Proof. The Gaussian–chi representation is the classical tridiagonal construction; see (Dumitriu and Edelman 2002, 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 \[R=(2+d^2-A)^{-1},\qquad a_i=(d^2+i/n)^{1/2},\] the positive walk expansion gives \[ (R^p)_{ii}\le C_p a_i^{1-2p},\qquad p\ge1. \tag{12}\] Indeed the positive series \[(R^p)_{ii} =\sum_{j\ge0}\binom{p+j-1}{j} (2+d^2)^{-p-j}(A^j)_{ii}\] 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 \[C_p\sum_{j\ge0}(j+1)^{p-3/2} e^{-c(d^2+i/n)j}\le C_p a_i^{1-2p}.\] 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 \(d=n^{-1/3}\sqrt\ell\): \[ \|R^{1/2}\Delta R^{1/2}\|=o(1) \quad\hbox{with probability tending to one}. \tag{13}\] The block-energy estimate below follows the partial-sum and summation-by-parts method used in (Ledoux and Rider 2010, Proposition 7 and Lemmas 8–9); see also (Ramírez et al. 2011, 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 \[u^{\mathsf T}(2+d^2-A)u =\sum_{i<n}b_i(u_{i+1}-u_i)^2+ \sum_i(2+d^2-b_i-b_{i-1})u_i^2.\] It controls \[\sum_i a_i^2 u_i^2+ \sum_{i<n/2}(u_{i+1}-u_i)^2.\] 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 \[\left|\sum_{i\in B}\zeta_i u_i^2\right| \le C\epsilon\left( a_B^2\sum_{i\in B}u_i^2+ \sum_{i,i+1\in B}(u_{i+1}-u_i)^2\right).\] 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 (13). 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 (13) gives \[\mathop{\mathrm{Tr}}(2+d^2-J)^{-1} =\mathop{\mathrm{Tr}}R+\mathop{\mathrm{Tr}}R^2\Delta +O\bigl(\mathop{\mathrm{Tr}}R\Delta R^2\Delta\bigr).\] The quadratic remainder is nonnegative, so its expectation controls it by Markov’s inequality. Its centered-noise expectation is \(O(d^{-2})\), by (12), 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 \[Q(d)=\sum_{i,j}\rho_i\rho_j^2|\widetilde\Delta_{ij}|^2.\] It is nonnegative and decreases with \(d\), regardless of the signs in \(\Delta\). For the linear term \(L(d)=\mathop{\mathrm{Tr}}R^2\Delta\), \[d\,\partial_d L(d)=-4d^2\mathop{\mathrm{Tr}}R^3\Delta.\] 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 \[\frac{(nd^3)^{1/4}d^{-2}}{\sqrt{n/d}} =(nd^3)^{-1/4}\longrightarrow0,\] 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 \[n^{-1}P_n''-zP_n'+nP_n=0.\] Therefore \(b_A=P_n'/(nP_n)\) obeys \[b_A^2-zb_A+1=-b_A'/n.\] The right side is \(O((nd)^{-1})\) by (12). 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 (11).

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 \[\frac{(-1)^{p-1}\Delta_{hd^2}^{p-1}(t+d^2)^{-1}} {(p-1)!(hd^2)^{p-1}} =\prod_{j=0}^{p-1}(t+(1+jh)d^2)^{-1}\] For an eigenvalue \(v\) of \(J\), take \(t=2-v\). The upper-edge bound already proved in (11) 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 \[(1+2(p-1)h)^{-(p-1)}(t+d^2)^{-p} \le \prod_{j=0}^{p-1}(t+(1+jh)d^2)^{-1} \le (t+d^2)^{-p}.\] The same comparison holds for semicircle depths \(t\ge0\). For fixed \(h>0\), (11) 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 \(p\)th 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 \[\nu_{n,d}=\frac1{nd^3}\sum_i(1+t_i)^{-2}\delta_{t_i}.\] 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 \[\int (1+t)^{-j}\,d\nu_{n,d}(t) =\frac{d^{2j+1}}n\mathop{\mathrm{Tr}}(2+d^2-J)^{-j-2}.\] 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 \[\pi^{-1}\sqrt t(1+t)^{-2}\mathbf1_{\{t>0\}}\,dt.\] 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 (11). ◻

Proof. After diagonalization and normalization it is enough to consider \[\sum_i a_i(g_i^2-1)+2\sum_i b_i g_i,\qquad a_i\ge0, \qquad 2\sum_i a_i^2+4\sum_i b_i^2=1.\] 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 \(\mathbb P(W\le a_i/2)\), 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 \[\bigl(1+4(\max_i a_i)^2t^2\bigr)^{-c/(\max_i a_i)^2}.\] The same proof permits an \(o(1)\) displacement of the target in standard-deviation units. ◻

Proof. The Gaussian shell formula gives \[\log Z_{\mathrm o}(0) =\frac{n(\lambda^0-1)}2 -\frac12\log\det(\lambda^0-J) +\log\frac{\text{length density for }N(0,D^0)\text{ at }n} {\text{length density for }N(0,I)\text{ at }n}.\] 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 (11). At this new argument, concavity of log determinant gives the upper bound \[\log\det(z-J)\le\log\det(z-A)-\mathop{\mathrm{Tr}}(z-A)^{-1}\Delta.\] The linear correction has mean \(O(1)\) and standard deviation \(O(\sqrt{\log n})\), by (12).

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, \[\frac{z-1}{2}-\frac12\int\log(z-v)\,\mathrm d\rho_{\mathrm{sc}}(v)\] 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 (14) 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 \[D^2D^0=\delta^{-2}(D^0-D)-\delta^{-1}D^2\] together with (11) and \(\mathop{\mathrm{Tr}}D^0=n\) gives \[\frac{\mathbb E\|D(rg+sX)\|^2}{nr} =\frac1{2\sqrt\gamma}+\frac1{2\gamma^{3/2}} +O_\gamma(r)+o_n(1), \qquad \frac{n-\mathop{\mathrm{Tr}}D}{nr}=\sqrt\gamma+O_\gamma(r)+o_n(1).\] 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 \[\frac{s^2}{t}\left(\frac1{(\delta+t)^2}-\frac1{\delta^2}\right) =-\frac{s^2(2\delta+t)}{\delta^2(\delta+t)^2}.\] 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 7. Normalize the replica integral bounds from Section 6.1 using (18). To handle large overlaps between replica groups after normalization, put \(q_\eta(U)=\mu_h^{\otimes2}(|R_{12}|>\eta)\). The global pair bound gives \[\mathbb E_U[e^{2F}q_\eta]\le e^{-c_\eta n}.\] 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 (19), 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, \[\mathbb E_U[e^F\mathbf1_E]\le (\mathbb E_Ue^{2F})^{1/2}\mathbb P_U(E)^{1/2}\] 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 \(F=\delta k\): \[\mathbb E_U[e^Fq]\le e^{\delta k}\mathbb E_Uq+ (\mathbb E_Ue^{2F})^{1/2}\mathbb P_U(F>\delta k)^{1/2}.\] 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. 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 \[m'(\lambda)=-\operatorname{Tr}C_\lambda^2.\] 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 \[\frac{f(x_1)}{f(x_0)} =e^{|\lambda|L/2}\frac{f_\lambda(x_1)}{f_\lambda(x_0)}.\] 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 4 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 \[|\mathbb E e^{it\|C_\lambda^{1/2}g\|^2}| \le (1+c t^2)^{-cn}, \qquad \sup_x f_\lambda(x)\le Cn^{-1/2}.\] 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 \[\prod_i(1+4t^2a_i^2/\sigma^2)^{-1/4} \le\left(1+\frac{4t^2a_{\max}^2}{\sigma^2}\right) ^{-\sigma^2/(8a_{\max}^2)}.\] 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, \[|m(0)-x_0| \le L+|m(0)-m(\lambda)| \le L+\int_0^{|\lambda|}C L^2\,du\le C_1L.\] ◻

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 \(B\), \[\int_G\int B(gz,gz')\,d\mu\,dg =\int\mathcal A_B(s)\,d(T_\#\mu)(s),\] 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 \[\frac{n+S+S'+R}{4},\quad \frac{n+S-S'-R}{4},\quad \frac{n-S+S'-R}{4},\quad \frac{n-S-S'+R}{4}.\] 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 \[\eta_i=S\alpha_i+\xi_i,\qquad S=C^{1/2}E(E^TCE)^{-1/2},\qquad S^TS=I,\] 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 \[\|\Sigma-I\|_{\rm HS}\le CP,\qquad \|\Sigma-I\|_1\le C\sqrt{nq}\,P,\qquad \|\mu\|\le CP,\] 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 \[\begin{align*} \mathbb E_{\Sigma,\mu}H={}& \operatorname{Tr}(D\Delta_2D\Delta_1) +\operatorname{Tr}(DR^TDR^T)+(\operatorname{Tr}(DR^T))^2\\ &+2s\operatorname{Tr}(DR^T)+s^2 +\mu_1^TD\Delta_2D\mu_1+\mu_2^TD\Delta_1D\mu_2 +2\mu_1^TDR^TD\mu_2. \end{align*}\] All zero-mean terms except the trace square cost a polynomial by Hilbert–Schmidt bounds. The exceptional term satisfies \[|\operatorname{Tr}(DR^T)|^2\le\|D\|^2\|R\|_1^2\le CnqP.\] 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 \[|\det(I-2iQ)|^{-1/2}\le C(1+c|v|^2/N)^{-c'N}.\] 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 \[\|\Sigma_{12}\|\le CN^{-1/2}(|v|+|u|),\] 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 (40) 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 (37), the derivative bounds following (40), and the mixed local-limit calculation therefore give precisely this cancellation for the smooth saddle ratios. Lemma 17 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 \[t_{ij}=n^{-1}(z_i-P\delta_i)^T(z_j-P\delta_j).\] 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 (43) supplies a fixed positive fraction of orientations with full mass at least \(Z_*/2\). The simultaneous covariance and mean bounds in (36) 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 \[ D_*\le \sum_{i=1}^l|\vartheta_i|^2 +\sum_{i<j}|t_{ij}/q|^2\le4D_*. \tag{44}\] 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 (44). 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 (37) cost at most \(CN D_*^{3/2}+o(n)\). At fixed projections the unweighted reference conditional Gram density has exponent \[-\frac n2\sum_{i<j}\frac{t_{ij}^2}{B_iB_j} +O(ND_*^{3/2}), \qquad B_i=b-2\sqrt q\,(\delta_i)_1-|\delta_i|^2,\] 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 \[\log\mathbb E\exp(\xi^TR\xi/2) =\tfrac12\operatorname{Var}(\xi^TR\xi/2) +O(\|R\|_{\mathrm{HS}}^3),\] 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 \[C\bigl(1+(q\sqrt{D_*})^{-1}\|R\|\bigr) \|R\|q^3D_*.\] Here we used the response bounds (40) 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 \[\|R\|_{\rm HS}\le Cq\sqrt{D_*},\qquad \frac1n\sum_{\text{sites}}\sum_{i=1}^l|d_i^0|^2 \le Cq^3D_*.\] The latter inequality is (40) 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 \[\Pr_{\rm pl}(\mathcal G\cap\mathcal R_s^c)\le n^Ce^{-a_0k},\] 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 \[\int_{\mathcal G\cap\mathcal R_s} \mu_{h_s}^{\otimes j}(E_j)\,d\Pr_{\rm pl} \le n^Ce^{-a_0k},\qquad j=1,2,\] after decreasing \(a_0\) if necessary. Conditional Markov on this retained domain gives \[\Pr_{\rm pl}\!\left\{\omega\in\mathcal G\cap\mathcal R_s: \mu_{h_s}^{\otimes j}(E_j)>e^{-a_0k/2}\right\} \le n^Ce^{-a_0k/2}.\] 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 \[n^{-1}z_i^Tz_j=t_{ij}+\delta_i\cdot\delta_j,\] their statistics force the total in (44) 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 (44). 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 (45) and Paley–Zygmund give \[G\{L_P\ge-1\}\ge(1-e^{-1})^2e^{-\alpha k} \ge e^{-2\alpha k}\] 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 \[ G\{L_P<-\beta k/4\}\le e^{-c\beta k} \tag{47}\] 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 (2013, 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 (47). Consequently, if \(P(A)\le e^{-\beta k/2}\), then \[G(A)\le e^{-\beta k/4}+e^{-c\beta k}.\]

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 (45) 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 \[\operatorname{Var}_P\phi-\operatorname{Var}_P K_{T'}\phi =\sum_{j=0}^{N_t-1}\mathbb E_{x\sim P} \operatorname{Var}_{K_{\Delta t}(x,\cdot)} (K_{j\Delta t}\phi).\] 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 \[(2+o(1))\Delta t\,e^{-v} \int\|\nabla\phi\|^2dP\] 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 (47). 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 (45) bound the exceptional probability by \[e^{\alpha k} \bigl(\operatorname{poly}(n)e^{-c\beta k}\bigr)^{1/2}.\] 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 21.

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 \[Q_s^+=\mathbf1_{F_s^+}+\Pr_s(F_T^+) +\int_s^T(1+b^{-C})\Pr_s(F_b^+)\,db.\] 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 \[8a<\min\{a_{\rm stat},a_*, c_{\rm init}/s_0^{3/2},c_+/s_0^{3/2},1\},\] 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 \(s_0,L_0\).

Proof. Set \(L=n^{2/3}\). The spherical raw second moment about \(m_o\) already satisfies the bound in (51). The hypotheses of the polylogarithmic edge comparison, namely \[\lambda-\lambda^0\le n^{-2/3}\log^C n, \qquad \|h\|\le\sqrt n\,r\log n,\] 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 \[\prod_{a=1}^{p/2} \left(\frac{(x_{2a-1}-m_o)^T(x_{2a}-m_o)}L\right)^2.\] The sign of that term is \((-1)^{p-|B'|}\).

Proof. We give the endpoint changes to Lemmas 3.1–3.4 of (OpenAI 2026a), 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 \[2zT_g v-z^2\chi_Av^2, \qquad 0\le v\le\|A\|^{1/2},\qquad T_g=n^{-1/2}\|A^{1/2}g\|.\] Indeed the increment difference between the comparison process \(2n^{-1/2}\|w\|g^{\mathsf T}w\) and \(w^{\mathsf T}Ww\) is \[\frac2n\left[(\|w\|^2-\|w'\|^2)^2 +2(\|w\|\|w'\|-w^{\mathsf T}w')^2\right]\ge0.\] 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 \[0\le d=z\sqrt{\chi_A}v\le\sqrt{\|A\|\chi_A} \le\sqrt{1-\kappa}.\] 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 \[K_A=I+A^{1/2}S_A^{-1}A^{1/2}(zW-z^2\chi_AI),\qquad K_AA=A^{1/2}S_A^{-1}A^{1/2}.\] 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 \[\mathbb E(WK_A)_{ii}=zr k_i+O(n^{-1}),\qquad k_i=1+z^2a_i h k_i+O(n^{-1}).\] 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 \[\left|z^2n^{-1}\sum_i a_i^2k_i\right| \le(1+C\delta+C/n)n^{-1}\sum_i a_i^2 \le(1+C\delta+C/n)(1-\kappa)\le1-\kappa/2\] 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 \[\mathbb E(UQ)_{ii}\,n^{-1}\mathbb E\operatorname{Tr}V;\] 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 25 along \(L_z=I+z^2bV-zWV\). On its path event, \[\frac{\mathrm d}{\mathrm dz}\frac1n\log\det L_z =2zb\,\frac{\operatorname{Tr}(K_V(z)V)}n -\frac{\operatorname{Tr}(WK_V(z)V)}n.\] 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 \[D(P\Vert N(d,\nu)) \ge u(C-b)-\frac{k u^2}{2}-b[-\log(1-u)-u].\] 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 \(b=1-q<1\), \[h(u)>0\quad(u\le q,\ u\ne0),\qquad h(q)\ge q^3/6.\] 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 \[2D(P\Vert N(d,\nu))\ge C^2/k-1-\log(C^2/k).\] Subtract the target and write \(f(C)=C^2/k-1-\log(C^2/k)-(C-b)^2/(1+k)\). Direct differentiation gives \[f(C_0)\ge-2q^3/3,\qquad f'(C_0)\ge\frac{2q}{1+q},\qquad f''(C)\ge1.\] Consequently \[f(C_1)\ge q^3\left[-\frac23+ \frac{2}{\sqrt3(1+q)\sqrt q}+\frac16\right]>0.\] 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 \[F'(v)=\mathbb E_{N(v,v)}[(1-m^2)(1-m)]\in[0,1) \qquad(v>0).\] 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 \[G_{s,\sigma}(y)= \prod_{i=1}^n\varphi_{d,\nu}(y_i)\,\exp\{n\psi\}.\] 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 \[\limsup_{n\to\infty}n^{-1}\log\int_{P_y\in\mathcal C}G_{s,\sigma}(y)\,\mathrm dy \le\sup_{P\in\mathcal C}\{\psi-D(P\Vert N(d,\nu))\}\] 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 \[\widetilde F_h(y)=y-h+(b(y)I-J)m(y),\qquad \widetilde{\mathcal H}(y,A)=I+A^{1/2} \left(b(y)I-J-\frac{2m(y)m(y)^T}{n}\right)A^{1/2}.\] Let \(E\subseteq\mathbb R^n\times\operatorname{Sym}_n\) be Borel and independent of the observation \(h_s\). For \(\sigma>0\) set \[I_{s,\sigma}(E)=(2\pi\sigma^2)^{-n/2}\int_{\mathbb R^n} \mathbf1_{\{(y,J)\in E\}} \exp\left[-\frac{\|\widetilde F_{h_s}(y)\|^2}{2\sigma^2} +\frac{nb(y)^2}{2}\right]\,\mathrm dy.\] The exact Gaussian reduction, Lemma 5.3 of (OpenAI 2026a), is valid for every positive \(j\) and here reads \[\mathbb E I_{s,\sigma}(E) =\int\sqrt{\frac\nu{\nu+q}}\,G_{s,\sigma}(y) \mathbb P\bigl((y,J)\in E\mid Wm+\sqrt{s+\sigma^2}\,g'=y-d\mathbf1+bm\bigr)\,\mathrm dy,\] where \(d=s+M\), \(\nu=s+\sigma^2+q\) and \(G_{s,\sigma}\) is the explicit scalar weight in Lemma 27. 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 27 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 \[\|A\|\chi_A\le(1+\eta_0)(1-q_0+\epsilon_A) \le1-q_0/2.\] 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 \[\begin{split} P_uWP_u&=P_uW_0P_u,\\ P_uWu&=\bar w+\tau P_uW_0u,\\ \bar w&=\sqrt{q/\nu}\,Z_1-(a/\nu)u, \qquad \tau^2=(s+\sigma^2)/\nu,\\ u^{\mathsf T}Wu&=\alpha_c =2(a+bq)/(\nu+q) +\sqrt{(s+\sigma^2)/(\nu+q)}\,\ell_0. \end{split}\] Writing out the three blocks gives the exact finite-rank difference \[\begin{split} W-W_0={}&u\bar w^T+\bar w u^T -(1-\tau)\{u(W_0u)^T+(W_0u)u^T\}\\ &+\{\alpha_c+(1-2\tau)\ell_0\}uu^T. \end{split}\] 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 \[U=\mathcal Y(u),\quad T_1=\mathcal Y(W_0u),\quad \overline W=\mathcal Y(\bar w),\quad X_1=\mathcal Y(\mathbf1/\sqrt n).\] The exact whitened Hessian is \[\begin{split} S_A^{-1/2}\widetilde{\mathcal H}(y,A)S_A^{-1/2} ={}&I+(b-\chi_A)S_A^{-1/2}AS_A^{-1/2} -X_1X_1^T-2qUU^T\\ &-\bigl[U\overline W^T+\overline WU^T -(1-\tau)(UT_1^T+T_1U^T)\\ &\hspace{35mm} +\{\alpha_c+(1-2\tau)\ell_0\}UU^T\bigr]. \end{split}\] For each fixed \(y,A\), the vectors \(\mathbf1/\sqrt n,u,Z_1\) are deterministic relative to \(W_0\). The bilinear tests in 25 give \[\begin{split} \mathcal Y(p)^T\mathcal Y(r)&\simeq p^TAr,\\ \mathcal Y(p)^TT_1&\simeq\chi_Ap^TAu,\\ \|T_1\|^2&\simeq\chi_A+\chi_A^2u^TAu \end{split}\] 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 \[\begin{split} \beta_U &=2q-\frac{2a}{\nu}+\frac{2(a+bq)}{\nu+q} -2(1-\tau)\chi_A+(1-\tau)^2\chi_A\\ &=q\left(2-\frac{\chi_A}{\nu} -\frac{2(a-\nu b)}{\nu(\nu+q)}\right) \end{split}\] 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 \[M_1=\sqrt q\,\mathcal Y(u),\qquad G_1=\mathcal Y(Z_1)/\sqrt v.\] 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 \(H\): \[C=\begin{pmatrix}1&0&0\\0&2-b/v&1\\0&1&0\end{pmatrix},\qquad H=\begin{pmatrix} b&e&-2e\\e&e&b-3e\\-2e&b-3e&b/v-2b+6e \end{pmatrix}.\] 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 \[\det(I-HC)=(q+2e)(q+e)^2>0.\] 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 \[n\left[\mathfrak h(\alpha)+\alpha\log(C/\delta)+o(1)\right], \qquad \alpha=\epsilon_A^2/\delta^2,\] 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 \[E^{\rm mat}=\left\{(y,J):\exists A\in\mathcal D_{\eta_0},\ \|A-V_y\|_{\rm F}\le\epsilon_*\sqrt n,\ \lambda_{\min}(\widetilde{\mathcal H}(y,A))\le c_E\right\},\] 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, \[\mathbb EI_{s,\sigma}(E^{\rm mat})\le e^{-a_0n}\] for some \(a_0>0\).

The stricter witness event uses \[\|\widetilde F_{h_s}(y)\|\le\rho\sqrt n,\qquad \|A-V_y\|_{\rm F}\le\tfrac12\epsilon_*\sqrt n,\qquad \lambda_{\min}(\widetilde{\mathcal H}(y,A))\le\tfrac12c_E,\] 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 \[\|V_{y'}-V_y\|_{\rm F}\le2\|y'-y\|,\qquad |b(y')-b(y)|\le2d_y,\qquad \left\|\frac{m(y')m(y')^T-m(y)m(y)^T}{n}\right\|\le2d_y.\] 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 \[r_{\rm loc}\le \min\{\epsilon_*/4,\ c_E/[12(1+\eta_0)]\}.\]

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, \[\|\tanh h_s\|\le\|m(y)\|+\|h_s-y\| \le C\sqrt{nq(y)}+\|\widetilde F_{h_s}(y)\|.\] 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 \[q(y)\ge q_{\rm vol}:= \left(\frac{v_-}{2(K+2)}\right)^2>0.\]

At such a witness the Jacobian is exactly \[\widetilde F_{h_s}'(y) =I+(bI-W)V_y-\frac{\mathbf1\mathbf1^T}{n}V_y -\frac{2mm^T}{n}V_y.\] The last two terms have rank at most two and norm at most three. Since \(b\le1-q_{\rm vol}\), Lemma 26 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 (OpenAI 2026a), 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 \[\frac{\varepsilon_1}{2} +C\left[\frac\rho\sigma+\frac\sigma\gamma+ \left(\frac\rho\sigma\right)^2+ \left(\frac\sigma\gamma\right)^2+ \frac\sigma{\sqrt\gamma}\right]+o(1).\] 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 11, 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 \(y\), \[\frac1n\log\frac{G_{s,\sigma}(y)}{G_{s,0}(y)} =-\frac12\log\frac{\nu_\sigma}{\nu_0} +\frac{\sigma^2\mathfrak v}{2\nu_0\nu_\sigma} +\psi(\nu_\sigma)-\psi(\nu_0), \quad \psi(\nu)=\frac{(a-\nu b)^2}{2\nu(\nu+q)}.\] 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 27 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 \[J_\sigma|_E=W_E+ee^T,\qquad P_EJ_\sigma u =\frac q{\nu_\sigma}\frac{P_E(y-d\mathbf1)}{\sqrt{nq}} +\frac M{\sqrt q}e +\sqrt{\frac{s+\sigma^2}{\nu_\sigma}}\,w,\] 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 \(\mathcal G_{\rm spec}(J_\sigma)\), \[\lambda_j(J_\sigma)\ge\lambda_j(W_E) \ge\lambda_{j+2}(J_\sigma), \qquad 1\le j\le n-2.\] 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, \[\frac{\operatorname{Tr}R}{n} =1+O(\sqrt{\Delta_{\rm sp}})+o_n(1),\qquad \frac{\operatorname{Tr}R^j}{n} \asymp_j\Delta_{\rm sp}^{3/2-j}\quad(j\ge2).\] 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 (34). The other needed margin uses the actual full spectrum: \[(b+b^{-1})I-J_\sigma\succeq(1-o(1))q^2I, \qquad b+b^{-1}=2+q^2+O(q^3).\]

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 (34) 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 \[\int_{\mathcal S}G_{s,\sigma}(y)\, \mathsf Q_{y,\sigma} (\mathcal G_{\rm spec}(J_\sigma)\cap\mathcal F_y)\,dy,\] 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 (34).

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 26. 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 \[\Pr\!\left\{\sup_{t_j\le t\le t_{j+1}} \frac{\|B_t-B_{t_j}\|}{\sqrt n}>\kappa_0\right\} \le \exp\{-n(c\kappa_0^2/\Delta-C)\}.\] 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 (OpenAI 2026a), 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 \[d_i g(x)=\frac{g(x)-g(x^{(i)})}{2x_i},\qquad dg=\nabla g=(d_i g)_i.\] 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 \[m^0=x,\quad b_0=0,\quad h^1=h+J^\circ x,\qquad m^l=\tanh h^l,\quad b_l=n^{-1}\sum_i(1-(m_i^l)^2),\quad R_l=m^{l-1}-m^l,\] \[h^{l+1}=h+J^\circ m^l-b_lm^{l-1}.\] 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 \[H_1=J^\circ,\quad M_0=I,\quad M_l=\operatorname{diag}(1-(m^l)^2)H_l,\quad H_{l+1}=J^\circ M_l-b_lM_{l-1}.\] 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 (OpenAI 2026a). 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 \[w_i^a=\sum_{j=1}^v \phi_i^{a,j}((h_i^l)_{l\in I_a},\theta)s_i^j +\sum_{b<a}\psi_i^{a,b}((h_i^l)_{l\in I_a},\theta)y_i^b.\] 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 \[y^a=J^\circ w^a -\sum_l\left(n^{-1}\sum_i\partial_lw_i^a\right)m^{l-1} -\sum_{b<a}\left(n^{-1}\sum_i\partial_{y^b}w_i^a\right)w^b.\] 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 \[F_h(h^k)=J^\circ R_k-b_{k-1}(R_{k-1}+R_k) +(b_k-b_{k-1})m^k, \qquad |b_k-b_{k-1}|\le2\|R_k\|/\sqrt n.\] Choose \(\rho\) so the displayed recurrence puts \(\Omega_{2\rho}\) strictly inside the root lemma’s residual region. Consequently on \(\Omega_{2\rho}\), \[\|h^k-y_*\|+\|m^{k-1}-m_*\|\le C\rho\sqrt n, \qquad |1-b_{k-1}-q_*|\le C\rho, \qquad q(h^k)\ge c_*.\] These constants are independent of \(k\). Define \[\Phi(z;r,a)=\int_0^1e^{a(1-u^2)/2} \frac{\cosh(u(z-r))}{\cosh z\cosh r}\,\mathrm du, \qquad a=1-b_{k-1}-q_*,\qquad A_i=\Phi(h_i^k;(y_*)_i,a).\] Differentiation in \(u\) proves the exact identity \[[(z-r-a\tanh z)-a\partial_z]\Phi(z;r,a)=\tanh z-\tanh r.\] 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, \[ \|A-V_{h^k}\|_{\mathrm F}\le C\rho\sqrt n, \quad |\chi_A-b_k|\le C\rho, \quad \|A\|\le e^{C\rho}, \quad \|A\|\chi_A\le e^{C\rho}(1-c_*+C\rho)\le1-c_*/2. \tag{57}\] 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 \[w=w_0+A(y^{\mathrm{imp}}-c_1m_*),\qquad y^{\mathrm{imp}}=J^\circ w-\chi_A w-c_1m^{k-1},\qquad c_1=n^{-1}\sum_i\partial_k w_i.\] 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 \[\left[I-A(J^\circ-\chi_AI) +\frac1n A(m^{k-1}+m_*)\ell^{\mathsf T}\right]w =A\left[u_0-(m^{k-1}+m_*)n^{-1}\sum_i(p_0)_i\right].\] Put \[\mathcal L=\chi_AI-J^\circ+ (m^{k-1}+m_*)\ell^{\mathsf T}/n.\] 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 (54), 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 \[(I+A\mathcal L)^{-1} =I-A^{1/2}(I+A^{1/2}\mathcal LA^{1/2})^{-1} A^{1/2}\mathcal L.\] 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 \[S_A(1;J^\circ)\succeq cI.\] 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 \(v\), \[\sum_{i,j}v_i^2|d_j\psi_i|^2 \le\|v\|^2\max_i\sum_j|d_j\psi_i|^2\le C.\] 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 (57) 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 \[\nabla w=A\nabla y^{\mathrm{imp}}+ \operatorname{diag}(\partial_k w)H_k+E_1, \quad \nabla y^{\mathrm{imp}}=(J^\circ-\chi_AI)\nabla w-c_1M_{k-1}+E_2.\] 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 \[\sum_{i\ne j}|P_{ji}|\,|d_js_i|\,Q_{ij} \le\|P_{\mathrm{off}}\|_{\ell^4} \|\nabla s\|_{\mathrm F}\|Q\|_{\ell^4}\le C, \qquad Q_{ij}=\sum_l|d_jh_i^l|.\] 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 \[\mathfrak S_{\mathrm{imp}} =K_A(z;J^\circ) \{\operatorname{diag}(\partial_k w)\mathfrak H_{h^k} -z^2c_1A\mathfrak S_{h^k}\}.\] 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 25. 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 \[\left|\operatorname{Tr}\bigl(\operatorname{diag}(u) (Q-\mathsf P(Q))\bigr)\right| \le \|u\|_2\, \|\operatorname{diag}(Q-\mathsf P(Q))\|_2\le C.\] 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 \(\partial_k w\): \[\|U\|+\|\nabla U\|_{\mathrm F} +\|\operatorname{diag}\nabla U\|_1\le C.\] 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 (OpenAI 2026a) to coefficient one. Put \[\mathcal E_h(G)= \bigl(\mu_h(G^2)+\mathcal D_h(G)\bigr)^{1/2}, \qquad \mathcal D_h(G)=\mu_h\sum_i(1-(m_i^1)^2)(d_iG)^2.\] If \(U\) is a terminal vector of an ordinary recipe admissible from level \(p\), then \[|\mu_h[G R_p^TU]|\le C\mathcal E_h(G)\] 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 \[|\mu_h[f^2H_0R_p^TU]| \le C\bigl(\mu_h(f^2)+\mathcal D_h(f)\bigr).\] Only ordinary recipes are used in this last version.

The induction can be checked directly. Its base identity is \[\mu_h\sum_i(x_i-m_i^1)GU_i =\mu_h\sum_i(1-(m_i^1)^2)d_i(GU_i).\] 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 \[R_p^{\mathsf T}m^{p-1} =n(b_p-b_{p-1})-R_p^{\mathsf T}m^p.\] 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 \[\mathcal T_k= \{h^k-h-(J^\circ-b_{k-1}I)m^k\}^Tw -(b_k-b_{k-1})nc_1.\] For every \(G\) supported on \(\Omega_\rho^{(k)}\), \[|\mu_h[G\mathcal T_k]|\le C\mathcal E_h(G).\] Indeed, on the buffer the exact expansion is \[\mathcal T_k= R_k^Ty^{\rm imp}+(\chi_A-b_{k-1})R_k^Tw -b_{k-1}R_{k-1}^Tw-c_1R_k^Tm^k.\] 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 \[\mathbb E_{\nu_h}(T_l/\sqrt n)^2 \le\frac{C_l}{\sqrt n}\left(1+\frac{\mathcal D_h(f)}N\right).\] 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 \[C_k=\vartheta\!\left(\frac{\|R_k\|^2}{\rho^2n}\right) \vartheta\!\left(\frac{\|R_{k-1}\|^2}{\rho^2n}\right), \qquad \widehat C_k=C_k\prod_{i=2}^{k-1}(1-C_i), \quad 2\le k\le d_{\rm sel}.\] Pointwise, \[0\le\widehat C_k\le C_k\le1,\qquad \operatorname{supp}C_k\subseteq\Omega_\rho^{(k)},\qquad \|dC_k\|+\|d\widehat C_k\|\le C_\rho/\sqrt n.\] 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 \[D:=1-\sum_{k=2}^{d_{\rm sel}}\widehat C_k =\prod_{k=2}^{d_{\rm sel}}(1-C_k)\in[0,1].\] The scalar moment bounds and a finite union bound give \[\mu_h(D>0)\le C_\rho/n, \qquad \nu_h(D>0)\le\frac{C_\rho}{\sqrt n} \left(1+\frac{\mathcal D_h(f)}N\right).\] 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 \[h^k-h-(J^\circ-b_{k-1}I)m^k =h^k-y_*-a m^k-J^\circ p_k+(1-q_*)p_k.\] Using the implicit equations and the scalar identity for \(\Phi\) then yields the exact identity \[\begin{split} \mathcal T_k={}& \sum_i\bigl[(h_i^k-(y_*)_i-a m_i^k)w_{0,i} -a\,\partial_k w_{0,i}\bigr]\\ &+(1-q_*-\chi_A)p_k^Tw-c_1R_k^Tp_k. \end{split}\] 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 \[|(1-q_*-\chi_A)p_k^Tw| =\frac{|L_*|}{\sqrt n}|p_k^Tw| \le C_0\rho |L_*|.\] 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 \[[(z-r-a\tanh z)-a\partial_z]\partial_r\Phi =\Phi-\operatorname{sech}^2r.\] 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 \[\mathcal E_h(C_k^2L_*)\le C_{\rho,k} +\|C_kL_*\|_{L^2(\mu_h)}.\] 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 \[\|C_kL_*\|_{L^2(\mu_h)}\le C_{\rho,k}.\]

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 \[|\mu_h[\widehat C_kG(m^k-m_*)^Te]| \le C_{\rho,k}\mathcal E_h(G).\] 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 (55). 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 (55) and its coarser form, only the ordinary diagnostics are required throughout. ◻

Proof. The signed curvature inequality, Lemma 8.1 of (OpenAI 2026a), 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 \[\mu_h[(-Lf)^2]\ge(1-Cr_4(U))\mathcal D_h(f) -\mu_h[a^{\mathsf T}Ua+C|a|^{\mathsf T}Q|a|].\] 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 \(z\), \[\mathbb P(|z^{\mathsf T}J^\circ z|>t)\le2e^{-nt^2/4},\qquad \mathbb P\left(\sum_{l\in S}(J^\circ_{il})^2>t\right) \le e^{-nt/4+|S|\log2/2}.\] 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, \[\|(J^\circ)_{SS}\|\le\varepsilon/2, \qquad \max_i\sum_{l\in S}(J^\circ_{il})^2\le\varepsilon \quad(|S|\le\alpha n).\] 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 \[h_{T+\lambda}=TX+\sqrt T\,g+\lambda RX+R^{1/2}B_\lambda,\] 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 \[\sup_{0\le\lambda\le1}\|h_{T+\lambda}-TX\| \le C(1+\sqrt T)\sqrt n\] 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 \[a^{\mathsf T}Ua+C|a|^{\mathsf T}Q|a| \le(1+C)\{\varepsilon+M(2\sqrt\eta+\eta)\}q_f.\] 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 23. 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 \[ h\le H_0=\mu(f^2\log f^2)\le2h+\log100. \tag{62}\] 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 \(h\).

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 \[g_j=\min\{(f-2^j)_+,2^j\},\qquad A_j=\mu(g_j^2),\qquad p_j=\mu(f>2^j).\] For a scalar \(u\ge0\) the same truncations satisfy \[ \frac{u^2}{4}\le \sum_{j\in\mathbb Z}\min\{(u-2^j)_+,2^j\}^2 \le u^2. \tag{74}\] 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 \(u\ge v\ge0\), \[d_j:=\min\{(u-2^j)_+,2^j\} -\min\{(v-2^j)_+,2^j\}\ge0, \qquad \sum_jd_j=u-v.\] 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 \[ \sum_j\mathcal D_J(g_j)\le\mathcal D_J(f). \tag{75}\] 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 \[\sum_{j\,\mathrm{discarded}}A_j \le\sum_j\frac{4^jp_j^2}{100} \le\frac1{50}\sum_jA_{j-1}=\frac S{50}.\] Every remaining normalized piece \(g_j/\sqrt{A_j}\) has squared supremum at most \(4^j/A_j\le100/p_j^2\). Proposition 31, on its single event valid for all test functions, therefore gives \[\begin{split} \mathcal D_J(f) &\ge c_\omega n^{-2/3-\omega} \sum_{j\,\mathrm{remaining}}A_j\log(1/p_j)\\ &\ge\frac{49}{200}c_\omega n^{-2/3-\omega} \log(1/p)\,\mu(f^2). \end{split}\] Renaming the constant proves (73).

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 \[\mathcal D_J(u)\ge\mathcal D_J(u_+)+\mathcal D_J(u_-) \ge c_\omega n^{-2/3-\omega}\log2\,\mu((u-m)^2) \ge c_\omega n^{-2/3-\omega}\log2\, \operatorname{Var}_\mu(u).\] 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 (83) give \[\log\mathbb E_{\mu_{p,h}}e^{\pm t u^\top(X-a_p)} \le 2\varepsilon_v k_p+\tfrac12t^2u^\top R_pu+C_{\rm dens}.\] Jensen’s inequality in the two signs yields \[|\langle u,P_pm_p(h)-a_p\rangle| \le \tfrac t2u^\top R_pu+ \frac{2\varepsilon_v k_p+C_{\rm dens}}t.\] 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 \[\frac{\|P_pm_p(h)-a_p\|}{M_p} \le C\theta+\frac{2\varepsilon_v+C_{\rm dens}/k_p}{\theta}.\] 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.

Proof. The same extension has a positive convolution margin because \[\|\Delta^{1/2}L_p\Delta^{1/2}\| \le(1+\gamma^4/100)(1-\gamma^4)<1.\] All inverse bounds in the shell estimates and the upper and lower covariance transports remain bounded by fixed powers of \(\gamma^{-1}\). Thus those deterministic estimates apply with enlarged constants. Formula (118) still gives the stated radial deficit, while the other blocks tend to zero with the input errors. The proof uses no additional orientation event and does not extend the probabilistic induction statement beyond ratio \(\gamma\). ◻

Proof. The static saddle bounds give \(\kappa_R,C_R>0\), depending only on the field radius, such that \[-1+\kappa_R\le\alpha/c^2\le C_R.\] Choose \(b_1\) so that \(\max(1,C_R)b_1^2<\sigma/2\). Then \(|\alpha|<\sigma p^2/2\) throughout the ratio interval. Choose fixed numbers \(e,w>0\), also satisfying the tail and scalar-margin bounds below, with \[C(w+\delta)\ll e\ll\kappa_R(\gamma b_1)^2, \qquad w<\sigma/4,\] and define nested shells \[S_j^\alpha=\{y\in H_p:|\alpha_p(y)-\alpha|\le t_jwp^2\}, \qquad (t_0,t_1,t_2)=(1/2,3/4,1).\] They lie inside the parent invariant annulus. The radius derivative computed above gives buffers comparable to \(wp^2M_p\) between them. Set \[L_\alpha=\bigl((p^2+\alpha)^{-1}+ep^{-2}\bigr)I_{H_p}.\] On \(S_2^\alpha\), the invariant’s positive-part bound gives \[K_p(y)\preceq \bigl((p^2+\alpha)^{-1}+C(w+\delta)p^{-2}\bigr)I_{H_p} \prec L_\alpha.\] This proves the upper-support inequality for pairs in \(S_1^\alpha\) whose joining segment remains in \(S_2^\alpha\).

For longer pairs, freeze the own saddle at the support point and use (83), the value comparison, and the gradient comparison as in (104). If the buffer divided by \(p^2M_p\) is \(\beta\asymp w\), the excess over the frozen \(\alpha\) quadratic is at most \[\frac{\|y-z\|^2}{2p^2} \left[Cw+ 2\left(\frac{2\varepsilon_v+C_{\rm dens}/k_p}{\beta^2} +\frac{\varepsilon_g}{\beta}\right)\right].\] Take \(\varepsilon_v+C_{\rm dens}/k_p\ll ew^2\) and \(\varepsilon_g\ll ew\). The bracket is smaller than \(e\), so the upper-support inequality holds for every pair in \(S_1^\alpha\). Define \[\bar\phi_{p,\alpha}(y)=\inf_{z\in S_1^\alpha} \left\{\phi_p(z)+\nabla_{H_p}\phi_p(z)^\top(y-z) +\tfrac12(y-z)^\top L_\alpha(y-z)\right\}.\] The argument of (105) shows that this function agrees with \(\phi_p\) on \(S_1^\alpha\), has at most quadratic growth, and satisfies \(\nabla^2\bar\phi_{p,\alpha}\preceq L_\alpha\) distributionally. Its dependence on the child field is through the scalar \(\alpha\), determined by the eigenvalues and \(\|h\|\). No child angular refinement enters its construction. For this evaluation hold the selected extension fixed when differentiating its convolution at \(h\). Different fields may select different extensions; no derivative of this selection, or of its parameter \(\alpha\), enters the Hessian formula.

We verify the complementary moments before comparing covariances. Let \(B_\alpha\) denote the logarithm of the frozen norm density at a field on the sphere \(\|y\|=\ell_p(\alpha)\). It is constant on that sphere. For \(d(y)=|\|y\|-\ell_p(\alpha)|/(p^2M_p)\), the nearest support point on the sphere gives \[\bar\phi_{p,\alpha}(y)-q_{p,\alpha}^{\rm fr}(y) \le B_\alpha+k_p\bigl(\varepsilon_v+ \varepsilon_gd(y)+\tfrac12ed(y)^2\bigr) \le B_\alpha+k_p\bigl(\varepsilon_v+ \varepsilon_g^2/e+ed(y)^2\bigr).\] The canonical calculation leading to (108) now gives, with constants depending only on the bounded child saddle range, \[\Pr_{\mathsf G}\{d(Y)>v\}\le2e^{-c_Rx^2k_pv^2}, \qquad v\ge C_R/(x\sqrt{k_p}).\] Require also \(e<c_R(\gamma b_1)^2/4\). Normalize the extended transition on a shell \(|\alpha_p(Y)-\alpha|\le\tau_0p^2\) with \(\tau_0\ll\gamma b_1w\). Its canonical probability tends to one. The off-saddle identity (93) and centered density bounds give a logarithmic normalization loss at most \(\varepsilon_vk_p+C\tau_0^2k_p+O(1)\) relative to \(B_\alpha\). Choose the comparison errors and then \(n\) so that \[2\varepsilon_v+\varepsilon_g^2/e+C\tau_0^2+O(1)/k_p \ll (\gamma b_1)^2w^2.\] Leaving \(S_1^\alpha\) requires \(d(Y)\ge c w\). Integrating the Gaussian distance tail against the upper support therefore bounds every fixed polynomial moment of \(1+d(Y)\) on this complement by \(C_j e^{-a k_p}\), for a fixed \(a>0\), uniformly in \(x,h\).

The true transition needs its own complementary estimate. On each fixed bounded parent field range, static comparison and the frozen numerator density upper bound give \[\phi_p(y)-q_{p,\alpha}^{\rm fr}(y) \le B_\alpha+\varepsilon_vk_p+C_{\rm dens}.\] The preceding normalization shell and Gaussian distance tail control the bounded part of the complement. For the remaining tail, sample the child spin by tilting the parent zero-field spin law with \(\exp\{h^\top X+X^\top\Delta X/2\}\), then set \(Y=h+\Delta X+N(0,\Delta)\). The tilt normalizer is at least one by zero-field symmetry and \(\Delta\succeq0\). With \(v=\|P_pX\|/M_p\), its numerator is at most \(e^{C_R k_p(1+v^2)}\). The parent projection tail \(e^{-ck_pv^6}\) survives this factor, and the observation noise has a Gaussian tail in parent field units. Increasing the fixed outer radius therefore makes the outer contribution and all its fixed polynomial moments exponentially small. This argument uses the true parent projection tails, not the extension’s Hessian bound.

The two unnormalized transition laws agree on \(S_1^\alpha\). Their normalization ratio and their complementary first and second moments thus show that their means and covariances differ exponentially in parent scaled units, as in the earlier transition comparison. Converting a covariance error in units \((p^2M_p)^2\) to child Hessian units multiplies it by \[c^2\|\Delta_U^{-1}\|^2(p^2M_p)^2 =\frac{x^2k_p}{(1-x^2)^2}.\] The exact formula (109) therefore leaves an error in \(c^2K_c\) tending to zero uniformly on the fixed ratio interval.

Finally put \(a=\alpha/c^2\) and \(q=(1+x^2a)^{-1}+e\). On \(U\), the increment is \((1-x^2)p^2I\), while the global support curvature is \(q/p^2\). Impose also \(e\le\kappa_R(\gamma b_1)^2/[4(1+\sigma/2)]\). Then the convolution has a strict margin. Brascamp–Lieb, first for smooth approximations and then by passage to the limit, and (109) give \[c^2K_c^{\rm ext} \preceq \frac{1+(1+x^2a)e} {(1+a)-(1-x^2)(1+x^2a)e/x^2}\,I_U \preceq \frac{8}{3\kappa_R}I_U.\] The denominator is at least \(3\kappa_R/4\), and the numerator is at most two. The exponentially small replacement error is absorbed by enlarging this constant once. This proves (133) with no dependence of its coarse constant on \(x\) or the sharp tolerances. All shell widths, comparison errors, and the ratio interval are fixed before \(n\to\infty\); no estimate here requires taking \(x\to0\) while keeping these choices fixed. ◻

Proof. We isolate the two places in the companion matrix proof (OpenAI 2026a, sec. 3) where the temperature gap is used. The first is a lower bound on the symmetric inverse along its scalar parameter path; the second is a coefficient below one in the loop equation. The restricted diagonal condition supplies both margins at the endpoint. Once these margins are available, truncation and chaining give the simultaneous assertion by the same finite recursions.

For a deterministic \(A\), the Gaussian process comparison in its Lemma 3.2 bounds the relevant supremum by \(2d-d^2\), where now \[0\le d\le \sqrt{\langle A\rangle\|A\|} \le\sqrt{1-\kappa}.\] It therefore gives the fixed margin \(c_0=(1-\sqrt{1-\kappa})^2/4\) along the whole path \(0\le z\le1\), after concentration and a fixed mesh of that path. The constants are uniform over deterministic \(A\) in (138). Next, the loop equation for \(u(z)=n^{-1}\mathbb E\operatorname{Tr}(K_A(z)A)-\langle A\rangle\) has linear coefficient \[z^2 n^{-1}\sum_i A_{ii}^2\mathbb E K_A(z)_{ii}.\] When \(u\) is small, the diagonal expectations are \(1+O(u)+O(n^{-1})\). Since \(\langle A^2\rangle\le\|A\|\langle A\rangle\le1-\kappa\), this coefficient stays below \(1-\kappa/2\). Continuity from \(z=0\) then gives \(u(z)=O(n^{-1})\) throughout the path. All subsequent noncrossing loop recursions are finite and unchanged.

For clarity, the simultaneous assertion does not condition on a randomly chosen diagonal. Use the globally truncated inverse from (OpenAI 2026a, sec. 3.2), with a smooth cutoff equal to \(x^{-1}\) on a neighborhood of \([\min(c,c_0)/2,C]\). Its word maps and their increments have Frobenius Lipschitz constants \(C\) and \(C\|\theta-\theta'\|_\infty^{1/2}\), respectively. Cover the restricted parameter set by meshes with representatives in that set, or in \(\mathcal A_{\kappa/2}\) when a buffer is needed. At resolution \(2^{-l}\) there are at most \(\exp(C(l+1)n)\) representatives. Gaussian concentration and chaining bound the centered word increments by \(C\sqrt{l+1}\,2^{-l/2}\), with summable failure \(Ce^{-c(l+1)n}\). The deterministic diagonal expectation error is \(O(n^{-1})\) coordinatewise by the preceding loop calculation. This proves the word assertion even for diagonals selected from the observed matrix. The bilinear assertion follows by applying concentration to each fixed truncated bilinear form, exactly as in the cited lemma.

For the determinant take \(A=V\). The fixed-diagonal margin and loop calculation give \[\mathbb E\frac1n\log\det(L^{\mathsf T}L+\gamma I) \le\langle V\rangle^2+C_\kappa\gamma+O(n^{-1}), \qquad L=I+(\langle V\rangle I-W)V.\] The regularized normalized log determinant is \((n\gamma)^{-1/2}\)-Lipschitz in Frobenius norm, so its concentration has rate \(n^2\) at fixed tolerance. A fixed mesh of the restricted diagonal class costs only \(\exp(Cn)\); use its slight enlargement for continuity. The bounded-rank perturbation changes the answer by at most \(\|E\|\sqrt{\operatorname{rank}E/(n\gamma)}\). This proves the stated uniform determinant bound. Removing the GOE diagonal changes the truncated words by a bounded Frobenius amount and the stability matrices by \(o(1)\) in operator norm, as in the companion proof. ◻

Proof of Proposition 37. For fixed \(j<1\) the root assertion is (OpenAI 2026a, Lemma 5.7); its proof is deterministic and allows any fixed \(j>0\). It gives a bounded inverse for \(F'_h\) and \(\|y-y_*\|\le C\|F_h(y)\|\) whenever \(|F_h(y)|_n<2\rho\); the later uses lie in a strictly smaller residual region. The last estimate in the proof of (OpenAI 2026a, Proposition 7.2), before centering its test function, is \[ \big|\mathbb E_{\mu_h}[G(x-m_*)^{\mathsf T}e]\big| \le C\left(\mathbb E_{\mu_h}G^2+\mathcal D_h(G)\right)^{1/2}, \qquad \|e\|\le1. \tag{139}\] Taking \(G=1\) gives the first assertion in (136). For covariance, apply the centered statement of that proposition to \(G=e^{\mathsf T}x\), with \(e\) a unit eigenvector for the largest covariance eigenvalue \(\lambda\). Since \(\mathcal D_h(e^{\mathsf T}x)\le\sum_i e_i^2=1\), it yields \(\lambda\le C\sqrt{\lambda+1}\), hence \(\lambda\le C'\). The square-density estimate is exactly (OpenAI 2026a, Proposition 7.3), in the present rate-one-per-site normalization.

The endpoint word event is useful only if the diagonals actually produced by the posterior proof belong to its restricted class. We therefore inspect the implicit recipe, rather than substituting the endpoint into the companion statement. The root comparison places its primary vectors near the same root throughout a buffered residual region. This controls both the largest diagonal entry and the average diagonal entry, giving the strict product margin required by the endpoint inverse.

Its primary iterates are \[m^0=x,\quad h^1=h+J_*x,\quad m^l=\tanh h^l,\quad b_l=1-|m^l|_n^2,\quad h^{l+1}=h+J_*m^l-jb_lm^{l-1}.\] Write \(R_l=m^{l-1}-m^l\) and \(\Omega_{2\rho}=\{\max(\|R_k\|,\|R_{k-1}\|)\le2\rho\sqrt n\}\). The root comparison gives, independently of \(k\), \[|h^k-y_*|_n+|m^{k-1}-m_*|_n\le C\rho, \qquad |1-b_{k-1}-q_*|\le C\rho.\] The only inverse diagonal in the implicit recipe is \[A_{ii}=\Phi(h_i^k;(y_*)_i,a),\qquad a=j(1-b_{k-1}-q_*),\qquad \Phi(z;v,a)=\int_0^1 e^{a(1-u^2)/2} \frac{\cosh(u(z-v))}{\cosh z\cosh v}\,du.\] Its secant-slope representation at \(a=0\) and its bounded derivatives give, on the entire buffer, \[\|A\|\le e^{C\rho},\qquad \langle A\rangle\le1-q_*+C\rho, \qquad n^{-1/2}\|A-V_{h^k}\|_{\mathrm F}\le C\rho.\] Choose \(\rho\) after \(q_{\min}\) and the stability margins so that \(e^{C\rho}(1-q_{\min}+C\rho)\le1-q_{\min}/2\). Thus all these diagonals lie in the interior of \(\mathcal A_{q_{\min}/4}\). At \(h^k\), \(\langle V_{h^k}\rangle=b_k\), so the same Frobenius bound gives \[|\langle A\rangle-b_k| \le n^{-1/2}\|A-V_{h^k}\|_{\rm F}\le C\rho.\] Removing the negative rank from (135) and then changing the scalar coefficient from \(jb_k\) to \(j\langle A\rangle\) therefore gives \[I+j\langle A\rangle A-A^{1/2}J_*A^{1/2} \succeq (b-jC\rho\|A\|)I .\] Choose \(\rho\) to retain a fixed positive margin. This is the symmetric matrix for the actual recipe inverse \(K_A(1)\); the path \(0\le z\le1\) is used to establish its formal word prediction. Every later auxiliary recipe is ordinary; the closed differentiated formulas contain at most this one inverse.

These bounds must also hold at the finitely many neighboring spin configurations used by the residual identities. The recipe depth is fixed before the dimension grows. Consequently the total Euclidean change across its required flips is bounded independently of the dimension, while the residual buffer has radius proportional to its square root. Choosing the dimension last keeps every required coefficient inside the same restricted diagonal class.

The finite-flip buffers require no uniform bound over all depths. After fixing a recipe and its depth, let \(Q\) be the largest number of successive spin flips appearing in its residual induction, and let \(C_Q\) bound the one-flip Euclidean changes of all involved primary vectors. Both are finite by (OpenAI 2026a, Lemma 6.1). Taking \(n>(2QC_Q/\rho)^2\) places every required neighborhood of \(\Omega_{3\rho/2}\) inside \(\Omega_{2\rho}\). All inverse coefficients there retain the preceding trace margin. This is the buffer argument in the released version of (OpenAI 2026a, sec. 6.3).

The order of choices is consequently: the root and diagonal stability margins; the restricted diagonal margin; a sufficiently small \(\rho\); the finite residual-selection depth; the resulting finite recipes and word length; and finally \(n\). The residual-selection depth may increase when the accuracy in (137) decreases. None of these choices depends on \(f\) or on a directional seed. Ordinary words have no restricted inverse parameter at all. The proof of (OpenAI 2026a, sec. 6 and Propositions 7.2–7.3) now applies term by term using Lemma 38, proving the endpoint version and its claimed uniformity.

For a compact coefficient interval strictly inside \((0,1)\), choose the same strict diagonal margin throughout and include \(\beta\) as one more bounded scalar in the parameter meshes. The additional mesh cardinality does not change their exponential-in-\(n\) entropy, and all Lipschitz and loop constants have the same margin. The word and posterior conclusions are therefore simultaneous on that interval. In the restricted endpoint case the same argument applies to a sufficiently small coefficient neighborhood that preserves (138). ◻

Proof. We adapt the conditional Gaussian, scalar-rate, and local-volume proof of (OpenAI 2026a, sec. 5). The adaptation discards a small-\(q\) region by an observation estimate, rather than using a temperature gap. Here are the details needed at the endpoint.

First the scalar entropy inequality in (OpenAI 2026a, Lemma 4.1) extends to \(j=1\) when \(q>0\). Its proof uses \(k=jq/s\le1\) and \(B=j(1-q)<1\), which remain valid. In its large-\(C\) case the quantity \(C^2/(jk)\) is still strictly greater than one, since \(C>1\). The strictness and equality statements are therefore unchanged. The consistent Gaussian laws at \(j=1\) are \(N(s,s)\) with \[s=t+\mathbb E_{N(s,s)}\tanh Y.\] Indeed both scalar one-Lipschitz comparisons in the proof of (OpenAI 2026a, Lemma 4.2) become strictly contracting on every nondegenerate compact interval: their Gaussian derivatives are strictly less than one. More explicitly, if \(f(s)=\mathbb E_{N(s,s)}\tanh Y\), then \(f(0)=0\), \(0\le f'(s)<1\) for \(s>0\), and \(f(s)\le1\). Thus \(s-f(s)\) is strictly increasing from zero to infinity. The solution is unique, equals zero at \(t=0\), and is continuous by this monotonicity and compactness. The second scalar comparison forces equality of the consistent mean and variance because its Lipschitz inequality is strict between distinct nonnegative arguments. These facts replace the global contraction constant \(j<1\); no bound involving \((1-j)^{-1}\) is used.

At \(t=0\), restrict the scalar-rate integral to \(q\ge q_0/2\). Its variance is then bounded below, and no consistent equality law lies in this set. The compactness and Gaussian-tail argument of (OpenAI 2026a, Lemma 5.4) gives a strictly negative exponential rate, including small residual smoothing. The determinant diagnostic is only required on this retained set, where \(\langle V\rangle=1-q\le1-q_0/2\), so Lemma 38 supplies it. A small normalized ball around a witness with \(q\ge q_0\) stays in the retained set. The local-volume lower bound of (OpenAI 2026a, Lemma 5.6) is deterministic and has no subcritical restriction. It turns the negative integral rate into exclusion of approximate zeros.

On \([t_b,T]\), very small \(q\) is excluded first. A sufficiently accurate approximate root satisfies \(|y-Y(t)|_n\le C\sqrt q+\rho\). Since \(\tanh\) is one-Lipschitz, this would force \(|\tanh Y(t)|_n\) to be small. For each fixed spin the coordinates of \(tX+B(t)\) are independent Gaussians with means \(\pm t\) and variance \(t\), so their bounded squared hyperbolic tangents have an empirical average bounded below, with exponential probability uniformly on the compact time interval. A fixed time mesh and Brownian maximal bounds give the same conclusion along the path. We may therefore retain \(q\ge q_{\min}>0\) with slack.

On this retained set, the scalar equality curve and all scalar-rate arguments are compact and nondegenerate. We make the conditional matrix law explicit. During the Gaussian reduction use the temporary planted interaction \(\widetilde J=W+\mathbf1\mathbf1^\top/n\) with full GOE \(W\). For fixed \(y\), let \[\begin{gathered} b=1-q,\quad M=\langle m\rangle,\quad d=t+M,\quad s=t+\sigma^2+q,\quad S=s+q,\\ a_y=\langle(y-d\mathbf1)m\rangle,\qquad z_{\rm cond}=y-d\mathbf1+bm . \end{gathered}\] The matrix factor in (OpenAI 2026a, Eq. (5.8)) is the Gaussian conditional law given \[Wm+\sqrt{t+\sigma^2}\,g'=z_{\rm cond}.\] Here \(g'\) is independent standard Gaussian and \(\sigma>0\) is the residual smoothing. Put \[u=\frac{m}{\sqrt{nq}},\qquad Z_1=\frac{y-d\mathbf1}{\sqrt{ns}},\qquad P_u=I-uu^\top .\] Writing \(W=P_uWP_u+uw^\top+wu^\top+\alpha_c uu^\top\) with \(w\perp u\), the elementary conditional Gaussian formulas give a representation using a fresh GOE \(W_0\): \[\begin{split} P_uWP_u&=P_uW_0P_u,\\ w&=\sqrt{q/s}\,Z_1-(a_y/s)u+ \sqrt{(t+\sigma^2)/s}\,P_uW_0u,\\ \alpha_c&=\frac{2(a_y+bq)}S+ \sqrt{(t+\sigma^2)/S}\,u^\top W_0u . \end{split}\] This is (OpenAI 2026a, Eq. (5.12)) at \(j=1\).

For the moment fix an admissible diagonal \(A\), deterministic after \(y\) has been fixed. Set \[\chi=\langle A\rangle,\qquad S_A=I+\chi A-A^{1/2}W_0A^{1/2},\qquad Y_A(v)=S_A^{-1/2}A^{1/2}v .\] Lemma 38 supplies a positive lower bound for \(S_A\). Its fixed-diagonal bilinear estimates apply to the three deterministic vectors \(\mathbf1/\sqrt n,u,Z_1\), whose norms are bounded by the retained second-moment bound. Their transformed Gram entries are close to \(v^\top Av'\) for the corresponding pair \(v,v'\); the extra vector \(Y_A(W_0u)\) has mixed entries close to \(\chi v^\top Au\) and squared norm close to \(\chi+\chi^2u^\top Au\). These are Gram comparisons under the fresh \(W_0\) law. At the limiting scalar equality law with \(A=V\) and \(\sigma=0\), the whitened finite-rank test has determinant \[(1-b+2g_1)(1-b+g_1)^2>0, \qquad g_1=\mathbb E_{N(s,s)}[\tanh^2Y\,\operatorname{sech}^2Y].\] The rank perturbation has at most one possibly nonpositive direction; the positive determinant excludes that direction, exactly as in (OpenAI 2026a, Lemma 5.5). Compactness gives a uniform margin. For \(A\) close to \(V\), choose \(\eta\) and its normalized Frobenius tolerance so that \(\|A\|\langle A\rangle\le1-\kappa\) on the retained range. To include data-dependent \(A\), keep \(y\) fixed and round the entries exceeding a fixed sup-norm tolerance on their exceptional coordinate set. That set has fraction at most \(\epsilon_A^2/\zeta^2\), and the logarithm of the number of choices is at most \[n\{H(\epsilon_A^2/\zeta^2) +(\epsilon_A^2/\zeta^2)\log(C/\zeta)+o(1)\}.\] Choose \(\zeta\) after fixing the vector-norm bounds, Gram tolerance, and fixed-diagonal exponent, then \(\epsilon_A\) so that this counting rate is below that exponent. For this fixed \(y\), the approximants are deterministic, so a union bound makes the conditional matrix comparison simultaneous in \(A\). The omitted scalar mass has norm at most \((1+\eta)|\chi-b|\le(1+\eta)n^{-1/2}\|A-V\|_{\rm F}\) and is included in the reserved margin. The probability bound is uniform in the retained \(y\)-data, as required inside the scalar integral. Enlarging the retained ranges slightly before the local-volume step preserves all these margins.

The preceding calculations are under the usual size-biased planted law. The critical zero-field partition comparison in Proposition 33 gives \(\log Z-\log\mathbb EZ=o(n)\) in probability. Thus negative exponential rates transfer to typical original disorder: restrict to \(Z/\mathbb EZ\ge e^{-\delta n}\), choose \(\delta\) below the rate, and use Markov’s inequality. A finite time mesh and the Brownian maximal bound supply the whole path. Finally, \(F_h\) is one-Lipschitz in its field and its stability matrix does not use \(h\); coefficient and diagonal changes are uniformly continuous under the norm bounds. Reserving slack in all thresholds proves the neighborhood and perturbation assertions. ◻

Proof. There are two different uniformity statements here. The frame estimates hold on one event for all coefficient vectors in the specified balls, so they admit shifts selected after the frame and noise are seen. The projection lower bound instead holds in probability for each fixed center, with a constant independent of that center. A spin sampled before the fresh noise can therefore be averaged into either statement. There is no simultaneous lower bound over every possible center.

The frame estimates are simultaneous coefficient-ball estimates. The shift contributes at most \(Ct\) to each unfiltered band norm; after a high-band inverse it contributes at most \(Ct/d\). Thus every shift in the stated ball is already included in their deterministic domains. The Gaussian band norms and the norms of their finite smooth filtered families have exponential tails at the specified fixed scales. If \(p(W)\) is the conditional noise failure probability and \(\mathbb E_Wp(W)\le e^{-2c_tn}\), Markov gives \(\Pr_W\{p(W)>e^{-c_tn}\}\le e^{-c_tn}\). This gives the asserted disorder event and conditional bounds, without any reference to a test function. The same argument integrated over a fixed time interval gives the integrated conditional statement, when that is all that is needed.

For (145), restrict to the nondegenerate band of rank \(r_t\ge cnt^{3/2}\). The Gaussian integral of a Euclidean ball is maximized when its center is the mean; alternatively, exponential Markov applied to \(\exp(-\lambda\|g+v\|^2/t)\) gives a product bounded by \((1+c\lambda)^{-r_t/2}\), since the noncentral exponential factors are at most one. Taking the ball radius squared to be a sufficiently small multiple of \(tr_t\) proves the claim. Finally, condition on each spin and then average this bound over its law. No union bound over spin configurations is involved, and no initial density bound is used. ◻

Proof. We first control the nonlinear source on the coordinates where a root is large, uniformly over every approximate root in the specified region. This gives initial localization bounds for the root and its derivative tests. A second gain is then necessary: an error of order \(q^2\) would still be as large as the spectral separation. The later high-rotation calculation provides that gain before we invert on the low bands. Lemma 39 at zero, applied with the small field as an additional residual, first makes \(q\) as small as required. The field equation gives \(|y|_n\le C(\sqrt q+\sqrt t)\). Put \[ s=q+\sqrt t,\qquad a=s^{7/5},\qquad S=\{2-W\le a\},\qquad u=s^{1/10},\qquad P_0=\mathbf1_{\{2-W\le u\}}. \tag{148}\] We only invert on \(S^\perp\) until the lower bound for \(q\) proves separation. Dyadic bands there have gaps between \(a\) and a constant. In all estimates below \(o_n(1)\) errors at fixed scales are chosen after the smallest scale and may be below any specified power of \(s\).

The derivative estimates below test vectors \(w\) normalized by \(|w|_n\le1\). For a diagonal \(A\) allowed by the stability test, put \(v=Aw\) and \(\dot r=-2\langle mv\rangle\). Equation (155) will specify the equation these tests satisfy; the present notation lets us account for their clipping errors before that calculation. The scalar variance profile satisfies \(V'_0(z)=-2\tanh z\,\operatorname{sech}^2z\). The cutoffs below are applied to recipes built from this profile, including \(V'_0(y)w\).