Universality of critical quench autocorrelations in the Sherrington–Kirkpatrick model

Proof. We use the eigenvalue-only event of [20, Proposition 3.1]. For every deterministic L_n\to\infty with L_n\le\log n, that proposition gives positive denominators and, uniformly for L_nn^{-2/3}\le u\le M with fixed M,

The errors have deterministic uniform bounds on an event of probability at least 1-Ce^{-cL_n^{1/4}}.

For (3.22), take \mathcal S_n^{\rm glob} to be the separate source event with L_n=\sqrt{\log n}. Its probability tends to one, and the edge and norm assertions follow directly from (3.23). For every fixed u>1, the first resolvent converges uniformly there to

The norm bound makes the empirical spectral measures a compact family. Every weak subsequential limit therefore has the displayed transform for every u>1, and uniqueness of the Stieltjes transform identifies it with \rho_{\rm sc}. A contradiction by subsequences gives this weak convergence uniformly over the source event. Intersecting with this event does not change the tight-threshold convention.

Let K_n\to\infty be arbitrary and set L_n=\min\{K_n^{1/3},\sqrt{\log n}\}, changing finitely many initial values if necessary. For np^3\ge K_n, L_nn^{-2/3}=o(p^2). Thus, for every fixed v>0,

uniformly in the scale range, and d_1/p^2\ge-o(1). For u\ge cp^2, the counted denominators satisfy 0<d_i+u\le2u. Consequently

Summing on dyadic bands gives the second bound in (3.21); the last band above u=4 costs O(p) by \|W\|\le3, and obeys the same bound.

For fixed v,w>0, the transform difference is exactly

To identify the limit including the tail, put \omega_p(\mathrm dx)=(1+x)^{-2}\nu_p(\mathrm dx). These finite measures have bounded mass, the tail bound (3.21), and negative support in an interval shrinking to zero. Every weak limit \omega is therefore supported on [0,\infty). Taking w=1 in (3.24) gives

Letting v\to1 gives \omega([0,\infty))=1/2, and hence

Uniqueness of the Stieltjes transform identifies \omega(\mathrm dx)=\pi^{-1}\sqrt x(1+x)^{-2}\mathrm dx. The continuous distribution function of the corresponding limit of \nu_p gives uniformity in endpoints on compact intervals. Failure of uniformity over p would provide a sequence of scales to which this same compactness argument applies, a contradiction.

The clipped second trace has limit

For the first trace, subtract an ordinary resolvent before integrating:

The difference kernel is O(x^{-2}). Its limiting integral is 4/(3\pi)-1, obtained by substituting x=t^2 on each of (0,1) and (1,\infty).

These assertions hold for every arbitrarily slowly divergent K_n. If (3.20) failed, one could choose K_j\to\infty and n_j\to\infty with failure probability bounded away from zero and embed K_{n_j}=K_j in a divergent deterministic sequence. The preceding uniform conclusion contradicts that subsequence.□

Proof. Put t=u/p^2 and \mathfrak d_p(t)=(n-\Tr R_{p,tp^2})/(np). For y=\|h\|/(p^2M_p), the saddle equation becomes

Lemma 3.2 gives \mathfrak d_p(0) arbitrarily close to b_0, and

It follows by integration that \mathfrak d_p(-1+\kappa)<0 for a sufficiently small fixed \kappa>0; no saddle can lie there or below. On t\ge0, \mathfrak d_p(t)\ge b_0/2, so (3.26) bounds t above when y\le R. The right side of (3.3), viewed as a function of its parameter, is strictly decreasing and continuous, tends to infinity at -p^2, and tends to zero at infinity. This also proves existence and uniqueness. Denominator comparison gives \Tr\widehat R^2\asymp n/p, \|m\|^2\le C_Rnp, and \|\widehat R\|\le C_Rp^{-2}, hence \sigma^2\asymp n/p. For a sufficiently small fixed s_0, both \mathfrak d_p(t) and \mathfrak d'_p(t) are bounded above and below on |t|\le s_0. Differentiating \ell_p(tp^2)=p^2M_p(1+t)\sqrt{\mathfrak d_p(t)} proves the length and derivative bounds.

We record the density argument used here and later. For independent Y_i=a_i+\sqrt{\upsilon_i}G_i, suppose

and at least ck_p variances are at least cp^{-2}. The Gaussian quadratic formula, as in [19, Section 15.3, Equation (15.5)], gives

Expansion of the centered log characteristic function through order two gives

Indeed its remainder is at most

The band in (3.27) bounds the modulus in these units by (1+ct^2/k_p)^{-c'k_p}. Its real logarithm is at most -c_0t^2 up to a small fixed multiple of \sqrt{k_p}, and its integral outside that interval is exponentially small in k_p. Fourier inversion and (3.28) give

At the saddle \E\|Y\|^2=n, proving the lower and upper density bounds. At any mean, the same characteristic majorant gives the upper bound Cp^2/\sqrt{k_p}.

For replicas let T_t=\diag(t_1,\ldots,t_{m_0}) and H_i=C_i^{1/2}T_tC_i^{1/2}. The modulus of the row characteristic function is at most \det(I+4H_i^2)^{-1/4}; its extra factor from a nonzero mean is at most one. The singular values of H_i are at least cu^{-1} times those of T_t, including when the entries of T_t have different signs. Independence of the L rows bounds the joint characteristic function by \prod_j(1+ct_j^2/u^2)^{-L/4}. Its integral proves (3.25). This is the argument of [19, Section 15.3, Equation (15.6)]; it explains the independence requirement on the spectral rows.□

Proof. We give the scale check in the polar argument of [19, Section 15.3, Equations (15.5)–(15.8)]. For an interaction A, write R_{A,u}=(2+u-A)^{-1} and

Coarea in squared radius gives, exactly,

where f_{A,u,h} is the density of \|N(R_{A,u}h,R_{A,u})\|^2. In the capped case take A=W_p and start at its saddle u=\alpha. For u\ge p^2,

The first term is (3.23); the second compares the clipped active denominators with d_i+u, using \dim\cH_p\le Ck_p. For u in a bounded range in p^2 units, the upper bound is C_Rnp. Integrating (3.31) yields

where the first constant is independent of the field radius.

For a rowwise quadratic test \mathcal T of m_0 replicas and \lambda>0, let f_{\mathcal Q}^{u,\lambda} be the norm density under the normalized Gaussian tilt e^{\lambda\mathcal T} of m_0 independent copies of N(R_{p,u}h,R_{p,u}). Exponential Markov and polar coarea give

This compares norm densities before and after the normalized tilt; it does not divide by the probability of a shrinking shell.

For a pair take \mathcal T=\pm X^{\mathsf T}X', b=nw, u=\iota w^2, and \lambda=c'u, with c'>0 a sufficiently small absolute constant. We will choose \iota>0 small and then the lower multiple of w/p large, so that u\gg p^2. The Gaussian formula gives

Indeed \Tr R_{p,u}^2\le Cn/\sqrt u and \|R_{p,u}h\|^2\le C_Rnp^5/u^2; the pair quadratic has zero first trace in the centered Gaussian formula. For \mathcal T=\|P_pX\|^2, b=nw, its extra first trace costs only Ck_p. The choice of c' preserves positive precision and covariance comparable to u^{-1}I_{m_0} on ck_p active rows. Lemma 3.3 bounds the density ratio in (3.33) by C_{m_0}(u/p^2)^{m_0}.

For uncapped zero field start at u=p^2. The resolvent gives n-\Tr(d+p^2)^{-1}\asymp np, a leading variance comparable to p^{-2}, and variance O(n/p) for the other squares. Chebyshev’s inequality puts the residual squared length to be supplied by that coordinate in [cnp,Cnp] with probability 1-C/k_p. On this interval its Gaussian-square density is at least c\sqrt{p/n}e^{-Ck_p}. Convolution gives the same lower density scale, with this extra exponential factor. The uncapped resolvent also gives (3.32), and the tilted upper density has the same band bound.

The logarithm of the resulting probability is at most

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

Proof. Fix errors in Lemma 3.2 for which the number of points with d_i\le p^2 is between c np^3 and Cnp^3, and apply it at the countable scales np^3=2^jK. For any finite list, finite edge-coordinate convergence transfers the inequalities, with slightly enlarged constants, to counts of limiting points below (2^jK)^{2/3}. A fixed small buffer in the endpoints may be used; the limiting process has no point at any of these countably many fixed endpoints. Increase the finite list and use continuity of probability for decreasing events. The uniform finite-n event shows that all the limiting inequalities hold with probability arbitrarily close to one when K is large. Letting that failure probability tend to zero shows that almost surely, after a finite threshold,

on dyadic L, and hence on all large L by monotonicity. Invert these inequalities, add b, and adjust the bounds for the finitely many remaining positive numbers g_i+b. The least integer threshold and rational bounds give measurable choices.□

Proof. We derive this form for cells centered at lattice points from the two-replica calculations in [19, Section 15.4, Equation (15.10)] and [21, proof of Lemma 3.2]. Set

Uniform Stirling bounds on |r|\le\rho_0 put p_n(r) between fixed multiples of n^{-1/2}e^{-nI(r)}. The same bounds for the gamma ratio, and the bounded logarithmic oscillation of f_n on I_r, put its cell integral between fixed multiples of n^{-1/2}e^{-nJ(r)}. Since J(r)-I(r)=O(r^4), this proves (3.36). The assumed oscillation of the positive angular weight gives (3.37) by integration. The final assertion follows directly from the cell endpoints.□

Proof. We adapt the two-replica calculation of [21, Lemma 3.2, Equations (3.3)–(3.4)], whose stated observation range starts at r_n. The estimate below the source range is proved here. Conditional on \rho=n^{-1}X^{\mathsf T}X', the Haar angular integral for a cube pair is the spherical angular integral. On the ordered orthonormal sum-and-difference frame (q,q'), its exponent is

On a fixed |s|\le2\rho_0<1, the spectral norm bound and the field bound \|h\|\le R\sqrt n\,p^{5/2} give |\partial_s\Psi_{p,h}|\le C_Rn. Thus the positive angular weight e^{\Psi_{p,h}} has a fixed logarithmic oscillation on every width-2/n cell, as required in (3.37). Retain |\rho|\le C'r_n, choosing C' larger than the lower tail constant in Lemma 3.4 times C_1. On dyadic bands between C'r_n and a small fixed \rho_0, the weighted cell comparison costs e^{C+Cn\rho^4}, while Lemma 3.4 gives Ce^{-cn\rho^3}. The spherical band endpoints move by at most 1/n, negligible compared with r_n. Decreasing \rho_0 makes the latter pay the former and leaves Ce^{-cnr_n^3}. At overlaps bounded away from zero, the global spectral law, edge limit, and norm bound in (3.22) supply the global inputs for the uncapped zero-field angular comparison of [19, Section 15.4]. Indeed, for every fixed w>2, they give

interlacing gives the same per-site limit for every orthogonal-hyperplane compression. The cited deficit-filling lower bound then gives the full critical free-energy normalization by taking n\to\infty at fixed w>2, followed by w\downarrow2. For clarity, the normalized zero-field contribution being bounded is

Here \boldsymbol\lambda=(\lambda_1,\ldots,\lambda_n) is fixed, W=U\diag(\boldsymbol\lambda)U^{\mathsf T}, and \mu_W is the zero-field cube Gibbs law for this W. The comparison in [19, Section 15.4, “Zero field and overlaps bounded away from zero”], together with the lower normalization in its Section 15.3, gives \mathcal M_n(\rho_0;\boldsymbol\lambda)\le \exp\{-\kappa_{\rho_0}n+o(n)\} for a fixed \kappa_{\rho_0}>0, along the retained spectral sequences with the preceding per-site errors uniform. This is a consequence of that proof; the norm bound supplies continuity at the endpoints. Changing to W_p,h changes weights and log normalizers by O_R(np^2+np^{5/2})=o(n), so that region still contributes e^{-cn}.

On retained cells of width 2/n, Stirling’s formula for the binomial overlap mass and the spherical density proportional to (1-\rho^2)^{(n-3)/2} gives relative error

This is the cell estimate in the cited replica proof, now with cutoff C'r_n at every p. More precisely than the fixed bound above, the oscillation of (3.39) on one cell is at most

which is uniformly bounded. The normalized spherical integral of this expression on retained cells is O(r_n). Indeed,

The diagonal terms cost O(r_n). For the cross term, tilt a Gaussian pair at the cap saddle by \pm cp^2X^{\mathsf T}W_pX', with c small enough to preserve a fixed precision margin. The Gaussian logarithmic moment is O_R(k_p), since \Tr R_{p,\alpha}^2=O(n/p) and \|R_{p,\alpha}h\|^2=O_R(np). The replica density bound (3.25) costs a fixed factor. Hence

Integrating from a sufficiently large multiple of np gives \E_{\rm sp}|X^{\mathsf T}W_pX'|\le C_Rnp\le C_Rnr_n. This full probability integral also bounds its unnormalized restriction to retained cells.

The bounded oscillation permits the mean value theorem for exponentials. After integration and summation over disjoint cells, the error is the preceding integrated O(r_n); no supremum over frames is taken. Slightly enlarged boundary cells are paid by the pair tail. Thus

Use E_UL_p(h)=1, nr_n^4=n^{-1/3+4\cdot10^{-4}}, and nr_n^3=n^{3\cdot10^{-4}} to obtain (3.38).□

Proof. On dyadic p and v, the spherical tail and (3.7) give

Markov’s inequality with half the exponent has total failure at most

The discarded L_\circ<1/2 event has probability o(1) by (3.5). Monotonicity of P_p, and a fixed-factor change in M_p, extend the uncapped result to every p,v.

The posterior tails follow deterministically. For q\ge p of bounded ratio, let B_\theta lie between B_p and B_q. It is positive, supported on \cH_q, and has norm at most Cq^2. Restricting the uncapped normalizer to \|P_qX\|\le VM_q gives

for a fixed large V. The zero-field capped law is even, so its normalizer for a linear field is at least one. On the shell \|P_qX\|\asymp vM_q, a field of norm at most Rq^2M_q costs at most Rk_qv. This and the denominator cost are absorbed by the uncapped e^{-c k_qv^6} tail for large v. This proves the posterior tail, including the interpolated assertion. The spherical argument is identical. The first two projection moments under all these posteriors are consequently bounded by C_RM_q and C_RM_q^2.

For nearby scales and fields, integrating first derivatives of the log partition function gives

and likewise for \phi^{\rm sp}. Indeed, \|B_q-B_p\|\le Cq^2|\log(q/p)| on \cH_q; its derivative is controlled by the second projection moment and the field derivative by the first. Fields in the smaller space are compared in \cH_q.

Cover K\le k_p\le C_0\sqrt{\log n}, with fixed C_0 large enough to overlap the source cutoff. Net scales from above and their spectral field balls at resolution \eta/(C_R(1+C_0\sqrt{\log n})) in the units of (3.41). Conditional on the spectrum, these are deterministic nets. Since \dim\cH_p\le Ck_p, their total size is

At a net point (3.38) and Chebyshev give \Pr_U(|\log L_p(h)|>\eta/2)\le C_\eta n^{-1/4}. The union bound and (3.41) yield

with failure n^{-1/4+o(1)}, in addition to the projection event. For k_p\ge c\sqrt{\log n}, the cap and interpolation results of [21, Proposition 2.6, Equation (2.11), and Lemma 2.7] apply as stated, since C_1r_n is eventually below their fixed upper cutoff. Together these prove the value comparison at every scale.

For the mean, use this comparison on a larger ball. For unit v\in\cH_p, freeze \alpha=\alpha_p(h). The polar formula and Lemma 3.3 give

The cube bound has the additional 2\epsilon_vk_p. Jensen with t=\theta p^2M_p, in the direction of the projected mean discrepancy, gives

Choose \theta from the required mean tolerance, then impose a smaller value tolerance and increase K.

It remains to prove the Hessian assertion on the low range. Put \widehat R=R_{p,\alpha_p(h)}, m=\widehat Rh, and \sigma^2=2\Tr\widehat R^2+4m^{\mathsf T}\widehat Rm. For a unit v\in\cH_p and |\theta|\le1, the shift h+\theta pv fits a fixed larger ball because p/(p^2M_p)=k_p^{-1/2}. Freezing \alpha, the exact polar formula is

The shifted mean m_\theta=m+\theta p\widehat Rv satisfies

The leading term is uniformly bounded. The density estimate (3.29), at this argument and at zero, gives

This is uniform in h,v,\theta. Cauchy–Schwarz and \sigma^2\ge4m^{\mathsf T}\widehat Rm imply 0\preceq\Gamma_{p,h}\preceq C_RI. Equation (3.42) changes the log moment by at most 2\eta, so the cube error is at most \delta=C_RK^{-1/2}+2\eta.

To justify differentiating this approximation, suppose a real Z obeys

The values at \pm1 bound \E e^{|Z|}, and hence bound the fourth derivative of \psi(\theta)=\log\E e^{\theta Z} on [-1/2,1/2]. Taylor’s formula gives

The assumed error changes this quotient by at most 2\delta/t^2. With t=\delta^{1/4}, this proves |\Var Z-a|\le C_C\sqrt\delta. Applying it to Z=pv^{\mathsf T}(X-m), for every unit v, yields

On the annulus put t=\alpha/p^2, A=(n-\Tr\widehat R)/(np), B=(p/n)\Tr\widehat R^2, and e=h/\|h\|. Since \widehat R|_{\cH_p}=p^{-2}(1+t)^{-1}I and m is parallel to e,

The trace estimates and denominator comparison give A=b_0+O(\epsilon_{\rm sp}+s_0) and B=2b_0+O(\epsilon_{\rm sp}+s_0), so this is 1/2+O(\epsilon_{\rm sp}+s_0). All other quadratic directions are bounded above by (1-s_0)^{-1}. Choose the spectral accuracy and s_0 small, then \eta small and K large in (3.44). This proves the annular curvature and radial deficit. On the entire ball, the saddle gap and the same estimate give C_Rp^{-2}, with C_R fixed before the annular tolerance. The source curvature assertion covers the higher range.□

Proof. By (3.5), under \mathbf P the spectral spin is spherical. Its Gaussian representation is that of [21, Proposition 2.3]. The annulus calculation in [21, Lemma 3.1, Equation (3.1)] is stated for the source observation range beginning at r_n; the tight-threshold dyadic estimates below extend that calculation to the scales used here. Set D_i'=d_i+b_nn^{-2/3}=n^{-2/3}(g_{i,n}+b_n). Before conditioning, let Y_i be independent N(0,(D_i')^{-1}); conditioning on \sum_iY_i^2=n gives the spherical law. The definition of b_n gives \min_iD_i'\ge n^{-2/3}, while (3.45) supplies the fixed leading variance lower bounds. The density of n^{-2/3}\sum_iY_i^2 at n^{1/3} is exactly q_n(D_n), and hence is bounded below on the chosen spectral set. The same set and the active count give

Indeed the first J_\chi terms are bounded by one, and the remaining terms are bounded by C_\chi i^{-2/3}.

Let \vartheta=p^2+b_nn^{-2/3} and, before conditioning, write h_{p,i}=B_{p,i}Y_i+B_{p,i}^{1/2}Z_i. The quadratic form obtained from \|h_p\|^2-\vartheta^2\|Y\|^2 has mean -\vartheta^2\Tr R_{p,b_nn^{-2/3}}. On an active coordinate the coefficient of the standardized Y_i^2 term is

the cross coefficient is O(p^3/\sqrt{D_i'}). Equation (3.48) and the capped second trace bound therefore show that the matrix of this quadratic form, in the standardized (Y,Z) variables, satisfies

The Gaussian quadratic-form moment formula for its centered version bounds a deviation of p^2k_p^{9/10} by

This remains true after norm conditioning, with a constant loss. Indeed use the same exponential tilt, with parameter at most one eighth of \|\mathsf A_p\|^{-1}. Its joint covariance and its Y-marginal covariance stay comparable to the originals. A fixed number of the leading marginal variances bounds the density of n^{-2/3}\|Y\|^2 from above by a constant, through the characteristic formula (3.27). The original density at n^{1/3} is bounded below on the chosen spectral set. The polar density ratio therefore costs a constant.

After conditioning, the deterministic target for \|h_p\|^2 is \ell_p(b_nn^{-2/3})^2\asymp p^2k_p. The preceding error is a relative O(k_p^{-1/10}). It first places the inferred saddle in a fixed annulus. On that annulus \mathrm d(\ell_p(u)^2)/\mathrm du\asymp k_p. Hence

Since b_nn^{-2/3}/p^2=O(k_p^{-2/3}), this proves the fixed-scale version of (3.46).

For a whole interval use a mesh of spacing k_q^{-3} in \log p. Its size is polynomial in k_q. Except with Ce^{-ck_q} probability, the spherical projection at the upper endpoint is at most CM_q, by Lemma 3.4. The signal increments on a mesh gap are then at most Cq^2M_qk_q^{-3}. The noise increments have covariance norm O(q^2k_q^{-3}). A constant-resolution net of the sphere in O(k_q) dimensions and the Gaussian maximal inequality show that their norm on every gap is less than q^2M_qk_q^{-1/10}, except with Ce^{-ck_q} probability. The target length is Lipschitz in the same scaled units: between activations differentiate the capped trace, using the active count and second trace; the target is continuous at activations. These observations interpolate the fixed-scale bound. The polynomial mesh size is absorbed by decreasing the positive exponent, proving (3.46).

For a fixed \zeta, it suffices by inclusion to treat \zeta below a fixed small annulus width. The stronger exponent follows from a different Gaussian reference, following the source-range fixed saddle test in [19, Section 19.2, Equation (19.9) and its proof]. Fix a small \xi>0, to be chosen after \zeta, and let

Increase K so that d_1\ge-\xi p^2/2. The vector a_{p,\xi}=R_{p,\xi p^2}h_p is centered Gaussian with covariance

The resolvent and clipped traces give

For a fixed multiple of np, exponential Markov with both signs of a sufficiently small multiple of \xi p^2 therefore gives a deviation cost e^{-c_\zeta\xi k_p}, once \xi is sufficiently small. The centered logarithmic moment costs O(\xi^{3/2}k_p), whereas the Markov gain is a fixed multiple of \zeta\xi k_p.

We spell out the norm-conditioning cost. The deficit n-\Tr R_\xi is between fixed multiples of \sqrt\xi\,np, after increasing K for this fixed \xi. A leading variance is comparable to (\xi p^2)^{-1}; the other squared norms have variance at most Cn/(\sqrt\xi\,p). Chebyshev places the residual length in a fixed interval comparable to \sqrt\xi\,np when K\gg\xi^{-3/2}. On this interval the leading Gaussian-square density is at least

This is a lower bound for the original norm density by convolution. Under the quadratic tilt used above, the marginal covariance of Y retains ck_p independent spectral rows with covariance comparable to p^{-2}; the tilt keeps the joint covariance comparable. Equation (3.25) bounds its norm density by C_\xi\sqrt{p/n}. Thus conditioning costs at most C_\xi e^{C\xi^{3/2}k_p}.

Since R_{p,0}h_p=(1+\xi)R_{p,\xi p^2}h_p, compare the saddle length at the two boundaries \alpha=\pm\zeta p^2. The derivative bounds in (3.26), and monotonicity beyond those boundaries, put \{|\alpha_p(h_p)|>\zeta p^2\} inside the relaxed squared-length deviation just bounded. Choose \xi so that its O(\sqrt\xi) target error fits the \zeta tolerance and C\xi^{3/2}<c_\zeta\xi/2. The conditioned deviation then has probability C_\zeta e^{-c_\zeta'k_p}. The same k_q^{-3} mesh proves (3.47).

Finally take q_j=2^j(K/n)^{1/3}. By (3.7) and Markov, the ordinary whole-interval probability is at most Ce^{-c k_{q_j}} outside disorder probability Ce^{-c k_{q_j}}, after decreasing c. Their sum is

Use (3.5) for L_\circ<1/2 and increase K. Every scale lies in one such interval and k_p\asymp k_{q_j} there, proving (3.13) simultaneously in p.□

Proof. Put t=p^2, and include U in the observation filtration, so that every disorder event is measurable at time zero. The exact likelihood identity (3.6) yields

where \mathcal B is the continuous cube innovation martingale with \mathrm d\langle\mathcal B\rangle_t=P_{\sqrt t}\,\mathrm dt; the coefficients lie in the active space. This is the exact filtering calculation in [21, proof of Lemma 3.5, after Equation (3.7)]; it follows as well by differentiating the posterior partition function, including the interaction change -\mathrm dB_{\sqrt t}. There is no jump in \log L at activation.

At a deterministic endpoint and for H>0, (3.6) gives

For a fixed R_0 containing the path annulus, put \mathcal U_p=\{\widetilde h:\|\widetilde h\|\le R_0p^2M_p\}. Lemma 3.7 gives

The field in the inner expectation is deterministic. Lemma 3.9 excludes the complementary endpoint fields with probability Ce^{-c k_q^{a_{\rm an}}}.

On an enlarged interval write \mathcal J_I=\int\|a_p\|^2\,\mathrm d(p^2) and M=\int a_p\cdot\mathrm d\mathcal B_{p^2}. On the two endpoint events just given, M+\mathcal J_I/2\le2H. If \mathcal J_I\ge8H, then M\le-\mathcal J_I/4. The positive exponential supermartingale \exp(-M/4-\mathcal J_I/32) bounds this event by e^{-H/4}. Taking H=Ck_q^{1/2} proves (3.49).

Between activations the two filtering equations give, with \Delta_t=K_{\sqrt t}-K_{\sqrt t}^{\rm sp},

Choose a smooth nonnegative cutoff \chi(t), equal to one on the target interval, zero at the ends of a fixed enlargement, and with |t\chi'(t)|\le C. The integrated Itô identity is

Here \Delta P_j is the jump of P_{\sqrt t} at the activation time t_j. The sum has the favorable sign. On \mathcal G and the whole-path annulus event, the coarse bounds give \|tK_{\sqrt t}\|+\|tK_{\sqrt t}^{\rm sp}\|\le C. Thus the left side on the target interval is bounded by C\mathcal J_I+C|\mathcal M|.

For concentration, stop only the stochastic integral at the first annulus failure, and set its integrand to zero on \mathcal G^c. The latter event is measurable at time zero. The stopped bracket is at most C\mathcal J_I, and the stopped integral agrees with \mathcal M on \mathcal G and the whole-path annulus event. The exponential bracket inequality, (3.49), and (3.46) consequently bound the target integral of \|p^2(K_p-K_p^{\rm sp})\|_{\rm F}^2\,\mathrm d\log p by Ck_q^{1/2}, with the required exceptional probability intersected with \mathcal G. The full identity (3.52) was not stopped, so it has no terminal boundary term.

On the shrinking annulus, rotational invariance inside the flat cap gives one spherical transverse covariance eigenvalue \kappa_\perp and one radial eigenvalue. The frozen linear moment bound at shifts \pm pe gives

Since \kappa_\perp=(e^{\mathsf T}m_p^{\rm sp})/\|h_p\| and \|h_p\|\asymp p^2M_p,

The scaled radial covariance is bounded by the spherical version of (3.44). There are O(k_p) transverse dimensions and one radial dimension, so

Combine this with the preceding cube–sphere covariance estimate, divide by k_q\asymp k_p, and decrease a_1>0 if necessary. This proves (3.50).□

Proof. We verify the deterministic extension proof of [21, Lemma 4.3], following [19, Proposition 16.1], with the new lower threshold. On \{|\alpha_p|\le3s_{\rm aux}p^2/4\} take the infimum, over z, of

Choose s_{\rm aux}\ll\eta,\gamma^2. The length bounds in Lemma 3.3 give a fixed \beta_{\rm buf}>0 such that a segment of length at most \beta_{\rm buf}p^2M_p starting in this smaller annulus remains in the full annulus. For such pairs the supporting inequality follows by integrating the coarse Hessian bound. For \|y-z\|\ge\beta_{\rm buf}p^2M_p, freeze the spherical saddle at z. The polar linear-moment bound gives the spherical quadratic support with Hessian (p^2+\alpha_p(z))^{-1}I, its saddle gradient, and a bounded norm-density error. The static value and mean estimates add 2\epsilon_vk_p+\epsilon_gM_p\|y-z\|. Relative to \|y-z\|^2/(2p^2), the total residual is bounded by

Choose s_{\rm aux}, then \beta_{\rm buf}, then \epsilon_v,\epsilon_g so that this is less than \eta, and finally increase K. Every support lies above \phi_p on the smaller annulus. Its infimum is finite and semiconcave with the stated upper Hessian, and it agrees there by choosing z=y. The length derivative gives \dist(\mathcal A_p,(\mathcal A_p^+)^c)\ge c_{\rm buf}p^2M_p.

For the transition tails freeze the child saddle \alpha=\alpha_q(h_q), and use as reference the Gaussian transition tilted by the parent frozen quadratic q_{W_p,\alpha}. Writing R_u=R_{u,\alpha}, Gaussian convolution gives

The child saddle equation shows that \E\|h_p\|^2=\ell_p(\alpha)^2, exactly. The field covariance has norm at most C\gamma^{-2}p^2. Gaussian concentration of its norm therefore gives

The reference radius lies strictly inside \mathcal A_p: |\alpha|\le s_{\rm aux}q^2\le4s_{\rm aux}\gamma^2p^2. The length derivative gives a fixed positive gap in units p^2M_p to the boundary of \mathcal A_p.

Choose z_0\in\cH_p with \|z_0\|=\ell_p(\alpha) and calibrate the reference quadratic by

The active covariance is scalar, so the added density depends only on \|z_0\|; the additive constant leaves the reference transition unchanged. On bounded scaled radii, the noncentral upper norm-density bound and the static value comparison give \phi_p(y)-q^\sharp(y)\le\epsilon k_p+O(1), with any prescribed fixed \epsilon>0. Outside a sufficiently large bounded radius, the zero-field projection tail gives

This follows by maximizing the exponent uv-cv^6 after integrating the tail against a linear field of scaled length u. Its growth is strictly subquadratic. For the extension choose the support centered at the point of radius \ell_p(\alpha) in the direction of y. At distance vp^2M_p, its excess over the reference is at most

This uses the static mean error for the linear term and the difference of the two quadratic coefficients for the last term. Choose \eta+s_{\rm aux}\ll\gamma^2, and then \epsilon small compared with the squared fixed gap to \mathcal A_p^c. Outside \mathcal A_p these errors are smaller than the reference Gaussian deviation cost.

The normalizing integrals relative to the reference are at least e^{-\epsilon'k_p} for any needed fixed \epsilon'>0. To see this, integrate over a thin fixed shell about the reference radius, contained in \mathcal A_p. Its reference probability is bounded below when K is large. On that shell the frozen density formula (3.31) bounds the off-saddle logarithmic density loss by C\tau^2k_p+O(1) when the shell width is \tau in scaled units: the first parameter derivative vanishes at the saddle and the second is one quarter of the norm-square variance O(n/p). The static lower value error adds only \epsilon_vk_p. Take \tau and \epsilon_v sufficiently small. Both potentials agree on this shell. Dividing the preceding upper tail integrals by this lower bound proves the stated disagreement tails. Polynomial moments are absorbed by the same quadratic reference tail and the subquadratic bound at large radii.□

Proof. Fix a deterministic nonactivation scale p on the retained spectral set. For h\in\cH_p, write D_p(h)=p^2K_p(h)-I_{\cH_p}. Define a Borel predicate \mathcal V_p(W_p,h) to be false unless h\in\cH_p, the saddle \alpha_p(h) exists, and h\ne0. Otherwise, with e=h/\|h\|, it requires

This condition contains no restriction to \mathcal G. At a nonactivation scale, P_p is the spectral projection of W_p at its flat eigenvalue 2-p^2, and R_{p,u}=(2I-W_p+uI)^{-1}. Thus the predicate, the saddle, and the covariance used here are determined by (p,W_p,h). Their defining formulas are jointly Borel away from activation scales. In the rest of the proof these objects are evaluated at h=h_p, and that argument is suppressed.

On \mathcal V_p, use blocks relative to \cH_p=\operatorname{span}\{e\}\oplus(\cH_p\cap e^\perp), put z=(D_p)_{\perp e}, and define on \cH_p\cap e^\perp

Extend E_p by zero on the radial direction and on \cH_p^\perp. Off \mathcal V_p, set both E_p and \widehat s_p to zero, before any restriction to \mathcal G. Covariance positivity and (3.57) give -I\preceq D_p\preceq\epsilon_{\rm aux}I and (D_p)_{ee}\le-0.1 on \mathcal V_p. Maximizing the quadratic form of D_p over its radial coordinate, and then taking the positive part on e^\perp, gives

on \mathcal V_p, with D_p\preceq E_p understood on \cH_p. For the norm bound, the maximizing radial coordinate of a unit transverse vector is bounded because the radial deficit is fixed and D_p is bounded; compare the resulting vector with D_p\preceq\widehat s_pI. For the trace, the rank-one correction has bounded trace, and \Tr(((D_p)_{\perp\perp})_+)\le \sqrt{\dim\cH_p}\|D_p\|_{\rm F}; the displayed trace bound also uses \dim\cH_p\le Ck_p from the retained spectral set. These are the algebraic bounds of [21, Lemma 4.4, Equation (4.4)]. In particular \widehat s_p\le\epsilon_{\rm aux} and \|E_p\|\le C\epsilon_{\rm aux} everywhere, including on disorder failures. Include C\epsilon_{\rm aux}\ll\gamma^2 in the initial auxiliary choice.

Apply Brascamp–Lieb’s covariance inequality [4] to the transition with \bar\phi_p. After whitening, its Hessian has a fixed positive margin because \Delta\preceq(1-x^2)p^2I and \eta\ll\gamma^2. If \bar K is the extended Hessian, the covariance is at most the transition expectation of

Mollify the semiconcave extension to apply the smooth inequality and pass to the limit using the positive margin. Lemma 3.11 replaces the extended Hessian and transition by K_p and the true transition on agreement. Its polynomially weighted tails control first and second moments on the complement. In field units the covariance error is O(p^4M_p^2e^{-c k_p}); after (3.54) and multiplication by q^2, the remaining polynomial factor in k_p is absorbed by decreasing the exponent.

On \mathcal G, the agreement annulus lies in \mathcal V_p, so (3.59) gives K_p\preceq p^{-2}(I+E_p). Put A=I-\Delta/p^2 and S=p^{-2}\Delta^{1/2}E_p\Delta^{1/2}. Then A\succeq x^2I and \|A^{-1/2}SA^{-1/2}\|\ll1, so

The baseline A^{-1}, substituted in (3.54), is exactly q^{-2}I_{\cH_q}. For the correction, the two outer factors \Delta A^{-1} are the scalar \Delta_q/x^2 on \cH_q. Thus, after multiplication by q^2, the first insertion is at most 2x^{-2}P_qE_pP_q. We obtain on \mathcal G, for a child in its annulus,

Only \Delta_q is inverted outside the covariance in (3.54); directions with arbitrarily small new parent precision cause no loss.

We next compress E_p before imposing \mathcal G. The assertion is trivial when E_p=0; otherwise e is defined by \mathcal V_p. The retained spectral set supplies ck_p\le\dim\cH_p\le Ck_p and \dim\cH_q\le Cx^3k_p; after increasing the threshold, \dim(\cH_p\cap e^\perp) is also comparable to k_p. Condition under \mathbf P on the spectrum, W_p in physical coordinates, the physical field h_p, and its spectral coordinates. Equation (3.6) makes the remaining rotation in e^\perp\cap\cH_p exactly Haar. The operator E_p is fixed under this conditioning: its definition uses only these parent data. For a uniformly rotating unit vector v in this space, the Gaussian representation of the uniform sphere gives

Indeed the numerator is a positive Gaussian quadratic form, and the Gaussian squared norm in the denominator is at least a fixed fraction of the dimension except with e^{-c_0k_p} probability. A fixed-resolution net chosen in the spectral child unit sphere before the remaining Haar rotation has at most \exp(C\dim\cH_q)\le\exp(Cx^3k_p) points. Project each rotated net vector onto e^\perp. Conditional on the parent data, it has a fixed norm at most one and a uniformly rotating direction. With t=C_2x^3k_p, a sufficiently large C_2 pays for the net, and a sufficiently small \gamma pays for the second exponential term. The net estimate for a positive quadratic form gives

outside a set of unconditional \mathbf P-probability Ce^{-c_\gamma k_p}. This is the compression argument of [21, Lemma 4.5, Equation (4.6)]; no restriction to \mathcal G was made in the Haar law.

Average (3.60) over I_q=[q/(2\gamma),q/\gamma], with normalized logarithmic measure. On \mathcal G outside the failure event in (3.50), (3.59) and Cauchy–Schwarz give

Here \operatorname{Av} denotes that normalized logarithmic integral. On its failure the same averaged trace is bounded, because E_p is bounded everywhere and \dim\cH_p\le Ck_p. Its contribution on that failure, with \ind_{\mathcal G} inserted, has expectation at most Ce^{-c k_q^{a_1}}. The failures of (3.61) are charged in the same way by the bounded \|E_p\|. These estimates are applied to the whole parent-scale integral before its conditional expectation. For every nonnegative error cost Y,

because \mathcal G is measurable from W.

The norm term in (3.61) has coefficient Cx^{-2}x^3\le C\gamma. The initial choice of \gamma makes this coefficient \theta<1/2. We have proved, on \mathcal G,

for some \beta>0. Off the child annulus the left side is zero. The bounded definition of E_p is essential here: no cost polynomial in n is charged on a failed parent interval.

Use a barrier B k_q^{-\alpha}, with 0<\alpha\le b and B large. Since k_p=k_qx^{-3},

It absorbs the deterministic error in (3.63). Taking positive parts, applying conditional Jensen, and using (3.62), we obtain

For each deterministic nonactivation scale r_n\le p\le C_\gamma r_n, use [21, Proposition 4.2, Equation (4.2)] within its stated range. Under the biased law \mathbf P, it gives p^2K_p\preceq(1+C(p^{a_0}+k_p^{-a_0}))I outside a set whose intersection with the full event \mathcal G_n^{\rm FUA} has probability at most Ce^{-n^{\xi_0}}. Since \mathcal G\subseteq\mathcal G_n^{\rm FUA}, the same failure bound holds after intersection with \mathcal G. Here pk_p=np^4\to0, so p^{a_0}\le k_p^{-a_0}, and k_p\asymp n^{3\cdot10^{-4}}. By taking \alpha\le a_0 and a sufficiently small positive terminal exponential power, the same bound for A(p) is Ce^{-c k_p^{\beta_0}}.

Every parent generation multiplies k by at least \Lambda=(2\gamma)^{-3}>1. Iteration, including the terminal bound, gives for a suitably decreased \beta_*>0

Markov’s inequality at the barrier costs only a factor k_q^\alpha, absorbed by the exponential. On \mathcal G, (3.57) holds throughout the full auxiliary annulus, and (3.46) removes the annulus truncation. Decrease to a common positive exponent in the barrier and probability. This proves (3.56).□

Proof of Proposition 3.1. Lemmas 3.2 and 3.3 prove the spectral and saddle assertions. Lemmas 3.4, 3.8, and 3.9 prove the static, curvature, projection, and ordinary annulus assertions. Assign a fixed fraction of the requested failure probability to each of their spectral and disorder restrictions, and take their finite intersection. The length bounds give a deterministic R_{\rm ann} containing the full annulus; include a further cube-cap request for the static comparisons on that ball, retaining the previously chosen width s_0. We select one additional event, independent of f, for the variance.

In the reduction above take \eta_p=Ck_p^{-a} and let \mathcal N be the jointly measurable field set where p^2K_p(h)\npreceq(1+Ck_p^{-a})I_{\cH_p}. Set its indicator to one at activation scales, which have zero scale measure for each W. At every deterministic nonactivation scale its evaluation at h_p is the failure event in (3.56). Use Z_p and the common cost \rho_p from (3.16); both are independent of f.

Under \mathbf P, conditional Jensen and the spherical identity give

by the uncapped spherical projection tail. Cauchy–Schwarz and (3.56), absorbing k_p in the exponential, imply

The left side is an ordinary path expectation averaged with the Haar bias. Take q_j=2^j(K/n)^{1/3} up to r_n. Fubini, (3.64), and L_\circ\ge1/2 bound the expected ordinary cost on [q_j,2q_j], intersected with \mathcal G, by Ce^{-c k_{q_j}^a}. Markov and

give an event of arbitrarily large limiting disorder probability on which, simultaneously in j,

Increase K and use (3.5) for the single discarded event. This event is independent of f.

The conditional-variance estimate of [21, Proposition 4.1, Equation (4.1)] holds on an ordinary event of arbitrarily large limiting probability, simultaneously for all real cube functions and scales in its stated interval [r_n,p_*]. Its remainder is Ce^{-n^{\xi_0}}\|f\|_\infty^2. We use it only at r_n, for 0\le f\le1:

Intersect this event with the preceding ones. In (3.19) with terminal scale r_n, its amplified energy is Cp^{-2}\cE(f). Its amplified remainder is bounded by Ce^{-c k_p^a} after decreasing a>0, since k_p\le k_{r_n}=n^{3\cdot10^{-4}}. For the other errors use (3.65). The first partial dyadic interval has k_{q_j}\asymp k_p; the rest increase geometrically. Hence

Applying (3.19) proves the first inequality of (3.14). It extends to activation points by continuity of V_p in B_p.

For the second inequality impose the additional fixed annulus estimate with \zeta_*=\min\{\zeta,s_0/2\}. On that annulus the coarse covariance and (3.17) contribute CV_p. For its complement, Cauchy–Schwarz and conditional Jensen give

Here \mu(Z_p^2)\le C holds at every scale on the chosen uncapped cube projection event, and the ordinary annulus estimate is also simultaneous in scale. Weakening the exponent if needed proves the second inequality of (3.14), on the same event for every f.□

Proof. Fix first one instance of Proposition 3.1, with \epsilon_{\rm sp}\le1, K\ge1, and a variance-remainder exponent 0<a\le1, decreasing the exponent if necessary. Its constants are fixed on a disorder event of arbitrarily high limiting probability. Choose M>1 sufficiently large that

We retain this variance estimate when imposing further cap tolerances below; intersection with finitely many additional events does not change its already chosen exponent a. Choose \delta>0 so small that M\delta/3<10^{-4}. This leaves every fixed factor buffer about the scales below within r_n=n^{-1/3+10^{-4}} for large n. Use the same fixed small \gamma>0 as in the covariance iteration of Section 3. Choose the bridge tolerance \epsilon>0 after M, with 2(M-1)\epsilon/3<\kappa.

For a small fixed c_0>0, to be chosen using only the projection-tail constants, define

Increasing K_0 puts these scales above the threshold of Proposition 3.1. Suppose, for a contradiction, that \Tn\cE(f)/N<L^{5/3-\kappa}. If F_R(h)=\mu_{R,h}[f] and V_R=Q_\mu[\Var(f\mid h_R)], that proposition gives

The estimate is uniform in n,L,f in the present range, because R^2\Tn=L^{2M/3}.

Consider the joint ordinary law Q_\mu of the spin and its observation path through R. Change this law to Q_{\rm spin} or Q_{\rm obs} by the respective densities

For K_0 large, A\ge N/2, so both densities are at most 2e^L. For any of these laws, Q|_p denotes restriction to \sigma(h_q:0\le q\le p), always excluding the spin. Define, for p<R,

The first entropy is on the full joint law. Since its likelihood is measurable from h_R, it equals the relative entropy on the terminal observation sigma-field. Both terms are finite by the likelihood bound, and the conditional entropy chain rule identifies H(p) with the expected entropy of the terminal observation tilt remaining after the prefix through p. To compare the two tilts, note that Q_\mu[fF_R]=A, and hence

Cauchy–Schwarz bounds the L^1(Q_\mu) distance between the two densities by 2\sqrt{2V_R/N}. This bound persists after projection to any part of the observation path. If two projected densities u,v\le2e^L have L^1 distance d, then

Indeed x\log x has derivative bounded by 1+L+\log2 on [1,2e^L], while its absolute value is bounded on [0,1]. It follows from (4.9) that the two relative entropies, both before and after any projection, differ by o(L), uniformly as K_0 increases.

We bound the information revealed at p_1 under the spin tilt. The field h_{p_1} is a sufficient statistic for the path through that scale. Let Q_0 be the output of the same Gaussian channel with zero signal. Convexity of relative entropy and evenness of \mu give

These formulas are on the range of B_{p_1}, so zero precision directions cause no difficulty. Subtraction bounds D(Q_{\rm spin}|_{p_1}\Vert Q_\mu|_{p_1}) by half the sum of the two quadratic moments. The spectral bound d_1\ge-\epsilon_{\rm sp}p_1^2 gives B_{p_1}\preceq(1+\epsilon_{\rm sp})p_1^2P_{p_1} \preceq2p_1^2P_{p_1}. The projection tail in Proposition 3.1, integrated against a density at most e^L, gives

For example, integrate \min\{1,e^LCe^{-c k_{p_1}v^6}\} in the variable v^2. The information is therefore at most

This is the proportional-cap argument in [19, Equation (19.12)]. Choose c_0 so that this last bound is less than L/8. The full spin-tilt entropy is at least L/2. The two continuity comparisons in (4.11), combined into one uniform o(L), therefore give

We follow that remaining entropy up to a fixed fraction of R. For p<R, let \mathsf T_{p,R}^h be the ordinary conditional law of h_R given h_p=h. It has density proportional to

where the Gaussian is restricted to the range of its covariance. Let \widetilde{\mathsf T}_{p,R}^h be its tilt by F_R(y)^2, and put

Observation sufficiency gives the prefix density \dd(Q_{\rm obs}|_p)/\dd(Q_\mu|_p)=A_p(h_p)/A. The conditional chain rule in (4.10) now reads

Write \Delta=B_R-B_p. On the currently active subspace \cH_p, \Delta=(R^2-p^2)I. Differentiating its Gaussian convolution therefore gives

In the parameter p^2, the likelihood martingale A_p(h_p)/A has innovation coefficient (A_p/A)\delta m_h. The Gaussian likelihood formula for its relative entropy and the chain rule yield

This identity holds almost everywhere and in integrated form across activation points. One direct justification applies Itô’s formula to (u+\epsilon)\log(u+\epsilon) of the bounded likelihood, then lets \epsilon\downarrow0; zero-rate directions are omitted. This is the observation entropy identity [19, Lemma 19.2, Equation (19.6)], applied here to a terminal observation tilt instead of a spin tilt.

The ordinary log moment generating function of S, for v\in\cH_p, is exactly

Suppose that the Hessian of \phi_p in this subspace is at most C_{\rm curv}p^{-2}I on all shifts of length at most p\sqrt{2l_p(h)/C_{\rm curv}} from h. The entropy variational inequality, tested against \ip vS, then gives

To check the finite shift radius, the centered log moment generating function is at most \frac12(C_{\rm curv}p^{-2}+(R^2-p^2)^{-1})|v|^2. Testing at length \sqrt{2l_p/(C_{\rm curv}p^{-2}+(R^2-p^2)^{-1})} in the mean-shift direction proves the inequality by contradiction; this length is no larger than the assumed radius.

We next account for observations where that curvature bound is not available. The entropy bound l_p\le-\log A_p, together with A\ge e^{-L}/2, implies

Under the ordinary joint law,

with an independent standard Gaussian G_p on \cH_p. For p\le\gamma R, the projection tails of Proposition 3.1 and Gaussian concentration give, for each prescribed a_0>0, a fixed V such that

To see the common scale, write the observation noise using a standard Gaussian G'_p on \cH_p. Since B_p\preceq2p^2P_p,

The last coefficient is bounded on this range and \dim\cH_p\le Ck_p. Integration of the same tails allows any fixed polynomial moment as well. For a prefix event E, conditional Jensen and the full and prefix likelihood bounds at most 2e^L give

Indeed Jensen bounds |\delta m_h|^2 by twice the sum of the second moments under the two conditional laws. For their parts |S|\le VM_p use the prefix probability of E; for the remaining parts use respectively the full likelihood F_R^2/A and the prefix likelihood A_p/A. Finally p^2M_p^2=k_p.

Choose p_2 by k_{p_2}=C_2k_{p_1}, where C_2 will be fixed below. Once it is fixed, p_2<\gamma R for K_0 sufficiently large. Fix A_0>0, choose a tail rate a_0>c_0^{-1} with a fixed margin, and then choose V in (4.17). On [p_1,p_2] retain \|h_p\|\le Vp^2M_p and l_p\le L+\log2+A_0k_p. The trial shifts in (4.15), in units p^2M_p, then have length at most \sqrt{2(c_0^{-1}+A_0+1)}, after enlarging the ball-curvature constant to be at least one. Choose a fixed ball containing the retained fields and these shifts before requesting its curvature constant from Proposition 3.1. Equations (4.16)–(4.18) show that discarded observations contribute at most Ck_pe^{-c k_p}. Thus (4.14)–(4.15) imply

The logarithmic length of this interval is fixed.

For [p_2,\gamma R], make a fresh auxiliary cap and extension request with the same \gamma. Choose \eta\ll\gamma^2, then \epsilon_{\rm aux}\le\epsilon sufficiently small compared with \eta. Reapply the construction of Lemma 3.11 with these choices, and write s_0 for its auxiliary width. Let \mathcal G_{\rm ext}=\mathcal G_n^{\rm aux,ext} \cap\mathcal G_n^{\rm FUA} be the resulting event for the required static tolerances, radii, threshold, and scale buffers. Lemma 3.8 gives \mathcal G_n^{\rm aux,ext} with arbitrarily high limiting ordinary probability, and \mathcal G_n^{\rm FUA} costs o(1). Assign a further fraction of the failure probability and intersect with the retained variance event; its exponent a and constants are unchanged.

On the full annulus |\alpha_p(h)|\le s_0p^2, this request gives K_p(h)\preceq(1+\epsilon_{\rm aux})p^{-2}I \preceq(1+\epsilon)p^{-2}I. The same annulus specifies admissible child fields for the extension. Choose 0<\zeta<s_0 and a fixed b_{\rm rad}>0 so that shifts of length at most b_{\rm rad}p^2M_p from |\alpha_p(h)|\le\zeta p^2 stay in the full annulus. After \zeta is fixed, impose its ordinary annulus estimate from Lemma 3.9 as one further fixed request and intersection. Thus the ordinary failure probability is Ce^{-a_\zeta k_p}, with a fixed a_\zeta>0; only this probability bound is used from the further request. The trial shifts fit whenever l_p\le b k_p, where b=(1+\epsilon)b_{\rm rad}^2/2. Choose C_2 so large that

The likelihood bound and (4.16) give exponentially small tilted probabilities for both discarded events. Using (4.18) with a further fixed S-tail threshold proves

The extra coefficient has bounded integral,

In both intervals the integrating factor charges a fixed power of p/p_1, whereas k_p=np^3 grows as its cube. Consequently the weighted errors are bounded by a polynomial in k_{p_1} times (e^{-c k_{p_1}}+e^{-c k_{p_2}}). They are o(L) after increasing K_0. The first interval costs a fixed multiplicative factor. We obtain

It remains to bound this entropy by the energy of f. For a child field satisfying |\alpha_{\gamma R}(h)|\le s_0(\gamma R)^2, the fresh request to Lemma 3.11 supplies an extension of the parent potential with upper Hessian (1+\eta)R^{-2}I. Put

The extension agrees with the true potential on \mathcal A_R^+. The distance from \mathcal A_R to (\mathcal A_R^+)^c is at least c_{\rm buf}R^2M_R. Both transition laws assign \mathcal A_R^c probability Ce^{-c k_R}, also with fixed polynomial moments of \|h_R\|/(R^2M_R). The precision increment satisfies \Delta\preceq(1-\gamma^2)R^2I. On its range the extended transition, denoted by \overline{\mathsf T}, has negative log-density Hessian at least

The Bakry–Émery logarithmic Sobolev criterion [1, Corollary 5.7.2] gives

It applies to the semiconcave extension by mollification.

Choose a smooth cutoff \chi equal to one on \mathcal A_R and supported in \mathcal A_R^+. The fixed buffer and the radial saddle derivative in Proposition 3.1 give |\nabla\chi|\le C/(R^2M_R). Apply the preceding inequality to \chi F_R. Its cutoff-gradient term is at most Ce^{-c k_R}, because R^2|\nabla\chi|^2\le C/k_R and the transition tail has that exponential rate. On agreement the ratio of the two transition densities is constant; their normalization and their exponentially small tails make that constant 1+O(e^{-c k_R}). To compare the entropies, use \Ent_{\mathsf T}(g)=\mathsf T[g\log g]-\mathsf T[g]\log\mathsf T[g]: x\log x is bounded and Hölder continuous on [0,1], and 0\le F_R,\chi\le1. The comparison, with a possibly smaller positive exponent, proves

Failure of the child condition is contained in the ordinary \zeta-annulus failure, since \zeta<s_0. Its probability is Ce^{-c k_R} after changing c, since k_{\gamma R}=\gamma^3k_R. On it the conditional entropy of F_R^2\le1 is at most 1/e. Average (4.20) under the ordinary child law and use (4.13). The tower property, A\ge N/2, and the two inequalities of (3.14) at R give

Here 0<a\le1 allows the transition error to be included in the displayed remainder, and k_R=L^M makes it C e^{L-cL^{Ma}}. The error is negligible compared with (4.19), even if the latter is a negative power of L. Comparing the two bounds and using \Tn p_1^2=(c_0L)^{2/3} gives

The choice 2(M-1)\epsilon/3<\kappa contradicts the assumed upper bound for K_0 large. All ball radii, tail rates, annulus tolerances and factor buffers have been fixed before this last increase of K_0 and before n tends to infinity.□

Proof. The first inequality is the product calculation of [19, Equation (19.18)], applied at each chosen endpoint. Its precision is t'I+W for the corresponding fixed t'>3. The posterior interaction is then a scalar diagonal, hence its spin law is a product with fields h_i. Tensorizing the two-point log-Sobolev bound gives

The field tail of [19, Equation (19.10)], on the common event for the chosen endpoints, restricts \|h\|_\infty\le C\sqrt{\log n+L}. Its ordinary complement has probability at most e^{-2L}n^{-10}, with polynomial moments; after the likelihood bound N_h/N\le e^L its entropy contribution is O(n^{-3}). Equation (4.24) gives the displayed bound.

For the second inequality, entropy tensorization at a product posterior and Jensen’s inequality for the logarithm give, for positive z,

In fact \Ent_\pi(z)\le\Cov_\pi(z,\log z) follows from \pi[\log z]\le\log\pi[z]. The posterior entropy formula is

It remains to compare the averaged modified form to its initial value. For a fixed site and fixed x_{-i}, let

be the two joint posterior masses. Their contribution to \cI_{\mu_h}(z) is

The second factor is nonnegative and independent of the observation. The function c(a,b)=ab/(a+b), extended by zero on the coordinate axes, is concave: in the positive quadrant its Hessian is -2(b,-a)(b,-a)^{\mathsf T}/(a+b)^3. Since Q_\mu[a_h] and Q_\mu[b_h] are the corresponding initial masses, Jensen’s inequality proves Q_\mu[\cI_{\mu_h}(z)]\le\cI_\mu(z). Take z=f^2+\epsilon and let \epsilon\downarrow0 to include zeros. This proves the second bound.□

Proof. We prove the stronger square-density residual estimate that replaces the linear energy dependence in [22, Lemma 6.5]. Use the zero-diagonal interaction J_*, so the cube coordinates in this proof are spins, not the scaled edge coordinates. For a spin flip \sigma^{(i)}, put

The primary iterates of [22, Section 6] are

For a fixed ordinary recipe admissible from level l, let U be a terminal vector in that recipe. We claim that, whenever \sup_\sigma(|H_0|+\|dH_0\|)\le C_0,

The constant has the dependence on the fixed recipe, depth, coefficient bounds and finite word diagnostics allowed in the source lemma, and is independent of f. The residual claim itself does not require root stability.

Here are the base case and the induction, including the point where the improvement occurs. The pointwise estimate [22, Lemma 6.4, Equation (6.20)] gives

It also gives the same bound for \widetilde U=H_0U: indeed

and Cauchy–Schwarz bounds the sum of the added diagonal derivatives. Write C_U for a bound on \|\widetilde U\|+\sum_i|d_i\widetilde U_i|.

Let p_i^\pm=\mu_h(\sigma_i=\pm1\mid\sigma_{-i}). Then m_i^1=p_i^+-p_i^- and the conditional variance is v_i^1=4p_i^+p_i^-, in particular v_i^1\le4p_i^\pm for each sign. Since a function of one spin is affine with slope d_if, the form is

The exact one-site integration by parts [22, Equation (6.31)] gives

Here f_{i,\pm}=f(\sigma^{i,\pm}), and d_i(f^2)=(f_{i,+}+f_{i,-})d_if, also for signed f. The first term is at most C_UN_h in absolute value. For the second set

At each configuration,

Weighted Cauchy–Schwarz bounds that second term by

The two separate bounds v_i^1\le4p_i^\pm imply

Since M_i is independent of \sigma_i, conditioning term by term gives

This proves (4.28) for l=1.

The finite induction in the source preserves this new right side. For clarity, let U be admissible from level l+1, put a_l=j(b_l-b_{l-1}), and define

Here |a_l|\le j. The integral formula bounds \Phi and every needed fixed-order derivative uniformly in its two real field arguments: the hyperbolic derivatives introduce bounded hyperbolic-tangent factors, and the exponential factor and its derivatives are bounded for |a|\le j. Together with the explicit level indices, these bounds make U' admissible from level l. The scalar identity in [22, Section 6] then gives the residual expansion below. Adjoin the auxiliary field

Here w^b,y^b are the finitely many earlier auxiliary sources and fields of the recipe; the partials differentiate their explicitly displayed local arguments, holding parameters fixed, as in [22, Definition 6.2]. Equation (6.33) of that source is

with the R_{l-1} term absent when l=1. All first-line terms are earlier residuals with ordinary admissible sources. Their scalar multipliers and their discrete-gradient norms are bounded by the fixed-recipe bounds. For q>l in the second line, use the source m^{q-1}/\sqrt n and multiplier j\sqrt n\,\theta_q; [22, Equation (6.21)] gives the same boundedness for that multiplier. Finally the exact identity

cancels the displayed -a_ln\theta_l. The remainder again uses an earlier residual, the source m^l/\sqrt n, and a bounded multiplier. A bounded multiplier \zeta with bounded \|d\zeta\| can always be absorbed into H_0, because

Every induction call therefore retains the original f,N_h,\mathcal D_h(f). There are finitely many calls with a smaller residual index. This proves (4.28) at every fixed depth.

We spell out its consequences for selection and the mean. Put S_l=R_l^{\mathsf T}m^l/\sqrt n. The primary derivative bounds in [22, Lemma 7.1] give |S_l|\le2\sqrt n and \|dS_l\|\le C_l. Apply (4.28) with U=m^l/\sqrt n and H_0=S_l/\sqrt n. Division by N_h\sqrt n yields the strengthened form of [22, Equation (7.1)]:

For a small fixed \rho>0, choose a fixed depth d with \lfloor d/2\rfloor\rho^2/4>2, and then e_\rho>0 with 2de_\rho<1. The telescoping identity

shows that \max_{l\le d}|S_l|/\sqrt n\le e_\rho forces a consecutive pair R_{k-1},R_k to have both norms at most \rho\sqrt n/2. Otherwise the disjoint pairs would make the sum of squared norms exceed 2, whereas telescoping bounds it by 1+2de_\rho<2.

As in that source lemma, choose smooth C_k\in[0,1], equal to one on this smaller pair region and supported where both norms are at most \rho\sqrt n. The primary derivative bounds give \|dC_k\|\le C_\rho/\sqrt n. Put

Positive deficit implies \max_{l\le d}|S_l|/\sqrt n>e_\rho. Chebyshev’s inequality and (4.29) give

The cutoffs are admissible bounded multipliers in (4.28).

Choose \rho\le\epsilon with (\|J_*\|+4j)\rho<2\rho_0. On the support of C_k, the recursion gives

so \|F_h(h^k)\|\le(\|J_*\|+4j)\rho\sqrt n. The root estimate of [22, Lemma 5.7],

and the Lipschitz property of \tanh therefore give \|m^k-m_*\|\le C_{\rm lead}\rho\sqrt n, with a constant independent of the depth. For a deterministic unit vector e, sum (4.28) for R_1,\ldots,R_k, with source e and multiplier \widehat C_k. Since \sigma-m^k=\sum_{l\le k}R_l, this bounds its reweighted mean against e by C_\rho(1+\sqrt{\mathcal D_h(f)/N_h}). The weighted root errors together cost at most C_{\rm lead}\epsilon\sqrt n, because the cutoffs sum to at most one. The remaining deficit costs at most 2\sqrt n\,\nu_h(D_{\rm sel}>0), controlled by (4.30). Sum over the fixed number of cutoffs, divide by \sqrt n, and take the supremum over e. This proves (4.27).□

Proof. We use the path and the field tests in [19, Section 19.6]; we verify that its entropy argument remains valid under (4.31). On source disorder events independent of the initial spin law, the calculation around Equation (19.20) there, using its Lemma 17.4, Equation (17.12), and Proposition 17.5, gives the following early field-test bound. On a fixed interval [t_*,t_b] of small positive scalar precision, the diagnostic and stability failures have integrated probability o(1), uniformly under every initial spin law. This follows there by conditioning on the spin: the centered Gaussian-noise diagnostics have integrated failure o(1), the additional spin shift has normalized norm O(t), and the Gaussian low-band small-ball estimate is uniform in its center. No factor e^L is used for this interval. From t_b until the interaction coefficient reaches a fixed globally controlled range, the buffered stability test fails anywhere on the path with probability at most e^{-a n} under the ordinary law Q_\mu, for a fixed a>0. This is the later field bound in [19, Lemma 17.3, Section 19.4, and Section 19.6 after Equation (19.21)]. The word diagnostics and stability constants on the retained regions are those of Lemma 4.5.

Here is the order of the fixed choices in that construction. Choose first a large scalar endpoint t_{\rm end} and a small interaction coefficient below which covariance is globally bounded. Choose t_b>0 for the small-positive-field estimate, and obtain the buffer and rate a for the later test. The norm bound and the fixed total duration give constants C_{\rm jump} and C_{\rm aft} for the jump loss divided by t_*n and the loss after failure divided by n, uniformly when 1-\beta'\le t_*/3. Choose a target drift budget \eta_1>0 so that

Next choose 0<t_*\ll t_b with C_{\rm jump}t_*\le\eta_1, and finally 1-\beta'>0 sufficiently small relative to these choices, including 1-\beta'\le t_*/3. Start with precision

which is positive when 1-\beta' is small enough. Increase scalar precision to t_{\rm end}I+(1-\beta')W, and then traverse

The latter rate 4I+W is positive on the norm event. The endpoint is (t_{\rm end}+4\beta')I+W, so its posterior is a product. Before the globally controlled portion, the posterior interaction coefficient is either \beta'<1 or in a fixed compact subinterval of (0,1). Reduce the available diagonal enlargement in (4.26) to 0<\eta<(\beta')^{-1}-1. Once these stability margins and C_{\rm lead} are fixed, choose the mean accuracy \epsilon in Lemma 4.5 so that its leading error, integrated over the bounded path and converted to a quadratic mean bound, is less than \eta_1. Then fix its residual tolerance, selection depth, finite word diagnostics and constant C_\epsilon, and finally take n large.

On a valid posterior, apply (4.27) to f and to 1. With a_u=\|\nu_u[X]-\mu_u[X]\|/\sqrt n, this gives

By (4.24) and Cauchy–Schwarz, the Q_\nu-mean of the square root, also with any validity indicator, is at most \sqrt{\cE(f)/N}. Under (4.31) the resulting error is

This bound is applied at deterministic observation times; inserting \ind_{\{u<\tau\}} for a stopping time only decreases its Cauchy–Schwarz bound. Since a_u\le2 always, a_u^2\le2a_u. The early invalid paths cost at most four times their integrated probability, hence o(1). Thus the normalized quadratic mean-shift integral, on valid paths and before any later stopping, is at most \eta_1+o(1) by the preceding choice of \epsilon.

Let u_g begin the globally controlled portion and let \tau be the first failure of the later buffered test in [t_b,u_g], with the immediate and no-failure conventions of the source. The initial jump leaks at most C_{\rm jump}t_*n entropy: compare to pure noise as above and use \|B_*\|\le Ct_* and \|X\|^2=n. The stopped form of (4.23), with the preceding integrated estimate, gives, with Q^\tau denoting the law of the stopped observations, excluding the spin,

For the event E of a failure before u_g, the ordinary probability is at most e^{-an}. Data processing for its indicator then gives

These are exactly the deductions in [19, Equations (19.21)–(19.22)], now with the enlarged energy range justified by Lemma 4.5. After a failure the remaining precision rate and duration are bounded, and both spin means have norm at most \sqrt n; the remaining loss per failed path is therefore at most C_{\rm aft}n. The total loss before u_g is bounded by

The fixed choices above make this less than \lambda n/4. The initial entropy is at least L/2\ge\lambda n/2, hence H(u_g)\ge\lambda n/4. In the remaining portion the spin covariance is bounded for every field. The linear entropy test then gives -\dot H\le CH; over its bounded duration it retains a fixed fraction of H(u_g). This proves the lemma.□

Proof of Proposition 4.2. Fix C_{\mathrm{lev}},\xi and a desired disorder confidence. Apply Lemma 4.3 with \kappa<\min\{\xi,\,5/3-1.662\}. After fixing its M, choose its \delta>0 as in that proof. For L\le n^\delta, (4.8) and (4.4) imply both claimed inequalities.

For n^\delta<L\le\lambda n, use the proportional branch of [19, Equation (19.16) and Section 19.4]. It gives, at the product endpoint,

Here \epsilon>0 is any fixed sufficiently small tolerance, and \lambda>0 is chosen afterward. We justify the exact test class being used. The starting dimension in that source is \max\{k_{\min},c_0L\}, with k_{\min}\asymp\sqrt{\log n}. It equals c_0L throughout the present range for large n. The proportional part of its initial jump estimate is the pure-noise argument reproduced in Lemma 4.3; it requires only the entropy at least L/2 and density at most e^L. The source’s support condition is used in its other, minimum-dimension branch. Its subsequent propagation uses the likelihood and conditional-entropy tails in [19, Equation (19.8)] and the ordinary path tests. These also require only the same density and entropy bounds. The continuation through its finite remaining scales uses those likelihood tails and conditional Pinsker bounds, then globally bounded covariance, and so preserves this test class. Thus (4.33) applies to (4.5) without a support restriction.

Since L\ge n^\delta\gg\log n, comparison with the first bound of (4.25) gives

The additive O(n^{-3}) is smaller than (4.33). The second product bound similarly gives

Choose \epsilon after \delta so that

Then choose \lambda below the finitely many fixed rates and shift buffers in that source argument. This proves the profiles in the moderate range.

Finally let \lambda n\le L\le C_{\mathrm{lev}}n. If the classical bound were false, then \cE(f)/N\le Cn^{1.162-2/3}. This meets (4.31) for a fixed positive \omega, whereas the first product bound would give

contrary to Lemma 4.6. If the modified bound were false, (4.4) would instead give \cE(f)/N\le Cn^{1-\xi}, so that lemma applies with \omega=\xi. The second product bound would give H_{\rm end}\le Cn^{1-\xi}=o(n), again a contradiction.

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

Proof. In logarithmic coordinates l=\log z, the logarithm of \mathcal U_d(e^l) is

Consequently u\le z, and u=z on the indicated level. This proves B\le\mu[u]\le1 and L=1.1d-\log\mu[u], with the stated upper bound from the lower bound on B.

Let E=\{\log z\ge0.7d\}. Since u\le e^{-0.1d}z on its complement,

Set e_d=e^{-0.1d}/B. Data processing for the indicator of E gives

Uniformly for d\ge v/2, the hypothesis on B makes e_d\to0 and 1+\log_+(d/v)=o(d) as v\to\infty. The last lower bound therefore exceeds L/2 for large v. The level is nonempty, so d\le C_an by (4.34); the bound on L, with C_{\mathrm{lev}}>2C_a+1, puts it below C_{\mathrm{lev}}n for large n. Its lower bound puts it above K_0 for large v.

Finally \mathcal U_d is nondecreasing and 2-Lipschitz as a function of z, while \log\mathcal U_d(e^l) is nondecreasing and 2-Lipschitz as a function of l. The product of their increments on an edge is at most four times the product of the increments of z and \log z. Summing proves (4.36).□

Proof. Put H=\Ent_\mu(z) and d_j=(H/2)(1.1)^j. The part of \mu[z\log z] with \log z<H/2 is at most H/2. Thus the levels [d_j,1.1d_j] account for at least H/2. If their tilted masses B_j were all less than c(H/d_j)^{1.1}, their contribution would be at most

Choose c small. One level therefore has B\ge c(H/d)^{1.1}. Apply Lemma 4.7 with v=H and r=1.1, followed by (4.6) with any fixed \xi<5/3-1.662. Homogeneity of \cI gives

The constants and the required lower threshold on H are uniform.□

Proof. First let 0\le g\le1 equal one on a nonempty set of mass a=e^{-L_a}, and let its support have mass at most e^{-L_{\rm supp}}. Take

Let S_i=\mu(g>i/K), T_i=\mu(g\ge i/K). Some i satisfies T_{i+1}\ge S_i^2; otherwise induction gives a\le T_K<e^{-2^KL_{\rm supp}}<a. For this i, with N_i=\mu[g_i^2],

Its logarithmic inverse mass L_i lies in [L_{\rm supp},L_a], and L_a\le C_an by (4.34). For L\ge K_0 large enough, the function

is decreasing within each range and has a downward jump at their boundary. Since \cE(g_i)\le K^2\cE(g), (4.7) implies

Absorbing the logarithmic denominator by lowering each exponent by 0.008 gives

Indeed L^{0.008}/(1+\log L)^2 has a positive infimum for L\ge K_0.

For arbitrary nonnegative g with the stated support, use \varphi_j=(g-2^j)_+\wedge2^j, j\in\mathbb Z, and let a_{j+1}=\mu(g\ge2^{j+1}). Omit empty upper level sets. The normalized layer 2^{-j}\varphi_j is one on that set and has the original support restriction. Its L_a=\log(1/a_{j+1}) is at least L_{\rm supp}. The right side of (4.39), divided by a, is at least \Phi_n(L_{\rm supp}): before the boundary it is at least L_{\rm supp}^{1.654}, and after the boundary it is at least n^{1.154\delta}, or at least L_{\rm supp}^{1.154} when L_{\rm supp}>n^\delta. For each oriented edge the nonnegative increments of the \varphi_j’s sum to the increment of g, so the sum of their squares is at most its square. Therefore

Pointwise, \sum_j4^j\ind_{\{g\ge2^{j+1}\}}\ge g^2/12. Integration proves (4.38).□

Proof of Theorem 4.1. Fix the parameters in the theorem and intersect the finitely many events needed for the preceding results. The finite heat-bath chain has a strictly positive kernel at every positive time. Hence w_s^\rho>0 for s>0, for every initial law, and symmetry of the generator gives

When the entropy exceeds the threshold of Lemma 4.8, its derivative is at most -c\Ent_\mu(w_s^\rho)^{1.662}. Integrating this differential inequality from any earlier positive time, and then letting that time decrease to zero, yields uniformly in the initial law

This proves (4.1).

For the kernel, put Q_s=\mu[(w_s^\rho)^2]=e^{L_s}. The atom bound gives Q_s\le e^{C_an}, so its logarithm is within the range where the support profile was proved. Whenever L_s is large, set g=(w_s^\rho-Q_s/4)_+. Its support has mass at most 4e^{-L_s} by \mu[w_s^\rho]=1, and

Its actual support logarithm is

The support is nonempty by the preceding square-mass bound, so (4.34) also gives L_{\rm supp}\le C_an. Since \dot Q_s=-2\Tn\cE(w_s^\rho), Lemma 4.9 and monotonicity of \Phi_n give

above a fixed threshold. A shift of the argument by \log4 only changes constants there. For any large K', the time allowed by this differential inequality above K' is at most

For completeness, below n^{\delta(1.154/1.654)} integrate L^{-1.654}. The following constant portion of \Phi_n, up to n^\delta, costs at most n^\delta/n^{1.154\delta}, and the integral of L^{-1.154} above n^\delta has the same order. Inverting (4.40) gives

uniformly even for point initial laws. Reversibility and two half steps show that

The preceding bound, with 1/0.654<1.54, proves (4.2) after enlarging the vanishing threshold.

We finish by proving (4.3). For the fixed integer m\ge1, let \Psi_m be the convex function on (0,\infty) determined by

It extends continuously to zero: in its twice-integrated formula the possible singularity is bounded by \int_0^1(1+|\log z|)^{m-1}\dd z<\infty. For a density z, put

There are constants depending only on m such that

Indeed for z\ge1, \Psi_m'(z)=((1+\log z)^m-1)/m. Integrating this on [1,z] gives upper and lower bounds of order z(\log z)^m when \log z is large; the remaining range contributes at most a constant times 1+z. On [0,1] \Psi_m is bounded.

For a strictly positive density z, define the corresponding dissipation \mathcal D_m(z)=\cE(z,\Psi_m'(z)). We claim that, for \mathsf M_m(z) above a fixed threshold,

Put V=\mathsf M_m(z) and use levels d_j=(V/2)(1.1)^j. If V is large, the part of \mu[z(\log_+z)^m] below V/2 is at most 2^{-m}V^m. Geometric summation as in Lemma 4.8 gives a level d\ge V/2 with

Indeed otherwise (1.1)^m\sum_jd_j^mB_j\le C_m c_m V^m would be too small. The test u=\mathcal U_d(z) from Lemma 4.7 therefore satisfies (4.5), and (4.6) gives

We check that the new dissipation controls this layer with its additional weight d^{m-1}. On an edge write a\ge b>0, x=\log a, y=\log b. Then

If x\ge0.4d, the last integral is at least c_m d^{m-1}(x-y). To verify this uniformly in y, use the top half of [0,x] when -x\le y\le x/2, the entire interval when y\ge x/2, and the negative half nearest y when y<-x; in each case a fixed fraction of the interval has |t| at least a fixed fraction of x. The two Lipschitz bounds in the proof of Lemma 4.7 then apply. If x<0.4d, both endpoints are in the lower branch of \mathcal U_d. On [b,a] its derivative is at most 2e^{-0.4d}, while its logarithm has slope two in log coordinates. Since d^{m-1}e^{-0.4d}\le C_m, the same conclusion follows using (1+|t|)^{m-1}\ge1. Thus in both cases

Summation gives \mathcal D_m(z)\ge c_m d^{m-1}\cI(u). Combining this with (4.43) and the lower bound on B, and using 2/3-\xi>0.1, gives

This proves (4.42).

Along the finite semigroup at positive times,

Equations (4.41)–(4.42) give, above a fixed threshold,

Integrating from an earlier positive time and letting that time decrease to zero yields V_m(w_s^\rho)\le C(1+s^{-m/(2/3-\xi)}). The comparison (4.41) proves (4.3). This argument allows initial zeros, because \Psi_m is continuous at zero and the kernel is strictly positive afterward.

The three conclusions (4.1)–(4.3) depend only on the off-diagonal couplings. Let \mathcal G_n^{\rm aug} be the final event in the augmented Gaussian space, and put

Then \Prob(\mathcal G_n^{\rm off})\ge \Prob(\mathcal G_n^{\rm aug}). For fixed off-diagonal couplings the three inequalities are deterministic, so their validity on an augmented realization in \mathcal G_n^{\rm aug} descends to \mathcal G_n^{\rm off}. Taking this as the theorem event proves the stated disorder quantifiers.□

Proof. Choose \xi>0 so small that

then choose a fixed integer m>2A/\beta. Apply (4.3) to the point initial law at \sigma and time s=d/\Tn. Set v=\Tn/d. Uniformly for 1\le d\le C_\star\Tn, so C_\star^{-1}\le v\le n^{2/3},

Markov’s inequality under the transition law P_d(\sigma,\tau)=\mu(\tau)p_d(\sigma,\tau) now bounds the left side of (4.44) by

The choices of \xi,m are fixed in that order before n. The statement for \sigma\sim\rho follows by averaging these row-specific probabilities.□

Proof. We first pass the semigroups on bounded sources, then construct the densities, and finally justify the sum over all edge coordinates.

Varying L^2 spaces. Choose a countable family \mathscr C of smooth compact cylinders with this approximation property: every smooth compact cylinder is approximated uniformly, together with its first derivatives, by elements of \mathscr C supported in one fixed compact coordinate set. Rational smooth approximations on rational boxes give such a family. The stated approximation and the definition of the closed form make \mathscr C dense in its form norm, and also dense in L^2(\Pi_g). For linear tests put \mathscr V=\operatorname{span}_{\mathbb Q}(\mathscr C\cup\{1\}). This is a countable family of bounded smooth cylinders, dense in L^1(\Pi_g) and L^2(\Pi_g). If \phi\in\mathscr V depends on the first d coordinates, let \phi_n=\phi(x_1,\ldots,x_d) on the finite cube.

The static cylinder and moment convergence of [21, Corollary 3.3] gives

On sequences bounded in L^2(\mu), the weak convergence of Section 2 can be tested on this countable family: the pairings with every \phi\in\mathscr V converge to the corresponding pairings with f if and only if the pairings with every smooth compact cylinder do. To pass from \mathscr V to an arbitrary smooth compact cylinder, approximate it uniformly on a common support by \mathscr C, and use the uniform L^2 bound to control the pairing error in both spaces. The converse follows from the same approximation, with cutoffs tending to one for the constant test. A sequence bounded in L^2(\mu) has a weakly convergent subsequence: diagonal extraction of its pairings defines a bounded linear functional on the cylinder span by (5.6), and the Riesz representation theorem supplies its L^2(\Pi_g) representative. The same argument gives

As in Section 2, the convergence is strong when the norms also converge. For such a sequence,

Consequently strong convergence is equivalent to approximation, in both spaces, by the same cylinders. In particular, the pairing of an L^2-bounded weakly convergent sequence with a strongly convergent sequence converges.

Resolvents from bounded optimizers. Set

For \phi\in\mathscr C and rational z>0, consider

The finite Markov property gives \|u_n\|_\infty\le z^{-1}\|\phi\|_\infty. The resolvent identity also bounds \|u_n\|_2 and \mathfrak q_n(u_n) uniformly. Thus these particular optimizers satisfy the polynomial supremum bound as well as the norm and energy bounds in the relaxed-energy theorem [21, Theorem 6.8]. That theorem applies to disorder-dependent tests under the class-uniform quenched convention recorded in Section 2. This proof invokes it only for the countably specified optimizer sequences, one for each \phi\in\mathscr C and rational z>0, on their common represented event. Its lower bound therefore gives

Here is the variational passage, including the norm conclusion. Let

and define v by the analogous supremum in L^2(\Pi_g). Take an optimizer subsequence attaining the upper limit and then a weak limit U. The norm lower bound and (5.8) imply

For the reverse inequality, take F\in\mathscr C. The recovery part of [21, Theorem 6.8] supplies, for each fixed recovery index j, polynomially bounded finite functions strongly converging to F, whose energies converge to \mathfrak q(F) in the order n\to\infty, then j\to\infty. Testing the variational supremum with these functions in exactly that order gives

The form density of \mathscr C allows the supremum over F, proving v_n\to v. Only the countably many sources in \mathscr C and rational z are used on the common source event. The limiting variational problem is strictly concave, so every weak optimizer limit is its unique optimizer u. Moreover,

Together with norm lower semicontinuity this proves strong convergence of u_n to u. This is the bounded-source version of the variational argument in [21, proof of Lemma 7.2].

The bound \|(z+\mathfrak A_n)^{-1}\|_{2\to2}\le z^{-1} and (5.7) now extend this resolvent convergence to every strongly converging input. No additional use of the lower energy bound is needed for that extension. To pass to semigroups, put R_n=(1+\mathfrak A_n)^{-1}, R=(1+\mathfrak A)^{-1}, and, for t>0, define on [0,1]

This function is continuous. Polynomial approximation of F_t and convergence of every power of R_n on strongly converging inputs prove

For the assertion with varying times, use

This proof identifies the limit of P_{tT_n} as \exp(-t c_*H_g) with the stated T_n=n^{2/3}.

Densities and the limiting kernel. On the represented sequence, Theorem 4.1, with fixed admissible auxiliary parameters, gives a finite random C and numbers \delta_n\downarrow0 such that \|P_{sT_n}\|_{1\to\infty}\le \mathcal K(s) for all s\ge\delta_n. Since w_{n,0} is a nonnegative density and the semigroup is self-adjoint,

Include the pairings of these densities with \mathscr V at each positive rational s in the representation. Every limit functional L_s obeys

Here \phi^-=\max\{-\phi,0\}. The bound follows from (5.10) and static convergence also for the continuous cylinders |\phi| and \phi^-. It extends L_s to L^1(\Pi_g). The second inequality proves positivity of the extension: if cylinders \phi_j converge in L^1 to f\ge0, then \Pi_g[\phi_j^-]\le\|\phi_j-f\|_1\to0. The extension is therefore represented by a density 0\le h_s\le \mathcal K(s) of mass one. In particular \|h_s\|_2^2\le \mathcal K(s). This also justifies normalization if only compact cylinders were initially retained: their cutoffs tending to one have complements bounded by \mathcal K(s) times the corresponding static probabilities. Measurability can be seen concretely. Take nested finite cylinder partitions which generate the product Borel sets and whose cell boundaries have zero \Pi_g measure; rational coordinate boxes have this property by the cylinder densities of [21, Proposition 2.3]. The value L_s(\ind_A) for each cell is a measurable limit of its values on fixed smooth approximations to \ind_A, which can in turn be approximated by \mathscr C. On each cell A put the ratio L_s(\ind_A)/\Pi_g(A), with value zero if the denominator is zero. These ratios are measurable, bounded by \mathcal K(s), and are the conditional expectations of h_s on the nested partitions. Their bounded martingale limit gives a measurable representative of h_s.

For positive rational s,t, reversibility and (5.9) imply for every \phi\in\mathscr C

We used the weak–strong pairing rule; the w_{n,s} are weakly convergent and bounded in L^2. Hence h_{s+t}=\Sg_t h_s. The closed cylinder-gradient form is a Dirichlet form: smooth normal contractions obey its contraction inequality, and closure preserves it. Constants belong to the form and have zero energy, by [21, Proposition 5.1]. Thus \Sg_t is Markov and preserves one. For a real s>0, choose a rational q<s and set h_s=\Sg_{s-q}h_q. The rational relation makes this definition independent of q. Strong continuity of the semigroup proves L^2 continuity in s>0, as well as the relation for all s,t>0. Positivity and mass one persist. Taking rationals q\uparrow s in the density bound, using L^2 convergence and an almost-everywhere subsequence, proves h_s\le \mathcal K(s). For deterministic dyadic rationals q_j(s)\to s, every cylinder integral at s is the limit of its measurable values at q_j(s). It is therefore jointly measurable in the environment and s. The same partition construction therefore gives jointly measurable representatives for the real-time family.

The finite convergence against cylinders also holds along every sequence s_n\to s>0. Indeed, for s\ge\delta>0, the identity w_{n,s}=P_{(s-\delta/2)T_n}w_{n,\delta/2} and spectral calculus give

Approximation of s by a rational time, followed by the weak convergence there and the L^2 continuity of h_s, proves the claim.

For completeness we obtain the kernel needed to sum coordinates. For nonnegative smooth compact cylinders \phi,\psi, the finite kernel bound yields

Passing to the limit using (5.9), and then approximating indicators in L^2(\Pi_g), gives the same inequality with \Pi_g[\phi\Sg_t\psi] in place of the left side. Define on rectangles the nonnegative functional

For a simple function on rectangles use common disjoint partitions of the two factors. The preceding rectangle inequality then bounds the functional on every nonnegative such function by \mathcal K(t) times its L^1(\Pi_g\otimes\Pi_g) norm. It extends to a positive bounded functional on that L^1 space, so has a density k_t^g with 0\le k_t^g\le \mathcal K(t). Self-adjointness makes the density symmetric. Its marginals are \Pi_g, since \Sg_t1=1. This proves (5.2); it does not require a joint pointwise version of the kernels for all t.

The tail of two equilibrium copies. Let x_a'=x_a(\sigma'), and set

To see why this equilibrium quantity controls the quench trace, define the relative transition kernel and the two-time density

The latter is the density of the two spins at times sT_n and (s+t)T_n relative to \mu\otimes\mu. On every compact subset of s,t>0, the warming bounds give a uniform bound on this density for all sufficiently large n on the represented sequence. Hence

The stationary trace has the same bound with d_{n,s,t} replaced by p_{n,t}. Thus it suffices to control V_{n,m} uniformly as m increases. We claim that the representation may be further extracted so that

Recall r_n=n^{-1/3+10^{-4}} from Section 3, and fix the spectrum in one of the tight spectral sets of [21, Proposition 2.3]. In this paragraph take its canonical parameters

Let \mu^{\mathrm{sp}} be the spherical Gibbs law, and L_\circ=Z_{\mathrm{cube}}/Z_{\mathrm{sp}}, with both partition functions relative to their normalized uniform measures. Average over the full Haar eigenframe at this fixed spectrum. Multiplication by L_\circ^2 cancels the two cube Gibbs denominators. In the uncapped replica comparison of [21, Lemma 3.2], the normalized contribution of pairs with |\rho|=|n^{-1}\sigma\cdot\sigma'|>C r_n is at most C e^{-c n r_n^3}. Since |Q_{n,m}|^2\le n^{2/3}, inserting this square in the discarded part costs at most C n^{2/3}e^{-c n r_n^3}.

On the retained overlaps, the same comparison can be used with this square inserted. Here are the required uniform details. Conditional on \rho, a Haar-rotated cube pair and a spherical pair have the same law of an orthonormal sum-and-difference frame (q,q'). Their spectral vectors can be written

The ratio of the discrete overlap mass and the spherical mass on a cell of width 2/n is at most \exp(C+C n\rho^4), by the replica comparison. This is bounded on |\rho|\le C r_n, because n r_n^4=o(1). The tilt exponent on this frame is

Its oscillation in a cell is bounded when \|\Lambda\| is bounded. If \rho,\widetilde\rho lie in that cell, the displayed vectors change in Euclidean norm by at most C n^{-1/2}. After multiplying by n^{-1/3}, each vector changes by at most C n^{-5/6} and has norm n^{1/6}. Projection onto the coordinates a>m therefore gives, uniformly in m,

The inequality (u+v)^2\le2u^2+2v^2 now bounds the inserted square by the corresponding spherical square and C n^{-4/3}, up to the preceding bounded comparison factors.

Under two independent spherical Gibbs laws, sign symmetry kills the cross terms in that square. The uniform coordinate moment bound in [21, Proposition 2.3] gives

Since \ind_{\{L_\circ\ge1/2\}}\le4L_\circ^2, the comparison just proved yields

The constants are uniform on the retained spectral set. The squared resolvent tails tend to zero as m\to\infty there, and n r_n^3 is a positive power of n. Also \Prob_U\{L_\circ<1/2\}=o(1), by [21, Lemma 2.5]. Markov’s inequality, followed by exhaustion of the tight spectral sets, proves for every \epsilon>0

The same estimates make V_{n,m} tight for every fixed m. Include these countably many quantities in the representation and extract coordinatewise limits V_m. The identity

shows that V_{n,m}, and hence V_m, decreases with m. Equation (5.14) forces V_m\downarrow0 almost surely. Since V_{n,m}\to V_m for each fixed m, this is (5.12).

The limiting version follows directly from the same sign symmetry and moments. With \ell_a^\circ=(g_a+b^\circ)^{-1}, for M>m,

Thus these interval sums have an L^2(\Pi_g\otimes\Pi_g) tail whose norm tends to zero.

Correlation limits and relaxation. Orthogonality of the eigenframe gives the exact finite identities

The preceding two-copy reduction and (5.12) give uniform finite correlation tails on every compact positive time set. For the limiting series, the corresponding two-time density is h_s(x)k_t^g(x,x') relative to \Pi_g\otimes\Pi_g; for the stationary series it is k_t^g(x,x'). Equations (5.1) and (5.2) bound these densities uniformly on compact positive time sets. The same Cauchy–Schwarz argument with (5.15) therefore gives the limiting uniform tail control.

It remains to pass each fixed summand. Static convergence of the mixed pairings x_a\phi and the second moment x_a^2 makes the finite coordinate x_a strongly convergent to the limiting x_a. If s_n\to s>0, then w_{n,s_n}x_a is weakly convergent to h_sx_a. To see this, test first against a cylinder times a smooth cutoff of x_a at height R, and use the convergence after (5.11), extended from \mathscr C by the common-support approximation above. For a bounded cylinder \phi, the omitted part of the pairing is at most

The fixed second moments are bounded and converge, so this error vanishes uniformly as R\to\infty. Also \|w_{n,s_n}x_a\|_2\le \mathcal K(\delta)\|x_a\|_2. The weak–strong pairing rule and (5.9) now give, whenever s_n\to s>0 and t_n\to t>0,

The same argument without w_{n,s_n} gives the fixed stationary summand. Each limiting summand is continuous. For t\ge\delta>0, the kernel bound gives

Thus L^2 continuity of h_s gives continuity in s, uniformly for t in positive compact intervals. Strong continuity of \Sg_t x_a, paired with h_sx_a\in L^2(\Pi_g), gives continuity in t. The uniform tail bound proves the asserted continuity and local uniform convergence of the full series. It also proves the two locally uniform finite limits in (5.3): failure of uniform convergence on a compact set would provide a sequence of time points with a convergent subsequence, contradicting the fixed-summand convergence, the uniform tails, and continuity of the limiting sum.

Finally, the kernel of H_g consists exactly of the constants, by [21, Proposition 5.1]. The spectral theorem and h_s=\Sg_{s-s_0}h_{s_0} give h_s\to\Pi_g[h_{s_0}]=1 in L^2(\Pi_g). For each fixed a, the last displayed bound implies, uniformly for t\ge\delta,

The correlation tails are uniform also for s\ge s_0, since \|h_s\|_\infty\le \mathcal K(s_0). Passing first through a finite sum and then letting its length tend to infinity proves (5.5).□

Proof. We will show that x_i^2>s^{13/20} has vanishing averaged probability under the limiting time-s law, while M_g<s^{14/25} also has vanishing averaged probability as s\downarrow0. Away from these two events, x_i^2/M_g is at most s^{9/100}. We obtain the numerator bound from the finite uniform-start dynamics, then bound the denominator directly under the conditioned edge measure.

The total mass in a block of finite modes. Fix \varepsilon>0, and fix the auxiliary log-moment parameters in Theorem 4.1 once and for all. Its entropy bound and Proposition 3.1 provide a quenched disorder event \mathcal G_{n,\varepsilon} of limiting probability at least 1-\varepsilon on which their constants and thresholds are fixed, the entropy bound holds simultaneously for every positive time, and the uncapped cube projection tails hold at every allowed scale. Separately retain a spectrum-only event \mathcal S_{n,\varepsilon} of limiting probability at least 1-\varepsilon for the band estimates in Lemma 3.2 and the tight leading-edge bound |g_{1,n}|\le C_\varepsilon. The latter follows from the GOE edge convergence in Section 2. The constants in the following calculations may depend on \varepsilon.

For 0<s<1, put m=\lceil s^{-5/4}\rceil. For n\ge m, set

Here x_a(v) is evaluated along the finite dynamics at physical time vT_n, and I=sR. Choose k_p=np^3=C' m, with the fixed C' sufficiently large. Take s small enough that C'm exceeds the fixed threshold, and then take n sufficiently large that m\le n and p\le r_n. The spectral band estimates imply that \dim\cH_p\ge m on \mathcal S_{n,\varepsilon}, so P_p contains the first m coordinates and g_{m,n}\le T_np^2=(C'm)^{2/3}. Together with |g_{1,n}|\le C_\varepsilon, this gives on that event

At a fixed s, the value of k_p is fixed as n\to\infty; this is why the extension to the lowest scales in Proposition 3.1 is used here.

On \mathcal G_{n,\varepsilon}\cap\mathcal S_{n,\varepsilon} put Y_p=\|P_p\sigma\|^2/M_p^2. Its projection tail is \mu\{Y_p>u\}\le C\exp(-a k_pu^3) for every u\ge u_0, with fixed u_0. Integration of this tail gives

The entropy inequality for any probability density w relative to \mu, followed by Hölder’s inequality, therefore gives

Since D_m\le n^{-2/3}M_p^2Y_p=k_p^{1/3}Y_p, the entropy part of Theorem 4.1 yields under the direct dynamical law at fixed disorder

This use of entropy is entirely relative to the ordinary Gibbs law \mu; the path in this expectation starts from \nu_n.

Let Q(v)=\int_0^vD_m(w)\,dw and define the event

Integrating (5.21) and applying Markov’s inequality at the integral and the endpoint gives, on \mathcal G_{n,\varepsilon}\cap\mathcal S_{n,\varepsilon},

Indeed m^{1/3}\le C s^{-5/12}, and both comparisons use the same two positive powers:

A stopped rotation at fixed spectrum. For this step fix only a spectrum satisfying (5.20). Let \E_\Lambda average over the full conditional Haar law of U and over the uniform-start heat-bath path. In particular, this expectation does not condition on \mathcal G_{n,\varepsilon}.

It is useful to specify the path likelihood. In normalized time v, take independent site clocks of rate T_n, a uniform initial spin, and an independent fair mark \eta\in\{-1,1\} at every ring, including rings at which the mark equals the old spin. This is a reference law; the marked path is then determined without U. If the ring is at site k, its heat-bath probability is

The likelihood of the dynamical path through s relative to the reference law is

There is no initial or clock factor depending on U. This likelihood fixes the marked spin outcomes while the matrix varies; it permits differentiating the smooth heat-bath probabilities. The absence of an initial score is the feature of the uniform start used in this rotation.

For i,j\le m, i\ne j, right-rotate these two columns with \dot u_i=u_j and \dot u_j=-u_i. Then

Writing \Delta_{ij}=\lambda_i-\lambda_j, diagonal deletion in the physical local field gives

Orthogonality and \sum_k u_{i,k}^2u_{j,k}^2\le1, together with |\Delta_{ij}|=T_n^{-1}|g_{i,n}-g_{j,n}| \le C n^{-2/3}m^{2/3}, imply

The final 1 is the contribution from deleting the diagonal.

In the filtration revealing U initially and then the marked rings, the logarithmic derivative of \mathcal L_U through v is

Under the dynamical law it is a square-integrable martingale. In fact each marked increment has conditional mean zero and conditional variance \operatorname{sech}^2(b_k)\dot b_k^2; its predictable bracket is

Define

The function Q is continuous, so Q(\tau)\le I. Stopped martingale isometry and (5.24) give

For a fixed marked reference history, D_m,Q,\tau, and G are exactly invariant under this rotation. Apply Haar invariance first to

Differentiating its integral with likelihood (5.23), and then projecting the terminal score onto the stopped filtration, gives

Here is a justification at the stopping boundary. The stopping rule is constant on each rotation orbit of a fixed marked history, so \dot F_{ij} differentiates only x_i x_j. At each fixed n and spectrum, the likelihood is bounded by 2^N, where N is the number of rings, and its derivative is bounded by this quantity times a constant (depending on n and the spectrum) times N. Poisson exponential moments justify differentiation of the full-time integral. The random variable F_{ij} is measurable at \tau, and optional projection of the square-integrable martingale replaces S_{ij}(s) by S_{ij}(\tau). This proves (5.26) without differentiating a law with a moving stopping time.

Since \dot x_i=x_j and \dot x_j=-x_i, this identity reads

Sum over j\le m, j\ne i. The pathwise bound

and Cauchy–Schwarz on the product of probability and index spaces, together with \sum_{j\le m} I_j\le I and (5.25), yield

This estimate does not require independence between the scores. To solve it, put

It gives y\le1+\sqrt{\alpha y(y+1+\kappa)}. Here I m^{1/3}=O(s^{4/75}), while \kappa\to0 as n\to\infty at each fixed s. For sufficiently small s and then sufficiently large n, \kappa\le1 and \sqrt\alpha\le1/2. Using \sqrt{y(y+2)}\le y+1 proves

Repeat (5.26) with F_{ij}=\ind_G x_i(s)x_j(s). This random variable is measurable at \tau: it is zero if \tau<s, and otherwise is measurable at s=\tau. The indicator G, including its boundary, is constant on each rotation orbit, so its differentiation contributes nothing. Write Z_a=\E_\Lambda[\ind_G x_a(s)^2] for a\le m. Now \sum_{a\le m} Z_a\le R and

Use (5.25) and (5.28) in the summed identity. Taking n large enough that m s n^{-2/3}\le I, we obtain

For the second inequality, z=mZ_i/R satisfies z\le1+\sqrt{\alpha z}, and hence z\le2+\alpha. As R/m\le C s^{18/25}, Markov’s inequality gives, at each of the retained spectra,

We now combine the two probability laws. Let \Prob_n include the Gaussian disorder, the independent diagonal and eigenvector signs, and the uniform-start dynamics. Write \Prob_{\mathrm{dis}} and \E_{\mathrm{dis}} for the completed disorder law and \E_{\mathrm{eig}} for the eigenvalue law. Then

The second term uses (5.22); the last uses (5.30) with its full conditional Haar average. The complements of \mathcal G_{n,\varepsilon} and \mathcal S_{n,\varepsilon} cost at most 2\varepsilon in the limit, so for each fixed i and all sufficiently small s,

This separation is essential: no frame-dependent quenched event was imposed inside the Haar identity. Sending s\downarrow0 and then \varepsilon\downarrow0 proves the finite numerator estimate.

For a fixed positive s, the cylinder convergence in Proposition 5.1 transfers this estimate to \rho_s(dx)=h_s(x)\Pi_g(dx). Indeed each coordinate marginal of \Pi_g has a density by [21, Proposition 2.3]; the same is true of \rho_s. The boundary x_i^2=s^{13/20} thus has zero limiting probability. Weak convergence of this coordinate marginal and bounded convergence over the represented environment yield

Only finite-coordinate convergence was used in this step.

The weighted denominator. Corollary 3.5 gives the two-sided bound in (5.16) for every admissible b, with measurable constants when b is measurable. It remains to choose b with D_g(b)>0. Start with a measurable admissible b_0, for example b_0=1+(-g_1)_+. The resolvent-difference identity in [21, Proposition 2.3] gives, for b\ge b_0,

The right side increases to +\infty as b\to\infty, because \sum_a(g_a+b_0)^{-1}=\infty. Choose the first integer increment b=b_0+j for which D_g(b)>0. This is measurable and admissible, so Corollary 3.5 gives (5.16).

Put \ell_a=r_a^{-1}. The uniform coordinate moment bound in [21, Proposition 2.3] gives

The first coordinate has no atom at zero, so M_g>0 almost surely. This proves that (5.17) is well defined under \Pi_g and under every \rho_s.

Let N=\lceil s^{-8/5}\rceil and consider the coordinates N\le a\le2N. Under the independent Gaussian law of the conditioning construction, write Z_a=\sqrt{\ell_a}\xi_a with independent standard normals \xi_a. Equation (5.16) implies

The chi-square lower-tail bound shows that this is at least c_gN^{-1/3}, except on an event of probability at most C_g e^{-c_gN}.

The same estimate survives the norm conditioning with a fixed cost. Leave a sufficiently large fixed head of m_0 coordinates outside this block. Let q be the density of \sum_a(Z_a^2-\ell_a), and q_{m_0}^{\rm head} the density of its first m_0 terms. The tail disintegration in [21, Proposition 2.3] states that

The head density is bounded by Fourier inversion once m_0 is large enough, and q(D_g(b))>0. Hence the ratio in (5.33) is bounded by a finite g-dependent constant. For all sufficiently small s, the block lies outside the head and

It follows that

Using (5.1), for almost every represented environment we obtain

because 8/5>77/50. These probabilities are bounded by one, so the convergence also holds after \widehat{\E}.

On the complement of the two exceptional events in (5.32) and (5.34),

This tends to zero and proves (5.18).□

Proof. The closed form of H is \cE_g(F,G)=\Pi[\nabla F\cdot\nabla G]. Its domain is the closure of the smooth compact cylinder functions, contains the constants and all coordinates, and satisfies \nabla x_i=e_i [21, Proposition 5.1]. Smooth contractions on the core extend by closure, so this is a Dirichlet form. The same approximation, with smooth clipping to preserve a common bound, gives the chain and bounded product rules. In particular the gradient vanishes almost everywhere on a set on which the function is constant: apply the chain rule to smooth truncations in successively smaller neighborhoods of the constant, and use closedness. These rules show strong locality.

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

The diffusion theorem and locality criterion for quasi-regular symmetric Dirichlet forms now give a continuous associated process [18, Theorems IV.3.5 and V.1.11]. Its Fukushima decomposition is supplied by [18, Theorem VI.2.5]. In particular, the coordinate martingales M^i are continuous and have brackets

The bracket identity is the energy-measure identity for the form, given by [13, Theorem 5.2.3 and Equation (5.2.34)] after the quasi-regular regularization of [18, Chapter VI]. To apply its bounded formula, first take a bounded smooth clip u_N=c_N(x_i). For every bounded form-domain test f, the chain and product rules give

Choose the clips with |c_N'|\le1 and c_N(x_i)\to x_i in form norm. Passing to the limit in the energy measures and polarizing gives the displayed coordinate brackets. The multidimensional Lévy characterization, for every finite set of indices, gives M^i=\sqrt2W_i with the asserted countable family of Brownian motions. Brownian motion here is relative to the filtration of the process, so its future increments are independent of the initial sigma field. The usual form exceptional set has \Pi-measure zero; it can be discarded for w\Pi. These facts also give continuous coordinate paths.

We verify the generator used for \theta_i. The finite-coordinate density formula for \Pi, on any finite set J, is

Here q is the density of the full centered Gaussian square sum, and q_{J^c} is the density after the coordinates in J are deleted [21, Proposition 2.3]. Both densities exist, q(D)>0, and q_{J^c} has bounded derivatives of any prescribed finite order. For the last assertion, leave sufficiently many fixed indices outside J; the product of their characteristic-function moduli is bounded by \prod (1+4t^2r_j^{-2})^{-1/4}, which is integrable even after multiplication by the prescribed power of |t|. Fourier inversion then gives the assertion. In particular (6.7) is bounded by a constant times the product Gaussian density.

The set R=0 is polar. Indeed a smooth cutoff equal to one on \{R\le\epsilon\}, zero on \{R\ge2\epsilon\}, and with derivative O(\epsilon^{-1}), has squared form norm O(\epsilon^m+\epsilon^{m-2}), by (6.7). Letting \epsilon\downarrow0 proves zero capacity. A diffusion starting outside its exceptional set never hits a polar set, which proves the stated pathwise positivity of R.

For a cylinder test whose index set J contains \{1,\ldots,m,i\}, integrate by parts in (6.7). The derivative of the factor q_{J^c} is radial in the J coordinates. Its pairing with the gradient of \theta_i vanishes, because \sum_{j\in J}x_j\partial_j\theta_i=0. Consequently the nonpositive generator on this function is

This integration by parts is legitimate across R=0. More explicitly, cut off at R=\epsilon and at large finite-coordinate radius. The function and its first two derivatives near zero are bounded by C_i(1+|x_i|)(1+R^{-3}). The products with derivatives of the scale-\epsilon cutoff satisfy the same bound on R\asymp\epsilon. Their squared integrals near zero vanish for m=12>6. Terms with a derivative on q_{J^c} are bounded by the same Gaussian integrals with polynomial factors, since its first derivative is bounded. Gaussian tails handle the outer cutoff. Thus the cutoffs converge in the form norm, proving \theta_i belongs to the form domain, and the weak generator pairing passes to the limit. The displayed \cL\theta_i is in L^2(\Pi) by the same bounds. Extending the pairing from the core shows \theta_i\in\operatorname{Dom}(H) and H\theta_i=-\cL\theta_i.

A direct differentiation gives, with the head vector extended by zeros,

The Fukushima drift is the time integral of this L^2 generator. Its martingale has brackets with the coordinate martingales equal to 2\int_0^t\partial_j\theta_i(X(v))\,dv, and self-bracket 2\int_0^t|\nabla\theta_i(X(v))|^2\,dv. It therefore equals \sqrt2\sum_j\int_0^t\partial_j\theta_i(X(v))\,dW_j(v): the difference has zero bracket. Only the head and index i occur in this finite sum. Integrability follows, for instance, from

because \|\widetilde S_vw\|_\infty\le\|w\|_\infty. This proves (6.6). Countability permits a common null set for all indices.□

Proof. The local integrability follows directly from

For L_N=\sum_{i\le N}r_i^{-1}E_i, put

The rational function F_N is strictly increasing between its negative poles, since F_N'(z)=\sum_{i\le N}2(z+2r_i)^{-2}>0. Its zeros interlace those poles and are strictly negative, using D>0. Its reciprocal is therefore a positive constant plus a sum of positive multiples of (z+a)^{-1}, a>0: the residue at a zero is 1/F_N'(-a)>0. Repeated rates can be grouped before this argument.

Write these sums as Stieltjes transforms of positive measures. Their measures weighted by (1+a)^{-1}, with the constant as mass at the point at infinity, have total mass F_N(1)^{-1}\le D^{-1}. Weak compactness on [0,\infty] and monotone convergence of F_N(z) give

By (6.1), splitting the sum at r_i=z gives

For sufficiently large z, the part with r_i\le z is comparable to \sum_{r_i\le z}r_i^{-1}\asymp\sqrt z, and the other part to z\sum_{r_i>z}r_i^{-2}\asymp\sqrt z. Positivity and continuity adjust the constants on the remaining compact interval z\ge1. Hence the transform tends to zero at infinity and c=0. It is bounded as z\downarrow0, so \nu has no atom at zero. The function

has the required Laplace transform by Tonelli’s theorem. The weighted integrability of \nu implies smoothness for t>0, and the representation makes K positive and decreasing.

The upper bound in (6.14) follows from

For the lower bound choose a fixed small \eta>0. When t\le\eta, (6.21) bounds \widehat K(\eta/t) below by c_g\sqrt{t/\eta}, whereas the upper bound just proved bounds its integral over [0,t] by C_g\sqrt t. Choose \eta so that the first constant is larger than twice the second. The integral over [t,\infty) is at most \eta^{-1}t e^{-\eta}K(t), proving the lower bound for t\le\eta. Positivity and monotonicity adjust the constant on [\eta,1]. Finally

For r_it\le1, integration of K(v)\le C_gv^{-1/2} gives the first bound in (6.15). For r_it>1, write K_i(t)=\int_0^t K(t-v)e^{-2r_iv}\,dv and split at t/2. The first half is bounded by C_g/(r_i\sqrt t); the second by C_g\sqrt t\,e^{-r_it}, which has the same bound. For the derivative use K_i'=K-2r_iK_i. When r_it\le1 the preceding bounds suffice. When r_it>1, write

On v\le t/2 the derivative bound on K makes the integrand difference at most C_g v t^{-3/2}; on v>t/2 the exponential factor pays the integral of K. This gives |K_i'(t)|\le C_g/(r_it^{3/2}). It also proves (6.16), uniformly away from zero.

The remaining estimates follow by summing (6.15). For example (6.1) gives

The restriction on \xi is exactly integrability at infinity after the change of variable v=r_it: the comparison integral there is \int_0^\infty v^{1/2+2\xi}(1+v)^{-2}\,dv. This proves (6.18). If h<t, integrate the bound on K_i' between t and t+h, and then use the same sum with \xi=0; this gives (6.19) and (6.17). If h\ge t, bound the two values separately. Finally the locally integrable function

has Laplace transform 1/z, and hence equals 1 for almost every t>0. Its causal distributional derivative gives (6.20).□

Proof. The bracket estimate and the Burkholder–Davis–Gundy inequality give

For the last inequality split at t/2; on the second half use v^{-1/2}\le C t^{-1/2}, and on the first use e^{-2r_i(t-v)}\le e^{-r_it}.

Write p_i=y_i^2/(2r_i). The bracket of the first series, at a fixed time t, is bounded pathwise by

by (6.17) and \sum_i p_i=1. This proves its bound in (6.24). In its increment, the old portion of the bracket is at most

and the new portion is bounded by C\int_t^{t+h}v^{-1/2}\,dv\le C\sqrt h. For the displayed integral, if t\le2h discard the minimum; if t>2h, split at t-h and at t/2. These bounds prove the first increment estimate.

For the second series, the martingale moment inequality followed by Minkowski’s inequality bounds its L^j norm squared by

We used (6.23), the deterministic bound on q, and (6.18) with \xi=0. For the increment use (6.19) on the old portion: its squared norm is bounded by

The new portion has the same bound by integrating (t+h-v)^{-1/2} from t to t+h.

All these estimates hold for partial sums. At each fixed time their tails tend to zero in L^j: for the first series use \sum_{i>N}p_i r_iK_i(t-v)^2\to0 and its constant bound inside the bracket; for the second use the summable bound just displayed. Dominated convergence applies in both cases. Taking j>4, (6.25) gives uniformly small moduli of continuity, by a dyadic chaining estimate (equivalently the Kolmogorov continuity criterion). Fixed-time convergence on a finite grid then proves convergence in probability in C([0,1]). The same chaining has a convergent series \sum_{k\ge0}2^{k/j}2^{-k/4}, and bounds the L^j norm of the supremum for j>4. Taking a higher order gives the asserted bound for every finite order.□

Proof. We now justify the infinite passage in (6.12). The constraint says R_N(x)=\sum_{i>N}(x_i^2-r_i^{-1}) \Pi-almost surely. The conditioned edge law describes the tail, after any sufficiently large fixed head, by a bounded density relative to its product Gaussian law [21, Proposition 2.3]. Consequently, for N beyond that head,

Indeed the Gaussian tail is centered and has variance 2\sum_{i>N}r_i^{-2}.

The density of X(t) is bounded by \|w\|_\infty for all elapsed t\ge0. The constraint therefore holds for almost every elapsed time almost surely, and (6.27) gives

The continuous process f is bounded on [0,T] almost surely. First restrict to a bound for that supremum, and then let the bound increase. It follows that fR_N(X)\to0 in probability in L^1(0,T). Also

Convolving (6.12) with K, its left side therefore tends to f in local distributions by (6.20). One can see this directly by writing the difference as the derivative of K*(L_N-L)*f; its undifferentiated factor tends to zero in L^1_{\mathrm{loc}} by (6.28). The convolved residual also tends to zero in probability in L^1_{\mathrm{loc}}, since K is locally integrable.

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

For a finite stochastic sum, stochastic Fubini gives

It can first be used with QY_i stopped at a finite bound. For such a bound a sufficient stochastic Fubini integral is at most

Removing that bound by localization gives the displayed identity as local distributions.

To specify the limit without a formal exchange of infinite stochastic sums, for C_0>0 set

By Lemma 6.4, the partial sums of \mathsf G_{q_{C_0}}^y+\mathsf N_{q_{C_0}} converge in probability in C([0,1]) on the whole probability space. Before \zeta_{C_0}, (6.10) gives Y_i(v)=e^{-r_iv}y_i+Z_i^{q_{C_0}}(v), so these clipped partial sums agree there with the convolved finite stochastic sums.

Fix a deterministic C_0>0 and a rational a\in(0,1]. On the whole probability space, the clipped partial sums converge in probability in C([0,a]), and the convolved residual K*(fR_N(X)) converges in probability to zero in L^1(0,a) as shown above. We may therefore choose a deterministic subsequence N_j\uparrow\infty along which both convergences hold almost surely. Only then restrict to the event \{a<\zeta_{C_0}\}. On this event, passage in (6.12) proves (6.26) as a distribution on (0,a). Take C_0 through the positive integers and a through the rationals in (0,1], and intersect the corresponding whole-space probability-one convergence events. The continuous versions then give equality for 0<t<\min\{\zeta_{C_0},1\}, and by continuity at a positive hitting time \zeta_{C_0}\le1 and at t=1 when \zeta_{C_0}>1. The positive continuous Q is finite at zero, and \sup_{v\le1}v^{1/4}Q(v)<\infty almost surely. A sufficiently large integer C_0 therefore covers the whole interval. This proves the claimed identity and its precise interpretation.□

Proof. By (6.16), 2r_iK_i\to K uniformly on a fixed compact positive interval, and these functions are uniformly bounded there. In the convex combination A_y=\sum_i p_i\,2r_iK_i, the contribution of a fixed finite set tends to zero in probability and the contribution of its complement is uniformly close to K. This proves (6.29).

For the stochastic assertion, its squared norm is at most

For every v<t, (6.15) implies r_iK_i(t-v)^2\to0, whereas (6.17) bounds it uniformly. Escape of p_i makes the expression in brackets tend to zero in probability, and its uniform bound gives convergence of its expectation. Dominated convergence against v^{-1/2} proves the assertion.□

Proof. Choose a fixed smooth increasing function f that is zero on (-\infty,1] and one on [2,\infty). For each nonempty A_j(d) use the factor f(-\log\mu^{\otimes j}(A_j(d))/a_j(d)), and omit empty events. Multiply these factors and a smooth function of n^{-6}\sum_e\theta_e^2 that is one below 1 and zero above 2. This gives (7.2) and a plateau equal to one when each probability there is at most e^{-2a_j(d)} and the size statistic is at most one.

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

We prove the Gaussian assertion with the diagonal and eigenvector sign randomization fixed as in Section 3. On spectral events of limiting probability as close to one as desired, Lemma 3.4 gives, for the uncapped zero-field spherical law and a sufficiently small fixed \rho_0>0,

where one can take any permitted reference scale with np^3 above the tight threshold. The smallest pair threshold r_2(d) in (7.1) is \gg n^{-1/3}, so such a p is available below all the nonvacuous shrinking pair thresholds. In a band z<|\rho_2|\le2z, truncated at \rho_0, we apply the weighted two-replica overlap-cell consequence of Lemma 3.6. On the retained event \|W\|\le3, the logarithm of the positive zero-field angular weight has overlap derivative O(n), uniformly over the orthonormal two-frame. It therefore oscillates by at most a fixed constant on each cell of width 2/n. Summing the weighted cell inequality over the band enlarges its spherical endpoints by at most 1/n and costs at most \exp(Cnz^4+C\log n) after increasing C. For the pair thresholds w at issue, z\ge w\gg n^{-1/3}, so the enlargement is absorbed by decreasing the tail constant c_0. Shrink \rho_0 so that this cost is bounded by \exp(c_0nz^3/2+C\log n). Summing the dyadic bands from w to \rho_0 therefore bounds the Haar mean of the restricted cube partition function, divided by the squared spherical partition function, by \exp(-c_1nw^3) for all the thresholds in question. Here nw^3\ge c n^{6\beta} absorbs O(\log n). The likelihood ratio L_\circ=Z_{\rm cube}/Z_{\rm sp} is at least 1/2 with probability tending to one on these restrictions, by Equation (3.5). Markov’s inequality consequently proves the pair plateau, because

For overlaps bounded away from zero, the fixed-overlap estimate in [21, Lemma 8.2] gives an e^{-c n} tail with Gaussian probability tending to one. This pays 2a_2(d)=2n^{1-3\beta}r_2(d)^3=o(n) for any such nonvacuous threshold.

Here is the corresponding four-replica calculation. If r_4(d)<1, then r_2(d)\le n^{-14\beta/3}=o(1) and r_2(d)^2/r_4(d)\le n^{-6\beta}. For one row of four independent uniform signs, let Y\in\{-1,1\}^6 be its six pair products and let H be its four-product. For n independent rows write \bar Y=n^{-1}\sum_iY_i and \bar H=n^{-1}\sum_iH_i. Put \Lambda(v)=\log\E e^{v\cdot Y}, and let \Prob_v be the row probability with density e^{v\cdot Y-\Lambda(v)}, with expectation \E_v. For several rows the same notation denotes its product probability. Around the origin,

Indeed the six coordinates of Y are centered and have identity covariance, while \E H=\E(HY)=0; Taylor’s theorem on this finite space gives both uniform remainders. If the empirical pair vector is the lattice value u with |u|\le\sqrt6 r_2(d), put E_d=\{|\bar H|>r_4(d)\}. The change of measure gives

Under \Prob_u the rows remain independent. Since |\E_uH|\le r_4(d)/2 for large n, Hoeffding’s inequality [15] bounds the last probability by 2e^{-nr_4(d)^2/8}. Together with the expansion of \Lambda, this gives

For completeness, the reverse Gram cost is also explicit. Put G_{aa}=1 and G_{ab}=u_{ab}. The Gram density of four independent uniform points of the sphere of radius \sqrt n is a polynomial normalization times \det(G)^{(n-5)/2} on positive correlation matrices. For |u|=o(1), \log\det G=-|u|^2+O(|u|^3). A cell of width c/n about G thus has probability at least

At a fixed spectrum with bounded norm, let F\in\R^{n\times4} be a uniform orthonormal frame and put

The exponent has Lipschitz constant O(n) in G, so the integrand changes by at most a factor e^{O(1)} across the cell. The integral of \mathcal A_\lambda against the spherical Gram probability equals one. Restricting it to the cell therefore bounds the Haar mean at Gram G by

Multiply this by (7.3) and sum the at most (n+1)^6 possible Grams. The remaining error Cnr_2(d)^3+C\log n is negligible relative to nr_4(d)^2, since

On L_\circ\ge1/2, division by the cube partition function changes the four-replica numerator by at most 16. Markov’s inequality proves its plateau e^{-2a_4(d)}, since nr_4(d)^2/a_4(d)=n^{12\beta}. A union over O(\log n) scales is harmless. Finally, the Gaussian mean of n^{-6}\sum_e\theta_e^2 is O(n^{-5}). The size plateau follows from Markov’s inequality. Enlarging the spectral restrictions completes the proof that \chi_n=1 with probability tending to one.□

Proof. Let \psi be smooth and nondecreasing, zero on (-\infty,2], with 0\le\psi'\le1 and \psi'=1 on [3,\infty). Put

For l\le3N the first inequality in (7.5) is immediate. For z=l/N\ge3, \psi(z)\ge z-3, so l-a_N(z)\le3N; the same inequality follows from W_N\le e^{-a_N}. If l\le N/2, then a_N(l/N)\le-(\log n)^2/2, proving the plateau.

All l-derivatives of a_N(l/N) of positive order at most K are bounded: the first is (\log n)^2/N+3\psi'(l/N) and the others are 3N^{1-k}\psi^{(k)}(l/N). Derivatives of the logistic function have a factor W_N(1-W_N) times a bounded polynomial in W_N, which proves the first part of (7.6). If W_N\ge n^{-L}, then a_N\le L\log n+O(1)<(\log n)^2; hence l/N<2 for large n. In this region a_N is affine, and its slope in l is (\log n)^2/N. The same derivative formula proves (7.7). It also proves the second part of (7.6) for l/N\le2. In the remaining region W_N\le e^{-(\log n)^2}, so the first global bound with its logistic factor gives the second bound there as well, since N is polynomial in n. Finally the two logarithms are bounded by C(1+|a_N(l/N)|). Since \psi is Lipschitz and (\log n)^2/N\le1 for large n, this proves (7.8).□

Proof. Positivity and domination by p_d follow by induction and the semigroup property of P_d. The cap is (7.5). At a split the two children have equal lengths and use the same rule, so their kernels are identical. The convolution of this symmetric kernel with itself is symmetric; multiplication by the endpoint function W_N(\log\bar k_I) preserves symmetry. This starts from detailed balance of p_d.

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

Proof. The log-moment assertion of Theorem 4.1 says that, for each fixed integer p\ge1 and sufficiently small fixed \xi>0, on disorder restrictions of arbitrarily high limiting probability,

Choose \xi so that \xi/(2/3-\xi)<\beta/2. Division by N(d) shows that the right side is at most C_{p,\xi}n^{-\beta/2}N(d) uniformly in this range: the largest possible extra factor is T^{1/(2/3-\xi)-3/2}=n^{\xi/(2/3-\xi)}. Markov’s inequality therefore gives

By (7.10), \bar k_I\le p_{|I|}. At each check, its failure probability under an ordinary path is thus bounded by the last display plus e^{-(\log n)^2/2}. The probability of a first failure is at most the sum of these probabilities, because all preceding weights are at most one. There are O(T) nodes. Choose p large after A and use Lemma 7.1 for the plateau. All estimates were uniform in the starting state and in d, which proves the asserted uniformity.□

Proof. The tree is laminar: every check whose interior intersects the interior of I is a descendant or an ancestor of I. Disjoint siblings may share an endpoint, but their factors are fixed by the exterior and boundary data. The ancestor factors, the top factor, and the external test are fixed by the same data. The remaining factors are exactly V_I. The Markov property gives (7.11) as the conditional law, and then gives (7.12) and the conditional independence. This proves the resampling assertion as well. It applies to tree nodes; an arbitrary subinterval can have crossing checks.

At a fixed start x,

Integrate the node first in the tested history and discard all other weights. The bound on \Phi and the lower bound on its mass show \mathsf Q_\Phi\{\bar k_I(X_a,X_b)<\delta\}\le n^{M_++M_-}\delta. This proves the endpoint assertion by choosing \delta=n^{-L}.

Let \mathcal F_t^\to contain just the marked prefix of the node from a to t. For t-a\le2d/3, its conditional density is exactly

The interval remaining after t contains a proper node of length at least c_1d. Keep its checks and discard all others in the numerator. By Lemma 7.3, that numerator is at most \mu(y)e^{C'N(d)}. This proves (7.13). Detailed balance for marked paths includes self-resampling rings, and the proper-descendant weights are invariant under reversal because equal-length children use identical symmetric kernels. Reversal therefore sends \mathsf C_I^{x,y} to \mathsf C_I^{y,x} on the reflected node, proving the last assertion. No reversal of the entire stopped forest is needed.□

Proof. First use deterministic times with m larger than a fixed constant. Inside each gap choose a contained proper checked node of length comparable to m. They are disjoint. Dropping all other weights and applying Lemma 7.3 shows that the joint law of the observed states, given any initial state, is bounded by their product \mu law times \exp(C_2N(m)). This estimate counts the retained weights without conditioning on survival. Apply (7.2) at a dyadic scale comparable to m, first to all relevant pairs, and then to the quartet on the event that its pairs obey the same threshold. The ratios a_2(m)/N(m)=n^{2\beta} and a_4(m)/N(m)=n^{7\beta} tend to infinity. Their exponents are at least a positive power of n, so they pay the cap and any polynomial test density with any prescribed inverse-polynomial precision. A vacuous threshold or bounded m gives the bound directly from |\rho_j|\le1.

Use a time mesh of sufficiently small inverse-polynomial spacing. The probability that any mesh cell has two rings is arbitrarily small after paying the polynomial test density: under ordinary paths the total clock rate is n, and the union bound is O(n^2T) times the spacing. A single ring changes either overlap by at most 2/n per affected time. This is smaller than the displayed thresholds, whose smallest quartet value has order n^{-1/2+10\beta}. A union over the polynomially many grid tuples proves the simultaneous assertion. A resampled interior has the same tested marginal by Lemma 7.6; a union proves the last claim for any fixed number of copies.□

Proof. The size support gives \sum_e|\theta_e|\le Cn^4, hence \min_x\mu(x)\ge e^{-Cn^4}. By Cauchy–Schwarz, each absolute one-site field is at most Cn^{7/2}. On a leaf, prescribe one ring of each site in successive subintervals of length 1/n, with its sign set to the desired endpoint, and no other rings. The clock event has probability e^{-n|I|}n^{-n}, and each prescribed mark has probability at least \tfrac12e^{-Cn^{7/2}}. Thus every leaf transition probability is at least e^{-Cn^5}, uniformly in its endpoints. Its logarithmic density has absolute value at most Cn^5. The last assertion of Lemma 7.2 and the kernel recursion propagate this property up the tree: composing two positive kernels costs a fixed multiple of the previous logarithmic bound plus a polynomial, and applying W_N has the same property. The depth is O(\log n), so all these absolute logarithms remain polynomial, with a fixed exponent. In particular \mu(y)\bar k_I(x,y)\ge e^{-n^{C_0}} for a fixed C_0.

The total number of ordinary rings on a node is Poisson with parameter n|I|\le CnT. Its tail above any sufficiently large polynomial is smaller than e^{-2n^{C_0}} times any inverse power of n. Dividing by the last lower bound and using V_I\le1 proves polynomial conditional count moments for every boundary. Each fixed Taylor coefficient of the ring product is a polynomial of fixed degree in its number of rings, since each single-ring coefficient in (7.16) is bounded. This proves the claimed raw moments, both conditionally and under the polynomially tilted tested laws. The copy statement follows from their preserved marginals.

At a leaf, (7.17) has an empty product; the preceding count bound and the bounded coefficients of (1+u\mu q_e)/(1+u q_e(y)) give the coarse bound for its pre-density. Suppose normalized coefficients of order l\le K at one tree level are bounded by D^l. For a composition, division by its base value writes its ratio as a conditional average of the product of the two child ratios and \mu_u(z)/\mu(z). Its order-l coefficient is therefore at most (C_K(D+1))^l. Taylor’s chain rule for W_N(\log\bar k_I^u) uses the global ratios (7.6); coefficients of the logarithm are polynomials in nonconstant pre-ratio coefficients, each product having total order l. The same bound applies to either survival or failure ratios and to the post-density. Starting with a polynomial D at the leaves and iterating D\mapsto C_K(D+1) for O(\log n) levels proves these coefficient bounds. Formula (7.17) itself follows by inserting the path likelihood and then dividing by \mu_u(y)/\mu(y) in the definition of the pre-density.

To check the last forest assertion, fix its designated external endpoints. The interiors of its maximal roots are the corresponding conditional laws (7.11), and their top survival or failure factors are fixed. Expand a coefficient of total order at most K by the product rule. A differentiated node contributes one of the normalized ratios just bounded; the raw ring factors have the conditional polynomial moments proved above. There are polynomially many nodes and at most K positive orders, so the number and magnitude of the resulting terms are bounded by n^{C_K}. Stationary initialization, if present, contributes only the bounded coefficients in (7.15). Integrating this conditional bound against the nonnegative endpoint test and its base endpoint measure proves the asserted factor of the base mass.□

Proof. Include a possible initial density z_0=d\mathsf Q|_{\mathcal F_0}/ d\mathsf P|_{\mathcal F_0}. With bounded integrated rates, the marked point-process likelihood formula [16, Theorem 5.1] gives

The product allows zero factors; a mark with b=0 has no mass under \mathsf Q. The product-exponential following \sqrt{z_0} is the stochastic exponential for intensities r\sqrt b relative to r. Its expectation conditional on the initial data is at most one, as follows by integrating the successive conditional waiting and mark densities, also when a mark factor is zero; this is the nonnegative likelihood supermartingale of [16, Proposition 4.3]. Cauchy–Schwarz gives \mathsf E_{\mathsf P}\sqrt{z_0}\le1. Thus the right side has \mathsf P expectation at most one. For general ratios, stop only when the integrated reference, changed, and square-root rates reach a finite level. Let Y=\sqrt Z e^{\mathcal J/2} on \{Z>0\}, with extended values, and set Y=0 on \{Z=0\}. The localized likelihood expressions converge to Y on \{Z>0\}; assigning zero on \{Z=0\} can only decrease their lower limit. Fatou’s lemma gives \mathsf E_{\mathsf P}Y\le1. No positivity restriction is imposed on b in this localization. Since \mathsf Q\{Z=0\}=0, this expectation bound also makes \mathcal J<\infty \mathsf Q-almost surely. On the event in (8.3), \sqrt Z e^{-\mathcal J/2}\le e^{(B-A)/2}. Multiplying this by Y and integrating under \mathsf P proves the inequality.

For a tested forward law, \log Z\le(M_++M_-)\log n by Definition 7.5. For a conditional portion, (7.13) gives \log Z\le CN(|I|)+L\log n outside an arbitrarily small boundary event. Since N(|I|)\ge c n^\beta, applying (8.3) with these bounds proves (8.4) after increasing C.

Let C_{i,H} be the number of rings at site i in a unit block H, and \Lambda_{i,H}=\int_H\sum_\zeta r_{i,\zeta}b_{i,\zeta}\,dt. The exponential count supermartingale [16, Proposition 4.3] gives

with the conditional version when the boundary is fixed. Ordinary unit counts have every fixed moment bounded; paying a polynomial test density and using the preserved copy marginals shows C_{i,H}\le n^{\eta/2} with arbitrarily high inverse-polynomial probability, simultaneously on any polynomial collection of blocks. The last exponential bound then gives \Lambda_{i,H}\le n^\eta with the same precision. Its tail inequality also bounds every moment of \Lambda_{i,H} by a constant times a higher moment of 1+C_{i,H}. The conditional assertions here are first averaged over the boundary law. Finally, (\sqrt b-1)^2\le b+1 bounds \mathcal J by the sum of the new and ordinary integrated intensities. The moment assertions follow, using the polynomial count moments from Lemma 7.8.□

Proof. The order-l prefix coefficient is a martingale starting from zero. At a ring its jump is \sum_{h=1}^l\gamma_{h,e}\ell_{l-h,e}([\inf I,t)); the prefix factors are predictable and every \gamma_h is centered. The martingale square-function inequality [5, Theorem 1.1], applied at successive ring times, and Hölder’s inequality bound its L^p supremum by a constant times

where C_{e,I} is the number of rings at the two sites. It is Poisson with mean 2d, whose fixed moments are O(d^p) for d\ge1. Induction in l, with the same assertion at the higher moment order 2p, proves (8.6).

Detailed balance for marked paths states that the forward path measure multiplied by its initial stationary mass equals the reversed measure multiplied by its final stationary mass. Apply this once at the base matrix and once using (7.15), and divide. This gives (8.7), including at self-resampling rings. The Taylor coefficients of the two endpoint factors in that identity are bounded through order K. Thus the reversed bounds follow from the forward ones. Choose p sufficiently large in (8.6) to pay the polynomial test density, the number of entries and intervals, and the desired inverse-polynomial precision. This proves (8.8); resampling preserves the marginal. On a unit block a fixed-order coefficient is a fixed-degree polynomial in the two-site count, which gives the last assertion in the same way.□

Proof. For each site write \alpha_i for the signed measure in the first integral with X_j omitted. Split it according to b\le2 and b>2. On the first set |b-1|\le C|\sqrt b-1|, so Cauchy–Schwarz and \sum_\zeta r_{i,\zeta}=1 bound the square integral of its absolute density by C\mathcal J_{H,i}. On the second set |b-1|\le C(\sqrt b-1)^2, so its total variation is at most C\mathcal J_{H,i}. On each unit block this variation is also bounded by a constant times the new and ordinary integrated site intensities, hence by n^\eta on the event of Lemma 8.1. If a_{ik} is the sum of the two variations on the kth unit block, these facts imply

In particular the large ratios have been treated by total mass, not by a pointwise square-integrability assertion.

First suppose all times in H have age at least a fixed multiple of d from the forward start of its checked root. Squaring the contribution of a fixed site i and averaging its other endpoint gives the kernel n^{-1}\sum_jX_j(t)X_j(s)=\rho_2(X(t),X(s)). By (7.14), on unit block indices this is bounded by n^{O(\beta)}(1+|k-k'|)^{-1/2}. Adjacent blocks use the bound one. The \ell^1 sum of this kernel over one index is O(d^{1/2}), so

Omitting j=i adds at most n^{-1}(\sum_k a_{ik})^2, which is absorbed by the right side since d\le CT. Summing over the contributing site and dividing by \binom n2 gives, using (8.11),

For the fourth power use the kernel n^{-1}\sum_j\prod_{h=1}^4X_j(t_h)=\rho_4(X(t_1),\ldots,X(t_4)). The age assumption and (7.14) bound its unit-block kernel by a constant times

For each chosen pair the convolution is at most Cd^{1/4}\sum_k a_{ik}^2; the other two \ell^1 sums contribute at most d\sum_k a_{ik}^2. Thus the fourth-power integral is at most n^{O(\beta)}d^{5/4}(\sum_k a_{ik}^2)^2. The omitted-diagonal term is n^{-1}(\sum a_{ik})^4 and is again absorbed, since d^{3/4}/n\le Cn^{-1/2}. Now sum over i and divide by \binom n2. Since \sum_i\mathcal J_{H,i}^2\le\mathcal J_H^2, this gives

The fourth root is the second bound.

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

Proof. For l=1, (8.9) is the representation, and Lemma 8.3 gives both bounds. For l\ge2, expand a raw coefficient by the smallest dyadic node containing all its positive-order ring insertions. If all insertions lie in a leaf, keep that leaf term. Otherwise, at that unique node J the term is \ell_{j,e}(J_{\mathrm{early}})\ell_{l-j,e}(J_{\mathrm{late}}) for some 1\le j<l. This is an exact partition of the ring expansion, including rings carrying Taylor order larger than one.

On a leaf, expand according to the last ring and its assigned order as in (8.9). The remaining prefix coefficients are predictable and are bounded by a power allowance on this unit block, by Lemma 8.2. The unit-block conclusion of Lemma 8.3 applies. At a split of length y, the earlier coefficient is known before the later child and is at most n^\eta y^{j/2} by (8.8). Apply the inductive representation to the later child, then include this earlier factor by its supremum norm. At any one scale the later children are disjoint and number at most Cd/y. Consequently

For the L^2 or L^4 bound the power before this summation is y^{(l-1)/2+\gamma}, with \gamma=1/4 or 5/16. Relative to the claimed right side, this scale thus has the factor

It is geometrically summable for l\ge2. The leaf terms cost at most C\sqrt{d\mathcal J_H} with the respective factor n^{-1/2} or n^{-1/4}, and are also bounded by the claimed expression. All compensations used the expectation identity with the earlier coefficient still predictable. After absolute values, each representation vector is bounded by a fixed polynomial in raw clock counts and prefix coefficients, and in \int_H\sum_{i,\zeta}r_{i,\zeta}|b_{i,\zeta}-1| \le\int_H\sum_{i,\zeta}r_{i,\zeta}b_{i,\zeta}+n|H|. For the sharper representations below, an earlier endpoint projection adds a conditional coefficient with the uniform coarse bound of Lemma 7.8. That lemma, the intensity moments in Lemma 8.1, and Hölder’s inequality therefore give polynomial fixed moments for the representation norms, including a fixed number of copies. The conditional exceptional-set rule then applies to each of the polynomially many terms and proves the stated probability formulation.□

Proof. All conditional estimates in this proof are first established under the tested boundary law followed by its interior sampling. We then use the conditional exceptional-set rule following Lemma 7.8. The polynomial coefficient and representation moments proved in Lemmas 7.8 and 8.4 permit any required stronger working precision. This yields the stated high-probability bounds on conditional expectations, without asserting them for every possible boundary.

Order one and its stronger L^4 bound. The infinity bound follows at once by applying (8.8) inside (8.12) and expanding the bounded endpoint denominator. For R_1, split instead at a variable v in the middle third of I. The pathwise reversal identity gives

Average this identity uniformly over that middle third. Compensating the two ring terms in the parent forward and reversed portions uses bounded deterministic time weights, common for the other-site index. By (8.4) and (8.10) their L^2 cost is at most

For the boundary average Q_e=(3/D)\int_{a+D/3}^{a+2D/3}q_e(X_v)\,dv, the exact edge sum \sum_{i<j}q_{ij}(X_v)q_{ij}(X_w) =\tfrac12(n^2\rho_2(X_v,X_w)^2-n) gives

The last step uses the square of (7.14), whose kernel is (1+|v-w|)^{-1} on this middle third, and absorbs \log(2+D). This proves the L^2 bound for R_1. Interpolation with its infinity bound gives the ordinary L^4 estimate.

We will retain a stronger intermediate L^4 improvement. Its ring terms can be bounded directly by the second estimate in (8.10), giving

If D\ge Tn^{-0.03} this is at most n^{-0.02+O(\beta)}. The boundary term has infinity norm at most one and L^2 norm \lesssim_\beta D^{-1/2}, so its L^4 norm is even smaller. Hence

For bounded D, where the conditional prefix estimate was not invoked, Lemma 8.2 gives all the ordinary bounds after a constant change.

Joint induction for the basic bounds and the sharper raw L^2 representation. We next prove the L^2 and L^4 bounds by induction on l, together with (8.21). Assume (8.19) in orders below l\ge2. To prove the raw improvement when d\ge Tn^{-0.02}, keep the basic representation of Lemma 8.4 on split scales y<dn^{-0.02}. The scale factor in that lemma’s proof is at most (y/d)^{1/4}, so these scales gain n^{-0.005}. Their leaf remainder has a still larger gain.

On a larger split, project the earlier raw coefficient before compensating anything in the later child. In the expectation at this stage, the remaining raw insertions are outside the earlier child. By Lemma 7.6, for any such exterior-measurable integrand H_e the exact identity is

For a prefix law this follows by first conditioning on the full exterior and then taking the prefix marginal. The new factor is known at the start of J_{\mathrm{late}}. Its L^4 norm is at most n^{O(\beta)}y^{j/2-1/2} by the induction hypothesis and (8.13); for j=1 the term q_eR_0 is simply bounded by one. Apply the basic L^4 representation to the later child and pair these factors in L^2. Compared with the basic split L^2 estimate the gain is

Indeed the new power of y is j/2-1/2+(l-j-1)/2+5/16=l/2-11/16, while the old one is l/2-1/4. Here y\ge dn^{-0.02}\ge Tn^{-0.04}, so the last ratio is at most n^{-0.024+o(1)}. Summing these scales and the smaller scales proves (8.21), with ample room for the stated exponent. The identity (8.24) was applied while the later coefficient was still raw. It is not applied to a drift from an earlier compensation, which could retain the earlier interior in its intensities.

Consider now (8.15). For each linear raw term, use (8.16) and apply Lemma 8.4 on the corresponding half in the parent oriented filtration with just x,y fixed. After the compensation identity, use the triangle inequality and the pathwise norm bounds. The action bound for that parent portion is \mathcal J\lesssim_\beta N(D), so its L^2 cost is

For the product terms in (8.15), the induction hypothesis and Hölder give |R_aR_b|_2\le |R_a|_4|R_b|_4\lesssim_\beta D^{(a+b)/2-1}. The terms with q_e have the still smaller exponent (l-1)/2-1. Using the conditional exceptional-set rule proves the L^2 bound at order l. Interpolating it with the infinity bound proves its L^4 bound.

For clarity we track the low-order margins needed below. When D\ge Tn^{-0.019}, a half has length at least Tn^{-0.02} for large n. Use (8.21) on the two raw order-two terms. Its product term uses (8.23) on both children. Thus

where the second bound follows by interpolation. Similarly, if D\ge Tn^{-0.018}, the raw order-three terms use (8.21); their product terms use (8.23) and (8.25) on the children. The terms with q_e have an additional factor D^{-1/2} relative to the target bound. This gives

These deductions use the induction hypothesis at strictly smaller orders, so the proof of (8.19) and of these improvements is complete.

The raw L^1 representation. With the basic conditional bounds established, we prove (8.22) by induction on l. In every split of the smallest-node decomposition, first apply (8.24). In the term R_j of c_j=R_j+q_eR_{j-1}, use its L^2 bound y^{j/2-1} and the basic later-child L^2 representation. The power before the summation at scale y is y^{l/2-5/4}. The term q_eR_{j-1} has an extra y^{-1/2} when j\ge2 and is bounded the same way. Summing the square roots of the actions at this scale contributes \sqrt{d/y}. Thus the sum over scales is bounded by

with a logarithm allowed at an equality of exponents.

When j=1, the remaining term in c_1 is q_e alone. It is known before the later child. Use (8.22) inductively for the later order l-1; order one follows from the L^2 estimate in Lemma 8.3. Its scale sum is at most

which is no larger than the claimed expression. Leaf terms cost C\sqrt{d\mathcal J_H/n} as in Lemma 8.4. This proves the ordinary bound at every order. Throughout, the earlier conditional coefficient is left inside expectation until the later compensation is performed, where it is predictable.

At order four the leading scale contribution, divided by its value at y=d, is (y/d)^{1/4}. If y<dn^{-0.01} this gains n^{-0.0025}. If y\ge dn^{-0.01} and d\ge Tn^{-0.006}, then y\ge Tn^{-0.016}. For an earlier order j=2 or 3, its R_j term has the improved L^2 bound (8.25) or (8.26) on that child. Its q_eR_{j-1} term has the extra y^{-1/2} just noted. For j=1 with its R_1 term, the later order is three, and (8.21) applies there. Constant halves are absorbed by the slack in these thresholds. Finally the leaf and q_e-alone branches cost only O(\sqrt d) in place of d^{3/4}. Each case gains at least n^{-0.001+O(\beta)}. This proves the additional factor in (8.22).

Conditional L^1 bounds. Apply the L^1 representation to each raw term in (8.15), using the order of conditioning in (8.16). Since

the two raw order-three terms are \lesssim_\beta D^{-1/4}, and raw order l\ge4 is \lesssim_\beta D^{(l-4)/2}. For the product terms use the two L^2 estimates in (8.19). At order three they cost at most D^{-1/2} (and the term with q_e at most D^{-1}); at order l\ge4 they cost at most D^{(l-4)/2}, with an extra D^{-1/2} in the terms containing q_e. This proves (8.20).

If D\ge Tn^{-0.005}, a half satisfies the threshold for the improved order-four representation. In its product terms, R_1R_3 uses the improved L^2 bound for R_3, R_2R_2 uses that for R_2, and the terms with q_e already have the extra D^{-1/2}. The claimed factor n^{-0.0007} follows. The stated weaker late factors for R_1,R_2,R_3 follow immediately from (8.23)–(8.26). The conditional exceptional-set rule and a union over the specified nodes and fixed number of copies complete the proof.□

Proof. The additive precision in the lemma is useful for very small tested masses. To prove it, first suppose each z_\alpha\ge n^{-M} for a fixed M. Normalize each integral by z_\alpha and use the tested probability law. The estimates below then have the factor \prod_\alpha z_\alpha. All exceptional contributions may be made smaller than n^{-A'} for an arbitrary fixed A' by the precision convention in Definition 7.5 and Proposition 8.5. If some z_\alpha<n^{-M}, Lemma 7.8 and the polynomial path moments bound its unnormalized coefficients by n^{C(K,B)}z_\alpha. Choosing M after A discards all such terms at cost n^{-A}. We shall use this reduction throughout the proof.

Coefficients with fixed survival and failure weights. We start with the terms in which no w_I is differentiated. Recall from Proposition 8.5 that, for a node of length D,

When D\ge Tn^{-0.005}, the L^4 bound for R_1, the L^2 bounds for R_2,R_3, and the L^1 bound for R_4 have additional factors n^{-0.01},n^{-0.001},n^{-0.0007}, respectively. The notation in (9.4) records the length, with the node and its fixed endpoints understood.

For a last raw coefficient of order l in a forward tested history, the compensation representations of Lemma 8.4 and Proposition 8.5(ii), with \mathcal J_I\lesssim_\beta1 by (8.4), give the following costs on an interval of length d:

Each line allows a factor n^{O(\beta)}. These are estimates for the integrands in the representation after taking the compensator, with earlier measurable factors kept inside the expectation. They do not estimate the expected absolute value of an unprojected score. The representations used for different lines or their improvements may be different; the table does not assert simultaneous bounds on one family of vectors. The L^2 cost gains n^{-0.004} for l\ge2 and d\ge Tn^{-0.02}, and the L^1 cost at order four gains n^{-0.001} for d\ge Tn^{-0.006}.

Stationary initialization contributes

We need a gain when these are the only differentiated factors. If p is a nonnegative stationary initial density with \mu p=1 and \|p\|_\infty\le n^{C_0}, then the exact replica identity is

for every fixed A'. Indeed choose the first dyadic d_\circ\ge cT, so d_\circ\le2cT, r_2(d_\circ)\le c^{-1/2}r_2(T), and a_2(d_\circ)\ge c_1 n^{3\beta}. This scale is among the query scales in Lemma 7.1. Split at |\rho_2|=r_2(d_\circ); the complementary probability is \exp(-c n^{3\beta}), which pays \|p\|_\infty^2. This is the weighted replica identity used in the stationary comparison [21, Section 8]. In particular

The normalized initial marginal of a tested stationary history is such a p: its density is at most \|v_\alpha\|_\infty/z_\alpha. The coefficient of positive order j in the initialization factor is (-\bar q_e)^j+q_e(X_0)(-\bar q_e)^{j-1}. A numerator supplies at most one q_e(X_0) on each history. Thus if all positive orders come from initialization, at least one factor is either \bar q or the integrated vector \mu[pq_e]; (9.6) and boundedness of the other factors give a fixed power saving.

Suppose now a raw clock likelihood has a positive order. Each path is a forest with O(\log n) roots, or consists of the one or two complete legs. Chronologically earlier roots have lengths at least a constant times later ones. Independent paths can be interleaved preserving this property: merge their lists in decreasing order of the greatest remaining root length, which is comparable to the current root length on each list. Project every differentiated root preceding the last such root onto its endpoints. The conditional coefficient is

by (8.13). This projection is within the expectation, using Lemma 7.6; the endpoint factors of preceding roots are measurable before the last root. After expanding their two monomials, all initialization coefficients can therefore be bounded by one in this calculation.

If these preceding roots supply no positive-order R factor, the last root, of order l\le k and length d\le CT, uses the L^1 line of (9.5). Its first term is O(n^{-1/6}). Since T^{3/4}=\sqrt n, its second term is at most CT^{(l-4)/2} when l\ge3; when l\le2 it is at most T^{-3/4}, because d\ge1. For l<4 there is a fixed power saving. Put

For l=4 the bound saves n^{-3\varepsilon_0/4} if d\le Tn^{-\varepsilon_0}, while the late L^1 improvement saves n^{-0.001} on the remaining range. This proves the required bounds for these terms.

Otherwise, let p be the total order in the preceding positive-order R factors and D their greatest length. Use the L^2 bound on one factor of length D, the supremum bound on the other preceding factors, and the L^2 line of (9.5) for the last order l and length d. The result is

For p+l<4 this has a fixed power saving. For p+l>4 it is bounded by n^{O(\beta)}T^{(p+l-4)/2}. To check both assertions when p=1, write the length factor as (d/D)^{1/2}d^{l/2-3/4}T^{-3/4}; when l=1 it is even smaller, and otherwise its power of d is positive.

Only p+l=k=4 is a borderline case. Write D=Tx, d=Ty with y\le Cx. Apart from n^{O(\beta)}, the three possible factors in (9.8) are

If D or d is at most Tn^{-\varepsilon_0}, these bounds save at least n^{-\varepsilon_0/4}. Otherwise every contributing earlier root also has length at least a constant times Tn^{-\varepsilon_0}. If l\ge2, the late L^2 improvement in (9.5) applies. If l=1 and a preceding order is two or three, choose that factor for the L^2 bound and use its n^{-0.001} improvement. Using its length in place of the greatest length costs at most n^{\varepsilon_0/2}, since all those lengths lie between a constant times Tn^{-\varepsilon_0} and CT. Finally, if the preceding orders are (1,1,1), apply L^4 to all four factors. The three R_1 factors have their late improvements, and the last factor is at most n^{O(\beta)}T^{5/16}n^{-1/4}=n^{-1/24+O(\beta)}. These are fixed power savings in all cases. The O(\log n)^K choices of roots are absorbed into n^{O(\beta)}. This proves (9.2) for terms with fixed cutoff weights.

The exact expansion at a differentiated cutoff. For a node I=[a,b], use the pre-density \bar k_I and conditional interior law \mathsf C_I^{x,y} from Lemma 7.6. For the perturbation at e, put

All denominators are positive. Write W_N^+=W_N and W_N^-=1-W_N. For j\ge1, the scalar chain rule gives

Thus a positive-order cutoff coefficient supplies one positive-order derivative ratio of W_N^\sigma and a product of positive-order pre-density coefficients whose orders sum to its assigned order. We call this occurrence of a derivative ratio a cutoff mark at I.

The pre-density ratio has the exact representation

Indeed, the perturbed stationary probability is \mu_u(y)=\mu(y)(1+u q_e(y))/(1+u \bar q_e). Dividing the perturbed pre-truncation transition mass by its base value gives the conditional expectation of L_{I,e}(u)\prod_{J\subsetneq I}V_{J,e}^+(u). The further factor \mu(y)/\mu_u(y) gives (9.10); see also Equation (7.17). It is essential here that \bar k_I is a density with respect to the normalized stationary probability.

Each product of coefficients of H_{I,e} in (9.9) is represented using independent interiors with law \mathsf C_I^{x,y} conditional on the same endpoints. Recursively do the same for every positive-order descendant cutoff coefficient in (9.10). Given all already sampled paths and the endpoints of a new copy, its interior is sampled independently with this law. Ancestor and exterior factors depend only on data outside the copied interior. By Lemma 7.6, replacing that interior in the full tested history preserves its law. This remains true for a copy inside a copy by iteration. In particular the marginal estimates for each copy are those for a base tested path, averaged over its endpoints; no pointwise assertion for arbitrary endpoints is being made.

There is a useful order invariant in this expansion. Assign to each coefficient slot its positive Taylor order. Formula (9.9) splits an order into positive orders with the same sum, and expanding a slot by (9.10) has the same property. The total order in all unexpanded slots, raw likelihood slots, and endpoint or \bar q_e factors is therefore at most k at every stage. Cutoff marks record transferred orders; they do not add to this sum. There may be a long chain of cutoff marks carrying a single order.

A second invariant gives a projected factor when expansion terminates. For a positive-order pre-density slot of order r whose descendants receive no positive cutoff order, keep the endpoint denominator together with the conditional likelihood. Its coefficient is exactly

Each summand contains a positive-order R_l(I), or the order-one factor \bar q_e (when r=1 the second summand is \bar q_e itself). Induction from terminal slots shows that every terminal branch descending from a cutoff mark contributes a positive-order conditional coefficient or an \bar q_e factor. The endpoint denominator in (9.11) is never estimated separately. By (9.7), \bar q_e can be treated as an R_1 factor of length T: it satisfies all the corresponding bounds in (9.4), including the late L^4 improvement.

Summation over cutoff marks and termination of the expansion. Let \lambda_I=1-w_I for a survival cutoff mark and \lambda_I=1 for a failure cutoff mark. For any fixed \eta>0, Lemma 7.2 gives, outside an arbitrarily small inverse-polynomial contribution,

For survival this estimate is used on w_I\ge n^{-L}. The discarded event w_I<n^{-L} is paid from its included base survival factor, with L chosen arbitrarily large. For failure the estimate holds without this restriction. The global derivative-ratio bounds of Lemma 7.2 control exceptional contributions.

Here is a precise way to sum the choices of nodes in (9.12). On a normalized tested history with mass z_\alpha\ge n^{-M} and test bounded by n^B, put H=\sum_{I\in\mathscr S_\alpha}(1-w_I). Since \prod_I w_I\le e^{-H},

One possible failure adds at most one to this hazard sum. The same statement holds for the marginal of every conditional copy, by its law-preserving construction. To make these statements simultaneous, for each deterministic prefix of at most P+K choices of nodes carrying cutoff marks, sample the possible next copies conditional on that prefix. All continuations of a prefix share its already sampled paths. There are n^{O_K(P)} prefixes for fixed P. The arbitrary precision convention permits the hazard and moment bounds to hold for all of them simultaneously. This union uses the joint copy law and the marginal estimates for each indexed path.

All expectation and compensator identities used for this summation are established in their stated filtrations before restriction to the simultaneous event. At a fixed sampled prefix, let \mathcal I\ne\varnothing be the eligible next nodes on one shared path. For each I, let B_I\ge0 bound the absolute value of the resulting continuation integrand or its edge norm on the common sampled tree, including all later factors and gains but excluding the factor C n^\eta\lambda_I/N_I used to bound the current mark ratio. The bound B_I may depend on later sampled data. On this common realization,

On the simultaneous event, the parenthesis is at most n^\eta+1. Combining this with the current C n^\eta from (9.12) costs at most C n^{2\eta} while retaining N_I^{-1} with the continuation for the same choice. Apply this bound separately to eligible slots and paths, paying the fixed combinatorial multiplicity. Iteration gives a supremum over indexed full branches. On the simultaneous event, the score bounds together with the length estimates below control this supremum before integration. The hazards are scalars independent of e, so the same argument applies to edge-norm estimates after projection.

The indicator of the simultaneous event is used only to bound these resulting integrands. The complementary integrals are controlled by the polynomial moments in Lemmas 7.8 and 8.1 and in the proof of Lemma 8.4, choosing their precision after P. Thus no future-dependent hazard event is used as a predictable factor.

It suffices to stop after a fixed number P of cutoff marks. If this threshold is reached, leave the remaining pre-density slots unexpanded. By Lemma 7.8 and the fixed-order path moment bounds, their product, together with all other positive-order slots, is at most n^{C_{\mathrm{cr}}} in the normalized estimate, for a constant C_{\mathrm{cr}}=C_{\mathrm{cr}}(K) independent of P,\beta. Indeed the sum of their positive orders is at most K; the derivative ratios already extracted at cutoff marks have uniform global bounds. Every node has length at most CT, so N_I^{-1}\le Cn^{-\beta}. The contribution stopped at P cutoff marks, with an overshoot of at most K, is therefore at most

apart from exceptional precision.

The parameter choices can now be made in order. First K=24 is fixed. After choosing a sufficiently small \beta for the score estimates, take P>(C_{\mathrm{cr}}+3)/\beta, and then \eta\le\beta/(4(P+K)). For this fixed P, all scalar multiplicities from (9.9) and powers of \log n from partitions cost at most n^{\beta/2} for large n. Together with 2(P+K)\eta\le\beta/2, (9.13) is at most n^{-2}. Finally choose the exceptional-set and small-mass precisions. In a fully expanded term at most K positive-order factors use the score estimates. Their total exponent loss is thus C_K\beta independent of P.

The edge norms in fully expanded terms with cutoff marks. It remains to estimate terms which terminate before depth P. We first form a partition from the locations of their cutoff marks, before deciding which endpoints to condition on. Split each root into dyadic pieces only at nodes which properly contain a descendant carrying a cutoff mark. Make the same partition on a copy using its own cutoff marks.

Let m be the least length of a node carrying a cutoff mark, capped at T. In each fully expanded monomial, select one factor R_l(I) with l\ge1, or one factor \bar q_e, supplied by a terminal H_{I,e} slot through (9.11). Make this choice from the indexed monomial, hence the same way for every e. The slot was created by a cutoff mark at I, so the selected factor has length |I|\ge m in the R_l(I) case and assigned length T\ge m in the \bar q_e case. Every other terminal factor remains in the product. The gain from a shortest cutoff mark will be retained separately below. All partition pieces have length at least a constant times m, except possibly untouched original forest roots. Indeed, a split made to isolate a descendant M carrying a cutoff mark creates two children, each of length at least |M|\ge m; further splits satisfy the same statement with their own such descendants. A copy starts at a node carrying a cutoff mark, so it has no untouched smaller root. This also gives O_P(1+\log n) pieces. The untouched roots smaller than a sufficiently small constant times m can occur only in a stopped forest, and occur later than all cutoff marks on that path: the lengths of its original roots decrease chronologically. Choose that constant below the constants for complete legs, so no complete leg is included in this exception.

Projecting the raw order-one coefficient on a root of length D_1 would replace it by c_1=R_1(D_1)+q_{\mathrm{end}}. If paired with R_3(D_3), the R_3(D_3)R_1(D_1) summand would have L^2-times-L^2 cost n^{O(\beta)}(D_3/D_1)^{1/2} by (9.4), which can grow when D_1 is very short. We therefore reserve one allocation before projecting piece coefficients. An untouched original root I of length z<cm may carry a raw coefficient of order one while the sole other factor of positive order is a conditional coefficient of order three. This is the small-root (3,1) allocation. Its terminal state is not revealed in advance, and its complete raw coefficient \ell_{1,e}(I)=[u]L_{I,e}(u) is retained until forward compensation. The total order is four, so no other positive-order factor is present. The small root carries no cutoff mark and cannot supply the selected terminal factor; hence in this allocation the selected factor is the R_3 factor.

For every other allocation, condition on the relevant endpoints of nodes carrying cutoff marks and on the partition endpoints. The fresh piece interiors then have their conditional laws \mathsf C_J and factor after the endpoint variables and their shared identifications are fixed: lower weights stay inside their pieces, and every other included weight depends only on the conditioned endpoints. Shared exterior variables occur once; this does not assert independence of the completed copied histories. Whenever a raw coefficient on a piece is projected, it becomes c_l=R_l+q_{\mathrm{end}}R_{l-1}. Nonterminal denominators in (9.10) and initialization may contribute endpoint powers of absolute value one. At a terminal slot use (9.11) instead. The resulting terms are products of R factors, \bar q factors counted as R_1(T), and bounded endpoint factors. The sum of their positive orders is at most k, and the selected terminal factor remains in this product. All these conditional identities are integrated under the joint copy law when the sharp estimates are used; they give no uniform sharp bound at an arbitrary fixed endpoint graph.

For total R order at most four, (9.4) gives the following complete list. Unused Hölder exponents are filled with the constant function on the edge probability space.

Thus only (3,1) can cost more than a constant. If its order-one length D_1 is at least a constant times m, its cost is at most C(T/m)^{1/2}. Retain the gain from a shortest cutoff mark,

when the shortest node carrying a cutoff mark has length greater than T, the same formula holds up to a constant. The remaining gains are at most a constant times n^{-\beta} each and pay their hazard summations. If m\le Tn^{-\varepsilon_0}, (9.15) leaves at most (m/T) in the (3,1) case and (m/T)^{3/2} in all other cases, after the total n^\beta allowance for hazards and logarithms. This is a fixed power saving.

For the reserved small-root allocation, retaining \ell_{1,e}(I) keeps its R_1 and q_{\mathrm{end}} conditional terms together before a triangle inequality. The selected order-three factor is earlier on the same path or belongs to an independent original history. The endpoints of all nodes carrying cutoff marks on the same path precede I. Their conditional copies can be sampled before I, independently of its future given those already observed endpoints. Factors on other original histories can also be exposed first, because those histories and their external tests factor separately. This auxiliary sampling is fixed as part of the admissible law before choosing the earlier vector factor.

After projecting the earlier pieces, we return to the joint tested expectation with the terminal state of I unrevealed in advance, and compensate its raw order-one coefficient in the forward filtration. Under the forward tested law on this root, the external terminal and possible failure tests are included in its tilted intensities. The L^2 compensation in (9.5) is therefore valid with these earlier factors retained and gives n^{O(\beta)}z^{1/4}/\sqrt n. Paired with the order-three L^2 bound, the result is

This calculation is made before projecting the small root onto its endpoint, and the compensation identity precedes restriction to the hazard event. It proves the same early-m saving using (9.15).

For completeness, suppose m\ge Tn^{-\varepsilon_0}. The selected terminal factor has length at least m and is in the late range. Except for (3,1), the norms in (9.14) give a fixed improvement at that factor: a selected R_1 uses its late L^4 bound, a selected R_2 its late L^2 bound, and a selected R_4 its late L^1 bound. If the selected factor is the singleton R_3, then D_3^{-1/4}\le T^{-1/4}n^{\varepsilon_0/4}. For the regular (3,1) case, put \delta_0=0.005 and use the R_3 late L^2 gain n^{-0.001}. Since D_1\ge cm,

The two savings are n^{-0.00095} and n^{-0.00245}. In (9.16), the corresponding cases gain n^{-0.001} and n^{-\delta_0/2}, respectively. These margins, and the early m margin, remain positive after an exponent C_K\beta is included. Together with the fixed-weight calculation, they prove the low-order part of (9.2); one may take, for example, a=10^{-7}.

For k>4 the product estimates need no late gains or lower length bound. The following allocation of Hölder norms makes the power count explicit. A factor of order l\ge4 uses its L^1 bound and saves two powers of T relative to the supremum bounds with all lengths replaced by CT. If the largest order is three, pair it with an order two or three in L^2, or with two order ones using L^2,L^4,L^4; again the saving is two powers of T. A singleton order three is bounded by one. An order three paired with exactly one order one costs at most T^{1/2}, which is allowed because k\ge5. If all orders are at most two, two order-two factors use L^2,L^2, one order two and two order ones use L^2,L^4,L^4, and four order ones use L^4 four times. Each of these saves two powers of T; when fewer factors are present their product is bounded by one. All unused factors use their supremum bounds. Thus if the total R order is h\le k, the cost is at most n^{O(\beta)}T^{(k-4)/2}. This proves the high-order part.

Finally the same expansion with the supremum bounds in Proposition 8.5, the raw fixed-order path bounds, and \sum l\le k gives (9.3). The depth remainder is already smaller, and the exceptional and small-mass reductions at the start give the stated additive precision. Since the Taylor coefficients of \tanh\vartheta at zero are fixed constants, a derivative of order k in \vartheta is a fixed linear combination of coefficients of orders at most k in u. Applying this to each factor of a product leaves total u order at most the assigned total \vartheta order. The lower-order estimates fit the asserted bounds for that total order.□

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

for arbitrary fixed A'. We used N(cT)=O(n^\beta).

Apply the ordinary chain rule to \chi(n^{10}\sum\ell_q) and to F(\widetilde Y). A term has at most K positive-order history coefficients, with total order at most the differentiating order; repeated histories are represented by independent copies. The coefficients from derivatives of F are evaluated at the base \widetilde Y(J), hence are common across e, and their total absolute sums are bounded by (9.20). For loss factors, expand (9.17). The mass factors in Lemma 9.2, summed over all choices of queries and first-failure nodes and multiplied by their n^{10} factors, give at most

when there are r loss factors. Trace mass factors cost at most n^{O_K(\beta)}. The additive precisions in that lemma are chosen after the polynomial number of nodes and queries and after the factors n^{10K}, so their total is negligible.

If a static cutoff is also differentiated, its pointwise derivative bound from Lemma 7.1 multiplies these estimates. If only the static cutoff is differentiated, that bound is O(n^{-b}) for some b=b(\beta)>0, and its remaining outer factor is bounded. Consequently the fourth averaged derivative is at most n^{-a+C'_K\beta}(\log n)^{C'}+O(n^{-b}), higher averaged derivatives are at most n^{C'_K\beta}(\log n)^{C'}T^{(k-4)/2}, and the supremum bound is at most n^{C'_K\beta}(\log n)^{C'}T^{j/2}. These estimates hold everywhere: outside the indicated supports the corresponding product-rule terms vanish. Choose \beta after K so that C'_K\beta<a/2 and C'_K\beta<0.005, while satisfying the preceding sections’ smallness requirements. Absorb logarithms by a still smaller fixed power. This gives (9.21)– (9.23) for some a_0>0 and \delta<0.01.□

Proof. We use the Taylor replacement principle [6], in the finite mixture form of the stationary comparison [21, Section 8]. Put \nu_\lambda=(1-\lambda)\mathsf N(0,1) +\lambda(\delta_{-1}+\delta_1)/2 for 0\le\lambda\le1. Differentiating its finite product measure gives exactly

where \xi is a symmetric sign, Z is standard normal, and \mathbb E_{\lambda,-e} averages the other entries.

Independently of J_{-e},\xi,Z, sample V from \nu_\lambda and set J^0=(J_{-e},V). Taylor-expand at this completed matrix:

The remainder bound is global by (9.23), including every point of the segment. Conditional on J_{-e},V, the orders k\le3 cancel exactly between the two proposals. In fact

where the sum is empty for k\le3. Independence of the completed entry from both proposals is what permits this conditional cancellation.

For 4\le k<K, on |V|\le\log n the absolute value of (9.26) is at most C_K(\log n)^K. The complementary mixture tail, even with any fixed moment of V, is smaller than every inverse power of n. Its contribution is negligible by the global polynomial bounds (9.23). When e is uniform, J^0 has the full iid mixture law independently of e. Therefore (9.24)–(9.25) yield, uniformly in \lambda,

The remainder uses the bounded Kth moments of \xi,Z,V. By (9.21), the k=4 term tends to zero. For k\ge5, (9.22) and T=n^{2/3} give

The remainder is n^{2-K/6+\delta}=n^{-2+\delta} because K=24. Thus (9.27) tends to zero uniformly in \lambda. Integration over [0,1] proves the lemma.□

Proof of Theorem 1.1. In Gaussian disorder, Proposition 3.1 supplies the cap estimates used in Theorem 4.1. Apply that theorem to the initial law \nu_n. Proposition 5.1 then supplies, on a further represented subsequence of every sequence, a normalized positive-time density family for S_t^g=e^{-tc_*H_g}, together with locally uniform limits of both finite autocorrelations. Proposition 5.2 gives the averaged uniform-start escape estimate. The sequence extraction and fixed-g uniqueness in Theorem 6.1 identify every represented family with the same measurable family h^g for almost every g. The subsequence criterion gives the Gaussian joint convergence.

The density family is bounded at each positive time, hence belongs to L^2(\Pi_g); it is nonnegative, has mass one, and satisfies the semigroup relation. Its weak product-topology boundary is \delta_0 by Theorem 6.1. Proposition 5.1 proves the locally uniform convergence and continuity of the series defining B_g. Each finite partial sum is continuous in (s,t), and its evaluations at rational pairs are measurable in g by the density and semigroup constructions in Theorem 6.1. It is therefore Borel as a C_{\mathrm{loc}}((0,\infty)^2)-valued map, and so is its locally uniform limit.

Proposition 5.1 gives \|h_s^g-1\|_{L^2(\Pi_g)}\to0, and hence the required L^1 convergence. It also gives locally uniform convergence B_g(s,\cdot)\to A_g(\cdot) for almost every g on which the selected family is defined. This implies the stated convergence in probability over g.

Proposition 9.1 transfers the joint function law to Rademacher disorder. Its comparison uses the uniform initial law for each quench trace, so the limiting family is the one already selected in the Gaussian argument. The added Gaussian diagonal and the auxiliary eigenvector signs do not affect either finite autocorrelation. For both laws the finite dynamics use rate-one site clocks and are observed in the time unit T_n=n^{2/3}. The temporary substitution u=c_*s was reversed in the entrance proof, so the limiting semigroup retains exactly the coefficient c_* in (1.5), completing all the assertions.□