}+r_{ij}. Here L_0 is common constant/quadratic, C_* is common, A_i,A'_j are additive first projection terms, B_{ij} is the odd term just computed, and the retained small terms have size CqE_*^C/\sqrt N. The remainder has size CqE_*^Ce^{\zeta E_*^2}/N. With s_p=1,s_{g_0}=-1, \sum s_is_j=\sum s_is_j(C_*+A_i+A'_j)=0. Quadratic products from exponentiation cost q^2/N\le q/N.

The common projection multiplier differs from one by N^{-1/2}E_*^Ce^{\zeta E_*^2}, so its effect on the surviving odd term also fits the stated remainder. The projection estimate (11.2), the derivative bounds following (11.5), and the mixed local-limit calculation therefore give precisely this cancellation for the smooth saddle ratios. Lemma 11.3 passes from these ratios to the actual mixed statistic measures.□

Proof. There are three errors to account for: scalar remainders, the surviving odd term, and quadrature.

The scalar remainders cost (q/N)(nq) using the signed density bound on each mesh separately. After quadrature, the odd leading term costs (q/\sqrt N)(nq/\sqrt N). Quadrature itself costs (nq)/n=q\le nq^2/N. Removing smooth projection cuts is exponentially negligible since |W|\le C|\alpha| and the cuts start at |W| of order \sqrt N. Mollification is separately controlled by an absolute angular derivative bound and an arbitrarily high inverse-polynomial smoothing scale. The scalar saddle remainders never require placing |H| inside the angular integral.□

Proof. Use the root orbits, their three projection columns P, and the reference partition integral Z_* from the local comparison. We write \vartheta for the normalized projection coordinate, so that \delta=\operatorname{diag}(\sqrt q,q,q^{3/2})\vartheta, and put z=X-m, N=nq^3. For different replicas let

On this compact interval all polynomial bounds may depend on r_1.

First, outside Haar mass e^{-Bk}, with any prescribed fixed B, the full tilted spin integral is at least Z_*e^{-o(n)}. Indeed (11.8) supplies a fixed positive fraction of orientations with full mass at least Z_*/2. The simultaneous covariance and mean bounds in (11.1) give a bounded raw operator second moment of z on this compact interval. Jensen’s inequality, evaluated at such an anchor, bounds the loss under a rotation at fixed Frobenius distance by C\sqrt n+O(1): the linear energy increment pairs with a raw second-moment matrix of Hilbert–Schmidt norm O(\sqrt n), and the quadratic increment costs O(1).

Concentration of distance to the anchor set excludes distances greater than a sufficiently large fixed constant with probability e^{-Bk}. The determinant weight changes this estimate only by the already established arbitrarily small loss in its rate.

Fix a small absolute D_*>0. For any fixed replica number l, consider the unnormalized tilted spin integral on

Its Haar mean, divided by Z_*^l, is at most e^{-cD_*N}, where c>0 is independent of l, although the threshold in n may depend on l. We verify this uniformity because the replica number will depend on \chi.

Under the tilted reference, the projection coordinates \sqrt N\,\vartheta_i have bounded Gaussian covariance before length conditioning. Every linear combination with unit Euclidean coefficients of the off-diagonal coordinates nt_{ij}/\sqrt{n/q} is a centered Gaussian quadratic form whose matrix has bounded Hilbert–Schmidt norm and operator norm at most C/\sqrt N. These constants are independent of l. Gaussian exponential Markov bounds, followed by a fixed-mesh net, therefore give cost at least c_0D_*N for (12.1). The net and conditioning on the finitely many lengths cost only polynomial factors for fixed l; the latter follows either from the length-density bounds or from buffered shells.

The annulus penalty just obtained is under the physically tilted reference shell, with covariance C_o before length conditioning. The change of prior uses a different density: the unweighted o prior has uniform complement directions after its projections and lengths are fixed. The common angular energy integral is multiplied by the spin Gram density divided by this reference Gram density. We estimate this quotient using polynomial boxes and Chernoff upper bounds. The single-projection density ratios in (11.2) cost at most CN D_*^{3/2}+o(n). At fixed projections the unweighted reference conditional Gram density has exponent

up to polynomial factors. At the independent spin projection saddles write the site deviations as \xi_i+d_i^0, with independent centered bounded \xi_i.

Set R_{ij}=t_{ij}/(B_iB_j) and R_{ii}=0. The quadratic log moment satisfies

with an absolute constant when \|R\|_{\mathrm{HS}} is small. For completeness, decouple the off-diagonal quadratic form using a random bipartition. Jensen’s inequality bounds its exponential moment by that of four times the corresponding directed cross sum. Integrating one side by the bounded-variable Gaussian moment bound produces a squared linear-image penalty; introducing a white Gaussian and integrating the other side gives a uniform exponential moment for a small multiple of |\xi^TR\xi|/\|R\|_{\mathrm{HS}}. Taylor’s formula then gives the displayed third-order remainder.

The linear terms involving d_i^0 are treated by Hölder’s inequality with relative inflation O(q\sqrt{D_*}) in the centered quadratic moment. The remaining averaged log cost is at most

Here we used the response bounds (11.5) for d_i^0 and linear sub-Gaussianity. The site-averaged quadratic coefficient for each pair differs from B_iB_j by O(q^2). To see that the constant is independent of l, note on the annulus that

The latter inequality is (11.5) summed over replicas. The displayed log-moment bound is first applied at each site and then averaged. Its cubic remainder costs n\|R\|_{\rm HS}^3\le CN D_*^{3/2}, the variance-profile discrepancy costs nq^2\|R\|_{\rm HS}^2\le CNqD_*, and the displayed Hölder cost, after multiplication by n, is at most CNqD_*^{3/2}. These estimates use only sums of squares and the Hilbert–Schmidt norm. Their constants therefore do not grow with the fixed replica number. After the Chernoff subtraction, the total relative cost is bounded by CN D_*(q+\sqrt{D_*}), besides polynomial factors. Choosing first D_*, then the upper bound on q, absorbs this cost into c_0D_*N. Angular integrals on these boxes have controlled variation; their constants may depend on l,r_1.

The annulus rate is independent of the fixed replica count. This lets us turn a requested overlap accuracy into a sufficiently large, but still fixed, number of replicas. A first-crossing selection will then find an annulus subset without allowing a single large statistic to account for the whole crossing.

We need an exponential posterior bound for each individually large statistic, not just the polynomial cutoff loss used in the local comparison. Let \mathcal R_s be the domain where all root/scalar diagnostics retained for the local comparison hold. On this domain the root is unique and regular, q>0, and the three axes are nondegenerate. Its excluded complement, including nonunique-root fields, has separate charge

after taking the fixed a_0>0 below the preceding exclusion rates.

Only on \mathcal R_s, define E_1=\{|\vartheta_1|^2>D_*/4\} for one fresh posterior replica and E_2=\{|t_{12}/q|^2>D_*/4\} for two. The plant and retained root/scalar data remain fixed inside \mu_{h_s}^{\otimes j}. The scalar-root estimates after equivariant plant replacement, and the plant–replica and replica–replica overlap estimates, give

after decreasing a_0 if necessary. Conditional Markov on this retained domain gives

Relabeling E_1,E_2 and taking a finite union covers every individual statistic in any fixed replica family. The complement of \mathcal R_s remains the separate charge displayed above.

Suppose the conclusion of the lemma fails. Draw a sufficiently large fixed number l_{\rm pair} of disjoint violating pairs, depending on \chi. It uses 2l_{\rm pair} replicas; the preceding estimates apply to every subset with l equal to that subset’s replica count. Since

their statistics force the total in (12.1) to cross D_*. Requiring all l_{\rm pair} pairs costs at most l_{\rm pair}c n in log likelihood. Choose c so that l_{\rm pair}c n\le a_0k/4, possible uniformly on the compact interval because k\ge nr_1^3. The preceding exponential conditional bounds then make all individually large statistics, including those across pairs, negligible even after this conditioning. This choice explains the dependence of c on r_1,\chi, while the field-rate constant remains fixed.

A subset of the selected replicas then satisfies (12.1). Indeed take the first prefix crossing D_*. If its new row contributes too much, retain enough earlier indices to make that row contribution comparable to D_*; each of its terms is at most D_*/4, while the earlier internal sum is below D_*. A finite union over subsets, the denominator lower bound, and the preceding Haar estimate exclude the violating orientations at a rate bounded below independently of the number of replicas. Returning to the counting weight costs o(k) on this compact interval. The initially excluded orbits had a fixed small rate as well. This proves the lemma.□

Proof. We first transfer small exceptional sets from P to G. Since \int e^{L_P}dG=1, the second moment in (12.2) and Paley–Zygmund give

for sufficiently large \alpha k; enlarging harmless constants covers the other case. Also G\{L_P>\beta k/8\}\le e^{-\beta k/8}. Convexity implies that

for an absolute c>0.

Here is a proof that does not presume regular boundaries of the two level sets. Use Gaussian endpoint conditioning to drive two Brownian motions with the same noise into \{L_P\ge-1\} and a proposed low set of mass at least e^{-c\beta k}, with joint success probability greater than 0.8. Positive smooth approximations to the conditioning densities and energy stopping, in the entropy–drift representation of Lehec [25, Proposition 1 and Section 2.5], give total expected squared drift energy at most C(\alpha+c\beta)k.

Reflect the second control about the first. The third endpoint is the reflected point, so convexity forces its likelihood above \beta k/8 on the joint hit, for large k. Its relative entropy with respect to G is at most C'(\alpha+c\beta)k, by Girsanov and data processing. Binary entropy applied to \{L_P>\beta k/8\} contradicts this bound if c and \alpha/\beta are small enough. This proves (12.4). Consequently, if P(A)\le e^{-\beta k/2}, then

The likelihood comparison has transferred rare starting sets to the Gaussian reference. We now obtain contraction along typical Langevin paths and telescope short-time variances to get the gradient term. A final comparison through the Gaussian reference will remove the remaining terminal variance.

Run stationary Langevin diffusion for P, with generator \Delta+\nabla\log(dP/dg)\cdot\nabla, for T'=C_D\log n. Apart from probability \operatorname{poly}(n)e^{-\beta k}+e^{-n^2}, its log-density Hessian is at most -2I/3 all along the path. Here are the buffers needed for this path statement. Let M_n be a polynomial bound for the Lipschitz constant of the Hessian, supplied by the bounded-spin formulas for posterior derivatives, and take \rho=(100M_n)^{-1}. At every point of a polynomial time grid, stationarity and (12.2) give the upper bound -3I/4 outside total probability \operatorname{poly}(n)e^{-\beta k}. The drift has linear growth with a polynomial additive bound. Choose the grid mesh polynomially small enough that, on a polynomial ball, its drift and Brownian increments are less than \rho/4 between successive grid points. Gaussian increment bounds and the tails of the Gaussian mixture P make the remaining probability at most e^{-n^2}. The Hessian Lipschitz bound then preserves a strict upper bound below -2I/3 throughout the tube of radius \rho about the path. The fixed observation bound and fixed \beta give \beta k=O(n). Thus these Gaussian-tail errors remain negligible after the exponential reweightings below. The lemma is only asserted on this compact observation domain.

This tube also controls neighboring flows for that same future noise. Start a synchronous flow within \rho/4 of the path’s initial point and stop at the first time its distance from the path reaches \rho/2. Before that time, the mean Hessian along the line segment between the flows is at most -2I/3, so their distance contracts. It cannot reach the stopping boundary. Differentiating this stopped comparison and keeping a little slack gives Jacobian norm at most e^{-v/2} for every v\le T'. This is a neighborhood of a good random path, not a claim about every initial point.

Let K_v be the Langevin semigroup. Choose a polynomially small \Delta t and an integer N_t with T'=N_t\Delta t, adjusting T' negligibly. Stationarity gives the exact telescope

Every short transition K_{\Delta t}(x,\cdot) has Poincaré constant at most (2+o(1))\Delta t, uniformly in x. Indeed the global upper Hessian bound is polynomial, hence the flow Jacobian is 1+o(1) up to this time; the conditional heat variance identity integrated over the short interval gives the assertion.

For a variance term involving K_v\phi, Jensen in the common future noise bounds \operatorname{Var}_{K_{\Delta t}(x,\cdot)}K_v\phi by the average variance of y\mapsto\phi(Z_v(y)), where Z_v is that future Langevin flow. On a good path from x, cut this test off on the inverse-polynomial ball just described. Its gradient there is bounded by e^{-v/2}\|\nabla\phi(Z_v(y))\|. Crossing and cutoff-annulus errors are negligible by short-time Gaussian tails after making \Delta t small enough.

The good-path event is determined from the future path started at x, and the cutoff test has polynomial bounds on this ball. Bad paths, averaged over stationary x, cost only a polynomial times their probability. On good paths, drop the event indicator from the resulting nonnegative gradient integral. Then x\sim P, y\sim K_{\Delta t}(x,\cdot), and independent future noise make Z_v(y) stationary. Each variance term is therefore at most

plus an arbitrarily small inverse-polynomial error. Summing over the polynomial number of time steps preserves the negligible error. The gradient contributions sum to at most (2+o(1))\int\|\nabla\phi\|^2dP.

It remains to bound \operatorname{Var}_P K_{T'}\phi. Let b(x) be the conditional probability that the future path from x fails the preceding tube test. Its P mean is at most \operatorname{poly}(n)e^{-\beta k}+e^{-n^2}. Markov at e^{-\beta k/3}, using \beta k\gg\log n, leaves a set of P-mass at most e^{-\beta k/2}. Off that set, averaging the common-noise coupling gives local differences bounded by e^{-T'/2}\operatorname{Lip}(\phi) times distance, plus an arbitrarily small inverse-polynomial error. Transfer the exceptional starting set to G using (12.4). For independent g_0,g_1\sim G, use a polynomial mesh on \cos\theta\,g_0+\sin\theta\,g_1, 0\le\theta\le\pi/2. Every vertex has law G, and Gaussian norm tails put adjacent vertices within the local ball. A union bound over the vertices leaves Gaussian exceptional probability \operatorname{poly}(n)e^{-c\beta k}.

The local estimates telescope along this polygon. To reweight its endpoints to two independent P-points, Cauchy–Schwarz and (12.2) bound the exceptional probability by

It is smaller than any required inverse polynomial when \alpha/\beta is sufficiently small and \beta k/\log n is sufficiently large. Taking C_D large gives the desired residual variance bound. Absorb the remaining errors in the slack between 2+o(1) and 3.□

Proof. Each backward step first uses later observations to obtain coarse covariance and moment bounds. These control derivatives along rotation orbits, where the static majority estimates reset covariance to its sharp threshold. The initial positive-time band uses the same reset; its coarse bounds come from the separate fixed-positive-time and terminal inputs. We carry all three estimates together. First fix 0<\iota<\gamma'/2, with the target covariance slack \epsilon small enough for the variance calculation below. That calculation gives a constant C_\iota' independent of L_0 and of the relative step size. Choose L_0 large enough that C_\iota'L_0^{\iota-\gamma'/2}<1/4, and then choose s_0 small enough for the static medians and Lemma 12.1.

The quadratic-variance constant C_v and the auxiliary fourth-moment constants below may depend on these choices, but will not depend on the eventual relative step size. The initial band is [s_0/L_0,s_0]. Let F_b^+ denote the positive-time covariance failure, including the terminal gap failure when b=T, and define on this band

Here and below \Pr_s conditions on (J,h_s), and the fixed C dominates the weights in the moment transfers. Tonelli and Markov on the fixed interval [s_0/L_0,T] give a fixed c_{\rm init}>0 such that Q_s^+\le e^{-c_{\rm init}n} outside integrated mass e^{-c_{\rm init}n}. On this one event, the bounded-covariance martingale transfer to T and its gap bound give all the initial quadratic and fourth-moment bounds proved below. No conditional variance lemma is needed for this initial transfer. Orientation concentration will reset the covariance thresholds on the same band.

Let c_+>0 be a positive-time failure rate on [s_0,T], and let a_{\rm stat}>0 be smaller than the fixed root, orbit, diagnostic, and spherical-width exclusion rates at the accuracies used below. The compact bridge has its endpoint-independent rate a_*. Choose a>0 once, with

and small enough for the final orbit-gradient condition whenever its concentration exponent is at most 2a. That condition uses only the moment constants and the already fixed s_0,L_0.

A conditional event with room in its probability exponent. For s<s_0/L_0, put u=(1+d)s and t=L_0s, and suppose the induction estimates hold at all required times at least u. The active d will be one of two fixed values specified below; the rate a will not be decreased after those choices. Let F_b be the union of the three inductive failures at b\le s_0, and the positive-time covariance failure at s_0<b\le T. Include the terminal gap failure in F_T. Define

Choose the fixed exponent C large enough to dominate the polynomial weights in the variance and fourth-moment estimates below. All these weights are polynomial in n on the stated time range. Tonelli’s theorem and the deterministic-time induction give

Here \mathcal G is measurable at time s; the equality behind Tonelli therefore retains its indicator. Above s_0 we used the positive-time inputs on the fixed interval [s_0,T].

Set \beta=c'da, with a sufficiently small absolute c'>0. Markov at Q_s=e^{-\beta k} gives, before polynomial costs, the exponent

For small fixed d this strictly exceeds ak. The other Markov term is \operatorname{poly}(n)e^{-c_+n+\beta k}; the prior choice of a, and k\le ns_0^{3/2}, also place it strictly above the required rate. We next intersect this test with the likelihood and diagnostic events needed for the derivative bounds.

For sufficiently small d, the covariance hypothesis in (12.2) follows from Q_s\le e^{-\beta k}. The likelihood hypothesis follows from the exact Gaussian identity

The initial band is already fixed, so the back-step must be chosen without shrinking that band again. Two likelihood estimates provide this separation of choices: the spherical log-moment estimate near zero, and the compact overlap bridge on the remaining interval.

There are two parameter ranges. At sufficiently small r, use the spherical shift m_{\mathrm{sh}}=m_o. Choose d_{\rm near}>0 so that Cd_{\rm near}^2 is a sufficiently small multiple of d_{\rm near}a. Apply the off-diagonal replica log-moment bound (6.2) to this fixed two-replica test, with requested failure rate 4a. Given \varepsilon_{\rm like}>0, it gives

on a sufficiently small upper range. Choose \varepsilon_{\rm like}\ll d_{\rm near}a, then its upper endpoint r_c>0, and finally take K_{\log} large enough that C_2\log n\ll d_{\rm near}ak. Shrinking r_c alone does not absorb this logarithmic term. Only this selected test is needed, not simultaneous control for all prospective step sizes. The width exclusion has the larger rate fixed above. On the remaining compact interval use m_{\mathrm{sh}}=m and Lemma 12.1, with \chi\ll a.

Choose a second step d_{\rm comp}>0 on this compact interval. The good-pair contribution to the second moment is at most e^{d_{\rm comp}\chi k}, while the exceptional-pair contribution is at most

using \|X-m\|\le2\sqrt n. Choose d_{\rm comp} after the compact-tail constant c so that this term is negligible. Then \alpha\le 2d_{\rm comp}\chi for large n, while \beta=c'd_{\rm comp}a. In either range, \alpha/\beta in (12.2) is as small as required. Thus no choice of s_0 depends on a or on these small step sizes. Choose K_{\log} last to absorb the polynomial costs of Lemma 12.2.

For each active step let \mathcal L_s be its likelihood event just constructed, and let \mathcal D_s retain the diagnostics and root or spherical-width events used in its orbit comparison. The full event used from now on is

The first two factors are tests of (J,h_s), and are the only ones used to apply the conditional variance lemma under \Pr_s. The diagnostic factor may also use the plant and root-orbit data; it restricts the subsequent orbit integration without further conditioning the posterior spin law. Put

The preceding estimates give

All assertions remain integrated with \mathcal G imposed; none is a bound conditional on every spectrum. On the initial band define \mathcal C_s instead as \{Q_s^+\le e^{-c_{\rm init}n}\}\cap\mathcal D_s, whose excluded mass has a fixed rate larger than 4a.

Coarse covariance and a longer variance transfer. On \mathcal C_s, total covariance gives

Apply (12.3) to every linear projection of m_u^{\mathrm{post}}. Its derivative in g is \sqrt\tau\,\Sigma_u, so the second term is at most 3\tau\mathbb E_s\Sigma_u^2, up to an arbitrarily small inverse-polynomial operator error. Apply the same argument after multiplication by A_0. This gives both covariance bounds with fixed coarse factors.

For any smooth polynomially bounded test of the later field,

Here \iota>0 is arbitrarily small if \epsilon is chosen sufficiently small, and C_\iota is independent of L_0,d. To prove this, the posterior-expectation martingale on [u,t] has diffusion coefficient

This follows directly from the Gaussian likelihood of the future observation increment given X.

The observations have unit diffusion and posterior-mean drift in their own filtration. If R_b=\mathbb E_s\operatorname{Var}_b(\phi(h_t)), their martingale variance identity and Cauchy–Schwarz yield

In the covariance term use \|\operatorname{Cov}_b(X,Y)\|^2\le \|\Sigma_b\|\operatorname{Var}_bY, and absorb the mixed term by the slack in \iota.

Integrating backwards from R_t=0 gives explicitly

The last term is negligible on the conditional event. For the first interval [s,u], apply (12.3) to \mathbb E_u\phi(h_t) and use the same gradient estimate. It adds at most a fixed multiple of R_u+\tau\E_s\|\nabla\phi(h_t)\|^2, since \tau/u\le d. As t/u\le L_0, this proves (12.7) with a constant independent of L_0,d. When covariance is bounded on a fixed positive band beginning at s, this same martingale argument works directly from s up to T, with fixed constants in place of the displayed powers.

Quadratic and fourth moments. Let M=[H,J]=-[H,A_0], with \|H\|_{\mathrm{HS}}\le1, and put D_t=m_t^{\mathrm{post}}-m_s^{\mathrm{post}}. Total covariance gives \mathbb E_sD_tD_t^T\preceq\Sigma_s. At time t, the centered quadratic from time s splits into a centered quadratic there and the linear term 2D_t^TM(X-m_t^{\mathrm{post}}), plus its conditional mean. The averaged variance of this linear term is O_{L_0}(s^{-1-\gamma/2}). Indeed expand M=-HA_0+A_0H. On successful descendants the two terms obey

The norm bound at t is applied before averaging, and \E_sD_tD_t^T\preceq\Sigma_s is then used in the trace. No independence between D_t and \Sigma_t is needed. Polynomially bounded errors on failed descendants are negligible.

The conditional mean of the whole quadratic is \operatorname{Tr}(M\Sigma_t)+D_t^TMD_t. As a function of the field at t, its gradient is its conditional covariance, as a spin test, with X. On successful descendants its squared norm is thus at most (1+\epsilon)/t times its conditional variance. Elsewhere use the polynomial deterministic bound. Total variance, (12.7), and splitting sums with a factor two give

The coefficient of the first term relative to C_vs^{-1-\gamma'/2} is C_\iota'L_0^{\iota-\gamma'/2}<1/4 by our earlier choice. Choose C_v large. The strict inequality \gamma<\gamma' absorbs the second term. This closes the quadratic-variance induction.

On the initial band, apply the bounded-covariance version of the transfer to T. At T, the gap input bounds the centered quadratic variance because its single-site heat-bath energy is at most a constant times \operatorname{Tr}(M\Sigma_TM)+\|M\|_{\mathrm{HS}}^2, as follows by expanding a single spin flip. Coarse covariance on the initial band comes directly from the positive-time input.

We also obtain uniformly in unit e

Integrate the posterior-mean martingale from u to T, whose diffusion coefficient is \Sigma_b. The respective directional quadratic variations have second moments bounded by the squares of the integrals of Cb^{-2} and Cb^{-1-\gamma/2} on small scales; use bounded covariance later. These estimates follow from \|\Sigma_b\|^2 and \|\Sigma_bA_0\|^2\le \|\Sigma_b\|\|A_0\Sigma_bA_0\|, with deterministic polynomial bounds on exceptions.

Martingale fourth-moment inequalities apply. Terminal fourth moments follow by applying the gap first to a linear test and then to its centered square. Finally apply (12.3) to a centered linear projection of m_u^{\mathrm{post}} and to its square; their gradients involve \Sigma_u. This treats m_u^{\mathrm{post}}-m_s^{\mathrm{post}} without constants depending on d. On the initial band start the mean martingale directly at s.

Restoring the sharp covariance thresholds. On each root orbit, monitor the two covariance norms and the two mean deviations in the static comparison, in their own normalized units. Use tolerance \epsilon/2 for the sharp covariance median and \epsilon for its failure threshold, with fixed slack also for the other quantities. In particular the mean thresholds are r^{-1-\eta} and r^{-\gamma/2}, with an arbitrarily small \eta>0. Let a Lipschitz ramp be zero below the median thresholds and one when any failure threshold is reached.

On the conditional events already constructed, its gradient in the transition region is bounded by

To check this, a root-preserving rotation has score z^TMz/2, with m fixed. Its centered quadratic variance is bounded by (12.6). The additional linear term about m_s^{\mathrm{post}} has variance at most Cr^{-2-\gamma-2\eta}, by the coarse covariance bounds and the transition thresholds.

Choose \eta sufficiently small. Cauchy–Schwarz with the fourth and second moments just proved then controls derivatives of directional centered squares and means. The derivative of A_0 itself costs at most Cr^{-1-\eta} in these units. Outside these conditional events the ramp has a polynomial global gradient bound. This calculation does not differentiate the root.

When K_{\log}\log n\le k<\log^{10}n, use instead unconstrained orbits that co-rotate the field and matrix, centered at m_o. The polylogarithmic edge comparison and the spherical shell estimate give the required medians, after excluding the small-rate event on which the spherical width squared differs from s by more than a fixed relative tolerance \epsilon'\ll\epsilon. Here is the conversion from raw moments. In this band r=n^{-1/3}\log^{O(1)}n, so the unweighted error n^{1/2+o(1)} in Proposition 7.1 is o(s^{-1}). On the width event, (5.6) bounds the spherical raw moment about m_o by (1+O(\epsilon')+o(1))s^{-1}I. The cube covariance is bounded by its raw moment about m_o, giving the sharp covariance median. The weighted raw moment is at most n^{1/6+o(1)}, which is o(r^{-\gamma}) because \gamma/3>1/6. Jensen bounds the squares of the ordinary and weighted mean deviations by their respective raw moment operators. These give the two looser mean medians r^{-1-\eta} and r^{-\gamma/2}. All four statements concern a Haar majority on the orbit that co-rotates this field.

The extra linear score coefficient is

Since \|m_o\|=O(\sqrt k/r), its standard deviation has an additional multiplier at most C\sqrt k in the preceding bounds. Differentiating the center costs the same allowance. Thus D_{\mathrm{grad}} may be enlarged by a fixed power of \log n in this range. The unconstrained partition-ratio comparison (6.1) transfers the orbit weights.

The conditional derivative bound has exceptions whose masses depend on the orbit. We need a concentration estimate that is linear in those masses, so that it can be averaged over the orbit parameters. A Haar ramp zero on a fixed majority, with gradient at most D_{\mathrm{grad}} off a set of mass p, and polynomially bounded gradient everywhere, has plateau mass at most

provided D_{\mathrm{grad}}\sqrt{bk/n} is sufficiently small. Indeed split the upper ramp levels at tail probabilities successively halving from 1/2 to e^{-bk}.

If the plateau exceeds both this last probability and a large polynomial times p, truncate and rescale the ramp across one such quantile interval of width w and probability level p'. The entropy of its square is at least cp'\log(1/p'), whereas its gradient-square integral is at most

The rotation-group heat log-Sobolev inequality gives w\le CD_{\mathrm{grad}}/\sqrt{n\log(1/p')}.

Strict supports and inclusive plateaux handle atoms or zero-width intervals. Summing these widths bounds the full rise by CD_{\mathrm{grad}}\sqrt{bk/n}, a contradiction. In particular the orbit-dependent exceptional masses enter linearly and can be averaged without taking a fractional power.

Take b=a+\Delta/2<2a. The gradient condition uses only L_0,s_0,C_v, the moment constants and \gamma''<1, and holds by the earlier choice of a. It also holds throughout the polylogarithmic band despite the extra logarithms. On regular root anchors, the determinant-weight comparison at the concentration radius costs

so it does not require shrinking r after the rate has been chosen.

We display the two measure changes. Let p(\omega) be the orbit mass of \mathcal C_s^c, including its likelihood and diagnostic exclusions, and let h(\omega) be its ramp plateau mass. The quantile estimate and \int w_0\,d\pi\le n^C give

The first comparison in Section 10 bounds \int w_0p\,d\pi by the counting derivative-failure mass times e^{\xi k}, plus larger-rate exclusions. Returning the last display to the counting law by the reverse comparison costs a second e^{\xi k}. Thus the derivative term retains exponent at least a+\Delta-2\xi, and the direct plateau term retains exponent at least b-\xi, before polynomial factors. Choose 0<\xi<\Delta/8. The excluded diagnostic and base-orbit terms have already been assigned larger rates. Finally the degree identity charges excess root multiplicity to the excluded negative-Jacobian roots when returning from counts to planted fields. For unconstrained orbits choose partition-ratio accuracies relative to the small-range step size, then enlarge K_{\log}. Every reset cost and discarded event therefore fits within e^{-ak}. The same reset works on the initial band with its fixed constants and the larger initial-event rate already reserved; a is unchanged.

For clarity, the order of choices is: the target accuracies, \iota<\gamma'/2, and then large L_0; small s_0 for the static medians, the fixed outer cuts, and the compact bridge; then C_v and all moment constants, including those on the initial band; then a; then the two positive relative step sizes d_{\rm near},d_{\rm comp}; and finally K_{\log}. Put d_{\min}=\min(d_{\rm near},d_{\rm comp})>0. On successive backward bands with ratio 1+d_{\min}, every active u=(1+d(s))s lies in an earlier band, as does t. Induction on these bands proves (12.5)–(12.6) for each deterministic s. Tonelli supplies the time integrals used above from these pointwise statements; no union over uncountably many observation times is taken. No positive-time input is required uniformly down to an endpoint depending on a.□

Proof. Set L=n^{2/3}. The spherical raw second moment about m_o already satisfies the bound in (12.8). The hypotheses of the polylogarithmic edge comparison, namely

may be imposed with arbitrarily small inverse-polynomial failure by testing the spherical observation at a larger polylogarithmic edge width. We prove a higher-moment version of that comparison. Only unweighted raw second moments are needed here.

On an unconstrained Haar orbit divide every integral by Z_o(h). Let \Delta be either the signed cube-minus-sphere mass, or the signed matrix integral of (x-m_o)(x-m_o)^T/L, with this common denominator. For a fixed even integer p, expand \|\Delta\|_{\mathrm{HS}}^p (the mass case means |\Delta|^p). A subset B'\subseteq[p] specifies which of the p draws have sign priors; the other draws have spherical priors. The common kernel is respectively 1, or

The sign of that term is (-1)^{p-|B'|}.

Truncating the moment before annealing. In each term first restrict the internal sign overlaps Q_{ij}=x_i^Tx_j/n, i,j\in B', to |Q_{ij}|\le\eta_p, where \eta_p>0 is fixed and small. This avoids assuming an unrestricted high moment of the partition function. Outside exponentially exceptional orientations the truncated alternating sum differs negligibly from the desired power: the unnormalized large-overlap sign-pair integral is exponentially small by the global pair bound, while the total normalized cube mass is at most e^{c_{\mathrm{tiny}}n} by its first moment. Choose c_{\mathrm{tiny}} sufficiently small after p,\eta_p. The polynomial kernel costs can also be absorbed.

To discard the exceptional orientations inside an alternating sum, we need a bound on each term separately. Let T_{B'}(O)\ge0 be the normalized, individually truncated term at orientation O, including its mass or product-of-squares kernel. We claim

To verify this, bound the polynomial insertions crudely and integrate out the spherical priors. Two groups of sign vectors remain, with nearly identity internal Grams. If R is their matrix of cross overlaps, the empirical-site-type entropy cost is at least

Indeed let E be the first-group sign vector in the empirical site-type law, and let b_j(E) be the conditional bias of the jth sign in the second group. Then

by projection onto the first-group coordinates and their Gram bound. Conditional relative entropy dominates the sum of marginal conditional entropies. The binary entropy inequality I(b)\ge b^2/2+c b^4, followed by Jensen and the fixed number of coordinates, gives the displayed quadratic and quartic costs. On the other hand, diagonalizing the full Gram and applying the semicircle angular bound successively shows that the angular energy gain above the independent baseline is at most n/4 times the squared Hilbert–Schmidt norm of the full off-diagonal Gram, plus o(n). The cross block R occurs twice in that norm, so the angular gain is bounded above by

The leading quadratic halves cancel against the entropy cost. The remaining positive terms are C_p n\eta_p\|R\|_{\rm HS}^2+O_p(n\eta_p^2), whereas the quartic term is -c_p n\|R\|_{\rm HS}^4. Thus cross overlaps outside any sufficiently small fixed neighborhood of zero have a strict negative rate after choosing \eta_p sufficiently small. Within that neighborhood the small-Gram comparison and the spherical length-deficit tails give e^{o(n)}. This proves the second-moment bound. Cauchy–Schwarz applied to each T_{B'} removes an exponentially exceptional orientation set at cost e^{o(n)}e^{-cn/2}. It may therefore be removed before the signed sum, even though that sum need not be nonnegative on the exceptional set.

The same estimates restrict each term’s Haar integral, to arbitrarily high inverse-polynomial accuracy, to

For internal sign overlaps, use the small-Gram comparison until this window and choose its logarithmic power large enough that the length-deficit decay absorbs all losses. Once those overlaps are in the window, all overlaps involving spherical priors have the same tails: conditional on the internal sign Gram, the rotated prior is exactly the spherical prior conditioned on that Gram. Polynomially fine boxes have controlled angular variation on these nonsingular internal Grams. The central cutoffs may be smooth. The large-overlap truncation has therefore disappeared before the alternating cancellation below.

The finite-label expansion. The overlap window is now common to all terms. The only remaining prior ratio is the internal sign Gram mass, in lattice density units, divided by the spherical internal Gram density on the label set B'. We expand this ratio with every normalization factor included. Each coefficient must have an explicit finite set of replica labels: the alternating sum cancels exactly those terms that omit a label.

For B'\subseteq[p], put v=|B'|, d_v=\binom v2, and let Q_{B'} be the Gram with diagonal one. Zero-dimensional densities and empty products are one. Let \rho_{B'} be the preceding density ratio. To every fixed expansion order it has the form

Here the displayed coefficients are homogeneous of degree b in the off-diagonal overlaps.

Every degree-b rate coefficient is a sum of terms involving at most 6b replica labels; every amplitude coefficient of degree b and order n^{-j/2} is such a sum involving at most 6(b+j) labels. A term involving a label set S means a universal summand indexed by labels in S, including all internally summed indices, whose value is independent of whether further labels belong to B'. This qualification includes the normalized central density prefactors, rather than merely the visible overlap monomial. On (12.9), the expansion and its exponential can be truncated with uniform relative error O(n^{-A}), for any prescribed fixed A.

We prove these assertions, starting with the normalizations. For one sign site let Y=(\varepsilon_a\varepsilon_b)_{a<b,\ a,b\in B'}. If \delta(S) denotes the cut vector of a subset of labels, then Y=\mathbf1-2\delta(S). Its difference lattice is \Lambda_v=\operatorname{span}_{\mathbb Z}\{Y-\mathbf1\}. Distinguish one label, denoted i_\bullet in this lattice calculation. A basis consists of

Indeed

Ordering the edges incident to i_\bullet first makes the basis block triangular, so its covolume is

The sample overlap belongs to \mathbf1+n^{-1}\Lambda_v, whose cell volume is c_vn^{-d_v}.

Thus the sign density in lattice units is

The reciprocal lattice, using the 2\pi convention, is

Equality in the triangle inequality shows that the characteristic function of Y has modulus one precisely on this reciprocal lattice. The same is true under every sufficiently small real exponential tilt, whose support is unchanged. There is consequently a single central saddle on the reciprocal torus, and uniform contraction away from it. Boundary representatives of that one saddle are identified on the torus.

For the spherical density, use independent Gaussian coordinates and the one-site statistics

Their covariance is the identity in dimension d_v+v; the sign statistics also have identity covariance in dimension d_v. Denote the two log transforms by \Psi_G,\Psi_S, and their Legendre transforms by I_G,I_S. At target (Q_{ab},0), I_G=-\tfrac12\log\det Q_{B'}. If f^G is the full Gaussian statistic density, conditioning on the zero normalized-length targets gives exactly

There is no extra Jacobian because both densities use the same normalized length coordinates.

With z_n=n/2, their exact length-only factor is

The saddle densities have forms

Their amplitudes at zero target and leading order equal one.

Hence

Every power of n/(2\pi), and the entire lattice covolume, have cancelled before expanding. As a check, the exact spherical Gram density is

whose central factor has the same leading value (n/(2\pi))^{d_v/2}.

The sign transform through degree three is

Indeed the only nonzero third products are triangles, each with value one. Its Legendre transform therefore has quadratic term \tfrac12\sum Q_{ab}^2 and cubic term -\sum Q_{ab}Q_{ac}Q_{bc}. These agree with the expansion of -\tfrac12\log\det Q_{B'}. Thus the rate difference starts in degree four, as in (12.10).

Here is the label count for all subsequent terms. Each statistic coordinate carries at most two labels, including the single-label Gaussian length coordinates. A zero-tilt cumulant tensor with M coordinate slots therefore involves at most 2M labels, and its value is unaffected by additional replicas. Formal inversion of \nabla\Psi(t)=Q produces rooted trees with nonlinear vertex valences m\ge3. A degree-b Legendre coefficient satisfies \sum(m-2)=b-2. Since m\le3(m-2), it uses at most 3(b-2) cumulant slots, hence at most 6b labels. Setting the Gaussian diagonal targets to zero preserves this bound.

For the central amplitude, perform saddle Fourier inversion on the n^{-1/2} frequency scale against an identity-covariance Gaussian. This expands the determinant instead of leaving an implicit factor depending on dimension. A higher Fourier vertex of degree m\ge3 carries weight n^{-(m-2)/2}; at order n^{-j/2}, their total Fourier-slot count is at most 3j. A vertex from the perturbed Hessian has two Fourier slots and at least one tilt slot. If the initial Taylor expansion has T tilt slots, all its slots therefore number at most 3j+3T.

Substitute the rooted-tree expansion for the saddle, with a total of b target leaves. This adds at most 3(b-T) slots. There are thus at most 3(j+b) slots, or 6(j+b) labels. Gaussian pairings identify slots and introduce no further labels; unused coordinates integrate to one. This counts the determinant, inverse-Hessian contractions, and every higher correction. Odd j vanish by Fourier parity, although allowing them in the bookkeeping is harmless. The factor b_n^v obeys the same rule by multiplying its single-label expansions. Products and reciprocals of series with leading constant one preserve the bound, because the weights b+j add. This proves the asserted support rule even for coefficients that depend on |B'| after their label sums are evaluated.

For fixed p, the transforms are analytic near zero with uniformly positive Hessian there. The saddles on (12.9) lie in this neighborhood. Taylor expansion under Gaussian damping on logarithmic frequency balls gives any required inverse-polynomial relative error. The sign Fourier tails are negligible by the reciprocal-lattice contraction just proved; the Gaussian quadratic determinant formula controls its remaining Fourier tails. This proves the uniform saddle remainders. Moreover nQ^4=O(n^{-1/3}\log^{C_p}n)=o(1) on the window, so the exponential in (12.10) can also be expanded to arbitrary inverse-polynomial precision.

A common integration measure and high-order quadrature. Fix the orbit and let \mu_o be its normalized spherical Gibbs law, equivalently the Gaussian law below conditioned on length. Let K(x_1,\ldots,x_p) be either 1 or the product of squares displayed above, and write \mathcal U(x)=(x_a^Tx_b/L)_{a<b}. Define the nonnegative common kernel density G_p by

for Borel A in the interior Gram region. This definition includes the common normalization Z_o(h)^{-p}. We now bound the density and its derivatives.

Use p independent Gaussian representatives x_a=m_o+D^{1/2}\eta_a, with D=(\lambda-J)^{-1}, and impose all their lengths exactly. Fourier-invert the length statistics (\|x_a\|^2-n)/L and overlaps U_{ab}=x_a^Tx_b/L. For a symmetric Fourier matrix T, the tilted white covariance is (I-2iT\otimes D/L)^{-1}, up to fixed conventions on off-diagonal coefficients, and has norm at most one. The tilted mean is bounded by C_p\|T\|\|D^{1/2}m_o\|/L.

The polylogarithmic edge inputs give

with logarithmically bounded reciprocals of the exact length densities in these units. The spectral spread supplies arbitrarily many covariance eigenvalues of inverse-polylogarithmic size. As a nonzero T has an eigenvalue at least \|T\|/\sqrt p, the quadratic determinant is integrable against every fixed frequency power with only a logarithmic loss; the noncentral characteristic factor has modulus at most one.

The matrix-moment kernel is a product of squares of \eta_a^T(D/L)\eta_b with a\ne b. Its Wick expansion under this complex tilt also costs only powers of \log n and T. A contraction loop of length at least two is bounded by powers of \|D/L\| times \operatorname{Tr}((D/L)^2). A loop of length one uses an off-diagonal replica block of the tilted covariance, whose identity term vanishes; the resolvent identity supplies an additional D/L, giving the same bound with a frequency factor.

Open contractions end in the already bounded means. Differentiation of the resulting density adds only frequency powers. Consequently G_p(U) has all fixed derivatives bounded by powers of \log n; on the common smooth window their L^1 norms have the same bounds. This holds for both the mass and matrix kernels.

The density and all required derivatives are now controlled. This is what allows high-order quadrature on each subset’s actual measure. For the subset B', only its internal coordinates initially lie on a lattice. In U units that lattice has covolume c_v/L^{d_v} and a fixed-shape basis of size O_p(L^{-1}). Its mixed lattice–continuous integral is exactly the quadrature of \rho_{B'}((L/n)U,n)G_p(U), with that cell-volume factor. For every finite smooth expansion term, repeated quadrature after the fixed basis change, or Poisson summation, gives error O_{p,A}(L^{-A}\log^C n), uniformly over affine lattice shifts.

The relative expansion remainder also has a uniform absolute bound on the common window. Indeed the rate transforms agree through degree three, and their analytic amplitudes are one at zero, so for fixed p

Here \max|Q_{ab}|=O(n^{-1/3}\log^{C_p}n), so the exponent tends to zero uniformly over all subsets B'. Thus a relative remainder O_{p,A}(n^{-A}) is also an absolute remainder of that order. It requires no derivatives, but is still evaluated on its own mixed lattice–continuous measure. Apply the same high-order quadrature to the smooth windowed G_p itself. Its derivative bounds show that its mixed sum is at most \log^C n+O_{p,A}(L^{-A}\log^C n), uniformly in the affine shift, because its continuous mass has that bound. Multiplying this mixed-sum bound by the uniform absolute remainder controls the remainder on its actual measure. We may therefore convert all subset terms to one continuous integral, with arbitrarily high inverse-polynomial error and with all covolumes included. The different subset lattices need not have matching supports before conversion.

Cancellation and the probability bound.

Quadrature has put every subset term on one continuous integration measure. We can now take the alternating sum without identifying the original lattice supports. A surviving term must use every replica label, and the coefficient-count rule turns that requirement into a power of the dimension that grows with the moment order. A term in the fully expanded expression has rate factors of degrees b_1,\ldots,b_t\ge4, and an amplitude of degree b and order n^{-j/2}. Put W=b_1+\cdots+b_t+b+j.

Its coefficient involves at most 6W labels. On the window,

where the first estimate uses b_i/3-1\ge b_i/12 for b_i\ge4. The whole term thus costs at least n^{-W/12}, up to logarithms. A diagram whose label set S is a proper subset of [p] cancels exactly, since

Every surviving term has 6W\ge p, and is bounded by n^{-p/72} before logarithmic losses.

Choose the expansion order sufficiently large in terms of p and the desired error power. The logarithmic kernel bounds and integration, followed by an enlargement of the threshold in n, give the safe absolute bound n^{-cp}, for example with c=1/200.

We now return from the expansion to probability. The calculation bounds the Haar expectation of the truncated alternating power. The separate second-moment bounds remove the exponentially exceptional orientations even though the truncated sum may be signed there. On their complement it agrees with the nonnegative quantity \|\Delta\|_{\mathrm{HS}}^p up to negligible error. For the mass choose threshold n^{-c/2}; Markov then gives Haar failure at most n^{-cp/2}. For the matrix choose threshold one in Hilbert–Schmidt norm; its Haar failure is at most n^{-cp}. The planted likelihood R=Z(h)/Z_o(h) has bounded second moment by the two-replica comparison and pair tails. Cauchy–Schwarz therefore leaves planted failure exponents at least cp/4 and cp/2, respectively. A sufficiently large fixed even p gives any requested B, with room for the earlier width exclusions.

Let M_{\rm cube} and M_{\rm sphere} be the normalized raw second moments about m_o. On the retained event,

The spherical bound is \|M_{\rm sphere}\|\le CL\ell^C. Thus \|M_{\rm cube}\|\le CL\ell^C, and \Sigma_s\preceq M_{\rm cube}. This proves (12.8).□

Proof. We give the endpoint changes to Lemmas 3.1–3.4 of [28], including the uniformity needed below. The proof separates fixed parameters from adaptive ones. A strict norm–trace bound gives the deterministic-diagonal path margin and controls expectations. Chaining makes the word errors simultaneous over the diagonal parameters. An inverse chosen after observing the matrix is used only where its own stated positive margin holds. For \|p\|=1 and w=A^{1/2}p, Gaussian comparison bounds the expected supremum of zw^{\mathsf T}Ww-z^2\chi_A\|w\|^2 by that of

Indeed the increment difference between the comparison process 2n^{-1/2}\|w\|g^{\mathsf T}w and w^{\mathsf T}Ww is

Also \mathbb E T_g^2=\chi_A and \operatorname{Var}(T_g^2)\le2a_+^2/n, so replacing T_g by \sqrt{\chi_A} costs o(1) uniformly in A.

The supremum then has integrand 2d-d^2, where

Its distance below one is at least \delta_\kappa=(1-\sqrt{1-\kappa})^2. The supremum is a_+-Lipschitz in W for the Frobenius metric. Gaussian concentration, the GOE norm tail, and a fixed mesh in z give the asserted path margin, for example with c_\kappa=\delta_\kappa/4.

To prove expectations without conditioning on a matrix event, project W onto a fixed operator-norm ball and replace the symmetric inverse by a smooth bounded functional-calculus function agreeing with x^{-1} on an interval containing all spectra with either required margin. Use

This defines bounded truncated words. For fixed length their Frobenius Lipschitz constants in W are bounded, and the increment between parameter tuples at sup-norm distance \Delta has Frobenius Lipschitz constant at most C\sqrt\Delta.

These follow by the product rule, \|A^{1/2}-B^{1/2}\|\le\|A-B\|^{1/2}, and the Fourier formula for differentiating a smooth matrix function. Thus entry variances are O(n^{-1}), normalized-trace variances are O(n^{-2}), and fourth centered entry moments are O(n^{-2}). These bounds gain respectively \Delta,\Delta,\Delta^2 for increments.

The fixed-parameter agreement event has exponentially small complement. Gaussian integration by parts therefore gives, with errors O(n^{-1}) uniformly in i,z,A, the same loop equations as in Lemma 3.3 of the companion article. Writing a_i=A_{ii}, k_i=\mathbb E(K_A)_{ii}, r=n^{-1}\sum_i a_i k_i, and h=r-\chi_A, these equations are

All expectations here may use the bounded truncations. Their integration-by-parts error is exponentially small times a fixed polynomial in n, because both true and formal differentials are bounded and agree on the path event.

If |h|\le\delta, the second equation gives k_i=1+O(\delta+n^{-1}). Consequently

for fixed sufficiently small \delta. Averaging the loop equation with weights a_i gives |h|\le C/n. Continuity from h(0)=0 prevents an exit from |h|\le\delta. It follows that k_i=1+O(n^{-1}) and \mathbb E(WK_A)_{ii}=z\chi_A+O(n^{-1}).

For completeness, the higher-word recursion has no new stability requirement. Mark an ordinary matrix letter, integrate it by parts, and let its differentiated partner split the word as U\boxed W V\boxed W Q. The leading term is

the reversed contraction is n^{-1}\mathbb E(UV^{\mathsf T}Q)_{ii} and is O(n^{-1}). Covariance errors are O(n^{-1}) by the preceding variance bounds. If the derivative meets the inverse, use \mathrm dK_A[E]=zK_AAEK_A.

Splitting then puts one copy of K_A in each smaller word, whose total number of ordinary letters is reduced by one. Otherwise it is reduced by two. The base formal prediction is \mathsf P(K_A)=I: for each nonempty noncrossing-paired chain, summing over subsets of its originally adjacent pairs gives \sum_T(-1)^{|T|}=0. Induction proves the polynomial assertion and entrywise expectation error O(n^{-1}). Every inverse in this calculation has the original coefficient A; no enlarged diagonal parameter set is used. For a normalized trace, the noncrossing face formula has no marked starting position on the trace cycle. Rotating the word permutes its faces and contractions, proving the stated cyclic invariance coefficient by coefficient, also after the formal inverse expansion.

To make the result simultaneous, take sup-norm meshes whose points lie in \mathcal A_\kappa and in the bounded boxes for the other diagonal parameters. At scale 2^{-l} their cardinalities are at most \exp(C_L(l+1)n). The truncated estimates hold in the ambient box, so mesh increments are legitimate even though no convexity of \mathcal A_\kappa is assumed. The diagonal \ell^2 and off-diagonal \ell^4 seminorms of centered increments have expectations at most C2^{-l/2} and concentration scale C\sqrt{2^{-l}/n}.

A union bound bounds every increment by C\sqrt{l+1}\,2^{-l/2}, with summable failure probabilities Ce^{-c(l+1)n}. Sum the increments and the fixed-parameter biases. Sign conjugation makes all off-diagonal expectations zero. This proves the asserted simultaneous estimates for W. Since \|\operatorname{diag}W\|_{\mathrm F} is bounded except with exponentially small probability, the truncated Frobenius Lipschitz estimate proves the result for W^\circ as well. Restriction to the stated positive margin restores the exact inverse factors.

Finally sign conjugation and the loop equations give the diagonal expectations of G_A, G_AW and WG_AW. They are respectively A, \chi_AA and \chi_AI+\chi_A^2A, to O(n^{-1}) in each entry. Their truncated bilinear forms are uniformly Lipschitz. Gaussian concentration proves the last assertion for each fixed finite list of vectors.□

Proof. For deterministic V use Lemma 13.1 along L_z=I+z^2bV-zWV. On its path event,

The expectation is zb^2+O(n^{-1}), by the loop calculation. Using a fixed norm cutoff in the path event \mathcal E_V, integration from zero to one gives \mathbb E[\mathbf1_{\mathcal E_V}2n^{-1}\log\det L_1] =b^2+O(n^{-1}). The inverse norm there is bounded; regularization costs C_\kappa\gamma. Off this event, the regularized logarithm is bounded in absolute value by |\log\gamma|+C+2\log(1+\|W\|) and its expected contribution tends exponentially to zero. Hence the regularized mean is at most b^2+C_\kappa\gamma+o(1) uniformly in V.

The map M\mapsto n^{-1}\log\det(M^{\mathsf T}M+\gamma I) has Frobenius Lipschitz constant 1/\sqrt{n\gamma}. Gaussian concentration thus gives a tail 2\exp(-\gamma n^2u^2/4) at deviation u. Choose \gamma_0 so C_\kappa\gamma_0<\varepsilon/8. A fixed sup-norm mesh of the restricted diagonal set has size \exp(O(n)). On \|W\|\le R, a mesh spacing depending on R,\gamma,\varepsilon controls both the logarithm and b^2 to \varepsilon/4. For this last mesh step choose a possibly larger R so that its exceptional probability is at most e^{-(B+1)n}.

This choice does not change the preceding expectation estimate or its regularizer threshold. The n^2 concentration bound pays for this mesh for sufficiently large n. Finally a rank-r_0 perturbation changes the logarithm by at most M_2\sqrt{r_0/(n\gamma)}. This proves the statement, including its arbitrarily large fixed exponential budget. The exponentially small path exception was used only in the mean estimate; its exponent does not limit this final budget.□

Proof. Suppose the entropy is finite; then 0<q<1. Set k=q/\nu\in(0,1] and C=a/\nu. The change of variable T_u(y)=y-u\tanh y, u<1, gives

This identity is the entropy change-of-variables calculation in Lemma 4.1 of the companion article and uses no strict restriction on j. Put h(u)=u^2/2-b[-\log(1-u)-u]. Since b=1-q<1,

For negative u use -\log(1-u)-u\le u^2/2; for positive u\le q use the increasing series ratio and h(q)=\sum_{l\ge3}q^l/[l(l-1)].

If (C-b)/(1+k)\le q, substitution proves strict inequality unless C=b. Otherwise let C_0=1+kq and C_1=C_0+\sqrt{(1+k)/3}\,q^{3/2}. The choice u=q proves strictness for C_0<C<C_1. For C\ge C_1, variance comparison and Cauchy–Schwarz give

Subtract the target and write f(C)=C^2/k-1-\log(C^2/k)-(C-b)^2/(1+k). Direct differentiation gives

Consequently

Since f' is positive and increasing after C_0, this finishes the strict cases. If C=b, the target is zero and equality is equivalent to vanishing entropy. Gaussian integration by parts verifies equality for the stated Gaussian law.

For a fixed variance \nu>0, the function d\mapsto d-\mathbb E_{N(d,\nu)}\tanh y is strictly increasing, onto \mathbb R, and zero at d=0; its derivative is \mathbb E\tanh^2y>0. Thus its solution at s\ge0 is nonnegative. Let H_\nu(d)=\mathbb E_{N(d,\nu)}(\tanh y-\tanh^2y). Gaussian reflection gives H_\nu(\nu)=0 and \mathbb E_{N(d,\nu)}[\tanh y\,\operatorname{sech}^2y]\ge0 for d\ge0. Hence |H_\nu'(d)|<1 for every finite d\ge0: the upper bound is strict since \mathbb E\operatorname{sech}^2y<1, and the pointwise lower bound of (1-m^2)(1-2m) is strictly greater than -1. The strict derivative bound is uniform on the compact segment joining d and \nu. Consistency gives d-\nu=H_\nu(d) and therefore forces d=\nu.

Finally F(v)=\mathbb E_{N(v,v)}\tanh y satisfies F(0)=0 and

The upper bound uses the same reflection identity. Thus v-F(v) is strictly increasing from zero to infinity, since 0\le F(v)\le1. It has a unique inverse at every s>0, with s\le v_*(s)\le s+1. This proves continuity and the positive lower bound for q on each compact positive interval. At s=0 there is no equality law with q>0: its mean equation forces d=0, while \mathbb E_{N(0,q)}\tanh^2 y<q.

We specify why the integral consequences extend as stated. In the smoothed integral the parameters are d=s+M, \nu=s+\sigma^2+q, and the exponent is \psi=(a-\nu b)^2/[2\nu(\nu+q)]. With \varphi_{d,\nu} the density of N(d,\nu), the scalar weight meant here is

When \nu\ge\nu_{\min}>0, Cauchy–Schwarz gives the uniform envelope G_{s,\sigma}(y)\le\exp(Cn-c\sum_i y_i^2). On a fixed empirical second-moment ball, M,q,a,d,\nu,\psi are continuous for weak convergence, since only bounded or at most linearly growing functions are averaged. The entropy variational formula and a finite covering by fixed product-Gaussian neighborhoods give

for each compact parameter set \mathcal C; Gaussian density ratios on the second-moment ball tend uniformly to one at the exponential rate when their parameters converge.

The rate is nonpositive, and its zero set at \sigma=0 is exactly the curve just proved. Upper semicontinuity therefore makes the rate strictly negative on closed sets outside its prescribed neighborhood. Compactness preserves this negativity when \sigma is sufficiently small. The exterior of the ball has any required negative rate by the envelope. For the tail-square assertion insert \exp(\lambda\sum_i y_i^2\mathbf1_{|y_i|>R}) into each of the finite Gaussian comparisons, with fixed sufficiently small \lambda>0.

Its additional logarithmic moment tends uniformly to zero as R\to\infty. Removing the tilt on a positive tail-square level produces strict negative rate. These are precisely the three stated integral conclusions, now with no subcritical assumption.□

Proof. The scalar calculation separates the overlap from zero and places the retained parameters near the positive-time equality curve. This gives a norm–trace margin for nearby diagonals. We next check the conditional finite-rank matrix correction and make that check simultaneous over the diagonal neighborhood.

First retain the diagonal and gauge the planted spin to \mathbf1. Then J=W+\mathbf1\mathbf1^{\mathsf T}/n and h_s=s\mathbf1+\sqrt s\,g. Until the final diagonal-removal paragraph, use the temporary full-matrix residual and Hessian

Let E\subseteq\mathbb R^n\times\operatorname{Sym}_n be Borel and independent of the observation h_s. For \sigma>0 set

The exact Gaussian reduction, Lemma 5.3 of [28], is valid for every positive j and here reads

where d=s+M, \nu=s+\sigma^2+q and G_{s,\sigma} is the explicit scalar weight in Lemma 13.3. Here g' is a fresh standard normal independent of W. The identity follows directly by inverting \nu I+mm^{\mathsf T}/n; the determinant factor is \sqrt{\nu/(\nu+q)}.

On [s_1,T], discard a large second-moment ball’s exterior, a fixed neighborhood of the equality curve’s complement, and the scalar square-tail event. Lemma 13.3 gives strictly negative rates on these discarded parts and an arbitrarily small positive rate for the total integral. On the remaining parameters q\ge q_0>0. Choose \eta_0 and the diagonal Frobenius tolerance so small that

The restricted fixed-parameter and bilinear estimates now apply. We verify their conditional use and its positive margin.

Set u=m/\sqrt{nq}, Z_1=(y-d\mathbf1)/\sqrt{n\nu} and P_u=I-uu^{\mathsf T}. Under the displayed conditioning, using a fresh GOE W_0, put \ell_0=u^TW_0u and

Writing out the three blocks gives the exact finite-rank difference

The scalar \ell_0 has variance 2/n and is smaller than any fixed tolerance outside an exponentially small event.

Write S_A=S_A(1;W_0) and define \mathcal Y(p)=S_A^{-1/2}A^{1/2}p. Set

The exact whitened Hessian is

For each fixed y,A, the vectors \mathbf1/\sqrt n,u,Z_1 are deterministic relative to W_0. The bilinear tests in Lemma 13.1 give

for this fixed finite vector list. In particular \|X_1\|^2\simeq\chi_A, rather than b for an arbitrary A. These are comparisons of actual Gram entries, with a chosen fixed accuracy.

The comparison Gram for T_1 is represented by \chi_AU+\sqrt{\chi_A}N, where N is an abstract unit direction orthogonal to the other three images. This is a model of Gram entries, not a coupling with a new random vector. Set aside the small full-rank term b-\chi_A and the scalar \ell_0. Eliminating the N direction, whose Hessian pivot is the identity, gives the ideal coefficient

on UU^T in the form being subtracted. The final term in the first line is the Schur correction.

At the equality-law limit, \sigma=0, d=\nu=v=v_*(s), a=vb, and the limiting diagonal is A=V_y, so \chi_A=b. Only in this limit is \beta_U=q(2-b/v). Write b=\mathbb E\operatorname{sech}^2y and e=\mathbb E[\tanh^2 y\operatorname{sech}^2y] under N(v,v). Use the rescaled triple

The ideal remaining matrix is identity minus the form on (X_1,M_1,G_1) with coefficient matrix C and limiting equality-law Gram matrix H:

The Gram matrix here is the limit of the empirical A=V_y comparison data, not the actual finite-sample Gram. Its entries follow by Gaussian integration by parts; for example \mathbb E[m(1-m^2)]=e, v^{-1}\mathbb E[(y-v)m(1-m^2)]=b-3e, and the double integration by parts gives b/v-2b+6e. Elementary column operations yield

Subtracting the first vector alone is positive because its squared length is b<1.

The remaining two-vector coefficient block has only one positive eigenvalue. Hence the final matrix can have at most one nonpositive direction; its positive determinant rules that out. The preceding orthogonal-direction Schur complement is therefore positive as well. Continuity on the compact equality curve gives a fixed positive margin.

The fixed-parameter calculation is complete. To make it simultaneous, we first quantify its continuity. The margin survives small errors in all Gram entries, coefficients, and \chi_A-b. To check uniformity over nearby A, each tested coordinate product is bounded by C(1+y_i^2)/n. On |y_i|\le R_1, its change under A-V_y is at most C_{R_1}\|A-V_y\|_{\mathrm F}/\sqrt n; the remaining part costs the prescribed square-tail tolerance. The V_y-weighted coordinate products are bounded continuous functions, so their equality-law limits follow from weak closeness.

This proves the fixed-A conditional estimate with a uniform positive exponential rate. The path margin S_A\succeq c_\kappa I returns the whitened margin to \widetilde{\mathcal H}, after the small (b-\chi_A) term is absorbed. We pay for simultaneity only on the coordinates where a diagonal can move appreciably. On the bounded conditional-matrix norm event choose a fixed sup-norm mesh spacing \delta so the 1/2-Hölder error of \widetilde{\mathcal H}(y,A) is smaller than one quarter of the margin. Only \epsilon_A^2n/\delta^2 sites can differ from V_y by more than \delta. Mesh those sites and leave the others at V_y. The logarithm of the number of approximants is at most

where \mathfrak h(\alpha)\to0 as \alpha\downarrow0.

Use twice the Frobenius radius in the fixed-parameter estimate and then decrease \epsilon_A to pay this mesh. Thus the conditional matrix failure, integrated against G, has strictly negative rate. This explicitly extends the positive-time part of Lemma 5.5 of the companion article; its subcritical small-q argument is not used.

The estimates so far control a smoothed integral of bad configurations. To exclude a bad approximate zero, a witness must force that integral to be large enough. The deterministic local-volume estimate supplies this implication, with determinant, smoothing, and residual tolerances chosen in that order.

Let \mathcal D_{\eta_0} be the compact set of diagonal 0\le A\le(1+\eta_0)I. Let \epsilon_*>0 be the full diagonal radius retained by the conditional calculation, and choose c_E below half its matrix margin. Use the relaxed Borel set

intersected with the external spectral and norm restriction. Compactness of the diagonal range and continuity make this set Borel and independent of h_s. The scalar exclusions and the conditional matrix rate just proved give, for sufficiently small \sigma>0, uniformly on the fixed positive band,

for some a_0>0.

The stricter witness event uses

with A\in\mathcal D_{\eta_0}. It depends on the observation, unlike E^{\rm mat}. Keeping this same A, for d_y=\|y'-y\|/\sqrt n we have

Thus the Hessian changes by at most 6(1+\eta_0)d_y. Every point of the ball \|y'-y\|\le r_{\rm loc}\sqrt n lies in the same E^{\rm mat} fiber if

We next obtain the overlap lower bound needed for the determinant estimate. A uniform lower bound on q at any witness is available before using the determinant estimate. On the external norm event,

Independence of the planted observation coordinates and a fixed Bernoulli deviation bound give \|\tanh h_s\|\ge c\min(\sqrt s,1)\sqrt n outside a fixed exponentially small event, uniformly on compact time intervals. For instance, at small s, a fixed positive fraction of the standard Gaussian coordinates lie in [1/2,1], on which the hyperbolic tangent has magnitude at least c\sqrt s. On the fixed band let v_->0 be a common observation lower bound, and let the external matrix norm be at most K. Taking \rho\le v_-/2 forces

At such a witness the Jacobian is exactly

The last two terms have rank at most two and norm at most three. Since b\le1-q_{\rm vol}, Lemma 13.2 with \kappa=q_{\rm vol} gives its regularized determinant upper bound with any needed fixed failure budget. The deterministic local-volume lemma, Lemma 5.6 of [28], has hypotheses j,K<\infty and hence applies at j=1 without modification.

With the regularized Jacobian Q=\widetilde F_{h_s}'(y)^{\mathsf T} \widetilde F_{h_s}'(y)+\gamma I, its change of variable y'=y+\sigma Q^{-1/2}g proves a lower bound e^{-\varepsilon n} for this bad integral whenever the residual is sufficiently small and 2\sigma/\sqrt\gamma\le r_{\rm loc}. Its losses per spin are bounded by

Choose the target loss \varepsilon<a_0/2, then the determinant tolerance \varepsilon_1, then the regularizer \gamma, then \sigma, and finally the residual tolerance \rho.

Markov’s inequality now bounds the witness probability by e^{-(a_0-\varepsilon)n}, together with the assigned determinant and observation exceptions. For low overlap, choose q_{\rm rel} below the equality curve’s minimum q, use the relaxed set E^{\rm low}=\{q\le q_{\rm rel}\} with the same external restriction, and take the witness threshold q\le q_{\rm rel}/2. Since |q(y')-q(y)|\le2d_y, a radius r_{\rm loc}\le q_{\rm rel}/4 gives the same inclusion. The scalar negative rate then replaces the matrix rate in the same volume argument.

For the stronger small-time budgets use Proposition 8.3, without the determinant weight. Its scalar integral estimates permit any preliminary fixed exclusion budget A_{\rm pre}nr^3, restrict q\asymp_{A_{\rm pre}}r, and give the outer rate cnu^3 for u\ge Cr. Choose A_{\rm pre}>A_* to reserve rate for smoothing and local volume. For an upper-tail witness q>u, use the relaxed set E_u^{\rm up}=\{q\ge u/2\} and radius r_{\rm loc}\le u/4. Taking u\ge2Lr, with L from the scalar proposition, gives a preliminary rate proportional to n(u/2)^3.

We compare the actual smoothed scalar weight with the root weight. On a fixed band s\in[s_-,s_+], s_->0, put \nu_0=s+q, \nu_\sigma=\nu_0+\sigma^2, and \mathfrak v=\langle(y-d)^2\rangle. For fixed y,

On \langle y^2\rangle\le R^2, all numerators are bounded and \nu\ge s_-, so the absolute value is at most C_{R,s_-,s_+}\sigma^2. The uniform Gaussian envelope from Lemma 13.3 makes the exterior integral have any prescribed fixed negative rate when R is large. Choose R first, then \sigma so this logarithmic loss is smaller than one quarter of the reserved gap (A_{\rm pre}-A_*)r_-^3, or than the reserved fixed multiple of u_-^3 for a finite outer-threshold family. Polynomial factors are absorbed for large n. This preserves every scalar rate used here for the smoothed integral.

If a diagnostic itself uses \nu_\sigma, give its thresholds fixed slack first. On the same ball its scalar parameters change by O_R(\sigma^2). On the retained scalar region, q\ge c r_->0, and the conditional column coefficients have the same bound. The stricter smoothed bad set is then contained in the relaxed unsmoothed one. This supplies the required set comparison as well as the weight comparison. For the matrix estimate the spectral indicator must remain attached to the actual conditioned matrix. Denote that matrix by J_\sigma=W_\sigma+\mathbf1\mathbf1^T/n; the subscript records the conditioning, not a replacement of the external disorder. With u=m/\sqrt{nq}, E=u^\perp, and e=P_E\mathbf1/\sqrt n, its exact regression gives

where the GOE block W_E and w\sim N(0,I_E/n) are independent. Its scalar block is the regression displayed earlier. In decreasing eigenvalue order, compression and rank-one interlacing give, on \mathcal G_{\rm spec}(J_\sigma),

Thus every cumulative count changes by at most two, and the top edge of W_E is bounded by that of this same full matrix.

Here is the precise spectral information used by the stability calculation. Put \Delta_{\rm sp}=q^{8/5} and restrict R=(2+O(q^2)-W_E)^{-1} to depths greater than \Delta_{\rm sp}. The inherited edge counts and bulk semicircle limit give, uniformly for the O(q^2) shifts in the stability Schur calculation,

The first estimate follows by removing the edge contribution to the first trace; dyadic counts give the upper estimates for higher powers, and one band at depth comparable to \Delta_{\rm sp} gives their lower estimates. The fixed-rank count error contributes only O((n\Delta_{\rm sp}^j)^{-1}). Likewise the restriction to depths at most a fixed \zeta has first trace divided by n at most O(\sqrt\zeta)+o_n(1). In particular the first trace is normalized to one, as required for the cancellation in (9.2). The other needed margin uses the actual full spectrum:

Condition first on the spectrum of W_E. On every spectrum satisfying these inherited bounds, its Haar eigenframe and the independent Gaussian column obey the tests in (9.2) at any prescribed fixed rate in nr^3, before imposing \mathcal G_{\rm spec}(J_\sigma). The scalar diagnostics bound every regression coefficient uniformly on the fixed positive band. Choosing their error tolerances first and then \sigma small preserves the column tests and their q^2 Schur margin. Consequently the estimate is for

with the preliminary rate paying the scalar-weight and mesh losses. Here \mathcal F_y is failure of the required Hessian margin. No event for the fresh comparison matrix W_0 replaces the displayed indicator, and no division by its conditional probability is used. This argument transfers the stability calculation, not the sharper inverse-trace accuracy for K_o, which is not needed here. All smoothing choices are made on a fixed positive band before taking n large.

We spell out the additional diagonal uniformity in this use. At the given positive scale truncate A^{-1}-I above at the same level q^{1-\eta} used for V_y^{-1}-I, assigning that truncation value at A_{ii}=0. The truncated scalar function is Lipschitz on [0,1+\eta_0]; its negative part is at most \eta_0. Choosing \eta_0 small preserves the norm bound for the truncated perturbation. Shrinking \|A-V_y\|_{\mathrm F}/\sqrt n makes its normalized first two moments and all the fixed-axis quadratic and column tests as close as required to those in (9.2).

For axes involving y, first discard their square tails by the scalar diagnostics, then use bounded site profiles on the remaining sites. For u=m/\sqrt{nq} one can directly use \|(B(A)-B(V_y))u\|\le C\epsilon_A/\sqrt q. All scales here are fixed before n\to\infty. Thus the Schur calculation retains its margin with the prescribed rate. The original precision is larger. Its positive lower bound also gives a positive symmetric-normalization margin: combine it with the bound A^{-1}-CI, then take limits if some entries vanish. The preceding sparse diagonal mesh costs an arbitrarily small exponential rate and makes the estimate simultaneous.

The observation lower bound just used has a fixed rate independent of sufficiently small r, so it pays any specified A_*r^3 after shrinking the upper scale. The determinant event can have an arbitrarily large fixed rate by Lemma 13.2. Apply the local-volume argument in the same order, now taking its loss below the available A_*r^3 or cu^3 rate. This proves the small-time assertions, including their order of tolerance choices.

Finally remove the diagonal. On \max_i|J_{ii}|\le\delta_D the normalized residual changes by at most \delta_D, and the symmetric Hessian by at most (1+\eta_0)\delta_D. On fixed positive bands take a sufficiently small fixed \delta_D; its failure has exponential probability. In the small-scale assertions keep the vanishing-diagonal event as an external restriction and then, for each fixed band, take n large enough. For example, take \delta_D\le\rho/2 and (1+\eta_0)\delta_D\le c_E/4. Then a zero-diagonal witness with residual at most \rho\sqrt n/2, diagonal radius \epsilon_*/2, and Hessian threshold c_E/4 implies the full-matrix witness above.

For the compact path version write h_t=t\mathbf1+B_t in the gauged planted process. On a mesh interval of length \Delta, the coordinate Brownian suprema have a fixed exponential moment after division by \Delta. Independence across coordinates gives

Choose \kappa_0,\Delta so this bound has the required fixed rate and the Brownian increment plus the deterministic drift is at most \rho/4 in normalized norm. A finite union covers the fixed compact interval. A path witness with residual \rho\sqrt n/8 then has residual below \rho\sqrt n/2 at the left mesh endpoint. The Hessian, overlap, and diagonal predicate are independent of the field and persist there. This proves the compact assertion with its reduced residual threshold, also for a fixed finite family of scalar thresholds. All previous exclusions estimated failure intersected with the original spectral restriction, using interlacing rather than a new typicality assertion for a conditional largest eigenvalue.□

Proof. The root and inverse statement, Lemma 5.7 of [28], is deterministic for every finite positive j and applies here at j=1. Its root conclusion uses only the whole-region implication at A=V_y stated above; the overlap lower bound is used later only for the inverse recipe. Denote the unique zero by y_*, and set m_* =\tanh y_*, q_*=n^{-1}\|m_*\|^2. It gives \|y-y_*\|\le C\|F_h(y)\| when \|F_h(y)\|<\rho_0\sqrt n, with C=1+(\|J^\circ\|+3)/c_0. The strict residual inequality will be preserved by the inner buffer below.

We verify the precise diagnostic hypotheses of the companion residual proof. Fix J,h throughout this deterministic subsection; all spin expectations below are under \mu_h, unless a displayed square reweighting specifies otherwise. For a spin function define

For a vector function, (\nabla g)_{ji}=d_i g_j. These are discrete spin-flip derivatives, distinct from the formal site derivatives in the recipes. Put

For each fixed depth the primary differentiation estimates of Lemma 6.1 of the companion article need only ordinary words, with contraction coefficient one. Explicitly their formal derivatives are

The diagonal predictions of H_l cancel by inclusion–exclusion over originally adjacent pairs of each noncrossing pairing.

The word estimates give bounded operator, diagonal \ell^2, and off-diagonal entrywise \ell^4 norms. Taylor’s error in a coordinate difference is at most C|d_i h_j^l|^2; summing its square uses that \ell^4 bound. Differentiating b_lm^{l-1} adds a rank-one term of bounded Frobenius norm, since \|db_l\|\le C_l/\sqrt n. Thus actual derivatives differ from H_l,M_l by bounded Frobenius errors. Every single flip changes a fixed primary vector by at most C_l in Euclidean norm. None of these finite recurrences uses j<1 or convergence as l\to\infty.

We use the following finite recipe class, as in Definition 6.2 of [28]. Fix the primary depth and finitely many seeds s^1,\ldots,s^v\in\mathbb R^n with bounded Euclidean norms. The seeds may depend on the fixed (J,h) but are constant as functions of the spin configuration x. Let \theta(x) be a finite tuple of scalar parameters with \sup_x(|\theta_\alpha(x)|+\|d\theta_\alpha(x)\|)\le C for every component; the averages b_l may be among them. Given auxiliary fields y^1,\ldots,y^{a-1}, a source has the form

All index sets are finite. The site coefficients and every fixed derivative needed by the recipe are uniformly bounded on their argument ranges and on the line segments joining evaluations. Deterministic site labels are allowed with the same bounds. Define

The partial \partial_l differentiates only the explicit h_i^l argument, keeping \theta and all auxiliary arguments fixed; \partial_{y^b}w_i^a=\psi_i^{a,b}, and an absent partial is zero. A terminal vector has the same source form. When a source is reused in a later source or terminal vector, substitute its source formula without expanding the auxiliary fields.

A recipe is admissible from level p when every explicit primary argument in these source and terminal formulas has index at least p. This restriction does not apply to scalar parameters or to the correcting vectors m^{l-1} in the auxiliary formula. All constants may depend on the fixed recipe and its coefficient bounds. In particular, the finite residual inductions below may enlarge the recipe before its diagnostic length and the eventual dimension threshold are fixed.

Only the directional covariance argument uses an inverse recipe. Fix k\ge2 and \Omega_{a\rho}=\{\max(\|R_k\|,\|R_{k-1}\|)\le a\rho\sqrt n\}. When several indices occur we write this set as \Omega_{a\rho}^{(k)}; the superscript is suppressed while k is fixed. The exact recurrence gives

Choose \rho so the displayed recurrence puts \Omega_{2\rho} strictly inside the root lemma’s residual region. Consequently on \Omega_{2\rho},

These constants are independent of k. Define

Differentiation in u proves the exact identity

At a=0, \Phi is the secant slope of \tanh.

Its positive integral and the bounded derivatives of its logarithmic factors give 0<\Phi\le e^{|a|/2} and bounded ratios of every fixed derivative to \Phi. The derivative recipes evaluate their coefficients at nearby spin configurations as well as at the selected residual point. We therefore establish the diagonal estimates on the whole outer buffer. Once the finite recipe and its number of flips are fixed, the threshold in the dimension will ensure that every evaluation stays in that buffer.

The preceding root-distance estimates imply, on the whole outer buffer,

Choose \rho small enough for the last inequality and for all stable field tolerances. This choice precedes the depth and recipe choices. The inverse diagonal therefore belongs to \mathcal A_\kappa with \kappa=c_*/2 everywhere it will be evaluated.

For either w_{0,i}=A_i e_i, \|e\|\le1, or w_{0,i}=n^{-1/2}\partial_r\Phi(h_i^k;(y_*)_i,a), the implicit recipe is

In this site partial, y^{\mathrm{imp}},c_1 and a are held fixed. After this one implicit solve, y^{\mathrm{imp}} may be used as the first auxiliary in an ordinary recipe, with \sqrt n c_1 as an additional scalar parameter and m_*/\sqrt n as an additional seed. The corresponding coefficient and parameter bounds are required on the flip buffers below; no such bounds are imposed on the arbitrary exterior extension of the implicit solution. Put \ell_i=\partial_z\Phi/\Phi, u_0=w_0/A and p_0=A\partial_k u_0. Then \partial_k w=\operatorname{diag}(\ell)w+p_0 and elimination gives the actual finite-dimensional solve

Put

Because \|\ell+m_*\|\le C\rho\sqrt n, the two-sided normalized coefficient I+A^{1/2}\mathcal LA^{1/2}, where the displayed solve is (I+A\mathcal L)w=\cdots, differs in operator norm by O(\rho) from \mathcal H(h^k,A). The normalized coefficient need not be symmetric. The latter matrix has a fixed positive margin by (13.1), so a perturbation smaller than half that margin still has smallest singular value bounded below and a bounded inverse. The unnormalized inverse is bounded by

This proves \|w\|+\|y^{\mathrm{imp}}\|+\sqrt n|c_1|\le C, uniformly in k and the unit seed. Removing the negative rank term in \mathcal H(h^k,A) and replacing b_k by \chi_A also gives

This separate adaptive stability condition is essential; the norm–trace restriction alone was only used for fixed-parameter expectations.

Every coordinate-difference solve uses the same displayed coefficient matrix evaluated at the flipped configuration. All coefficient differences multiply vectors at the common starting configuration. Their squared column norms sum to a constant: for a coefficient \psi and a fixed bounded-norm vector v,

The averaged coefficients have differences of order n^{-1/2}; the factors m^{k-1}+m_* have norm at most 2\sqrt n.

Columnwise inversion therefore gives \|\nabla w\|_{\mathrm F}+\|\nabla y^{\mathrm{imp}}\|_{\mathrm F} +\sqrt n\|dc_1\|\le C_k, as in Lemma 6.3 of the companion article. If the finished finite recipe uses at most N_f successive flips, the primary bounds put every required flip-neighborhood of \Omega_{3\rho/2} inside \Omega_{2\rho} once n>(2N_f\max_l C_l/\rho)^2. Thus all four bounds in (13.4) and the adaptive inverse margin hold at every derivative evaluation. The recipe and its depth are fixed before this threshold on n is chosen.

We also fix the whole-cube convention for these local recipes. Keep the actual implicit solution y^{\rm imp},c_1 on \Omega_{2\rho}^{(k)}, assign arbitrary finite values to them outside that set, and define w and all later ordinary auxiliaries by their stated formulas. The implicit equation is required only on the outer buffer. Every scalar test in a local residual statement is supported on \Omega_\rho^{(k)}. Its discrete differences use only the fixed finite flip neighborhoods already placed inside the buffers, so the exterior values never enter a tested estimate.

We record the remaining closure of the residual proof to ensure that no other inverse diagonal is being assumed. Freezing parameters in formal differentiation yields

Solving this equation introduces exactly the one factor K=(I-AJ^\circ+\chi_AA)^{-1} multiplying \operatorname{diag}(\partial_k w)H_k-c_1AM_{k-1}+E_1+AE_2. Later auxiliaries are ordinary, so error propagation multiplies by ordinary words only.

The Taylor errors have bounded Frobenius norm and bounded trace against the required words. For the only non-nuclear product error, the decisive bound is

The diagonal part uses the primary diagonal \ell^2 estimate. All other dropped terms are bounded in nuclear norm, using \|BC\|_{\mathrm{nuc}}\le\|B\|_{\mathrm F}\|C\|_{\mathrm F}.

The formal trace cancellation in Lemma 6.4 of the companion article is algebraic. Its only cyclic node closes as

Both terms, including the averaged correction, must be retained. For each dependency path, inclusion–exclusion over its originally adjacent contracted pairs cancels every complete noncrossing pairing. There is no empty derivative path since seeds do not depend on x.

The contracted trace prediction vanishes as a formal series and is a polynomial by Lemma 13.1. Its evaluation at z=1 is therefore zero; this does not require convergence of the inverse’s formal series. Each resulting trace has either a bounded-norm small vector as a diagonal factor or a scalar O(n^{-1/2}). In the first case rotate the small diagonal to the beginning of the trace. Cyclic invariance of the normalized prediction preserves the canceled value, and Cauchy–Schwarz gives

In the scalar case the diagonal word error has trace at most C\sqrt n, canceled by its O(n^{-1/2}) coefficient. These bounds remain uniform after a norm-one diagonal test. Thus every small vector U in an ordinary recipe obeys the following bound on the whole cube, and every local recipe vector obeys it on \Omega_\rho^{(k)} with the stated flip buffers, including the vector \partial_k w:

This proves the derivative assertion with only the single inverse whose full buffer has already been checked.

We state the residual interfaces that now apply, specializing Lemmas 6.5–6.6 of [28] to coefficient one. Put

If U is a terminal vector of an ordinary recipe admissible from level p, then

for every scalar G. For a terminal vector of a local recipe admissible from level p, with implicit starting index k, the same inequality holds when p\le k and G is supported on \Omega_\rho^{(k)}, with the buffers active. For an ordinary recipe admissible from level p and a multiplier satisfying \sup_x(|H_0(x)|+\|dH_0(x)\|)\le C_0, the square-density version is

Only ordinary recipes are used in this last version.

The induction can be checked directly. Its base identity is

The preceding diagonal \ell^1 bound and Cauchy–Schwarz control it by C\mathcal E_h(G). For the induction from p to p+1 multiply the terminal vector by \Phi(h_i^p;h_i^{p+1},b_p-b_{p-1}), add one ordinary auxiliary, and use the displayed scalar identity for \Phi.

Here |b_p-b_{p-1}|\le1, so all required coefficient bounds hold. The term n(b_p-b_{p-1})\operatorname{av}(\partial_pU') cancels by

Only smaller residual indices remain; no further inverse is introduced. The square-density version follows from d_i(f^2)=2f\,d_if-2x_i(d_if)^2 and has right side C(\mu_h(f^2)+\mathcal D_h(f)). It uses ordinary recipes only. All these operations are finite, so one fixed diagnostic length covers the whole argument.

For the implicit recipe define the terminal residual

For every G supported on \Omega_\rho^{(k)},

Indeed, on the buffer the exact expansion is

The local weak rule applies to the first three terms. In the last one use target m^k/\sqrt n and multiplier \sqrt n\,c_1, whose size and discrete differences are bounded on the buffer.

We finish by spelling out selection and the uniformity in the tests. Put T_l=R_l^{\mathsf T}m^l/\sqrt n. The ordinary residual rule gives \mu_h(T_l^2)\le C_l and

The exact telescoping identity \|R_l\|^2/n=b_l-b_{l-1}-2T_l/\sqrt n supplies a pair of consecutive small residuals. More precisely, choose a fixed depth d_{\rm sel} with \lfloor d_{\rm sel}/2\rfloor\rho^2/4>2, and then a scalar tolerance with 2d_{\rm sel}\varepsilon<1. If \max_{l\le d_{\rm sel}}|T_l|/\sqrt n\le\varepsilon, the telescoping sum is less than 2, forcing a consecutive pair whose two norms are at most \rho\sqrt n/2.

Take a smooth \vartheta:[0,\infty)\to[0,1], equal to one on [0,1/4] and zero on [1,\infty), and set

Pointwise,

The derivative bounds follow from the primary differences and the finite product rule. Also C_k=1 when both residual norms are at most \rho\sqrt n/2, and

The scalar moment bounds and a finite union bound give

These are the conclusions of the companion Lemma 7.1, with the preceding derivation of its endpoint hypotheses.

Residual selection now turns the finite recipe estimates into posterior bounds. The covariance argument uses its one controlled inverse; the square-reweighted mean argument below uses only ordinary recipes. This distinction also separates their diagnostic requirements.

We make the terminal implication explicit. On the selected buffer put p_k=m^k-m_* and L_*=\sqrt n(\chi_A-(1-q_*)). Subtracting the root equation gives

Using the implicit equations and the scalar identity for \Phi then yields the exact identity

The final term is controlled by the local weak rule with target p_k/\sqrt n and multiplier \sqrt n\,c_1. The remaining scalar error satisfies pointwise

Here the root-distance and source-size bounds give C_0 depending only on the fixed norm and stability margins, not on k or the selection depth. The earlier choice of \rho includes C_0\rho<1/2.

For the source w_{0,i}=n^{-1/2}\partial_r\Phi, differentiation of the scalar identity in r, with a fixed, gives

Thus the sum in the terminal identity is L_*. Globally, |L_*|\le C\sqrt n and \|dL_*\|\le C_k. Use the local test G=C_k^2L_*, whose support lies in \Omega_\rho^{(k)}. The cutoff difference bound gives

The tested scalar error is at most C_0\rho\,\mu_h(C_k^2L_*^2). Absorbing it into the left side of the terminal inequality proves

For the source w_0=Ae, \|e\|\le1, the same sum is p_k^Te. Use the local test \widehat C_kG. Its support and difference bounds control its test norm by C_\rho\mathcal E_h(G). For the scalar error use \widehat C_k\le C_k, the last L^2 bound, and Cauchy–Schwarz. The terminal and weak residual inequalities give

Summing the ordinary weak rule with target e from level one through k also controls the tested (X-m^k)^Te.

The deficit costs at most 2\sqrt n\|G\|_2\mu_h(D>0)^{1/2}. This is the deterministic argument of Proposition 7.2, applied only after checking its hypotheses on the selected regions and verifying both of its actual source diagonals above. It yields the asserted covariance inequality after replacing G by G-\mu_h(G) and taking the supremum over e. For G=e^{\mathsf T}X, \mathcal D_h(G)\le1; taking a top covariance eigenvector gives \|\Sigma_h\|\le C(\|\Sigma_h\|+1)^{1/2} and hence a constant bound.

For the mean estimate, the ordinary residual rule suffices; no implicit source is needed. On each selected region m^k is within C\zeta\sqrt n of m_* if the residual threshold was chosen below a fixed multiple of \zeta. The ordinary square-density rule, summed with the constant target e through k, bounds the tested difference between X and m^k by C_\zeta(1+\mathcal D_h(f)/N). The deficit has the same bound. This proves the conclusion of the companion Proposition 7.3 relative to m_*.

Apply it also with f=1 and use the triangle inequality to obtain (13.2). If stability is absent, choose the iteration threshold so the exact formula for F_h(h^k) is at most \delta\sqrt n on each selected region. Then \|m^k\|\le\sqrt{nQ_\delta(h)}, and the same argument, once for \nu_h and once for \mu_h, proves the coarser assertion. This also covers an empty approximate-zero set, since then every selected cutoff vanishes and the displayed deficit estimate controls the entire measure.

All bounds use only norm bounds on seeds, so they hold for the whole Euclidean unit ball simultaneously. The finite recipe choices may depend on the requested scalar tolerance, but the diagnostic event quantifies over every bounded diagonal parameter after J is observed. It therefore does not depend on f,G, or the directional seed. For covariance choose the stability margins and c_*, then \rho and \kappa, then the selection depth and finite recipes, then their word length, coefficient bounds and truncations, and finally the size threshold guaranteeing every flip buffer. For (13.2) and its coarser form, only the ordinary diagnostics are required throughout.□

Proof. The signed curvature inequality, Lemma 8.1 of [28], is deterministic for arbitrary symmetric zero-diagonal interaction matrices with sufficiently small entries. For interaction B, write U_{ij}=\tanh B_{ij}, Q_{ij}=U_{ij}^2 (entrywise), r_4(U)=\max_i\sum_jU_{ij}^4, and a_i=v_i d_i f, where v_i is the conditional spin variance. It states

Its proof has no inverse-temperature parameter, so this is an exact companion input.

Here is the endpoint matrix event, with exponential rather than only asymptotic probability. Fix a curvature tolerance \varepsilon>0 so small that (1+C)\varepsilon\le1/8. Choose a fixed global bound M for the row-square sums and \|J^\circ\|, with exponentially small exception. Next choose an entry tolerance \delta>0 small enough for the curvature lemma and for M\delta/3\le\varepsilon/2 and CM\delta^2\le1/8. Gaussian tails give \max|J^\circ_{ij}|\le\delta outside probability e^{-c_\delta n}. For each fixed unit vector z,

A 1/4-net of a k-dimensional unit sphere has size at most 9^k.

Union these bounds over supports of size at most \alpha n and the 9^{\alpha n} nets. Since n^{-1}\log\binom n{\lfloor\alpha n\rfloor}\to\mathfrak h(\alpha), choosing \alpha sufficiently small gives, with exponential probability,

These are the net and row arguments of Lemma 8.2 of the companion article, now evaluated at variance 1/n.

They work simultaneously for all B=\theta J^\circ, 0\le\theta\le1. Indeed |\tanh z-z|\le|z|^3/3 gives \|\tanh(\theta J^\circ)-\theta J^\circ\|\le M\delta/3 by row sums; Q_{ij}\le(J^\circ_{ij})^2 and r_4(U)\le M\delta^2. By the preceding choice of \delta, these errors fit the \varepsilon and curvature budgets. We obtain uniformly in \theta bounded global norms of B,U,Q, small principal submatrix norms of U,Q on |S|\le\alpha n, and Cr_4(U)\le1/8. The bounds hold pointwise over spin configurations and require no union over configurations or initial laws.

The observation source law may differ from the Gibbs law whose gap is being estimated. Throughout this argument the posterior is formed from the Gibbs reference law. A norm estimate makes most coordinates strongly biased for every source configuration; the small-submatrix bounds control the remaining coordinates uniformly.

At time T the observation field is TX+\sqrt T\,g. During the extension it is

where B is a fresh standard Brownian motion independent of X,g. Updating the original Gibbs prior by this likelihood subtracts \lambda x^{\mathsf T}Rx/2 from its interaction; the diagonal is constant on the cube. This identifies its posterior with the stated reference law. The probability estimates below are uniform over the possibly different law generating the observation source X.

On the norm event \|R\|\le C, the perturbation to TX obeys

outside an event of probability e^{-cn}, independently of the law of X. The Gaussian norm tail controls g, and exponential Markov on the sum of the coordinate Brownian suprema squared controls B. Thus, for sufficiently large fixed T, fewer than \alpha n/2 coordinates have |h_{T+\lambda,i}|<T/2, uniformly in \lambda. For every configuration x simultaneously, \|(1-\lambda)J^\circ x\|\le M\sqrt n, so at most 16M^2n/T^2\le\alpha n/2 coordinates have interaction field larger than T/4. Consequently the set where the full conditional field has magnitude below T/4 has size at most \alpha n, for every x and \lambda.

Put q_f(x)=\sum_i v_i(x)(d_i f(x))^2 and \eta=\operatorname{sech}^2(T/4). On the complement of that small set, \|a_{S^c}\|^2\le\eta q_f, while \|a\|^2\le q_f. The small-submatrix and global bounds therefore give pointwise

With \varepsilon already fixed, choose T large enough both for the sparse-set bounds above and for (1+C)M(2\sqrt\eta+\eta)\le1/8. The displayed expression is then at most \mathcal D_h(f)/4 after averaging. The curvature inequality yields \mu_h[(-Lf)^2]\ge\mathcal D_h(f)/2. We finish by converting curvature to the gap.

By the finite-dimensional spectral theorem each nonzero eigenvalue of -L is at least 1/2. The kernel consists of constants because the Ising law is positive on every cube edge. Thus \operatorname{Var}_{\mu_h}f\le2\mathcal D_h(f), uniformly in \lambda and f. Applying this to unit linear tests also gives \|\Sigma_h\|\le2. This proves the proposition.□

Proof. Choose the covariance accuracy \epsilon>0 sufficiently small in terms of the requested \omega, and obtain the corresponding a,s_0,K_{\log} from Proposition 12.3. This same accuracy is used in all three entropy ranges below. Put h=\log(1/p). Entropy relative to the support and the upper bound on f^2 imply

On \|J\|\le3, every atom of \mu is at least 2^{-n}e^{-3n}, so h\le(3+\log2)n. We treat three ranges of h.

Moderate entropy. For constants K' and c' fixed in the order below, first suppose

The environment restrictions below are imposed on dyadic bins for h; using an endpoint of a bin in place of h only changes fixed constants. Fix the low-likelihood exponent C_2>0 and a desired negligible error order n^{-M}. Choose C_{\rm rate} with fixed slack beyond the likelihood exponent 4+C_2 and the subsequent Markov losses. The remaining constants will be chosen after the following static consequences are stated. For every required fixed rate C_{\mathrm{rate}}, the preceding static estimates have the following consequences. All integrated probabilities in this list use planted disorder and reference observations with the spectral restriction imposed; the first also draws two conditionally independent replicas from \mu_b.

1. 1.

   For 0\le b\le c_1t_h, where c_1\le1, a reference posterior pair has |\langle X^1,X^2\rangle|\le C_1nr_h, except with integrated probability at most e^{-C_{\mathrm{rate}}h}.

   Here C_1 need not increase as c_1 decreases.

2. 2.

   On each fixed-ratio window [c_1t_h,L_1t_h], with 0<c_1<L_1<\infty, the replica log-moment estimate (6.2) holds with time b and scale nb^{3/2}, with the required small off-diagonal tilts and any prescribed fixed relative error, outside integrated probability e^{-C_{\mathrm{rate}}h}.

3. 3.

   By choosing L_1 large enough, the sharp bound \|\Sigma_b\|\le(1+\epsilon)/b on [L_1t_h,s_0] has the same failure rate. Beyond s_0, use the bounded positive-time covariance, including the terminal extension. At E, the product field satisfies \max_i|(h_E)_i|\le C'\sqrt h with that rate, also when the exceptional probability is weighted by 1+\max_i|(h_E)_i|.

For completeness, the first assertion does not need a lower spherical width bound at very early times. Impose the upper width bound at scale t_h with an arbitrarily large fixed rate. The spherical length-deficit estimate then controls the pair integral relative to Z_o(h_b)^2, using (6.3), the field-norm bound, and the large-overlap comparison. Exact shell conditioning costs only a polynomial by the spectral and spherical estimates; translations improve the required small-ball bound.

The trace loss is estimated at shifts above the upper width bound. Orientations with Z(h_b)/Z_o(h_b)<e^{-C_{\mathrm{rate}}h} have at most that planted mass conditional on the eigen-coordinates, so division by the partition function is absorbed by an increase of the rate budget. The second and third assertions follow from the spherical width tails, (6.2), (12.5), and the positive-time inputs. For the last field maximum, Gaussian tails and a union bound apply; under planting, the RX term has Gaussian site marginals with bounded translates. Gaussian moment bounds give the assertion with the extra maximum-field factor.

Complete the fixed parameter choices before constructing the disorder event. Obtain C_1 at the chosen rate and then take c_1 small enough for (14.7); C_1 does not increase under this choice. Next enlarge L_1. At the start of its sharp window the covariance exponent is

so L_1 can pay the required rate. Then choose the fixed replica number and its relative accuracy for the intermediate window. Finally decrease c': a positive-time exception e^{-cn} has exponent at least (c/c')h, and this choice also keeps L_1t_h\le s_0 and all scales in their allowed ranges. Increase K' to absorb polynomials and the logarithmic threshold. We may require all discarded errors below to be O(n^{-M}) for the fixed sufficiently large M chosen above.

With these choices fixed, choose the disorder event on which the support profile will hold. Entropy levels have been placed in dyadic bins; in each bin integrate the conditional exceptions over its three time windows. This construction uses the bin and fixed tolerances, before any test function is chosen. For a field failure B_b, write q_J(b)=\mathbb P_J^\mu(B_b) for its reference-field probability conditional on J. For a pair failure, q_J(b) denotes the conditional field average of its \mu_b^{\otimes2} probability. The ordinary high-probability lower bound Z(0)\ge n^{-C}e^{n/4}, together with \mathbb EZ(0)=e^{n/4}, transfers planted exceptions on the spectral event to ordinary exceptions at polynomial cost. Indeed, with L(J)=Z(0)/\mathbb EZ(0),

Integrate over each specified time window, apply Markov with unused rate slack, and take a union over the O(\log n) bins.

This produces one typical disorder event, independent of f, on which all required conditional time-integral exceptions hold. There is enough unused rate to absorb the later likelihood factors. The chosen K' makes their products with any specified polynomial in n and e^{C_0h} negligible, for each of the finitely many constants C_0 used below. Endpoint tests are treated in the same way.

On the early window, the pair bound implies

Indeed the square of a posterior mean is bounded by the posterior expectation of the absolute pair inner product. To apply this also to \nu_b^{\otimes2}, discard w_b<e^{-C_2h}: its probability under \nu is at most e^{-C_2h}, and its squared-drift contribution is at most 4ne^{-C_2h}. On the complement, the density of the \nu-field and two \nu_b replicas relative to the reference field and two \mu_b replicas is exactly

The ordinary posterior mean evaluated under the \nu-field law is controlled in the same way. Taking c_1>0 sufficiently small, (14.1), (14.5), and nr_ht_h=h now give

On the intermediate window, use l replicas in (6.2) with off-diagonal matrix entries \delta/l, where \delta>0 is sufficiently small and independent of l. The entropy variational inequality applied to \nu_b^{\otimes l} gives a multiple of b\|\nu_b(X)-m_o\|^2 bounded by a constant times \operatorname{KL}(\nu_b\Vert\mu_b) and the log-moment cost divided by \delta(l-1). Indeed the expectation of the tilt is exactly b\delta(l-1)\|\nu_b(X)-m_o\|^2/2, whereas the relative entropy is l\operatorname{KL}(\nu_b\Vert\mu_b).

The squared Hilbert–Schmidt norm of the tilt matrix is bounded by \delta^2, uniformly in l. Apply the same inequality to ordinary replicas, whose entropy cost is zero, and use \|a_b\|^2\le2\|\nu_b(X)-m_o\|^2+2\|\mu_b(X)-m_o\|^2. After averaging over the \nu-field law, this yields

Here \zeta>0 can be as small as desired by first taking a sufficiently large fixed l and then the relative log-moment error sufficiently small.

All exceptional field likelihoods are paid for by the same sup-norm bound. Gronwall on this fixed-ratio window, followed by a sufficiently small choice of \zeta, gives

The positive constant may depend on the fixed ratio L_1/c_1, which causes no loss of a power of n.

For later times let c_v be the good covariance bound: c_v=(1+\epsilon)/v up to s_0, and a fixed constant afterwards. Set M_v=\mathbb E^\nu\|\Lambda_v^{1/2}a_v\|^2. Conditional Jensen in (14.1), then Cauchy–Schwarz including its endpoint term, gives

We used 2H_E+\int_b^E M_v\,dv=2H_b.

The deterministic bounds \|\Sigma_v\|\le n, \|a_v\|\le2\sqrt n control the discarded integrals at polynomial cost. Squaring introduces only another negligible error, since all remaining factors are polynomially bounded on this range. As \Lambda_b=I below s_0, the coefficient in the resulting differential inequality for H_b is at most

and is bounded afterwards. Gronwall and (14.9) imply, with \kappa=(1+\epsilon)^2,

For a Bernoulli law with smaller atom z, the elementary two-point inequality is

One may first replace u by |u|. For comparable nonnegative values use \operatorname{Ent}(u^2)\le\operatorname{Var}(u^2)/\mathbb E u^2; if the values are not comparable, bound the entropy by the larger value squared times binary entropy and compare the difference of the values. For a spin of field v, \log(1/z)\le C(1+|v|). Tensorizing at the product endpoint and using (14.3) therefore gives

To justify the correlated exceptional term, put B_E=\{\max_i|(h_E)_i|>C'\sqrt h\}. Pointwise, \mathcal D_E(f)\le n\mu_E(f^2)=nw_E\le n\|f\|_\infty^2. Hence its contribution is at most

The weighted endpoint exception and the restricted supremum bound make this negligible. On B_E^c, pull out C'\sqrt h and use (14.3).

Combining (14.11) and (14.12) yields

At \kappa=1 the right side is c n^{-2/3}h^{7/6}. The initial choice of \epsilon in terms of \omega proves (14.4) in this range, since h\ge1 and the extra exponent of n can be less than \omega.

Small entropy.

In this range the test likelihood is only polynomially large, so the bottom-scale estimate can absorb its exceptional contribution. We use variance rather than entropy; the accumulated covariance bound has the leading exponent needed for the support profile. For h<K'\log n, use V_b=\mathbb E^\mu\operatorname{Var}_{\mu_b}(f) up to T. Cauchy–Schwarz on the support gives V_0\ge1-p\ge1/2. Filtering and the covariance bound imply

Here the exceptional errors can be made arbitrarily small polynomially: the test bound is polynomial in n, (12.8) has any required inverse-polynomial failure rate, and (12.5) is used only above a sufficiently large logarithmic threshold.

Specifically take b_*=(K''\log n/n)^{2/3}, with K'' enlarged as necessary. Below b_* use c_b\le Cn^{2/3}(\log\log(n+1000))^C from (12.8); its integral is o(\log n). Between b_* and s_0 use (1+\epsilon)/b, and then the bounded positive-time input. Thus

The same transfer and time-integrated Markov argument supplies one typical environment event for these assertions. Gronwall gives V_T\ge n^{-2/3-O(\epsilon)-o(1)}. The terminal gap and (14.3) give V_T\le C\mathcal D_J(f)+n^{-M}. The initial choice of \epsilon leaves enough power slack to absorb h\le K'\log n and obtain (14.4).

Entropy of order n. It remains to treat h>c'n. The maximum test likelihood no longer controls the fixed positive-time exceptions. Instead, we charge those exceptions to the relative entropy spent in changing the observation law. The resulting differential inequality forces that spent entropy to vanish at leading order. We show that, for every fixed C_*>0, with probability tending to one no test function in this range can have \mathcal D_J(f)\le C_*n^{1/3}. Taking C_*=1 implies the desired estimate because h\le(3+\log2)n.

The active class throughout this paragraph is still

We choose the diagnostics before selecting a violating function. Take a countable list of compact positive-time bands, integer rate multipliers A_*, the fixed dyadic outer-threshold grids from Section 13, and residual and mean tolerances tending to zero on each band. For each fixed finite collection, the integrated planted bounds, the size-bias lower bound, and Markov with unused rate slack give an ordinary disorder event of probability tending to one. Intersect it with the finite ordinary word diagnostics, the original spectral restriction used to define R, and the ordinary disorder event in Proposition 13.6. This intersection still has ordinary probability tending to one, and all its restrictions are independent of f. On this event the corresponding conditional reference-field bounds hold outside time sets whose Lebesgue measure tends to zero.

If the desired conclusion failed with probability bounded away from zero, diagonal selection would therefore give sizes tending to infinity, environments satisfying successively more of these finite collections, and a violating restricted function on each. All diagnostics in this selection precede that function. Work along such a sequence and define

These functions vanish at zero, are nondecreasing, and are 2-Lipschitz by (14.1) and \|a_b\|\le2\sqrt n. Passing to a subsequence, assume e_n\to e uniformly. We prove e=0.

For the fixed environment J and b\in[0,T], let B\in\mathcal F_b be an event of the observation history through b with \mathbb P_J^\mu(B)\le e^{-nw}, where w>0. The relative entropy of the restrictions of the two field laws to \mathcal F_b, conditional on this same J, followed by binary relative entropy, gives

Indeed this restricted field relative entropy is ne_n(b), and its projection to the indicator of B is at least \mathbb P_J^\nu(B)nw-\log2.

At small fixed positive times apply (13.1) with any fixed large rate multiplier A_*. Its bad event has w=A_*b^{3/2} in (14.14), after using rate slack in the environment transfer. On its complement, (13.1) implies the whole-region stability at A=V_y required by (13.2); that mean estimate uses no overlap lower bound. It makes \mathbb E^\nu[\|a_b\|^2/n] arbitrarily small as n\to\infty and then its tolerance tends to zero. To see the error scale explicitly, the additive energy error in that estimate has expectation at most

by (14.3). Since \|a_b\|/\sqrt n\le2, this error is charged linearly when estimating the squared drift.

On the bad event use the approximate-zero estimate (13.3). Write Q_b for the supremum of q(y) over the sufficiently-small-residual points occurring there. It yields, in squared units, an upper bound CQ_b plus the same vanishing error. For C\sqrt b\le u\le u_*, the outer estimate supplies \mathbb P_J^\mu(Q_b>u)\le e^{-cnu^3}. On a compact positive-time band, and with finitely many thresholds, one may use a single residual tolerance smaller than all those provided: an approximate zero at this tolerance remains one at each larger tolerance.

Put R_b=C'A_*^{1/3}\sqrt b, taking C' large enough for the outer range and the initial interval short enough that R_b\le u_*. If B_b is the stability failure, then 0\le Q_b\le1 gives

On a fixed compact band the moving thresholds are bracketed by the finite fixed dyadic grid described in Section 13, losing at most a fixed factor in their cubic rates. Binary transfer makes the first term at most CR_be(b)/(A_*b^{3/2}) in the limit, while the sum is at most

The final saturated tail uses the rate at u_* and contributes at most Ce(b)/u_*^3\le C_{u_*}e(b)/R_b^2. Thus no outer rate beyond the supported small range is used. Consequently, in integral form on every compact band in the initial interval,

The constant C is independent of A_*. To see this, first fix A_*, then take n\to\infty, and then send the mean tolerance to zero. The stability margins and the refined mean constants, which may depend on A_*, disappear in this order. The remaining coefficient comes from the absolute leading constant 2 in (13.3), the absolute outer-tail constants, and the fixed u_*.

Choose A_* so large that \theta=CA_*^{-2/3}<1. For 0<a<b in this interval, Gronwall and e(a)\le2a imply e(b)\le2b^\theta a^{1-\theta}; letting a\downarrow0 proves e(b)=0. On the remaining positive times, the fixed-rate version of (13.1), the good-field estimate (13.2), and (14.14) give e'\le Ce on each compact band. Therefore e=0 through T.

The selection at the start of this paragraph licenses these inequalities in integral form on every selected compact band. The omitted time sets have vanishing measure, and the uniform Lipschitz bound on e_n makes their contribution vanish. Thus the limiting conclusion e=0 applies to the selected sequence of violating functions.

As e(T)=0, (14.5) gives H_T\ge c'n-o(n). During the terminal extension the bounded covariance input holds with exponentially small exceptions even for an arbitrary input spin law. It therefore applies directly to the \nu-field law. The proof of (14.10) on this fixed interval gives H_E\ge cH_T-o(n)\ge c''n. At the product endpoint it is enough to use \max_i|(h_E)_i|\le C_B\sqrt n, with exception at most e^{-Bn} for any fixed B. This follows from \|R\|=O(1), \|X\|=\sqrt n, and Gaussian tails.

Choose B to pay the restricted bound f^2\le100e^{2h}. The deterministic estimate in (14.13), now with threshold C_B\sqrt n, and the corresponding weighted Gaussian tail control the correlated exceptional energy. The same product entropy estimate now gives

a contradiction. This completes all three ranges and the proposition.□

Proof. The restricted profile holds for every admissible test on one disorder event. We may therefore apply it to all dyadic pieces of an arbitrary function on that same event. Pieces too small to meet the likelihood bound will lose only a fixed fraction of the squared norm, and their combined heat-bath energy is bounded by the original energy.

It suffices to consider f\ge0: replacing f by |f| preserves its support and squared norm and cannot increase any conditional-variance form. For every integer j put

For a scalar u\ge0 the same truncations satisfy

For 2^m\le u\le2^{m+1} the sum is 4^m/3+(u-2^m)^2, which verifies both bounds; u=0 is immediate. Thus S:=\sum_jA_j is finite and lies between \mu(f^2)/4 and \mu(f^2).

The truncation increments also obey, for u\ge v\ge0,

The sum identity follows by integrating the indicators of the disjoint intervals (2^j,2^{j+1}). Hence \sum_jd_j^2\le(u-v)^2. Using the representation of conditional variance by two independent conditional samples gives

All sums are justified by nonnegativity and monotone convergence.

Discard indices with A_j<4^jp_j^2/100, as well as p_j=0. On \{f>2^j\} the preceding piece g_{j-1} is saturated, so A_{j-1}\ge4^{j-1}p_j. Since p_j\le p\le1/2, the discarded pieces satisfy

Every remaining normalized piece g_j/\sqrt{A_j} has squared supremum at most 4^j/A_j\le100/p_j^2. Proposition 14.1, on its single event valid for all test functions, therefore gives

Renaming the constant proves (14.16).

Finally, for an arbitrary real u choose a \mu-median m, so that u_+=(u-m)_+ and u_-=(m-u)_+ both have support mass at most 1/2. For each pair of values the squared increments of these two functions sum to at most the squared increment of u. Thus the support profile gives

This is the asserted gap bound.□

Proof. It suffices to prove the value comparison on a slightly larger prescribed ball, with an error \varepsilon_v k_p chosen below the requested value error. To see the mean implication, let u\in H_p be a unit vector and choose t>0 so that h\pm tu stay in the larger ball. The two value bounds and (15.8) give

Jensen’s inequality in the two signs yields

Set t=\theta p^2M_p, and take u along the projected discrepancy. The saddle separation gives p^2\|R_p|_{H_p}\|\le C, while p^2M_p^2=k_p. Hence

Choose \theta from the desired mean tolerance, then \varepsilon_v\ll\theta^2, and finally k_p large. The larger ball provides the same fixed margin at boundary fields. We now prove the value comparison and the stated test transfers.

Replica Gram comparison. Haar first moments agree exactly for cube and sphere. For l-th moments, l fixed, the angular average once the replica Gram matrix is given is again the same. At off-diagonal normalized overlaps bounded by small \rho the cube Gram law is bounded by the spherical law, with factor

using comparison cells of radius O_l(1/n). Indeed the off-diagonal products of the signs in a row form a bounded lattice vector in fixed dimension l(l-1)/2, with identity covariance. Exponential tilting gives near the origin a local upper probability bound C_l n^{-l(l-1)/4} times the Cramér exponential. The prefactor follows by Fourier inversion (off neighborhoods of the finite dual-lattice points in the torus one has strict decay, uniformly in small tilts). Its entropy rate agrees with -\frac12\log\det Q, Q the Gram matrix with diagonal one, up to and including degree three: the only nonzero third moments are triangles, giving the cubic term minus the sum of triangle monomials. The spherical density proportional to (\det Q)^{(n-l-1)/2}, by successive projection of unit vectors, has the reciprocal-width prefactor. Cells of sidelength comparable to 1/n can be taken disjoint. Cholesky coordinates on the angular integral have bounded derivatives near the identity so change of the exponential weights on the cells costs O_l(1) in logarithm. Inequalities on projected samples used as restrictions there can correspondingly be thickened by O_l(1/n) in unit-sphere norm.

For two replicas and \rho\le n^{-0.27}, the binomial masses equal spherical masses on overlap cells of width 2/n to relative o(1). At zero field the angular integrand is even increasing in absolute overlap (the sum/difference orthogonal coordinates have weights with coefficients 1+r,1-r); thus these cell comparisons can then use monotonicity instead of a constant-factor loss.

Zero field and overlaps bounded away from zero. Fix the eigenvalues and let \E_O denote Haar averaging of the eigenframe. At uncapped zero field, and also for a small zero-field cap with the bounded scalar modifications above and comparable saddle separation, we have the following formulas. Here A denotes the uncapped interaction J or the indicated capped interaction.

For the second assertion, outside n^{-0.27} up to a small constant, (15.10) costs less than the spherical tail gain on dyadic bands. For the uncapped sphere use the preceding overlap argument with lower-normalizer scale p taken minimal. At overlaps bounded away from zero the net exponential is strictly negative even with binomial weights. Here is one verification. In sum and difference orthogonal coordinates, upper exponents are bounded by the spherical free energies at 1+|r| and 1-|r|; restriction to the orthogonal hyperplane does not change the bound by interlacing. In the limit at zero cutoff the spherical upper bound F(t), exact at 1, has F'(t)=t/2 for 0<t\le1 and F'(t)=1-1/(2t) above 1. This follows by Gaussian change of density with precision t(w-J), giving per-site bound (tw-1-\log t-n^{-1}\log\det(w-J))/2, optimized at w=t+1/t below one and at w=2+0^+ above. The per-site density correction for the upper bound at strictly positive separation is o(1), also in the hyperplane; equality at 1 follows by the deficit-filling lower bound. Hence F(1+r)+F(1-r)-2F(1)<r^2/2, whereas the binomial cost is at least r^2/2. Continuity handles endpoints, and a small operator-norm change of the matrices preserves these strict bounds.

All errors giving convergence of the moment ratio are summable over our geometric list (Stirling errors negative powers of n; tails for |r|\ge n^{-0.27} exponentially small in a power of n). This proves zero-field concentration and transfers projection tails, by the first moment and Markov.

Positive moments and uniform upper bounds. The zero-field comparison supplies a denominator and removes pairs with macroscopic overlap. At a nonzero field, we first keep the replicas in the small-overlap region and compare their joint Gram law. The resulting high-moment estimates have two uses: they control the upper value of the partition function, and they transfer a posterior test whose spherical failure has a positive exponential rate. The latter use requires an additive estimate on discarded replicas after the denominator has been bounded below.

For bounded field lengths, high positive moments restricted to small overlaps among all replicas have normalized exponents as small as desired in units k_p, by (15.10) and the spherical tail at overlaps above C p; the relative error O_l(np^4) here suffices. The restriction discards a negligible relative contribution with high probability in the quenched model, uniformly: pairs at overlap above fixed small cutoff have exponentially small probability at cap zero field by the preceding argument, and the field costs only O(n p^{5/2}). One may take the upper scale small after fixing l.

Here is the normalized form of the test transfer used later. Fix the eigenvalues and a field u in the spectral copy of H_p; in physical coordinates J_p(O)=O J_p^{\rm diag}O^\top and h_O=Ou. Let E_u be one of the projected-configuration events described by the Gram-cell comparison, with fixed geometric slack. Write E_u^+ for its specified thickening: whenever a configuration satisfies E_u, all configurations in the same Gram cell satisfy E_u^+. Define the tested, unnormalized sum

Its ratio to Z_{p,O}(u)=Z_p(h_O) is the posterior tested probability. Assume that the spherical probability of E_u^+ is at most e^{-c k_p}. For l independent copies, let \mathcal Z_{E,l}^{\rm small}(O,u) be the sum defining Z_{p,E,O}(u)^l, restricted to pair overlaps below the fixed small cutoff. The Gram comparison and the spherical tail give

For fixed l, the exponent loss \eta_l>0 can be made as small as prescribed by decreasing the upper scale; this includes the O_l(np^4) Gram error. The remaining o_n(k_p) is taken afterward. The spherical denominator depends only on the spectral field u. On overlap bands larger than Cp, Cauchy–Schwarz divides the spherical test cost by two and the remaining spherical overlap cost pays (15.10). This is the source of the factor 1/2.

Let b_l(O,u) be the posterior l-replica probability that some pair exceeds the fixed overlap cutoff. Directly from the decomposition of the replica sum,

On the quenched high-probability zero-field event, the pair estimate above, followed by the bounded-field change of density, makes b_l exponentially small in n, uniformly after the upper scale is chosen small for fixed l. Once the value lower bound proved below gives Z_{p,O}(u)\ge e^{-\varepsilon_v k_p}Z_p^{\rm sp}(h_O), Markov’s inequality applied to (15.11) shows that

Choose c'<c/2 and \varepsilon_v<c/2-c'. Increasing the fixed replica number l, and only then decreasing the upper scale, gives any prescribed fixed Haar failure rate. The separately bounded event for b_l is added after normalization, as the displayed decomposition requires. The untested positive moments give the upper value bounds.

The reusable test statement takes the spherical rate as a hypothesis. Linear and quadratic projection tests, including bounded spectral transformations, supply that rate through (15.7) when their exponential tilt retains the required precision margin. When a later concrete test also uses Gaussian observation noise, each replica receives an independent noise mark. Its application must check the same cellwise inclusion with those marks fixed; a small spherical probability alone does not extend the statement to an arbitrary bounded test.

Upper value bounds extend uniformly by convexity from a fine constant mesh on a larger ball. The zero-field bound (and evenness) then gives a Lipschitz constant in scaled units on inner balls. Posterior tests with slack and positive-rate tails can likewise use nets, by restricting the change of density to bounded projection lengths first (the removed large-length tails follow from the zero-field version and absorption). We only use nets of size \exp(O(k_p)).

Angular lower bounds and gradients. Upper moments alone do not provide a lower value at every field direction. We first obtain a set of directions on each length sphere where the value is large and the projected gradient is close to its prediction. The scalar modification of the cap turns a zero-field partition comparison into an average over such a sphere. A second-moment estimate then controls how small this set of directions can be.

First, on each fixed length sphere in field space, up to arbitrarily small log losses in scaled units its angular partition average has the predicted lower bound with high probability. To see this at nonzero length add a scalar interaction on P_p to J_p so that, at the parameter \alpha predicted for the given length, the eigenvalue of R_p on P_p increases by \|a_p\|^2/{\rm Tr}P_p. This restores the trace to n, with comparable separation. Decouple the addition by a centered Gaussian. Under the integrated spherical weight the field length thus generated concentrates exponentially at the specified length (the unconditioned generated field using the modified saddle is Gaussian, with mean zero and expected squared length precisely \|h\|^2; the norm constraint has at most polynomial cost in k_p, by density bounds, or by a small quadratic tilt with the bounds above). By first moment and the zero-field two-moment argument at the modified cap the same shell has nearly all the integrated cube weight. Narrowing the shell to any fixed tolerance and using the scaled Lipschitz bounds gives the claim.

Angular second moments of Z/Z^{\rm sp} are at most e^{\eta_{\rm err}\,k_p} with high probability by the small-overlap comparison. Moreover the two-replica numerator cost is e^{-c_\tau k_p} for

as follows. In the two independent canonical Gaussians, the exponential factor e^{\pm\theta p^2(P_pX-a_p)^\top(P_pX'-a_p)} changes each two-replica precision by an off-diagonal block of size \theta p^2 on H_p. For sufficiently small \theta the cap band remains comparable to p^2I_2, and the Gaussian log moment is at most C\theta^2 k_p. Equation (15.7) then gives exponent -\theta\tau k_p+C\theta^2k_p+O(1); choosing \theta as a small multiple of \tau gives the stated spherical rate. The Gram comparison transfers it to the angular numerator. Polynomial weights in the scaled projections are handled by the projected norm tails.

Thus there is a set of angular measure at least e^{-\eta k_p}, for arbitrarily small prescribed \eta, where the lower value holds up to error \eta k_p and gradients differ from a_p in projection by any prescribed small fraction of M_p. Indeed discard the points with low value and those with large normalized mean discrepancy in the angular first moment (use the two-replica bound and Cauchy-Schwarz), then use the angular second moment. Losses in these arguments can be chosen in order after the tolerances. Length grids suffice. At these directions, the value and gradient estimates define supporting planes. Evaluating those planes at another field gives a lower bound there. The Gaussian moment-generating-function bound and the bounded saddle range show that the extra scaled loss is a small multiple of the distance plus a constant multiple of its square.

Refinement from a coarser cap. The remaining task is to reach every child field from the large-value sets just constructed. Freeze the child saddle parameter and use its canonical Gaussian observation as a reference bridge to a coarser cap. Under this reference law the parent length is predictable. Fresh Haar refinement makes the reference field close to the useful parent directions, and their supporting planes convert that distance estimate into a lower bound for the child value. The gain must pay for a net at the child scale; this is why the additional scale ratio is chosen small.

For this comparison it is useful to retain the exact normalization of the frozen Gaussian quadratic. For any admissible \alpha, define

The determinant is on the ambient space. Coarea in squared radius gives, for every such \alpha,

This identity also follows by writing the Gaussian density on \|x\|^2=n: its factor independent of x is (2\pi)^{-n/2}(\det R_{u,\alpha})^{-1/2} e^{-n(2+\alpha)/2-h^\top R_{u,\alpha}h/2}, and the squared-radius coarea factor is \pi^{n/2}n^{n/2-1}/\Gamma(n/2).

To obtain a uniform lower bound at c, write c=rp for a very small fixed ratio r, using an additional coarser cap list. Put \Delta=B_p-B_c, freeze \alpha=\alpha_c(h_c), and abbreviate R_u=R_{u,\alpha}, a_c=R_ch_c. Since R_c^{-1}=R_p^{-1}-\Delta, Gaussian integration gives

The corresponding probability law for the parent field is

Equivalently, sample X\sim N(a_c,R_c) without norm conditioning and set Y=h_c+\Delta X+N(0,\Delta). Then X\mid Y=y\sim N(R_py,R_p). Conditioning this joint law on \|X\|^2=n gives the spherical transition and the exact density ratio

The numerator here is generally evaluated away from the parent’s own saddle. Let \widehat\alpha=\alpha_p(y). From (15.15),

The first \alpha-derivative of q_{p,\alpha}^{\rm fr}(y) is \tfrac12(n-\Tr R_{p,\alpha}-\|R_{p,\alpha}y\|^2), which vanishes at \widehat\alpha. Its second derivative is \tfrac14\operatorname{Var}_{p,\alpha,y}(\|X\|^2). Thus, with d_\alpha=\alpha-\widehat\alpha,

On bounded ranges with comparable gaps this variance is O(n/p). Consequently a shell |\widehat\alpha-\alpha|\le\tau p^2 costs at most C\tau^2k_p in the logarithm. The centered parent and child densities have ratio of order \sqrt{p/c}, so their logarithmic cost is O_r(1). This is a statement about the ratio: each individual density contains the factor n^{-1/2}.

The true heat convolution, rewritten using \mathsf G=\mathsf G_{c,p}^{h_c}, is

Jensen and (15.15) therefore give

We must bound both expectations from below, including the reference fields outside any fixed shell. We first prove that the normalization term satisfies

Here R is the prescribed child field radius. Since r is fixed and k_c=r^3k_p\to\infty, this loss is o(k_c) even at the bottom scale.

Averaged normalization loss. The child saddle bounds give -1+\kappa_R\le\alpha/c^2\le C_R, with constants depending on R before r is chosen. Taking r small makes |\alpha|/p^2 small. Put

Then d_p\asymp k_p, v\asymp p^{-2}, D\asymp np, and \Tr R_p^2\asymp n/p. For the deficit, changing the parameter from zero to \alpha changes the trace by at most C|\alpha|n/p=O_R(r^2np), while (15.4) gives the positive deficit (b+o(1))np at zero. The same estimates hold in a sufficiently small fixed p^2-neighborhood of \alpha.

We need a density lower bound valid for every y\in H_p. Split the squared norm of N(R_py,R_p) as S_a+S_b, where the independent active and bulk parts have laws

Write D_b=n-\E S_b=D+d_pv\asymp np. Since \Var S_b\le Cn/p, Chebyshev gives

On this event t=n-S_b\in[D_b/2,3D_b/2]. Spherical integration in the active space, followed by Jensen with a uniform unit vector \omega\in H_p, gives

Here t/(d_pv) lies in a fixed compact subset of (0,\infty). Stirling’s bound for the chi-square density therefore gives f_{v\chi_{d_p}^2}(t)\ge C^{-1}(v\sqrt{d_p})^{-1}e^{-Ck_p}. Convolving over the bulk event proves

The child density is comparable to \sqrt{c/n}, and the noncentral upper bound from (15.5) bounds the parent density by C\sqrt{p/n} for every y. Their common n^{-1/2} factor thus cancels before taking logarithms. In particular the logarithmic ratio in (15.20) is integrable under the Gaussian bridge.

For its sharper average bound, set T=R_c-R_p and W=\|R_pY\|^2. The bridge and the child saddle equation give

Indeed \Tr R_c^2\le C_Rn/c, \|R_c\|\le C_Rc^{-2}, and \|a_c\|^2\le C_Rnc. Fix a sufficiently small \delta>0 and put \mathcal C=\{|W-D|\le\delta np\}. The Gaussian quadratic moment formula bounds \log\E e^{t(W-D)} by C_Rt^2n/c for |t|\le c_Rc^2. Exponential Markov in both signs consequently gives \mathsf G(\mathcal C^c)\le C_{R,r,\delta}e^{-a_{R,r,\delta}k_p}.

On \mathcal C, monotonicity of the saddle equation places \widehat\alpha=\alpha_p(Y) in the comparable-gap neighborhood above: the derivative of \Tr R_{p,s}+\|R_{p,s}Y\|^2-n has negative magnitude at least c_2n/p there. Integrating that derivative to each endpoint first locates the root, and then gives

Along the intervening parameter segment, the norm-square variance is at most Cn/p. The centered density bounds and (15.18) imply on \mathcal C that

For the complement, \|Y\|^2/(p^4M_p^2)\asymp W/(np) has bounded second moment. Cauchy–Schwarz and the preceding exponential tail show that the negative contribution allowed by (15.21) is at most C_{R,r,\delta}k_pe^{-a_{R,r,\delta}k_p/2}=o(1). This proves the lower bound in (15.20). Integrability and (15.17) identify its expectation with -D_{\rm KL}(\mathsf G\Vert\mathsf T_{c,p}^{\rm sp}), which proves the upper bound.

Averaging the parent supporting planes. On a finite length grid near the typical parent length, let \mathcal U be the closed set of fields z satisfying

Choose \varepsilon_a,\varepsilon_g using the preceding angular argument, so that these sets have angular measure at least e^{-\eta k_p} on each grid sphere, with \eta as small as required. The set \mathcal U uses only parent data and the prescribed radii, so it is fixed under the fresh refinement. These anchors lie in a fixed bounded parent field range. Convexity gives a lower supporting plane for \phi_p at z. On the spherical side, (15.8), frozen at the own saddle of z, gives

This holds for every y\in H_p: the denominator density is centered at z, and the numerator has the same covariance, so its noncentral upper bound is independent of y. Subtracting the two inequalities yields the global estimate

Put d(y)=\operatorname{dist}(y,\mathcal U)/(p^2M_p) and D_2=\E_{\mathsf G}d(Y)^2. Choosing a nearest anchor and averaging (15.22) gives

In particular this average includes all Gaussian length tails.

To bound this distance, put \ell_*=(p^2+\alpha)\sqrt D. The child saddle \alpha depends only on the eigenvalues and \|h_c\|, since h_c\in H_c; thus \ell_* is independent of the fresh orientation. The preceding variance estimate gives

Let the grid mesh be at most \xi p^2M_p. Under Haar averaging the direction of Y is uniform conditional on its length. Choose a grid radius within \xi p^2M_p of \ell_* and apply spherical concentration to its anchor set of angular measure at least e^{-\eta k_p}. The radial bound and the triangle inequality give

where O denotes the fresh refinement. Hence the Haar average of \sqrt{D_2} is as small as prescribed. Its Frobenius Lipschitz constant is at most C\sqrt r+O_r(k_p^{-1/2}). Indeed the reference field in parent units has mean norm O(\sqrt r) and covariance norm O_r(1/k_p); the Gaussian L^2 norm of its change between two rotations bounds the change of its L^2 distance to the fixed set \mathcal U. Haar Lipschitz concentration therefore gives D_2\le\epsilon r^3+o(1) except with probability \exp(-c_3\epsilon r^2k_p) (proper rotations suffice). The constants before choosing r depend only on the prescribed child field range. First choose \epsilon from the requested child value error \varepsilon_v, then choose r sufficiently small that the failure rate pays for the fixed fine child net, whose size is \exp(O(k_c))=\exp(O(r^3k_p)). Next choose \varepsilon_a,\varepsilon_g,\eta,\xi so that the Haar mean of \sqrt{D_2} is smaller than \tfrac12\sqrt{\epsilon r^3} and the coefficient \varepsilon_a+\varepsilon_g\sqrt{D_2}+CD_2 is a sufficiently small multiple of \varepsilon_vr^3 on the retained event. Then choose the upper cap scale sufficiently small for the preceding angular estimates, and finally take n large with all parameters fixed. Equations (15.19), (15.20), and (15.23) give the child lower bound throughout the net. Extend by the scaled Lipschitz bound. All finitely many grids and extra lists, with positive-rate failures per k_p or the summable zero-field errors above, retain high probability. This proves (15.9).□