Cutoff throughout the high-temperature Sherrington–Kirkpatrick phase

Proof. Use the backward martingale F(s,X_s)=S_{d-s}f(X_s), whose initial value is S_df(x) and terminal value is f(X_d). A spin flip has increment -2x_i d_iF, of magnitude at most 2\sqrt{b(d-s)}. Its instantaneous sum of squared jump sizes times rates is

The backward equation cancels the linear part of the exponential drift. The remaining jump compensator is controlled by the preceding square-jump bound and e^z-1-z\leq z^2e^{\abs z}/2. Integrating the resulting drift inequality and dividing by e^{\theta S_df(x)} gives (2.6). Expansion at \theta=0 gives the variance assertion. This argument used the flip-rate bound and the assumed gradient bound; it used no restriction on the temperature.□

Proof. For the upper bound, use the full density relative to equilibrium. Starting from x, the spins at time t are independent with means e^{-t}x_i. Against uniform measure,

Thus total variation tends to zero uniformly in x at t=(1+\epsilon)\log n/2. For the lower bound, a single statistic suffices. From the all-plus state, the normalized sum n^{-1/2}\sum_iX_i has mean \sqrt n\,e^{-t} and variance at most one; at equilibrium its mean is zero and its variance is one. At t=(1-\epsilon)\log n/2, the mean tends to infinity. Chebyshev’s inequality gives total variation tending to one.□

Proof. We first extend the weak residual identity to expectations under Q, then use first moments to select two consecutive small residuals, and finally compare the selected iterate with the stable root. The deterministic differentiation bounds are supplied by [21, Lemmas 5.7, 6.1, 6.3, and 6.4]. The conclusion Equation (3.7), including its square-root dependence on the energy, is the additional assertion proved in this argument.

Residual identities under the changed law.

The recursion uses Onsager corrections, as in Bolthausen’s TAP iteration [7]. Post-iteration Gaussian comparison and local convexity are studied by Celentano [8]; neither result substitutes for the buffered residual estimates used here. Define the primary fields by

and put R_l=m^{l-1}-m^l. At any fixed depth the derivative matrices of h^l,m^l have bounded operator norm and \norm{db_l}\le C_l n^{-1/2}. The small-vector constructions in [21, Definition 6.2] start with seeds of bounded Euclidean norm that may depend on M,h but are independent of the spin x. Their sources are finite sums of smooth site coefficients times seeds or earlier auxiliary fields, and are linear in those auxiliary arguments. The coefficients and their derivatives through every fixed order used in the construction are uniformly bounded. Scalar parameters \theta satisfy \sup_x(|\theta(x)|+\norm{d\theta(x)})\le C; they may include the averages b_l. A source U is admissible from level p when every explicit primary field in its site coefficients has index at least p. In particular, the seed (1,\ldots,1)/\sqrt n permits the source m^l/\sqrt n at level l. An ordinary construction starts with an empty auxiliary list. At stage r, write w^r=U for the new source and y^r for the output of the operation below; the auxiliary sum runs over earlier pairs (w^a,y^a), a<r. The formal partial \partial_lU_i differentiates the explicit primary-field argument h_i^l, holding the seeds, scalar parameters (including the averages b_l), and auxiliary arguments fixed. The partial \partial_{y^a}U_i differentiates the linear argument y_i^a with its coefficient held fixed. A partial in an absent argument is zero. The associated auxiliary operation is

These partials act on the displayed arguments of a recipe. They are distinct from the discrete derivative matrix (\nabla U)_{ij}=d_jU_i, which records the response of the whole recipe to spin flips. For every fixed construction the imported deterministic bounds are

They also hold for the local implicit construction used below, on its specified fixed flip buffers. These statements concern pointwise differences and have no dependence on the law under which they are tested.

Put D=D_P(Q) and, for a scalar test, let

We claim that the weak residual bound becomes

For a local construction the test is supported on its small-residual region, and the supremum bounds are only needed on the corresponding buffer. In particular no conditional regularity of Q is assumed.

At the first level, the only change from the equilibrium residual argument is the mismatch between the Q-conditional mean and the P-conditional mean. Conditional covariance under Q gives

This remains true when a conditional law of Q is a point mass. The product rule d_i(GU_i)=Gd_iU_i+(d_iG)(U_i-2x_id_iU_i) and Equation (3.9) bound the first term by CB(G). Equation (3.2) and Cauchy–Schwarz bound the second by 2C\norm G_\infty\sqrt D. This proves the base case.

For higher levels we reduce to lower residual indices by an exact cancellation. For a target admissible from level p+1, set

where

It satisfies [(z-r-a\tanh z)-a\partial_z]\Phi=\tanh z-\tanh r. Adjoin the auxiliary field y' of U' using Equation (3.8). With \alpha_l,\gamma_a belonging to U', the recurrence gives

The R_{p-1} term is absent at p=1. The l=p term cancels the last term because

After this cancellation, the remaining l=p term is -j(\sqrt n\alpha_p)R_p^{\mathsf T}(m^p/\sqrt n). Its target m^p/\sqrt n is admissible from level p and has bounded norm. For l>p, use the target m^{l-1}/\sqrt n in the same way. Thus every term has a lower residual index, a permitted target, and one of the bounded multipliers a_p,b_{p-1},\gamma_a,\sqrt n\alpha_l. Their difference bounds in Equation (3.9) and the product rule give B(\theta G)\le C B(G) for each such multiplier. Strong induction over finite admissible constructions therefore proves Equation (3.10). Each step adds finitely many operations and decreases the residual index, so a finite word length suffices. For a supported local test, a difference can be nonzero only on a one-flip neighborhood of its support; the imported buffers include every such neighborhood used in this finite induction.

Selecting two consecutive small residuals. The weak estimate now has the desired 1+\sqrt D cost. We preserve that cost in the selection step by using first moments rather than second moments. Set S_l=R_l^{\mathsf T}m^l/\sqrt n and \psi(s)=s/\sqrt{1+s^2}. The primary derivative bounds give \sup\norm{dS_l}\le C_l, while |\psi|,|\psi'|\le1 and s\psi(s)\ge|s|-1. Testing Equation (3.10) with U=m^l/\sqrt n and G=\psi(S_l) yields

The exact telescoping identity is

For a fixed small \rho>0, choose a fixed depth d_0 with \lfloor d_0/2\rfloor\rho^2/4>2, and then a fixed \varepsilon>0 with 2d_0\varepsilon<1. If every |S_l|\le\varepsilon\sqrt n, the sum of the squared residuals divided by n is less than two. Some consecutive pair consequently has both norms at most \rho\sqrt n/2. Let C_k be smooth cutoffs equal to one on this latter condition and supported where both norms are at most \rho\sqrt n. The primary derivative bounds give \norm{dC_k}\le C_\rho/\sqrt n. Combine the cutoffs in order so that earlier cutoffs reduce the weight available to later ones, and retain the unassigned deficit. Put

Markov’s inequality and Equation (3.13) give

Comparing the selected iterate with the stable root. Let r be the unique zero of Equation (3.3), whose existence and inverse bound follow from [21, Lemma 5.7], and put t_* =\tanh r, q_*=n^{-1}\norm{t_*}^2. On the buffered region of C_k, fix the primary fields and the root, and choose one of the two sources w_0 specified below. The implicit construction of [21, Lemma 6.3] supplies a vector w, an auxiliary vector y^{\rm imp}, and a scalar c_1 solving

Here w_0 is either Ae for a fixed \norm e\le1, or n^{-1/2}\partial_r\Phi(h_i^k;r_i,a) coordinatewise. In \partial_k w_i, the arguments y_i^{\rm imp},c_1,a are held fixed; in \partial_r\Phi, a is held fixed. With this convention the displayed equations form a linear system for (w,y^{\rm imp},c_1), whose unique solution is supplied by the cited lemma. That lemma also gives

after decreasing the common positive inverse margin if necessary. The distinction between its bounds matters for selecting parameters: derivative constants may depend on the fixed depth, while size and inverse constants do not. We will first control the susceptibility average with the second seed, then test the mean in direction e with the first seed.

Define

The algebra in [21, Lemma 6.6] expresses this as

Thus the already proved local residual estimate gives

This is the required arbitrary-law version of the terminal residual identity; the last term uses the multiplier \sqrt n c_1.

To extract a mean comparison from this terminal residual, write p=m^k-t_* and L_* =\sqrt n\{\chi-(1-q_*)\}. Subtract the root equation and substitute the displayed equation for Mw to obtain

The inverse bound and the two small residuals imply \norm p\le C\rho\sqrt n. Consequently the middle term has absolute value at most C_0\rho|L_*|, with C_0 independent of the selection depth. Also \sup\norm{dL_*}\le C_k. Because C_0 is independent of depth, we can choose \rho with C_0\rho<1/2 first, and only then choose d_0 and \varepsilon as in the selection step above.

For w_{0,i}=n^{-1/2}\partial_r\Phi(h_i^k;r_i,a) the first sum in Equation (3.16) is L_*: differentiate the identity following Equation (3.11) in r, holding a fixed. Test with G=C_k\psi(L_*). Its B-norm is bounded. The last term is controlled by Equation (3.10) with target p/\sqrt n and multiplier \sqrt n c_1. Absorbing the middle term gives

For w_0=Ae, the first sum is p^{\mathsf T}e. Test this time with \widehat C_k alone. Since \widehat C_k\le C_k, Equations (3.15) and (3.17) and the same residual bound imply

It remains to pass from the selected iterate back to the original spin. Sum Equation (3.10) with constant target e to handle x-m^k=\sum_{l\le k}R_l, and then sum over k. The omitted part has absolute value at most 2\sqrt n\,Q(\Delta>0)\le C(1+\sqrt D) by Equation (3.14). Taking the supremum over unit e proves \norm{\E_Qx-t_*}\le C(1+\sqrt D). Apply this also with Q=P, when D=0, and use the triangle inequality.□

Proof. We verify second-stage stability and its gap first. We then run the observation argument conditionally from each first-stage posterior to obtain its gap, and finally place all single-site cavities on one common disorder event.

Stability through the two stages. For the first stage, [21, Proposition 5.1 and Corollary 5.2] give Equation (3.5), with positive slack and a test-independent stopping rule, throughout the first stage up to any prescribed fixed b, outside an event of ordinary observation probability e^{-cn}. Intersect its matrix event with the finite word events required in Lemma 3.1 and in the directional estimate [21, Proposition 7.2].

On the second stage write u=1-r and use M=uJ, j=u^2\beta^2. The word bounds hold uniformly in u\in[0,1]. In fact their inverse factor is exactly

The other word factors only acquire powers of u. The fixed upper bound on A can thus be the same as for J. All derivative and implicit construction estimates used above are uniform for these parameters: the recurrences have bounded coefficients, j remains bounded away from one, and the inverse margins are fixed. At u=0 the measure is product; Equation (3.2) gives its mean bound directly.

To obtain a margin throughout the second stage, use the large field created in the first one. Gaussian maximal bounds on the two fixed Brownian intervals imply, outside Ce^{-cn}, that

for b\ge1, with a constant depending on K,k. One elementary proof of the bound for B uses a 1/2-net of the unit sphere and the scalar Brownian maximal inequality; the covariance norm of A_*^{1/2}W is at most K+k. This also gives the asserted exponential probability. If y has residual at most \rho_0\sqrt n in Equation (3.3), then \norm{y-h}\le(\rho_0+\beta^2+K)\sqrt n. Thus, by Equation (3.21), at most C n/b coordinates have |y_i|<b/2. Outside these coordinates V(y)_{ii}\le\sech^2(b/2). For a candidate A in Equation (3.5), an additional set of size at most \epsilon_A^2n/a^2 has |A_{ii}-V(y)_{ii}|>a.

The small principal submatrix bounds of [21, Lemma 8.2] apply also to J, because \norm{J-(\tanh J_{ij})}\to0 on that event. If S is the preceding exceptional set and A_{ii}\le2a off S, they give

Choose the principal-submatrix tolerance and a first so that the sum of these bounds is less than 1/2; choose an allowed exceptional fraction next, then b large and \epsilon_A small. The nonnegative term jb(y)A in Equation (3.4) can be kept or dropped. This proves Equation (3.5) with \kappa=1/2 on the second stage, after reducing the common margins if necessary. These choices are compatible with the first-stage theorem: choose this fixed b first and subsequently intersect and reduce its supplied margins.

The second-stage gap. The same large-field event also gives a gap at least 1/2 throughout this stage. The curvature bound must hold uniformly in the spin configuration. For that purpose combine Equation (3.21) with \norm{uJx}\le K\sqrt n. Only a small fixed fraction of the fields h_i+(uJx)_i can be smaller than b/4 in absolute value. Let v_i=\sech^2(h_i+(uJx)_i) and a_i=v_id_if. The signed curvature inequality of [21, Lemma 8.1] is

where H=-L_P, U_{ij}=\tanh(uJ_{ij}), (Q_0)_{ij}=U_{ij}^2, and r_4=\max_i\sum_jU_{ij}^4=o(1). On the weak-field set use the small principal-submatrix norms; on its complement use v_i\le\sech^2(b/4). If q_f(x)=\sum_i v_i(d_if)^2, each quadratic cost is bounded by [\varepsilon+K'(2\sqrt\eta+\eta)]q_f(x) with \eta=\sech^2(b/4) and a fixed K'. Choosing the same tolerances sufficiently small proves \E_P(Hf)^2\ge\mathcal E_P(f)/2. The finite-dimensional spectral theorem gives the gap 1/2. The matrix bounds are uniform in u: \tanh(uJ)=uJ+o_{\op}(1), while the nonnegative matrix (\tanh(uJ_{ij})^2) is dominated entrywise by (J_{ij}^2).

The first-stage posterior gaps. The remaining task is to obtain a gap at every first-stage posterior. We do this by conditioning the observation experiment from that posterior onward. The zero-field gap theorem does not by itself apply to these random fields. Let E be the event of a first-stage stability failure or failure of the terminal gap at time b. The preceding bounds and [21, Proposition 8.3] allow a fixed a_0>0 such that \P(E\mid J)\le e^{-2a_0 n}. For the observation filtration set q_s=\P(E\mid\mathcal F_s,J). The nonnegative martingale maximal inequality gives

Outside this event and E, fix a time s<b. Conditional on \mathcal F_s, the remaining experiment is an ordinary white observation starting from its posterior P_s, with horizon b-s\le b. Its future bad probability is at most e^{-a_0n}.

For every f with P_sf=0 and P_sf^2=1, set d=\mathcal E_{P_s}(f), and stop the remaining experiment at its next stability failure or at b, denoting this time by \sigma. Write N_t=P_tf^2 and V_t=P_tf^2-(P_tf)^2. Conditional variance decomposition gives \E[\mathcal E_{P_t}(f)\mid\mathcal F_s]\le d. At every retained stable posterior, the directional estimate of [21, Proposition 7.2] states, in squared form,

It uses the same stable-field implication and a sufficiently long finite set of word diagnostics. This homogeneous estimate, combined with the filtering variance identity, implies

Indeed the stopped expected variance has derivative -\E[\1_{t<\sigma}\norm{\Cov_{P_t}(f,x)}^2\mid\mathcal F_s], which is at least -C_{\rm d}(\E V_{t\wedge\sigma}+d).

The variance lower bound must be compared with the terminal energy. To control the variance lost on bad paths, change the input law to f^2P_s. The observation density is N_t and its posterior is Q_t=f^2P_t/N_t. Before stopping, Lemma 3.1 and conditional variance decomposition give

By Equation (3.19), the stopped observation entropy is at most Cb(1+d). Relative entropy between event probabilities gives

Here the bad event is measurable at \sigma: it is the stopping event before b, or the terminal gap failure at b. Its ordinary conditional probability is exponentially small by Equation (3.22). The changed probability in Equation (3.24) is derived anew for each f; it is not asserted to be exponentially small.

On good terminal paths V_b\le2\mathcal E_{P_b}(f); on bad paths V_\sigma\le N_\sigma. The stopped density identity and Equations (3.23) and (3.24) consequently give, for d\le1,

For all sufficiently large n this implies d\ge v_0/6; for d>1 the same lower bound is automatic. All constants are uniform in s and f. A short remaining horizon improves v_s and decreases the entropy integrals, so the estimate stays uniform as s\uparrow b. At s=b use the terminal gap directly. Alternatively one can first argue at rational times and then use continuity of the finite-state gap in the field.

One event for every cavity. We now transfer the matrix and observation estimates to all single-site cavities at once. This requires a common disorder event, rather than a union bound over unspecified o(1) disorder probabilities. Write J^{(i)} for a principal minor. Word diagnostics for this matrix are restrictions of those for J: insert the complementary coordinate projection among diagonal factors, and set the missing entry of an inverse diagonal to zero. All normalized traces can be retained with denominator n; equivalently the (n-1)-site variance parameter is j_i=\beta^2(n-1)/n. The identity j_i/(n-1)=\beta^2/n makes these the same contractions. Inserting these projections may increase the required word length by a fixed factor; choose that larger finite length in the full event. The event is already simultaneous in its diagonal parameters. The small-submatrix bounds pass directly to every minor.

For first-stage stability, take a candidate cavity vector y_{-i} and extend it by choosing y_i to solve the missing scalar equation F_{J,h}(y)_i=0. This equation is continuous with a bounded nonlinear part and hence has a solution. On the retained coordinates the residual changes by a vector of norm at most \norm{J_{-i,i}}+O(n^{-1/2})=O(1). Extend the candidate diagonal with A_{ii}=\sech^2 y_i. Its Frobenius discrepancy is unchanged. The principal block of the full Hessian differs from the cavity Hessian by O(1/n) in norm: the rank coefficient is exactly 2\beta^2/n in either dimension, and only the missing contribution to jb(y) changes. Start with smaller cavity residual and Frobenius thresholds and use the slack in the full stability event. It follows, uniformly in i, that full stability implies cavity stability with common positive margins for large n.

The comparison of sampling laws costs only \exp(C\sqrt n). Indeed the full marginal on x_{-i} is proportional to \mu^{(i)}(x_{-i})\cosh(J_i x_{-i}), and |J_i x_{-i}|\le K\sqrt n. More explicitly, the law obtained by sampling the cavity spin configuration and an independent uniform missing spin has density at most \exp(2K\sqrt n) relative to \mu. Extend its Brownian observation by an independent coordinate. Thus a full exponential observation failure remains exponential for the cavity experiment. The second-stage large-field and curvature arguments apply directly to the minors with the same constants. A union bound over the n cavities now loses only a factor n from exponential observation probabilities. This proves the claimed simultaneous statement on one common disorder event. It also proves a common equilibrium gap by taking s=0.

A test-independent stopping rule. To finish, all properties tested above depend only on the observed field and finite matrices. Stable-field failure is detected by the Lipschitz residual-distance stopping construction in [21, Corollary 5.2]; the gap is a continuous finite-matrix eigenvalue. Stopping at smaller stability margins or at a gap below half the established bound gives an observation stopping time independent of the test function, with the same exponential exceptional bound.□

Proof. We prove a Gaussian subset bound under the planted law, transfer it to stationary sampling using the partition function, and then convert the averaged bound to one at fixed disorder. The exponential slack will also make the cavity statement simultaneous. Let Z(J) be the partition function with counting measure on the cube. The annealed planted law has density e^{x^{\mathsf T}Jx/2}/\E_J Z(J) relative to Gaussian disorder times counting measure on x. Under it X is uniform, and, conditionally on X, the off-diagonal entries are independent Gaussian variables with mean \beta^2X_iX_j/n and variance \beta^2/n. Moreover

where R_n is the sum of n independent uniform signs. The last bound follows by introducing a standard Gaussian G and using \cosh t\le e^{t^2/2}: \E e^{\beta^2R_n^2/(2n)}=\E_G\cosh(\beta G/\sqrt n)^n. The gradient of \log Z in the independent standardized Gaussian coordinates has norm at most C\sqrt n. Gaussian concentration and Paley–Zygmund applied to Equation (3.26) imply

For clarity, Paley–Zygmund gives a fixed positive probability of \log Z\ge\log\E Z-\log2; concentration places its mean within O(\sqrt n) of this level. Concentration once more proves Equation (3.27).

We next estimate the fields under planting. At the observation endpoint

Under planting, conditionally on X, JX is a Gaussian vector with bounded covariance norm and a mean of bounded coordinate magnitude. On \norm J\le K, the last observation noise has conditional covariance norm at most K+k. Thus, for a fixed subset of size m, Gaussian norm tails, followed by a union bound over at most (en/m)^m subsets, give an annealed bound C_1e^{-c_1t} for violation of

simultaneously in I. This uses the deterministic inequality \norm{z_1+\cdots+z_4}^2\le4\sum\norm{z_a}^2, so independence of JX and A_*^{1/2}W_1 is unnecessary. Increasing C_2 gives any fixed desired exponential rate in t. Fix this increase once, before the quenched comparison below; it does not depend on \delta.

To return to the quenched experiment, note that the density of the planted joint spin-disorder law relative to stationary sampling with the original disorder is Z(J)/\E_JZ(J). Restrict to Equation (3.27) with, for example, \log Z\ge\log\E Z-\delta n/10. The preceding annealed bound, with ample exponential slack, therefore bounds the disorder average of the quenched failure probability at level s by e^{-3s} for s\ge\delta n, once n is large. Markov’s inequality and a union bound over the integer levels s\ge\delta n give a disorder exceptional probability O(e^{-c\delta n}) outside which the required conditional probability is at most e^{-s}. For a real s\ge\delta n\ge1, apply the integer estimate at \lceil s\rceil and double the threshold constant; then \lceil s\rceil\le2s and e^{-\lceil s\rceil}\le e^{-s}. Only the good disorder event depends on \delta.

For a cavity the same planted computation has parameter \beta\sqrt{(n-1)/n}<\beta and common constants. Alternatively extend its spin and observation experiment as in Proposition 3.2; the additional \exp(C\sqrt n) factor is absorbed by the exponential slack for fixed \delta. Taking the union over cavities still gives probability tending to one. This argument avoids a union bound over unquantified convergence in probability in Equation (3.27): the full partition lower bound transfers to the minors with an O(\sqrt n) error. More explicitly,

These inequalities transfer the same full lower bound to every minor at once; the remaining exponential Markov bounds tolerate the union over i.□

Proof. We first bound the entropy of the observation law, then the conditional entropy left at its product endpoint. The latter bound contains a term involving the original entropy, which will be absorbed only at the end.

Entropy of the observations. Put D=D_\mu(\nu) and H=H_\mu(\nu), and change the input of the two-stage experiment from \mu to \nu. Let P_t,Q_t denote the ordinary and changed posteriors, and let \sigma be the first loss of the retained stability conditions, capped at b+1. By Lemma 3.1 and Equations (3.19) and (3.20),

The ordinary probability of \sigma<b+1 is at most Ce^{-cn}. Applying the binary entropy inequality to this stopped event gives

Continue the observations even on this event. The squared difference of posterior means is always at most 4n, and the precision and horizon are bounded. Hence the contribution after stopping in Equation (3.19) is at most another C(1+D). The relative entropy of the complete observation laws is therefore at most C(1+D).

Entropy remaining at the product endpoint. The observation channel is the same for the two input laws. Thus the entropy chain rule on the joint experiment reads

At the product endpoint, entropy tensorization gives

This version with all other coordinates conditioned follows, for example, by the entropy chain rule and convexity of relative entropy: conditioning on more coordinates can only increase the corresponding average conditional entropy relative to a fixed coordinate law.

For a two-point law p of least atom p_{\min} and any q,

Indeed for 0\le z\le1/p_{\min}, z\log z-z+1\le C(1+\log(1/p_{\min}))(\sqrt z-1)^2: the ratio has a finite limit at z=1, is bounded on [0,4], and for z\ge4 is at most C\log z. Sum this inequality with weights p and use \sum p(\sqrt{q/p}-1)^2=2(1-\sum\sqrt{pq}) \le2[1-(\sum\sqrt{pq})^2]. For the product law of logit h_i, \log(1/p_{\min})\le\log2+2|h_i|. Define the conditional energy contributions

They satisfy 0\le z_i\le1 and \E_{\mathbb Q}\sum_i z_i\le D by Equation (3.20). Consequently

The weighted field sum and absorption. The variables z_i record where the energy is spent. We use subset tails to bound their field-weighted sum without replacing the field by its maximum over sites. By Lemma 3.3 there is a nonnegative excess R(h), at least \delta n, such that simultaneously in all subsets

after a fixed increase of C. In particular \E_{\mathbb P}e^{R/2}\le C e^{\delta n/2}. The entropy variational inequality on the joint spin-observation laws, whose relative entropy is exactly H, gives

The same conclusion follows from the observation marginal by data processing. At this stage H is still unknown. Keeping it on the right side is legitimate; the square-root estimate below will allow us to absorb the resulting contribution.

Set z=\sum_i z_i. Layer cake for I_t=\{i:z_i>t\} and concavity of u\mapsto u\log(en/u) on [0,n] give

Thus, first by weighted Cauchy–Schwarz and then by Cauchy–Schwarz in the observation law and Jensen’s inequality,

Here u\log(en/u) is increasing on [0,n], allowing \E z\le D to be substituted. Insert this in Equation (3.32). The term C\sqrt{DH} is at most H/2+C'D; the term C\sqrt{D(\delta n+1)} is at most C'D+C'(\delta n+1). Absorbing H/2 proves Equation (3.28). All estimates and choices were uniform over \nu and the cavities.□

Proof. A defective logarithmic Sobolev inequality. We keep the dependence on the defect parameter explicit. For 0<\eta<1/2, the concave function u\sqrt{\log(en/u)} has at u=\eta n a tangent with slope at most \sqrt{\log(e/\eta)} and intercept at most \eta n/2. Thus Theorem 3.4, with \delta=\eta, implies, after normalizing f^2 to a density,

The energy of the normalized density square root lies in [0,n], as required by Theorem 3.4. This also proves Equation (3.35) for signed f, using \mathcal E_\mu(|f|)\le\mathcal E_\mu(f).

Hypercontractivity with the defect retained. Use the logarithmic-Sobolev-to-hypercontractivity argument of Gross [15], carrying the additive defect through the norm calculation. Let g_t=S_tf>0, p(0)=2, and p'=4(p-1)/A_\eta. Reversibility gives

For z,u>0, Cauchy–Schwarz on the interval between them gives

Apply Equation (3.35) to g_t^{p/2} and use this inequality edge by edge in the Dirichlet form. The energy terms cancel, leaving

Since p(t)=1+\exp(4t/A_\eta) and \int_0^t p'/p^2\le1/2, positivity and approximation give

A pointwise kernel bound and the choice of defect. On \norm J\le K, every spin Hamiltonian has absolute value at most Kn/2, so \mu_{\min}\ge e^{-(K+\log2)n}. Consequently

Self-adjointness and the semigroup property give

The equality is the elementary formula for the norm of a positive kernel with respect to \mu.

We now choose the defect for the prescribed fixed time d. Put t=d/2 and choose \eta=\exp(-a d^{2/3}), where a>0 is a sufficiently small fixed constant. For d large, A_\eta\le C\sqrt a\,d^{1/3} after adjusting the lower threshold for d. Hence p(d/2)^{-1}\le\exp(-c_1a^{-1/2}d^{2/3}). Choose a so that c_1a^{-1/2}>a. The logarithm in Equation (3.37) is then at most C+Cn e^{-a d^{2/3}}. For this fixed d its constant term is absorbed once n is sufficiently large, proving Equation (3.34). All ingredients were common to the cavities, with their dimension n-1; enlarging C gives the stated common bound.□

Proof. Intertwining. The exact product rule is

Use Lf=\sum_j(m_j-x_j)d_jf, d_i^2f=0, and d_id_jf=d_jd_if. Differentiation gives

Uniqueness of the finite-dimensional evolution proves intertwining.

Coefficient bounds with the variance factors retained. For i\ne j, the subtraction formula for \tanh gives

The remainder is uniform for |m_j|\le1 and \max|J_{ij}|\le1/10, including when the local conditional variance is small. Its maximum absolute row and column sums are o(1), because \sum_j|J_{ij}|^3\le\max|J_{ij}|\sum_jJ_{ij}^2. The last condition in Equation (4.1) proves Equation (4.3) and the first bound in Equation (4.4). Finally, |a_{ij}|\le C|J_{ij}|v_j and v_j=w_j(1+x_jm_j)\le2w_j give v_j^2/w_j\le2. This proves the row bound.□

Proof. We first prove the energy inequality, then represent each row evolution by a likelihood change, and finally truncate the expansion in the multiplication term.

Energy control. Put q=K_tp and \mathcal D(q)=\sum_jw_j\norm{d_jq}^2. The jump identity gives

The middle first-order term in absolute value is at most C\norm q\sqrt{\mathcal D(q)} by Equation (4.4). Young’s inequality absorbs one copy of \mathcal D(q); the multiplication term costs 2\norm a\norm q^2. Integrating the resulting inequality along an ordinary trajectory proves Equation (4.6). Retaining dissipation also gives

Likelihood representation and moments. To control the expansion tails, represent the evolution without its multiplication term by a change of the ordinary update rules. In row i replace, at attempts at j\ne i, the field H_j=(JX)_j by H_j-2X_iJ_{ij}; attempts at i are unmodified. The corresponding row generator is L-2x_i\sum_ja_{ij}d_j. If \ell_i[s,t] is its likelihood relative to ordinary attempts, its semigroup, including -I, is

The site-i spin is part of the ordinary trajectory; only its update rule is left unchanged in this representation.

The required moments come from a one-attempt calculation. Let b_H(\sigma)=e^{\sigma H}/(2\cosh H), \sigma\in\{-1,1\}, and r=b_{H+u}/b_H. Taylor’s formula and (\log\cosh)''\le1 show, for every fixed real p, that

For a rate-one clock at j, the compensator of the p-th likelihood moment is therefore at most C_pJ_{ij}^2; the largest entry is small. The exponential martingale formula gives, conditionally on any past,

where I is any fixed union of intervals and |I| is its total length. Negative powers are allowed. All fields and starting states are covered by the same constants.

Truncating insertions and then likelihoods. Rowwise Cauchy–Schwarz gives \norm{T_hp(x)}^2\le e^{Ch}S_h\norm p^2(x). Writing A for multiplication by a, the term with k insertions is

Repeated rowwise Cauchy–Schwarz and the Markov property bound its L^2 norm by e^{Ct}C^kt^k/k! times the terminal L^2 norm. Consequently

which is at most C_He^{-Ht} when A is sufficiently large. Once this insertion cutoff is fixed, only finitely many likelihood intervals remain in each term. For clipping, with any fixed r>0,

Apply this estimate at the first clipped interval, use the ordinary second-moment bounds elsewhere, and sum over at most At+1 intervals and the simplex. Increasing B after A proves the stated bound. For fixed t>0 and a prescribed error, choose a finite insertion cutoff making the factorial-series tail sufficiently small. Then choose a likelihood cap controlling the finitely many retained terms, using the displayed moment estimate. This convergence does not require t\ge1; that restriction was used only in the stated exponential-in-time tail formulation. Every estimate sums over columns, which proves the matrix version.□

Proof. Products. First assign independent column indices to every factor. Their rowwise tensor product is obtained from \bigotimes_lB_l by restricting its output to the diagonal copy of \R^n, an operator of norm one. Next require all occurrences of a slot to have the same index. The map

is an isometry, since the sets S_l cover I. This proves Equation (4.10). There is no absent, freely repeated column index that could contribute a dimension factor.

An entrywise square and arbitrary bounded coefficients. For the second assertion put A=|B_0|^2 entrywise. Its maximum row sum and maximum column sum are at most b_0^2, hence the rectangular Schur test gives \norm A_{\op}\le b_0^2. For any two matrices A,B with the same output rows, their rowwise tensor product satisfies

For example, factor it as D_B(A\otimes I), where (D_Bz)_i=\sum_b B_{ib}z_{i,b}; the norm of D_B is the displayed maximum. Apply this repeatedly to A and the absolute values of the other factors, whose row Euclidean norms are at most b_l. Restrict repeated column indices by the same isometry as before. Finally entrywise domination implies |u^tRv|\le C|u|^tP|v| for the nonnegative dominating matrix P. Taking unit-vector suprema proves the assertion.□

Proof. For spin inputs and fixed labels, the tensors are explicit: a spin input in slot a has T_{\{a\}}=I and every other nonempty tensor zero, while a fixed label has all nonempty tensors zero. Linear maps and sums preserve the claimed bounds. The remaining task is closure under smooth componentwise functions. We use an exact corner expansion so that no compact-range assumption is imposed on a linear query.

First consider F_i=\phi(z_i). Fix I, |I|=r, and a derivative tuple j_I. For every nonempty S\subseteq I put

If the indicated coordinate is flipped in each slot of B\subseteq I, the exact cube identity is

Temporarily regard the 2^r-1 increments as independent scalar variables. Repeated use of the fundamental theorem of calculus gives, for any set V of increments,

The empty term is \phi(z). Apply this with V=\{S:\varnothing\ne S\subseteq B\} and take the alternating sum over B\subseteq I defining T_I. A term survives exactly when

Indeed, otherwise summing over a missing slot cancels the term. The remaining normalization is 2^{-r} and a product of column signs.

Every surviving monomial thus contains enough increments to cover all derivative slots. Its coefficient may still depend on the derivative indices, so the product estimate alone is not yet sufficient. Expand that integral coefficient once at z:

This is an exact integral remainder. The first term is a bounded row coefficient times a product of tensors satisfying Equation (4.14), so Equation (4.10) applies. The remainder acquires a second copy of an increment already present in its monomial. It is therefore dominated by a product with one squared tensor factor, and Equation (4.11) applies, even though its coefficient depends on every derivative index. There are finitely many terms.

For a componentwise function of q scalar inputs z^1,\ldots,z^q, introduce increments t_{b,S} for every channel b and nonempty S\subseteq I. Apply the same anchored expansion in these q(2^r-1) formal variables. The surviving monomials again cover I, and expanding each coefficient once repeats an increment already in that monomial. Bounded mixed scalar derivatives through order q(2^r-1)+1 suffice at this gate. The same two estimates prove the claim. Induction over the fixed calculation completes the proof.□

Proof. We integrate one leaf at a time. The propagation bound controls the evolving derivative slot, while the mixed tensors retain the derivatives in every stored slot. After writing this as a finite linear recursion, we use one common truncation of each evolution operator.

Propagation with auxiliary derivative indices. The point-start bounds in Proposition 4.2 are

They amplify to any auxiliary Euclidean space by applying them to each component and summing squares. In particular the other derivative indices of T_IF can all be retained as Hilbert coordinates without a dimension factor. The truncation is one common linear operator, not a choice depending on an output test vector.

Integrating a leaf and identifying its parent. Integrate a final observed slot m from its parent slot p over an interval of length h. Let y denote all slots other than m. Before identifying the parent and the evolving starting state, set

The two occurrences of the parent are independent arguments of G. The copy identity at the start of this subsection gives

All other frozen-slot differences commute with S_h and K_h. For p\notin I, the resulting tensor is simply S_hT_IF restricted to z=y^{(p)}. If p\in I and I'=I\setminus\{p\}, its three terms are respectively

The sign multiplier in the last term acts on the derivative index. In the last two terms K_h acts only on the slot-m derivative coordinate; all other indices are auxiliary Hilbert coordinates. The restriction \mathsf R_{j_m=j_p} is a coordinate restriction within that Hilbert space, hence has norm one. It never identifies an output row with an input column across the chosen flattening.

We can now estimate the three terms without changing the tensor flattening. Fix a deterministic output vector c and apply the transpose of each tensor to c. Jensen for S_h and Equation (4.17) give

and C_I(H)\le C_I(F) otherwise. Taking the supremum over c after these deterministic inequalities proves the operator bounds. Only one difference per original slot occurs. Iterate this finite elimination along a path, or remove leaves of a finite causal tree. Other branches can be held as parameters during each leaf integration. If the original calculation does not use the joining state, adjoin it as a dummy slot. Its initial difference tensors are zero. Reversibility makes a stationary reverse edge another use of the same S_h.

One truncation for the complete derivative matrix. The leaf elimination is a finite linear recursion on the family of mixed tensors. Open that recursion and replace each K_h by the same chosen K_{h,M} at every occurrence with that interval. Ordinary S_h expectations may remain exact. Each step has the bounded norm in Equation (4.20); replacing one step costs at most the corresponding \varepsilon_{h,M} from Equation (4.17). Telescoping the finitely many steps gives Equation (4.15). This approximates the derivative of the exact conditional expectation. It does not assume that A_M is the derivative of a separately approximated scalar message, and no operator supremum is moved through an expectation.

Weighted errors and recombination. Finally, \norm{D_uE}_{\HS}^2=\sum_i|u_i|^2\norm{\operatorname{row}_i(E)}^2 proves Equation (4.16) pointwise. The recursion and the common truncation are linear in the endpoint observables; therefore actual and predictor terms can be recombined before the centered label is used. Opening their ordinary conditional expectations as causal branches is done only after these uniform norm errors have been removed.□

Proof. Forward evolution preserves the density bound. Reverse conditioning introduces denominators, so we first remove joining states where those denominators are too small. All forward marginals have relative density at most e^{o(n)}, since S_t is a positive L^\infty(\mu) contraction. A reverse conditional edge from time t to s has, relative to the stationary reverse edge, the factor h_s(X_s)/h_t(X_t). Discard endpoints where h_t<e^{-\delta n}. Their actual marginal probability is at most e^{-\delta n}, and their polynomially weighted probability vanishes. The remaining joint density on a tree with r reverse edges is at most \exp\{o(n)+r\delta n\}. Choose \delta<c/(2r+2) and compare probabilities. The same calculation after Fubini proves the integrated assertion. For a finite tree with more than one original marginal, apply this argument successively at each copying operation.□

Proof. We first compare absolute log densities on retained forward and reverse bins. We then retain the quadratic log-likelihood cost to obtain the sharper entropy estimate. In both arguments the exceptional events are discarded on the original forward marginals before forming the weighted conditional law.

The forward log-density bound. Write \Delta(u)=J(X_u-X_s) in a bin starting at s, and let N_{i,b} be its attempt counts. On a fixed mesh, Poisson exponential bounds imply, with exponential accuracy,

The log ratio at an attempt at i is bounded by 2|\Delta_i|. For a bounded predictable integrand the compensated attempt sum obeys

where V bounds its predictable bracket and b its jumps. Apply this after cutting |\Delta_i| at B, then use \sum_i|\Delta_i|\le\sqrt n\norm\Delta for its compensator. The bounded part costs O_T(\sqrt h)n. For the remaining part, \sum_i\Delta_i^2\le Chn bounds the compensator by Chn/B. Its individual jumps are O(\sqrt n), its bracket is O_T(n), and the displayed inequality makes a fixed positive multiple of n an event of superpolynomially small probability. This proves the forward density comparison.

Reversed bins. For a reversed bin separate the forward within-bin difference from J(X_{s+h}-X_s). The latter vector has norm O(\sqrt{hn}). Its entries larger than B occupy at most Chn/B^2 sites. Independently of how this set is selected, the sum of N_{i,b}^2 over such a set is bounded by the sum of the largest Chn/B^2 squared Poisson counts. For each fixed h, truncating counts at a level K gives

with superpolynomial accuracy, by first choosing K and then B. Cauchy–Schwarz bounds the across-bin excess by o_B(hn). The bounded part has exponential stationary tails by reversibility. Discard the excess on each actual edge in its original forward direction, before applying Lemma 4.7 to the bounded part. Summing the finitely many bins proves the first assertion.

The entropy of the retained actual tree. For the entropy assertion, first compare Q with P. Let c_n=o(n) bound the logarithms of the original forward marginal densities. Discard reverse joining states with marginal density below e^{-a_n}, where a_n=o(n) and a_n/\log n\to\infty. Their actual marginal probabilities are at most e^{-a_n} and remain negligible under the permitted polynomial weights. On the retained set, the reverse density factors h_s(X_s)/h_t(X_t) bound the total tree log density by C(c_n+a_n). Polynomial reweighting and retained event normalization add O(\log n). Thus H(Q\mid P)=o(n). The change of reference is exactly

Changing to the frozen reference. The root laws cancel in the last ratio, leaving the root-directed within-bin refresh log ratios. The absolute estimate already proved would lose the desired factor h. We instead keep the quadratic log-cosh remainder. Conditional on the old configuration, the Kullback–Leibler divergence between two refresh coins at fields H and H+u is at most u^2/2. Consequently its predictable integral over all bins is at most

on Equation (4.23). For the centered log likelihood M with field differences cut at B, stop when Equation (4.23) fails. Its jumps are bounded by CB and its predictable bracket is at most C_T hn: the conditional variance of a refresh log ratio is at most a constant times the squared field difference. The bounded-jump exponential martingale inequality therefore gives

Apply the entropy inequality \E_QM\le\theta^{-1}\{H(Q\mid P)+\log\E_Pe^{\theta M}\} with fixed positive \theta\le c/(1+B). At fixed h,B, the additional o(n) entropy costs o(n), and the centered term costs O_T(hn). The same estimate for -M gives the reverse bound. It remains to remove the field cutoff. For the unbounded excess use the preceding forward compensator bound and the reversed-bin count bound. At fixed h its normalized entropy contribution tends to zero with the cutoff; choose it below h before taking the next mesh limit. Conditioning on any retained event of polynomially significant probability adds O(\log n) entropy. All superpolynomial discards remain negligible under the permitted polynomial weights and polynomially significant conditioning. An arbitrary o(n)-entropy tilt would not have this property and is not being allowed here. The chain rule over the finite tree, with the same argument on every root-directed edge, proves Equation (4.22).□

Proof. We prove a two-point entropy estimate and then tensorize it. The small minority mass enters through the factor \log(1/p); keeping that factor is what makes the subsequent mesh refinement converge.

One coordinate. For a nondegenerate Bernoulli coordinate let its minority mass be p\in(0,1/4], put L_p=\log(1/p), and set \Lambda_p(t)=\log(1-p+pe^t)-pt. For |t|\le L_p/2,

Since \Lambda_p(0)=\Lambda_p'(0)=0, Taylor’s formula gives \Lambda_p(t)\le Ct^2/L_p in this range. Outside it, \Lambda_p(t)\le|t|\le2t^2/L_p. Jensen also gives \Lambda_p(t)\ge0.

If a two-point function G has values a,a+t, with the second value on the minority atom, direct calculation yields

Convexity at t, evaluated at 2t, gives t\Lambda_p'(t)\le\Lambda_p(2t)-\Lambda_p(t) for either sign of t. Consequently

This obtains the entropy inequality by convexity, without differentiating a bound on \Lambda_p. A deterministic coordinate contributes zero entropy.

Tensorization and the exponential-moment bound. Apply Equation (4.27) conditionally with G=\theta F and |t|=2|\theta d_iF|. Entropy tensorization gives

For \psi(\theta)=\log\E_\pi e^{\theta F} this bounds \theta\psi'(\theta)-\psi(\theta). Integrate the derivative of \psi(\theta)/\theta from zero to obtain Equation (4.25); use -F for negative \theta.

Changing the product law. Finally, the entropy variational inequality gives

Optimize over \theta>0 and apply the same argument to -F. This proves Equation (4.26).□

Proof. We first compare one terminal test under the actual and autonomous frozen laws. This requires two telescopes with a common ordinary semigroup reference. We then treat stored states by backward leaf elimination and smooth each fixed simulation.

The actual-law telescope. Choose a regular h-mesh augmented by the finitely many prescribed observation times, so that 0=t_0<\cdots<t_N=T, \delta_b=t_b-t_{b-1}\le h, and N\le C_T/h. First consider one terminal test F, with \sup_x\norm{dF(x)}\le B/\sqrt n, and put

The point-start bound in Proposition 4.2 gives \sup_x\norm{df_b(x)}\le C_TB/\sqrt n. Project Q and P_h to the mesh states and put q_b(\cdot\mid\mathcal F_{b-1}) =Q(X_{t_b}\in\cdot\mid\mathcal F_{b-1}), where \mathcal F_{b-1}=\sigma(X_{t_0},\ldots,X_{t_{b-1}}). The entropy chain rule and data processing give

The omitted initial-state entropy is nonnegative. Since f_{b-1}=\mathsf P_bf_b, the exact telescope is

Both kernels on the right are evaluated at the actual joining history; \mathsf P_b depends on it only through X_{t_{b-1}}. Insert \pi_b between them and apply Lemma 4.9 separately to q_b and \mathsf P_b. The second entropy has the uniform bound

The first inequality is data processing, the second is the refresh-coin log-cosh bound, and the last uses \E_x\norm{X_u-x}^2\le4nu and the bound on \norm J. In particular it holds for every joining-state law. Writing H_b for the sum of these two conditional entropies yields

Here dimension tends to infinity at fixed h.

The autonomous frozen-law telescope. Let \widehat P_h be the autonomous frozen simulation started from the same initial law Q_0. The second, zero-tilt telescope is

Its conditional kernels equal \pi_b, so their entropy relative to \pi_b is zero. The ordinary-kernel entropies still sum to at most Cn\sum_b\delta_b^2\le C_Thn, uniformly in Q_0. Consequently this second difference is at most C_TB/\sqrt{\log(1/h)}. Since \E_QS_TF(X_0)=\E_{Q_0}S_TF(X_0), the triangle inequality proves Equation (4.28), after enlarging C_T, for the difference between the actual and frozen tests.

Stored states and admissible conditioning. Stored parameters may be included by writing the test as F(y,x) with the same derivative bound uniformly in y. Disintegrate the reference tree at the joining state and the passive variables y. The active future must retain its ordinary, respectively frozen, Markov kernels under that reference law. Past variables and other branches conditionally independent at the joining state have this property. Conditioning on a later endpoint of the active branch generally does not. The conditional entropy of the active future is bounded by the total tree entropy by the chain rule, so both telescopes apply after averaging over (y,X_0), and the simulation retains their joint initial law. A backwards leaf elimination gives precisely these admissible disintegrations. For replicated predictions the other copies are passive and the test bound is uniform in their realized values.

Smoothing a fixed simulation. A fixed frozen simulation is a finite calculation using J, independent site clocks and uniforms, and scalar threshold maps. Replace each threshold by a smooth transition in a strip of width \varepsilon. Fresh uniforms put at most O(\varepsilon) mass in that strip. Induction over finitely many operations, using the bound on \norm J, shows that the normalized mean square error tends to zero. Derivative constants of the resulting simulation can depend on its fixed mesh and smoothing width.

Backward replacement and derivative control. Replace the last observed state first. The smooth simulation produces a new smooth test of earlier states. Fix that calculation and its finite derivative constants before choosing the accuracy at the next earlier replacement. At a branch, duplicate the branching state as an input to the child calculations. For replicated predictions, hold all other copies as passive parameters while replacing one copy; the conditional entropy chain rule bounds its cost. Thus its derivative constant depends only on its own currently fixed test, not on the accuracy subsequently selected for a passive copy. This backwards, nested construction avoids requiring constants uniform over all meshes.

For clarity, the derivative bound for a fixed smooth recipe follows without any probabilistic assertion. If a vector z has bounded half-difference matrix, then, off its diagonal,

The absolute row and column sums of the error are bounded by the squared operator norm of that matrix. Keep its diagonal exact. The Schur estimate in Lemma 4.3 and induction give a bounded derivative matrix. A normalized empirical inner product has difference norm O(n^{-1/2}) by the product rule. The remaining assertion concerns square tails of linear queries. Test such a tail with a smooth excess function whose second derivative is bounded and whose first derivative is at most C(1+|z|). The same Taylor calculation, using bounded column square sums, gives a difference bound independent of the tail cutoff. For each fixed simulation Gaussian regression gives square integrability and vanishing square tails. The uniform comparison just proved then transfers this assertion to the original paths.□

Proof. We first transfer exponential estimates to a Gaussian experiment in which the initial spin is planted. Under the annealed one-replica weight, the initial spin \sigma is uniform and the off-diagonal disorder has the form

where G is independent GOE. The correction from its Gaussian diagonal has norm tending to zero. Its mean shift on the diagonal has norm O(1/n). The Gibbs-to-planted comparison loses only e^{o(n)}: indeed

where R_n is a sum of iid signs. Gaussian concentration of \log Z and Paley–Zygmund then give \log Z-\log\E Z=o(n) in probability. Thus an annealed exponential failure transfers to an exponentially small Gibbs failure on a high-probability disorder event. This calculation explains why the planting comparison remains valid throughout \beta<1.

We can now describe exactly which randomness remains after a query. Let U have orthonormal columns, put P=I-UU^t, and suppose Y=GU has been revealed. Symmetry gives U^tY=Y^tU. The conditional matrix is

with G' a fresh GOE independent of the revealed sigma-field. To verify independence, test the GOE covariance

against a matrix supported in P\R^n and against each revealed column. The covariance is zero. Joint Gaussianity proves independence, and the displayed revealed part is the unique symmetric completion with those columns and zero P–P block.

Conditional Gaussian regression along iterative spin-glass calculations also underlies Bolthausen’s construction of TAP solutions [7, Sections 2–3 of the arXiv version]. Here it is used for finite causal path simulations and their extra recipe queries; the required empirical and operator conclusions are proved separately. The same formula applies adaptively. Given previous answers, normalize the residual of the next query to a unit vector q\in P\R^n. Its new perpendicular answer has law n^{-1/2}(P-qq^t)g, and its component along q has law \sqrt{2/n}\,zq, with g,z independent standard Gaussians. After this answer the remaining compression is again independent. Projection coefficients are empirical inner products of previous calculation vectors. Thus every fixed finite recursion reduces to iid site Gaussians and seeds, finitely many empirical scalar coefficients, and finitely many rank corrections.

It remains to control the finite recursion uniformly at a fixed accuracy. For bounded smooth tests, iid empirical concentration and Gaussian quadratic tails propagate through each successive query. If a query residual has length tending to zero, omit that query; the spectral bound makes the normalized square error vanish. Equivalently, fix a residual-length threshold, propagate its deterministic error through the finite recursion, and send the threshold to zero only after the dimension limit. Fresh uniform thresholds are continuity tests. They can therefore be smoothed and the smoothing subsequently removed without changing the limits. This proves deterministic empirical convergence with exponentially small failure at each fixed positive tolerance.□

Proof. We first construct the limiting inner product, then identify conditional means in its completion. Apply Proposition 4.10 and Lemma 5.2 to successively fixed smooth simulations. The comparison error in Equation (4.28) makes their empirical limits Cauchy. If a deviation had polynomially significant probability, conditioning on it would cost only O(\log n) extra entropy. The same comparison would then force the original limit on that event, a contradiction. Apply this argument to the weight-normalized tilted laws to obtain the weighted typical assertions. The limiting Gram matrices are positive semidefinite; their quotient by the null space, followed by completion, defines \mathcal H.

For the second step, let z be a path recipe and put m(X_0)=\E[z\mid X_0] on the stationary rooted tree. Replace the future by a fixed finite smooth simulation and average its seeds. An average of R independent seed copies has spatial L^2 error at most C/R from that simulated conditional mean. At each fixed R, it is a one-state recipe under the seed convention above. To compare it with the actual m, expand the squared error into \norm m^2, its cross term with the simulation, and the simulation’s squared norm. Realize these three terms on two independent branches sharing X_0, with the same nested approximation on both copies. Proposition 4.10, applied while the other branch is passive, controls each term and proves the conditional mean-square approximation. They also give

for distinct branches; here the z_i are written without the normalizing factor. Reversibility identifies stationary reverse predictors, and Lemma 4.7 plus the finite-simulation comparison gives the same pair centering on actual conditional trees.

The remaining assertion is orthogonality to a path recipe. Let u satisfy Equation (5.1) and use R centered replicas z^{(r)}-m. Their Gram matrix tends, conditionally in the typical weighted sense, to a bounded diagonal matrix. Bessel’s inequality therefore gives

All summands have the same expectation; divide by R, take dimension large, and then send R\to\infty. The m term vanishes by recipe approximation and Equation (5.1). For a matrix word Q, Q^tz is another path calculation with negligible square tails after clipping its linear outputs. The same argument applied to it proves the last assertion.□

Proof. Let c_j(x)=(1-x_j\tanh H_j)/2 be the flip rate. Its logarithmic field derivatives are uniformly bounded: the first is -x_j-m_j, and the next two are bounded functions of m_j. For row i the field perturbation at j is u_{ij}=-2x_iJ_{ij} on its specified interval mask. The flip likelihood has logarithm

Expand this logarithm in J_{ij}. The linear term is a centered score, the quadratic term is its bracket/compensator contribution, and the remaining terms have absolute expectation at most C_T\sum_j|J_{ij}|^3=o(1), uniformly in i. Higher moments are controlled by Equation (4.7). Both the score integrands and the coefficients of J_{ij}^2 are bounded smooth site functions. We may consequently cap the exponential at a fixed continuity value, or use a smooth cap, with arbitrarily small spatial square-mean loss.

Freeze the bounded integrands on short bins. A bin with two or more attempts at one site has probability O(h^2); summing its squared Poisson counts shows that ignoring all such bins costs O_T(h) in normalized mean square. On the remaining bins the spin values at the two ends determine whether a flip occurred. Bounded derivatives, Equation (4.23), count truncation, and predictable rate averaging show that freezing the field-dependent integrands also has vanishing normalized mean-square cost. Multiplication by J preserves that cost.

The preceding freezing estimate still leaves the outside row-spin factor x_i(s). Freeze this factor at the beginning of its bin. The resulting error is a vector martingale, with row-i bracket bounded by

The total bracket trace is O_T(hn) on the Poisson count event, while the operator norm of its bracket density is bounded by C\norm J^2. A vector jump is bounded in Euclidean norm. Apply the scalar bounded-jump inequality to its squared norm, stopped at a fixed multiple of \sqrt n: its jumps are O(\sqrt n) and its bracket is O_T(n). For every fixed tolerance exceeding the O(h) bracket trace, failure has superpolynomially small probability. Thus the fraction of rows with a significant score error tends to zero with arbitrary polynomial accuracy. Bracket terms need only absolute rate averaging. Use a finer inner mesh when approximating integrands inside a bin whose outside factor has already been frozen.

After these error estimates, the retained score can be written as a finite calculation from observed spins. On a bin [s,t], set X=X_s, Y=X_t, h=t-s, and freeze m=m(X) and v=v(X). The first log-rate derivative is -X_j-m_j; outside the negligible multiple-flip bins,

Since (-X_j-m_j)c_j=-X_jv_j/2, the vector of frozen row scores is

where products inside braces are componentwise. In particular the mixed query J(mXY) is retained. It need not be a finite sum of sitewise products of single-time outputs. Quadratic bracket terms are finite calculations using J\circ J and bounded site factors; this matrix has bounded operator norm and bounded absolute row sums. Their values are therefore uniformly bounded at each fixed mesh.

For the cap of the exponential, choose a smooth bounded function \psi of the logarithmic score, with globally bounded derivatives, which agrees with the desired clipped exponential outside an arbitrarily small smoothing interval. An outside spin multiplying an unbounded query is handled by its two sign cases, not by assuming the map (a,u)\mapsto\psi(au) has bounded derivatives for arbitrary bounded a and unbounded u. Explicitly,

For finitely many bins, split the finite sign pattern of all the frozen outside row spins. Each summand is a bounded product of spin selectors and a smooth cap applied to a fixed signed linear combination of the mixed J queries and the bounded quadratic terms. All scalar coefficient bounds are global, independently of the query values. Products of already bounded factors have smooth bounded extensions. Thus Lemma 4.4 applies to every fixed capped calculation, including all its mixed temporal slots.

Fix the mesh, smoothing, sign decomposition, and cap before taking the dimension limit. When these choices are subsequently refined, their derivative constants may grow. The refinement removes the spatial approximation errors established above; it makes no claim that derivatives of those errors converge. Observation times can be strictly inside their bins, and can be chosen one-sidedly at endpoints. For predictors first take these fixed calculations, then their exact stationary conditional means. Conditional expectation contracts spatial L^2; replicated pair tests and Proposition 4.10 give the one-state recipe approximation. When an exact predictor is differentiated, use its fixed multitime calculation on an added stationary causal branch and apply Lemma 4.5. The opened tensors are treated by Proposition 5.12. No score estimate on an artificial reverse branch with a tilted endpoint, and no derivative estimate for a tracial approximation error, is used.□

Proof. The norm proof has three stages: types of positive proportion, rare coordinate types, and a concentration estimate uniform in the type counts. We begin with the first stage. Every finite-dimensional Wick contraction uses the empirical trace \tau_n=n^{-1}\Tr, so that internal returning pairs cancel exactly. During the moment calculation, p_a means the empirical type proportion and \tau still means the limiting state. Suppose first that p_a\ge\alpha>0 for every type. If every limiting proportion is positive, this holds eventually after decreasing \alpha. For a homogeneous Wick string W_F of degree k, use the product-state norm

For k=0 the string is diagonal and has norm at most \alpha^{-1/2}\norm F_2, so the preliminary bound below is immediate. Assume henceforth in the moment calculation that k\ge1. Let r be even and consider n^{-1}\E\Tr((W_F^*W_F)^{r/2}). Gaussian pairings glue the rk sides of a polygon in either orientation. There are E=rk/2 paired edges; if V is the number of free index vertices, set e=E-V+1\ge0. A connected gluing contributes n^{-e} times a type average. The two GOE covariance terms are precisely the two possible band orientations, including their multiplicities for a loop.

Internal immediate untwisted returns cancel. Indeed selecting a specified pair of adjacent edges and applying their covariance gives exactly the trace contraction subtracted in the Wick definition. Inclusion and exclusion over disjoint pairs cancels every diagram with at least one such internal return. Hence every degree-one vertex of a surviving graph lies at a boundary of one of the r strings. There are at most r leaves, and the degree identity gives

We need a diagram count whose loss grows polynomially with the word degree. Suppress every chain of degree-two vertices, keeping one vertex for a pure circle. By Equation (5.10), the remaining core has s\le C(r+e) edges. Delete the e edges outside a spanning tree. Reversing local cyclic orders at vertices untwists all tree bands, and a rooted plane tree on s-e edges has at most 4^s encodings. Reinsert the deleted bands in ordered corner slots. Their endpoints, order, and twist signs have at most (Cs+C)^{Ce+C} choices. Subdividing the core edges and choosing the rooted boundary start adds a factor at most

Using s\le C(r+e), assign the O(r) core edges to the factor (C(1+k))^{Cr} and the additional O(e) edges to (Crk)^{Ce+C}. Thus the number of diagrams of excess e is at most

Loops, parallel bands, and the pure-circle case are included by the same encoding. No factor exponential in rk remains: tree-band twists were removed by vertex reversal, and only the e extra bands carry independent twist choices.

The count must now be combined with a bound for each type average. Delete e nontree edges. At most 2e strings cross these edges; bound each of them by

Every remaining string is a simple tree path, since a repeated tree edge would force an internal immediate return. To count repeated type variables, let c_v be the number of remaining string occurrences at vertex v, and let b_v count their endpoints there. Each internal occurrence uses two edge incidences and each endpoint uses one. As each paired edge has at most two occurrences, 2c_v-b_v\le2\deg(v), using the degrees of the original graph. Thus Equation (5.10) and \sum_v b_v\le2r give

Replace each occurrence beyond two by a fresh independent type variable. For a nonnegative summand the comparison costs at most \alpha^{-1} per replacement: an equality constraint can increase a product-state integral by at most the inverse of its smallest atom. Each remaining variable occurs at most twice, never twice within one tensor. Successive coordinate Cauchy–Schwarz therefore bounds the product integral by one \norm F_2 per surviving string. Altogether a diagram is bounded by

We can now sum the diagram bounds. Choose an even r of order \log n, keeping k,\alpha fixed. Equations (5.11) and (5.12) give

The sum is eventually at most two. Markov’s inequality for the trace power, increasing the constant in r/\log n if necessary, proves

The strategy of obtaining a polynomial word-length norm bound and then removing its loss by taking powers has an antecedent in Haagerup [17, Lemmas 1.3–1.4, pp. 283–286]. His setting is free-group convolution; the Wick/GOE diagram bound, rare-type treatment, and adaptive-label passage here are proved locally. To turn this preliminary estimate into the sharp comparison, decompose a degree-k polynomial into homogeneous Wick strings. Complete noncrossing pairing identifies the sum of their squared tensor norms with the vacuum square norm. Cauchy–Schwarz and Equation (5.13) therefore bound the matrix norm by C_\alpha(1+k)^{C+1}\norm{P(s,D)}. Apply that bound to (P^*P)^m, whose degree is 2mk. Since \|(P^*P)^m\|=\|P\|^{2m} in both representations, it gives

Letting m\to\infty removes the polynomial loss. For each fixed polynomial and each fixed power, there are only finitely many empirical contraction coefficients, and they converge to the coefficients formed with \tau. The difference of the resulting matrix polynomials is o(1) in operator norm on every fixed spectral ball. Thus empirical and limiting Wick conventions have the same norm limit. Dimension is sent to infinity before the power is increased; no rate for \tau_n\to\tau is required. This proves Equation (5.9) when all types have positive proportion.

The positive-proportion case is complete. We next keep the coordinate space of the rare types instead of discarding it. Write this space as R_n, with dimension m_n=o(n), and split

The internal rare block has norm o(1). Conditional on the bulk block and its deterministic diagonals, V_n has independent N(0,1/n) entries. For every fixed bounded bulk word A_n, Gaussian quadratic-form concentration, polarization, and a net of size \exp(O(m_n)) imply

Indeed a unit-vector quadratic form has variance O(1/n) and fixed accuracy tail e^{-cn}, while the net entropy is o(n). The normalized bulk traces converge by ordinary Wick pairing. Equation (5.14) holds jointly for every finite list of A, including A^*A'.

The remaining task is to turn the rare-coordinate Gram comparisons into an operator-norm bound. Take a subsequence realizing a limiting norm witness and form its Hilbert ultraproduct. The closed linear span of \{A V\xi:A\text{ a bulk word},\ \xi\in R\} is a reducing subspace for the bulk word algebra, and Equation (5.14) identifies it isometrically with L^2(\mathscr A,\tau)\otimes R through A\Omega\otimes\xi\mapsto AV\xi. Here \mathscr A is the bulk Gaussian/diagonal algebra, whose norm has just been controlled. On this subspace, V\xi=\Omega\otimes\xi and V^*(A\Omega\otimes\xi)=\tau(A)\xi. The orthogonal bulk complement has no coupling to R and obeys the bulk norm bound. On the reducing sum of R and its bulk cyclic space, every return through a bulk word is exactly its vacuum expectation, while the diagonal on R retains its actual type value. This is the representation induced from that base in Equation (5.8). Its polynomial norm is bounded by the faithful-base Fock norm. The norm on the orthogonal sum of the two reducing pieces is the maximum of their restricted norms, proving the rare-type assertion.

Both the positive-proportion and rare-type comparisons are now proved. To obtain the stated exponential rate, project GOE onto the operator-norm ball of a fixed radius K>3. The bare-GOE comparison already puts a median of \norm{G_n}_{\op} below 3 for large n. This norm is one-Lipschitz in Hilbert–Schmidt norm, so Gaussian concentration gives

A word of degree k with diagonal bound M satisfies, by telescoping,

where L depends on k,M, the coefficients, and the spectral cutoff, but not on \alpha, the type counts, or any compression rank. The projection is nonexpansive in Hilbert–Schmidt norm. Gaussian concentration for the clipped polynomial, combined with its median comparison to the limiting target, therefore gives the original polynomial the failure bound

For 0<a\le1, reducing the positive constant c absorbs the escape term and the prefactors into 2e^{-cna^2/L^2} for sufficiently large n. The preceding convergence puts the clipped median below the limiting norm target plus any fixed slack. These median comparisons are uniform over type counts: otherwise take a failing sequence, extract convergent proportions, and use the full-support or rare-type case just proved. One can realize this continuity on a fixed letter space \R^q with creation vector (\sqrt{p_a})_a and a faithful base; creation changes in norm by at most \norm{\sqrt p-\sqrt{p'}}_2. This completes the proof.□

Proof. The parameters are chosen by comparing the entropy of all allowed changes with a fixed Gaussian failure rate. First fix the word family, caps, bin resolution, spectral cutoff, and operator slack a>0. For one label, changing at most \delta n entries with at most q replacement choices gives

For finitely many labels add these entropies. At fixed positive \delta this is a small positive multiple of n, not o(n). Let c a^2/L^2 be the Gaussian rate from Lemma 5.5. Choose \delta so that the sum in Equation (5.15), divided by n, is below one quarter of that rate. The union bound then gives a common exponential event for every possible modification. Polynomially many type-count cases cost only o(n).

This common event is enough for tracial label approximations. Indeed, if n^{-1}\sum_i|d_i-\widetilde d_i|^2\le\eta^2, then at most \delta n sites satisfy |d_i-\widetilde d_i|>\eta/\sqrt\delta. Off those sites, binning and polynomial telescoping absorb the small uniform error. On those sites, the comparison for every modification has already been secured by the union bound.

Only after fixing these label approximations choose a different mesh for the path density. By Lemma 4.8, its specific log-density price \kappa can be made smaller than another quarter of the same Gaussian rate. A density tilt e^{\kappa n+o(n)} leaves a strictly positive exponential failure rate. Then take n\to\infty. The minimum positive type proportion never enters the rate: it only enters the preliminary moment constant used to establish the median. Finally let the sparse fraction, density cost, and operator slack tend to zero in this nested order.□

Proof. We must pass from the independent Gaussian comparison to the matrix left by the path queries. Fix the finite label calculations first. After conditioning on the frozen path data, the spin inputs at each mesh time are fixed, and the likelihood sees J only through its images on those input columns. Retain an orthonormal basis of their nonvanishing residual directions, together with the input seed columns and the finitely many adaptive queries required by the fixed labels. Formula (5.5) leaves an independent compressed Gaussian matrix. The removed part is a bounded sum of dyads of input and innovation vectors, of rank at most twice the number of retained directions.

Small query residuals are handled before any matrix inverse is needed. At the fixed density mesh, process the queries in causal order and omit residuals shorter than a fixed threshold \theta. Use the orthogonal projection of each omitted query when evaluating later fields. This still defines a probability scheme: every refresh uses a normalized Bernoulli probability depending on its past. In particular, no lower bound on the smallest eigenvalue of an arbitrarily long mesh Gram matrix is assumed. The omitted spin input has normalized length at most \theta. The spectral bound, finite-bin count truncation and Lemma 4.8 bound the resulting log likelihood error by \kappa_\theta n, with \kappa_\theta\to0 at the fixed mesh. Subsequent label queries with small residual can be omitted using normalized square-mean continuity. Thus nearly singular regressions cause a small density error, not an inverse-Gram operator factor.

Before applying a full-GOE word comparison, augment the residual PG'P by its missing Gaussian blocks, independently of the actual data. Conditional on the revealed columns and fixed labels, this gives a full independent GOE G' on an enlarged probability space. The path-density change depends only on the compressed residual; the independent added blocks have density cost one. A full-GOE operator comparison is applied only to this completed matrix. In particular, no unrestricted polynomial norm domination for PG'P by the full-GOE Fock norm is asserted.

We now apply the comparison on that enlarged space. The GOE map from standard Gaussian coordinates has Hilbert–Schmidt Lipschitz scale \sqrt{2/n}. For a Gaussian good set of probability at least 1/2, isoperimetry bounds the probability of Frobenius distance greater than r from the set by 2e^{-cnr^2}. Use as the good set the full-GOE event for the required fixed-word comparisons in Lemma 5.5. Under a pointwise density price e^{\kappa n+o(n)}, taking r>C\sqrt\kappa+a leaves an exponential error. Telescoping a bounded polynomial changes its operator norm by at most Lr. Both L and the Gaussian scale are independent of the number of queried directions; compressing a perturbation has norm at most its original norm. The countably many fixed approximations are taken in successive limits, not intersected as a dimension-dependent word family. Sparse label changes are covered by Lemma 5.6 on this same event.

Only the hidden word inner products remain to be compared with the actual compressed matrix. The discrepancy between the actual and completed matrices has range and corange in the span of the queried directions and finitely many added isotropic Gaussian columns. Expand each polynomial discrepancy as a sum of dyads. Its two vectors lie in the cyclic spaces of those directions under the actual bounded words and labels. The queried directions and their images are path vectors, so Lemma 5.3 makes them orthogonal to hidden endpoint vectors. For an added Gaussian column g of covariance at most I/n, condition on all actual data. For any actual bounded word Q and actual endpoint vector u,

The same estimate treats the finitely many added columns jointly and, by a further finite polynomial expansion, their word orbits. Thus the added cyclic directions vanish in hidden word inner products as well. The hidden vectors do not anticipate the added blocks: these blocks were sampled only after fixing the actual data. Removing them at this stage proves the comparison on the hidden complement that is required here, while retaining the rare diagonal base of the full-GOE model.

This gives a positive matrix-valued functional for every finite family of hidden vectors on the algebra generated by D and s, bounded by the norm in Equation (5.8). To represent creation as well as the self-adjoint sum, lift the word inner products to the algebra containing \ell. Use a norm-preserving Hahn–Banach extension of the positive functional from the unital algebra generated by D,s to the one generated by D,\ell. Positivity follows from preservation of the norm and its value at the identity. Apply this to matrix amplifications to retain all the joint vector states, and use the Gelfand–Naimark–Segal (GNS) construction. The resulting lifted vectors have exactly the original joint word inner products, while \ell^*D\ell=\tau(D)I holds in the ambient representation. If a finite star has a permutation symmetry one may average the extension over that finite group and take direct sums. This proves precisely the one-tree assertion; it imposes no symmetry joining independent choices at two different endpoints.□

Proof. First prove the entry predictions, and then strengthen the resulting Hadamard estimate to an operator bound. For independent GOE and deterministic binned diagonals, each entry of a fixed word is Frobenius Lipschitz on a spectral cutoff, with a dimension-free constant. Gaussian concentration and a union bound over n^2 entries therefore give uniform convergence to the entry expectations at every fixed positive accuracy. Conjugation by diagonal sign matrices makes all off-diagonal expectations zero. A fixed-degree Wick expansion of the diagonal entry leaves precisely its complete noncrossing contraction with the outer site type fixed. Every other pairing has at least one additional index identification and costs O(1/n), uniformly in the outer type, including a rare type. This proves Equation (5.3). The same expansion for the normalized trace averages the remaining outer diagonal and gives the vacuum trace rule.

For the off-diagonal Hadamard product, its expectation depends only on the two endpoint types. A Wick pairing connecting the two distinct fixed endpoints has at least one fewer free index than paired edges, so each mean entry is O(1/n). If E_a is a type projection, this mean matrix is, up to an O(1/n) diagonal correction, a sum of dyads

It is visible. We must still prove that the centered product is small in operator norm; entrywise smallness alone would not prove it.

We obtain the operator bound by treating diffuse and large coordinates separately. Start with unit vectors u,v satisfying \max(\norm u_\infty,\norm v_\infty)\le\zeta and consider u^t(Q\circ R)v. Perturbing G by \dot G gives terms of the form

The word differentials are Hilbert–Schmidt bounded by C\norm{\dot G}_{\HS}. Moreover,

Hence this bilinear statistic has Gaussian concentration rate \exp(-cn a^2/\zeta^2). Nets of the two unit balls at a fixed resolution have size \exp(C_an). Choose \zeta sufficiently small after the required accuracy a. A union bound controls all diffuse unit tests.

To cover all unit vectors, separate the coordinates of magnitude greater than \zeta. Each vector has at most \zeta^{-2} such coordinates. After removing the diagonal,

and the same holds for columns. Thus the portions involving the finitely many large coordinates have norm o(1) uniformly in their locations. The finite-net approximation is legitimate because the Schur bound gives a uniform a priori operator norm. Combining sparse and diffuse parts proves that the centered off-diagonal Hadamard product is o_{\op}(1).

Visible dyads do not spoil these claims. A bounded path vector b with uniform spatial square tails has \max_i|b_i|\to0, since

with dimension first and then M\to\infty. Thus a visible dyad has negligible maximum entry. Its Hadamard product with a bounded matrix is D_bQD_c and has norm at most \norm b_\infty\norm Q\norm c_\infty=o(1). Finite-rank approximation extends the argument to the visible closure.

Finally use Lemmas 5.7 and 5.6 to transfer the uniform Gaussian comparisons to actual path labels. Polynomial norm differences tolerate the small Frobenius comparison error. Uniform entry predictions retain the actual outer diagonal on rare sites. The dyads of binned type indicators are limits of bounded path-recipe dyads by tracial approximation. This proves the stated Hadamard reduction and diagonal formula on admissible trees.□

Proof. We prove closure by induction over the recipe operations. The spin input satisfies E(x)=I, and a query by J left-multiplies its difference matrix by J. For the nonlinear step, assume closure for z and apply a bounded smooth scalar function \phi. Off the diagonal, use z_i(x^j)=z_i(x)-2x_jd_jz_i in Taylor’s formula to obtain

The remainder’s maximum row and column absolute sums tend to zero:

and similarly for columns. Keep d_i\phi(z_i) exactly on the diagonal. The first two terms in Equation (5.16) are diagonal multiplications of E(z) and its Schur square; they belong to the asserted closure by Lemma 5.8. Their operator norms are bounded, and their off-diagonal maxima vanish. The exact diagonal is a bounded site label, obtained from z_i and its already constructed own-spin difference. Its smooth limiting versions follow by the same recipe recursion. This completes induction over every finite calculation. Finite sums and products use the exact product rule and the bounded Schur estimate; no unbounded derivative coefficient appears.

For column flips define \mathsf F(Q)_{ij}=Q_{ij}(x^j). Starting at the right end of a word, one has the exact recursions

The first uses that J is independent of x; the second is the identity f_i(x^j)=f_i(x)-2x_jd_jf_i(x) with its derivative index still tied to column j. These recursions express every column-flip matrix through bounded path words and Schur insertions. They preserve the closure just proved. The conditional derivative of a fixed smooth multitime calculation requires the full mixed-slot recursion of Lemmas 4.4 and 4.5. After its uniform tails are removed, the equality-network reduction in Proposition 5.12 expresses every opened term through the matrix closure just described. Thus the dynamic assertion uses that additional argument; it is not inferred from first partial Jacobian bounds alone.□

Proof. Only the increments need polynomial approximation; the base query values may remain unbounded. In the corner expansion from Lemma 4.4, the formal increments satisfy |t_{b,S}|\le 2^{|S|}C_S(z^b) and hence range in a fixed compact box. Each remainder has the form

Here z consists of the undifferentiated calculation values at the output site. The integral formula for R and the global scalar derivative bounds give a uniform bound and a uniform modulus of continuity in t, independently of z\in\R^q.

Apply multivariate Bernstein approximation on the increment box, treating z as a parameter. Given \eta>0, this gives a fixed finite polynomial

In the Bernstein basis the coefficients are values of R at fixed increment grid points. They are therefore bounded smooth functions of z with bounded derivatives of each required order. Conversion to the fixed monomial basis preserves these properties. This approximates only the increments, not the possibly unbounded z.

The error still contains the repeated factor t_\gamma in Equation (5.17). The repeated-square estimate in Lemma 4.3 therefore bounds its flattened operator norm by C\eta. Every retained monomial still covers the original set I of derivative slots. Carry out this operation at every nonlinear gate, and substitute the already constructed approximations of its input tensors. The product bound in Equation (4.10) makes each finite substitution continuous in those operator norms. Finite induction proves the asserted approximation.

We record the spatial structure of the resulting terms for the conditional derivative calculation. Unroll one term into a tree rooted at its output site. A linear gate gives a matrix edge. At a tensor monomial, take copies of the nonempty child tensors and join their output roots at an equality vertex labelled by b_\nu. Each spin input supplies an identity edge ending in a derivative leaf; retain this edge even if it could immediately be contracted. A constant input has no nonempty derivative term. Factors x^{(a)}_{j_a} from the corner increments become bounded labels at the corresponding leaves. The covering property ensures that every required derivative slot has a leaf. Several occurrences of one slot remain several leaves with the same slot name, to be copied by equality tensors when the temporal indices are attached. Thus no coefficient depending on two unrelated vertex indices remains.□

Proof. We first bound a network contraction, then show how to reduce the network while controlling each error. Put H=\R^n. For p,q\ge1, the labelled equality gate E^d_{p,q}:H^{\otimes p}\to H^{\otimes q} sends e_a^{\otimes p} to d(a)e_a^{\otimes q} and annihilates nonconstant input tuples. Its norm is \norm d_\infty. Count the external input as an incoming wire at the source and the external output as an outgoing wire at the sink. Each vertex now has both an incoming and an outgoing wire. Executing vertices in topological order, with tensor products on the other live wires, gives the deterministic bound

In particular, there is no free summation map with norm \sqrt n.

Call an internal edge e:u\to v safe if deleting it preserves the unique source and sink and makes u,v incomparable in the remaining directed order. Both endpoints then retain positive incoming and outgoing degrees. Process their predecessors first and process u,v jointly. If their remaining degrees are p_u,q_u,p_v,q_v\ge1, the joint gate with coefficient R is

and is zero on all other input tuples. Both the displayed input family and output family are orthonormal. Its norm is at most \norm{d_u}_\infty\norm{d_v}_\infty\max_{a,b}|R_{ab}|. Taking R to be the edge matrix restores the original network; taking its diagonal identifies a=b and contracts u,v. Hence

with new vertex label d_ud_v\diag(A_e). Its diagonal can be replaced by d_ud_v\delta_{A_e} using Equation (5.18). Contraction cannot create a directed cycle: such a cycle would yield a path from u to v or from v to u after deletion. It preserves the unique source and sink.

The norm estimate justifies pinching a safe edge. To ensure that such an edge is available, first consolidate parallel edges by Hadamard products and suppress every nonterminal vertex of indegree and outdegree one, using matrix multiplication with its diagonal label. Continue until neither reduction applies. If any internal vertex remains, the earliest one in a topological order has just one incoming edge, from the source. It must consequently have at least two outgoing edges, or it would have been suppressed. Choose the latest internal vertex u having at least two outgoing edges, and let v be its earliest immediate successor. The successors are distinct, so v is internal. It has outdegree one by the choice of u, and indegree at least two because suppression has ended. There is no alternate path u\to v: its first edge would go to another successor of u, later than v, which cannot subsequently reach v. Deleting u\to v therefore makes them incomparable and leaves positive incoming and outgoing degrees at both endpoints. This is a safe edge.

Pinch that edge and repeat the ordinary reductions. A pinching or series reduction decreases the number of vertices, and parallel consolidation decreases the number of edges. The procedure therefore ends after finitely many steps with one edge from source to sink. All newly formed edges belong to \mathcal A, so their own entry predictions justify every subsequent pinching. Summing the finitely many errors proves the claim. If source and sink have already been identified, acyclicity leaves only a diagonal equality gate; this case is immediate. Finally \max_{ij}|E_{ij}|\le\norm E_{\op} transfers the off-diagonal assertion through the accumulated errors.□

Proof. The first step removes errors that are uniform in the joining state. Use Equations (4.20) and (4.17) for the tensor approximations and the common K tails. At fixed accuracy, only finitely many terms and fixed finite-measure Duhamel integrals remain. Every K insertion history is a bounded finite path-matrix word, with a fixed cap on each likelihood label. Keep ordinary S expectations exact for now. On opening a later inner-product or replica test, they become ordinary causal branches and introduce no new spatial summation index.

We now identify the two trees whose gluing represents an opened term. The spatial recipe tree is the one constructed in Lemma 5.10. A second, temporal tree starts at the joining derivative coordinate: construct it by reading the finite conditioning recursion backwards. In Equation (4.19), the memory term keeps its current slot p. The active term replaces p by the newly introduced slot m, through the K path-word edge. The mixed term forks into p and m, puts the K edge on the latter branch, and puts the bounded sign -2x_p at the fork. The new slot m was absent from the current derivative subset. It therefore cannot reconnect to an existing temporal leaf. Repeating this construction gives a temporal tree with one terminal for each original derivative slot used in that term. Finite causal-tree conditioning has the same construction, with copies attached at the common joining vertex.

If a slot occurs several times in the recipe polynomial, append an equality fork to its temporal terminal, one outgoing branch for each occurrence. Glue the resulting temporal leaves to the recipe leaves in pairs, retaining the input-spin identity edges until this gluing is complete. Every leaf is paired. The covering property ensures that there is at least one leaf, and that both trees are connected. Orient the temporal tree away from the joining derivative index and the recipe tree toward its output site. A directed path can cross from the temporal tree into the recipe tree only at a paired leaf; it cannot cross back. Thus the glued network is acyclic, with its unique source the joining derivative coordinate and its unique sink the output site. Every vertex is on a path between them. The only free indices are these two external ones; all internal indices are ordinary equality contractions with no normalization factor.

The glued graph now has exactly the orientation required by Lemma 5.11. Its matrix algebra, including bounded visible dyads, is closed under ordinary products and transposes. Lemma 5.8 supplies the Hadamard reduction and entry predictions for each newly formed fixed expression. Applying the network lemma therefore reduces the opened term to the same path-matrix closure in operator norm. The bounded scalar coefficients, spin signs and diagonal predictions are path labels. Integrate its pointwise network bound over the fixed Duhamel parameter sets and use the integrated entry predictions. No worst auxiliary time or starting configuration is selected. The same argument applies to stationary reversed causal branches. Actual and predictor terms keep the same subdivisions and approximations before they are recombined and centered.

This proves closure for the opened histories used in the subsequent conditional tests. It does not assert a uniform-operator-norm approximation of an arbitrary conditional matrix average by finitely many sampled histories. Conditional expectations remain exact until those branches are opened. For a still-averaged off-diagonal maximum one may instead use conditional Jensen:

The uniform errors were removed before this typical network step. In particular, no spatial square-mean error has been substituted for an operator error.□

Proof. We first separate the two planted starting directions, then propagate that separation through the finite simulations. Plant the samples \sigma and \sigma'. Their overlap r=n^{-1}\sigma^t\sigma' has the independent-uniform-spin law tilted by \exp(\beta^2nr^2/2), up to a spin-independent constant. The Rademacher rate function satisfies I(r)\ge r^2/2. Since \beta<1, for every fixed \varepsilon>0 the probability |r|>\varepsilon is at most e^{-c_\varepsilon n}. Conditionally on the spins the Gaussian disorder has the two rank shifts \beta^2(\sigma\sigma^t+\sigma'\sigma'^t)/n. The two initial site spins are therefore asymptotically independent, and the two planted directions are asymptotically orthogonal.

Perform the finite Gaussian recursion from Lemma 5.2 for both causal simulations. Each query needed for spin and field histories is odd under global spin inversion of its own root; a score query for likelihoods is also odd. Fresh refresh uniforms can be reflected under this inversion without changing their law. Deterministic cross-overlap limits of an odd query from each different tree change sign under either inversion, and hence vanish. The Gaussian regression innovations for the two trees have zero cross covariance and are independent. By induction their site recursions are independent, because their initial spins and site seeds are independent as well. Likelihood labels are bounded functions of the resulting histories; their computation needs no additional noncentered Gaussian query. Thus their joint limits also factor. This induction does not apply to an arbitrary shared noncentered query such as J\1, explaining the restriction in the statement.

Transfer this factorization to the actual stationary paths using the two-replica planting comparison and then Proposition 4.10 and Lemma 5.4. For a bounded label b, represent the spatial mean of \E[b_i\mid J]^2 by the empirical product on the two independent roots. The factorization makes its limit (\tau b)^2, proving the claimed self-averaging of the site means.□

Completion of the proof of Theorem 5.1. The required empirical space and endpoint predictions are supplied by Lemma 5.3, together with orthogonality of hidden vectors. Lemma 5.4 puts the likelihood labels in this construction with their stated tracial convention. The norm comparison then follows from Lemmas 5.5, 5.6 and 5.7: the representation is on one tree, retains the rare diagonal base, and uses the Gaussian rate before paying the entropy and density costs. The entry and Hadamard assertions come from Lemma 5.8; the difference and column-flip closure is Lemma 5.9, with the opened conditional terms just reduced to that closure. Finally, Lemma 5.13 proves the restricted two-root independence assertion.

Every construction used the fixed countable dense recipe family and successive fixed approximations. Diagonal extraction therefore gives common high-probability disorder events for any finite initial list of tests. Lemma 4.7 permits polynomial weights and actual conditional copies. All superpolynomial path discards were made separately on the original forward marginals. These observations give the typicality convention and the order of limits in the theorem.□

Proof. Consider first a refresh at another site. The logarithmic perturbation is a Bernoulli likelihood increment with field displacement \theta aJ_{ij}. Its conditional exponential moment, at any fixed real order, is at most \exp(C_qJ_{ij}^2). Iteration over the attempts, followed by the Poisson exponential formula, gives e^{C_qd} because \sum_jJ_{ij}^2\le C and \max_j|J_{ij}|=o(1). The same calculation controls inverse moments. After its prescribed refresh the forced spin is predictable, so it does not change the calculation.

We next control the forced-coin factor. For f(y)=J_iy, Proposition 4.2 supplies a fixed-horizon bound on \sup_y\norm{dS_sf(y)}^2. Applying Lemma 2.2 controls every fixed exponential moment of f(Y(s_*))-\E f(Y(s_*)). Combine this with Equation (6.4) and Hölder’s inequality. The resulting constant may grow rapidly with fixed d. This step provides uniform integrability at fixed time, rather than a large-d rate.

For a positive V and A=\E[V\mid Z], Jensen gives

Choosing an order strictly above the desired one gives uniform integrability for the required powers of A. To identify those powers by replicated tests, take copies V^{(1)},\ldots,V^{(b)} that are conditionally independent given Z. They satisfy

Cap the V^{(a)} first and pass to the dimension limit with the cap and b fixed. Then let b\to\infty and remove the cap. The higher moments in Equation (6.7) justify this passage for positive and negative powers alike; in particular, a limiting A has no mass at zero. Hölder’s inequality in Equation (6.6) transfers the conclusion to the terminal density ratio.□

Proof. Begin with a finite-dimensional list and condition on Z. Diagonalize

where u_\alpha denotes a normalized list vector. Work first on the sets where the list norms are bounded. If \Pi_k projects on the first k eigenvectors, independence of two branches and \Tr T\le K give

Extract the laws of the finite coordinate lists together with their self-Grams, and then let k\to\infty. Count tails and the spin-norm bounds remove the preliminary restriction on list norms. The resulting directing law on lists in \ell^2 identifies the limiting cross-Grams. At every finite coordinate projection, the stated covariance difference is positive semidefinite, and this property persists in the limit. A common isonormal process on \ell^2, plus independent Gaussians with the covariance difference, realizes exactly these Gaussian Gram arrays. The conditional-average assertion follows from the replicated approximation in Lemma 6.2.□

Proof. Suppose a support point had squared norm greater than \delta. Continuity would give a sufficiently small ball about it in which every pair has inner product greater than \delta. The ball has positive probability, so independence makes this pair event have positive probability, contradicting the hypothesis. A countable base of balls shows, by separability, that the law is carried by its support. The same argument applies to the conditional law for almost every directing variable.□

Proof. Fix the specified data. The full list is then Gaussian, and the orthogonality identities make each score independent of the earlier scores and of every field already used to choose its sign. Its variance is b_h. It may remain correlated with fields at later coin times, which does not affect this chronological calculation. Generate the Gaussian list and the coins in time order: the sign a_h is measurable before revealing \widetilde Z_h, so

Their product has mean one. Using a_h^2=1 and multiplying by \exp\{(r^2-r)\sum_hb_h/2\} proves the bounds. Conditioning on future self-Gram data is legitimate here because the needed Gaussian orthogonality identities hold pointwise for each value of those data. At no point are the scores themselves conditioned on future observed fields.□

Proof of Theorem 6.1. The prefix is now controlled at powers close to one. Conditional Hölder inequalities will put every fixed larger power on the short suffix and on the last-spin factor. First retain schedules with at most Kd refreshes. In the limiting representation, factor

where the prefix ends at T=d-\sqrt d and U contains the suffix and the normalized last-spin factor, wherever that last refresh occurs. For each fixed q, positive and negative moments of the suffix likelihood are at most e^{C_q\sqrt d}. Apply the forward martingale calculation on the interval of length \sqrt d, and then use the fixed-d moment passage proved above. The last-field factor has bounded moments independent of d in this limiting calculation: its centered Gaussian field has variance at most 4\beta^2 by Equation (6.14), before the common and private parts are separated. Hölder therefore gives

The deterministic field means disappear from this bound through Equation (6.4).

For 0<\eta<1, conditional Hölder gives the two inequalities

For example, the second follows by writing W_{\mathrm{pre}}^{1-\eta} =(W_{\mathrm{pre}}U)^{1-\eta}U^{-(1-\eta)}. This second inequality supplies the inverse-moment bound: inverse powers of A are controlled by inverse powers of the prefix conditional average together with positive powers of a suffix moment.

By Lemma 6.5, each corrected prefix conditional average with power 1\pm\eta lies between e^{-C\eta d} and e^{C\eta d}, uniformly in the branch data and common field forcing. Jensen permits integration over the directing law and also taking inverse powers. The small common score and covariance-coupling errors discussed above are handled by Hölder, with a power change smaller than \eta. Combining Equations (6.17), (6.18) and (6.16) yields for every fixed p>0

This inequality concerns the moments after the dimension limit. Now send d\to\infty and then \eta\downarrow0. The rough full-horizon constants from Lemma 6.2 served only to identify these limiting moments; they contribute no additional term to Equation (6.19).

Hölder’s inequality in Equation (6.6) proves the required bound for every retained schedule. We next remove the schedule conditioning. Under the ordinary path law, the tower property gives

The conditional expectation averages the schedule with its ordinary conditional law given X_d. For r\ge1 or r<0, conditional Jensen therefore decreases the moment when the schedule is no longer observed. Control the omitted schedules by a higher raw likelihood moment, Hölder’s inequality, and the Poisson estimate

The raw bounds are uniform conditional on the schedule. Taking K large after the desired moment order makes this contribution smaller than any prescribed e^{-Hd}. The powers 0\le r\le1 again follow from Jensen. This completes the moment estimate.

It remains to obtain the stated exceptional sets, including for an integrated family of masks. Fix d, the schedule-count cutoff, the moment orders, and the comparison accuracies. Every row concentration constant above is then uniform in the mask parameters and deterministic schedule times. If \mathcal B(J,x,\alpha) denotes a failure of one of the finitely many bounded tests, its row-and-start bound has the form

on the common good cavity events. Apply Fubini at this point, before choosing the quenched event. Markov’s inequality gives integrated Gibbs bad mass at most e^{-cn/2} outside an event of probability Ce^{-cn/2}. If necessary, apply Markov once more to remove starts with excessive bad-schedule mass, and use higher rough moments to bound its contribution. The countable approximations are fixed successively, with arbitrary fixed slack, so their finite good events suffice before the dimension limit. Polynomial endpoint weights and an e^{o(n)} initial density preserve negligibility. At fixed times the remaining integrands are dominated by the rough moment bounds and the relevant mean-square vector budgets.

In a path expansion the order is therefore: fix the global time, hop cutoff, shapes, elementary caps, moment orders, and label bins; integrate over their deterministic simplex parameters; take dimension to infinity; then remove caps and approximation errors; finally increase the global time and decrease rate slack. Annulus and first-exceedance restrictions multiply a positive masked label by factors in [0,1]. Its positive moments are bounded by those of the unrestricted label; no inverse moment of a restricted label is used. A selected start or endpoint vector changes the weights in this integrated assertion and does not choose an exceptional mask. Likewise, good interior times can be chosen from a positive-measure set after an integrated estimate; a prescribed worst time is not being asserted to be good.□

Proof. For p\ge1, condition on both endpoints of the span and then apply Jensen to the left endpoint’s conditional law given the right endpoint. The resulting integral over ordinary starts is bounded by Theorem 6.1. A higher raw moment and Hölder’s inequality make the exponentially small bad-start contribution negligible. The Markov property and stationarity make conditioning at a later endpoint another L^p contraction. At the left endpoint, the bound follows directly from \E_xa_i\le\E_x\ell_i^M=1. Powers between zero and one follow by concavity. The raw bound is Proposition 4.2.□

Proof. The proof proceeds from a time-domain diagram identity to the transform bound, then identifies the covariance trace density and tests its edge. We first derive the diagram identity, making explicit the role of stationarity. Integration by parts and gradient intertwining yield

Here K_tI applies K_t to each constant coordinate vector. Insert the Duhamel expansion and use Theorem 5.1: visible finite-rank terms vanish in normalized trace, each hop contributes \beta sD_v, and the remaining trace is the sum of complete noncrossing pairings. Each resulting vertex carries the product of its diagonal waiting likelihoods, its arrivals v, and the bare factors e^{-h}. These are nonnegative. The root initially carries v, as in Equation (6.25).

The recurrence for P_k(s) removes exactly the adjacent returning contraction, making this polynomial the Wick string of k consecutive edges. Each of its edges must therefore pair into the Duhamel walk. Removing those k pairs exposes an ordered trunk of k+1 completely paired rooted time intervals. Each interval starts with a v weight, including the interval at the original root, and each trunk edge contributes \beta. Conversely, concatenate k+1 rooted complete diagrams and adjoin the outside Wick string. This recovers one and only one pairing, so no multiplicity factor appears. After contraction, distinct vertices contribute scalar traces. Stationarity makes each interval’s trace depend only on its duration. The separation into scalar factors comes from the noncrossing contraction, rather than from temporal independence of intervals along one spin history. The spin/site self-averaging assertion in Theorem 5.1 is what identifies each such scalar factor with its stationary trace when it is represented by independent samples. It is used only for spin, field, and likelihood labels.

The simplex integral over the successive interval lengths therefore gives the convolution identity

For the limiting argument, first keep time, Duhamel order, and likelihood caps fixed. Then Proposition 4.2 removes the caps and order cutoff in the matrix L^2 comparison, establishing the identity at each fixed t. Because the complete-diagram terms are nonnegative, we may also sum them monotonically when taking Laplace transforms.

To pass to transforms, we need a bound that is valid before the dimension limit and does not require a finite-dimensional resolvent at z. Positivity supplies it: for a positive matrix B and a self-adjoint Q, |\Tr(QB)|\le\norm Q_{\op}\Tr B. At finite n apply this to the positive matrix \int u e^{-tu}\mathsf M_n(\dd u) and to Q=P_k(J/\beta). The Wigner spectral bound and \sup_{[-2,2]}|P_k|=k+1 show that, after the dimension limit,

The same justification can be expressed through truncated integrals: integrate the finite-dimensional inequality over 0\le t\le T, take the dimension limit, and then send T\to\infty. The right-hand side is integrable for z<\lambda_{\rm sp}. Tonelli’s theorem applied to Equation (6.26) now proves Equation (6.23) and yields

Taking kth roots yields \beta\widehat R(z)\le1. Given z<\lambda_{\rm sp}, choose z' strictly between them. Since \widehat R is strictly increasing, \beta\widehat R(z)<\beta\widehat R(z')\le1.

Integrating Equation (6.26) at z=0 gives

Once again, justify the integral by first stopping at fixed T. The spectral gap bounds the omitted covariance tail by \norm{P_k(J/\beta)}_{\op}e^{-\gamma T}. To identify the resulting covariance density, use the orthonormality of the P_k for \rho_{\rm sc} and the generating function

the series converges uniformly because \beta<1 and |P_k(u)|\le k+1. This proves Equation (6.24). The measure identified here is positive: at finite n it is the trace of the positive covariance matrix against the spectral projections of J/\beta.

We next use this covariance density to bound the spin edge. Fix a real polynomial q. Weighting \mathsf M_n by q(J/\beta)^2 gives a nonnegative scalar spectral measure, dominated by \norm{q(J/\beta)}_{\op}^2\rho_n. All its limiting mass therefore lies in [\lambda_{\rm sp},\infty), even though the finite-dimensional spin measures may put small masses below that edge. Its first moment satisfies

because the energy matrix \int u\mathsf M_n(\dd u) is exactly D_{\E_\mu v}. Use Equation (6.24), and then the semicircle trace limit, to obtain

Alternatively, discard scalar spectral mass below \lambda_{\rm sp}-\delta before sending \delta\downarrow0. This gives the same inequality without requiring a uniform finite-n lower edge. Choose q_m(u)=((u+2)/4)^m. After normalization, the measures q_m^2\rho_{\rm sc} concentrate at 2. Letting m\to\infty in Equation (6.28) gives \lambda_{\rm sp}\le1-2\beta+\beta^2.

Finally, for 0\le z<\lambda_{\rm sp},

For z<0, \widehat R(z)\le\widehat R(0)=1 and \beta^2<1<1-z. These prove the second renewal inequality and complete Equation (6.22).□

Proof. The Fock comparisons in this proof first remove the terminal visible component, as in the endpoint projection argument used below for the predictor caps. Fix a finite family of endpoint recipes with nonsingular limiting Gram matrix, and let Q_m be their actual orthogonal projection on the corresponding Gram-good event, and zero off that event. Both Q_m and I-Q_m are contractions. For every fixed capped path word W, Cauchy–Schwarz bounds the removed pairing by \|Q_mW^tu\|_2\|p\|_2, which tends to zero at fixed m by Assertion (ii) of Theorem 5.1 and the fixed inverse Gram bound. The weighted typicality convention removes the exceptional Gram event. Exhaust the endpoint recipe completion, omitting null directions, and choose m=m(n)\to\infty sufficiently slowly that these errors vanish for every fixed word and (I-Q_m)p is hidden for every fixed endpoint recipe. Its second-moment and polynomial bounds follow from contraction. The projection depends only on the shared endpoint and its fixed seeds, so permutations of the replicated branches preserve it. This projection is used only for the Fock bilinear comparisons; we retain the original exact p for the Hessian estimates that follow.

Work on one finite tree and use the cut expansion behind Equation (7.3). Every cut away from the replaced factor puts that factor on a pure creation or pure annihilation side. Its trace two-norm then bounds the error, multiplied by the fixed bounds on the other factors and the two endpoint norms. Only the cut at the replaced factor needs another argument. At that cut, at least one distinguished centered factor lies on the pure side leading to its own centering endpoint.

Attach b conditionally independent copies at this endpoint. Lift this single replicated star to the Fock representation, and symmetrize by taking a direct sum over permutations of the copies. Repeatedly applying l^*D_al=\tau(a)I expresses inner products of pure-side strings as products of pair traces. For different copies, one factor is the pair trace of the centered label and has zero limit. With approximated predictors, bound its absolute value by \eta, tending to zero with the approximation. Write C_\alpha for the pure-side creation operator on copy \alpha, oriented toward the shared endpoint. The Gram bounds are

If v denotes the vector at the shared endpoint, symmetry of the lifted star gives

The vector at the opposite end may depend on its whole branch; Cauchy–Schwarz now bounds the squared bilinear contraction by

The fixed caps bound the diagonal Gram blocks. Apply Cauchy–Schwarz and take the limits n\to\infty, \eta\to0, and b\to\infty in that order to obtain zero. All cuts for this replacement are treated on the same star. The argument makes no simultaneous choice of creation operators on stars rooted at different endpoints.

A terminal insertion with small trace support has the same two cases. Every nonterminal cut pays its trace two-norm; at the terminal cut, the left distinguished factor lies on its centered pure side, so the replication argument applies. Iterate for finitely many replacements. When the replaced factor is an approximated distinguished label, keep its exact forward or backward subtraction throughout this step and use the other distinguished factor at the exceptional cut.

The word comparison retains every rare diagonal’s action on the base. Small trace is used on a pure side, and centering treats the remaining cut; neither step uses the false implication \tau(c^2)\to0\Rightarrow\norm{D_c}_{\op}\to0.□

Proof. First fix a tensor-network approximation from Lemma 5.10. Before opening paths, use the mixed-tensor budget to bound its operator errors, and remove those errors in the prescribed successive approximation limit. The temporal tree off the marked path consists of disjoint subtrees ending at derivative leaves of the chosen label. Each attachment has at least one such leaf. The corresponding maps are bounded tensor products and compositions of capped path matrices and equality copies; every equality copy has norm at most its fixed scalar label bound. Pull their adjoints into the row tensor of the label circuit. Repeated occurrences of a derivative index are handled by the norm-one equality embedding. The resulting tensor U_i on the r attachment indices satisfies

Use the operator norm of each flattened mixed tensor to bound each of its rows in Euclidean norm. The off-path maps act on disjoint groups of leaves, so their tensor-product operator norms give the stated estimate without a dimension factor.

Suppress vertices on the marked path with no label attachment. The matrices between its remaining consecutive attachments are bounded path matrices Q_1,\ldots,Q_{r-1}. For r\geq2, put

Then \Phi(M)_{i,b}=\sum_a M_{i,a}L_i(a,b). If r=1, this formula means L_i(a,b)=\1_{a=b}U_i(a).

The next step is to make every Q_\ell in Equation (7.14) diagonal. Control one replacement at a time: fix an edge e and put Q_e^{\rm off} there. With the endpoints fixed, Cauchy–Schwarz bounds the squared kernel by

where B_\ell=|Q_\ell|^{\circ2} entrywise. Every row sum and every column sum of B_\ell is at most \norm{Q_\ell}_{\op}^2. Writing \eta_e=\max_{a\ne b}|(Q_e)_{ab}|, we have B_e^{\rm off}\leq\eta_e^2\1\1^t entrywise. The second factor is therefore at most

Sum over the endpoints and use Equation (7.13). This gives

The estimate remains valid after any other edges have been made diagonal. Telescope over the finitely many edges and apply the entry rule in Theorem 5.1 to obtain \sup_i\norm{L_i-L_i^{\rm diag}}_{\op}=o(1). Its row action costs at most o(1)\norm{M}_{\HS}.

After these replacements, diagonal marked-path edges force all attachment indices to agree. The remaining action takes the form M\circ E, where

This E contains the entire label derivative subnetwork, including its memory copies. It is the two-terminal network obtained by tying all temporal attachment indices to the common source a, with the label output i as sink. Every derivative leaf is paired. Lemma 5.11 and Proposition 5.12 place it in the bounded path-matrix closure. The diagonal factors above may first be replaced by their uniformly accurate diagonal predictions. Consequently \max_i|E_{ii}|\leq C and \max_{i\ne a}|E_{i,a}|=o(1). The estimate

proves Equation (7.12). Here the entry estimates are used on fixed typical finite trees, followed by their integrated comparison. Their scope does not include uniformity over arbitrary states.□

Proof of Lemma 7.2. Fix a summand of Equation (7.6), with distinguished labels a=a_r-a_r^0 and b=a_q-a_q^1, where r<q. By Lemma 7.3, approximate its path labels by fixed bounded smooth multitime circuits, with every observed state inside the label’s time span. These circuits may contain mixed queries such as J[f(X_s)\odot X_r]; no representation as a sum of separated single-time products is required. The frozen-score construction of Lemma 5.4 puts its smooth caps in the class of Lemma 4.4; in particular, outside spins multiplying an unbounded query are handled by its finite sign-pattern decomposition. Before estimating derivatives, fix the circuit, its cap, and every observation time. Keep both distinguished vertices exactly conditionally centered for each approximant. Approximate the nonspecial endpoint factors by fixed smooth one-state functions before the derivative limits, and perform the field cutoff above. Its threshold and the suffix approximants remain fixed throughout the derivative comparisons.

For the exact right predictor, use a conditional path representation instead of requiring a regular one-state recipe. Represent its value by an independent ordinary stationary-kernel path from X_t, reading the multitime evaluations in reverse order. Differentiate by propagating the full mixed-slot recursion down this added causal branch. Its tensor and path-network conclusions are those of Lemma 4.5 and Proposition 5.12. Keep the exact conditional subtraction before opening any of these derivatives. The conditional estimates Equations (7.10) and (7.11) apply there as well. The error in replacing a tracial predictor by this exact prediction is handled before derivatives are opened, by Lemma 7.3.

Split the Wick string at a and b using Equation (7.4). The subtractions involving \tau(a) or \tau(b) vanish; errors from approximate trace centering are controlled in the unsplit Wick form by Lemma 7.3. Absorb the outer strings into the endpoint vectors. The left vector remains hidden by Theorem 5.1; the right vector has bounded mean-square unweighted Hessian by Equation (7.8) and Equation (7.9). It remains to prove

where Q is a positive-length Wick string with identity end labels. The left centered label a will give a martingale covariance. If its paired derivative reaches p, the backward centering of b will cancel the term. If the derivative first hits a tail label, the nonempty Wick prefix will supply the cancellation instead.

To prove the remaining bilinear limit, use conditional martingale covariance up to the last evaluation in a. For scalar future observables F,G, it reads

where F_s,G_s are the conditional future expectations with their already observed arguments held fixed. The integration stops once F is fully observed. To check the factor, a flip of j has rate w_j/2 and changes a scalar martingale by -2x_jd_jF_s; its product bracket is 2w_jd_jF_s d_jG_s. Since a has exact forward conditional mean zero, Equation (7.15) is the corresponding covariance, with u(X_0) fixed under the initial conditioning.

Equation (7.16) introduces a derivative index in addition to the sites of the Wick string. Propagate the two derivatives by the mixed-slot recursion and dS_h=K_hd, including every exact product and memory-copy correction. First conditionally average the factor differentiating a and remove its truncation with Equation (7.11). On the other side, each terminal tensor term contains p or dp exactly once, because p has only the terminal state slot. The bounded mixed label tensors and the Hilbert-space amplification of Proposition 4.2 therefore propagate its unweighted mean-square Hilbert–Schmidt budget. Tensor and propagation errors are removed in these norms before paths are opened. For the retained histories use Proposition 5.12 to reduce their complete derivative subnetworks to bounded path matrices. The two conditional future expectations in the covariance can then be represented by independent causal futures from their joining state.

We now split the retained terms according to the derivative’s first destination: either it reaches p before differentiating any tail label, or it first differentiates such a label. The two cases use different cancellations.

The derivative reaches p without differentiating a tail label. Expand each terminal mixed tensor by the exact product rule, and classify a term by whether any difference falls on a tail label. For a label c and a remaining factor q, we use

The term that bypasses c retains its unshifted value; classify the mixed correction with the first-hit terms. If no tail label is differentiated, the only differentiated state slot is the terminal slot of p. Any extra frozen-slot branch would differentiate a label, as would a same-slot correction containing both a label difference and dp; both belong to the second case. Thus the history in the present case is unbranched and follows the common K propagation through the entire observation grid. With the derivative as column index, its term is a finite sum or integral of

where A is a bounded path matrix containing the derivative propagation, the differentiated a-side, and the bounded rate diagonal. Condition first on X_t, so the terminal Hessian is fixed when we apply Cauchy–Schwarz. We must estimate the conditional matrix

Take two conditionally independent copies of the entire path system given X_t. Directly expanding its Hilbert–Schmidt square gives

Reversibility and the tree comparison identify this system as a causal tree rooted at X_t. Each conditional copy shares that one root; we do not introduce a new bridge conditioned on two endpoints.

Apply the Hadamard rule in Theorem 5.1. Operator-small errors and visible dyads have zero bilinear limit between these hidden vectors. The remaining diagonal is the product of the two diagonal predictions. The prediction of Q_1D_{b_1b_2}Q_2^t is zero. Indeed, the Wick strings admit no internal return, so the two equal-length strings must pair fully across each other. Their innermost contraction is \tau(b_1b_2). Under the stationary kernel, the two copies are conditionally independent and \E[b_{1,i}b_{2,i}\mid X_t]=0 at every site. The zero limiting pair trace transfers to the actual, possibly tilted, reversed tree by Theorem 5.1; exact finite-dimensional stationary centering is not asserted for that actual reversed kernel. Thus the right-hand side of Equation (7.18) tends to zero, and the Hessian budget proves that Equation (7.17) vanishes.

This cancellation requires common subdivisions and identical truncated propagation kernels for the actual and predictor terms. Keep every evaluation time in the subdivision, even when the factor evaluated there is absent from a particular term. In the present case no derivative falls on b, so its actual and predictor pieces have the identical matrix A. Recombine them into the exactly centered b before using Equation (7.18). Independent predictor simulation branches may then be used in the two copies. Separate truncations followed by separate Hadamard estimates would not establish the vanishing of \tau(b_1b_2).

A tail label is differentiated. Stop at the first such vertex from the left, including a product-rule term that differentiates the label while also continuing to the right. If an earlier contraction has made this vertex scalar, its coefficient is a deterministic limiting trace from the word calculus, not a state-dependent empirical trace, and is not differentiated. The remaining prefix retains its Wick inclusion–exclusion.

The stopping rule must be fixed before opening derivatives of the later suffix. The surviving actual labels have ordered spans with disjoint interiors: an adjacent return removes the interior excursion and merges the two neighboring labels, whose convex span contains no surviving vertex span in its interior. Repeating returns preserves this property. The endpoint predictions in the double-centered string occur only in its terminal suffix, apart from the initial predictions already absorbed into u. If the first differentiated actual label c_i has span ending at v, every later actual label is observed after v, and every later endpoint predictor is a function of X_t. For each fixed spatial suffix, form its exact conditional value

The Markov property makes the local observable exactly c_i(\text{path up to }v)q_i(X_v). Spatial matrix contractions in that suffix are included in q; they need not be diagonal.

Before reopening the suffix, we need its L^2 value and mean-square Hilbert–Schmidt derivative to be bounded. Prove both by working backwards from the bounds for p,dp, using the bounded mixed label tensors and exact product and copy rules at each ordered span. For the empty derivative-index set, this means multiplication by the bounded label c, with bound C_{\varnothing}=\norm c_\infty. The single terminal state slot contributes the value or the first difference of the current suffix once. Their tensor Hilbert–Schmidt norms are bounded by the row tensor bounds of the labels times the corresponding suffix norm. The Hilbert-space amplification of S,K and their common tails propagates these bounds to the preceding span, while a spatial matrix costs its bounded operator norm. For a terminal predictor, the ordinary one-state product rule uses its bounded conditional derivative operator. This finite backward induction uses no second difference of p in its original state slot and does not require q to be an exact gradient.

Now apply the local temporal recursion to c_iq_i, retaining q and dq as unopened terminal objects. A term that differentiates c either terminates in its label tensor, leaving q as a value, or has exactly one marked leaf carrying dq. Every other derivative leaf belongs to this single label c, including all its mixed-slot and memory-copy leaves. Thus the latter term has precisely the marked history in Lemma 7.4, illustrated in Figure 3. The later differentiated labels remain inside dq until after this local estimate. Conditioning in this order keeps all side leaves at the same spatial output site, which a simultaneous expansion of the later labels would not ensure.

For a first hit on a terminal predictor g(X_t), apply the exact one-state product rule to D_gq before opening the auxiliary causal branch representing dg. Its terminating term uses the whole two-terminal matrix dg; its continuing correction is the one-attachment case of Lemma 7.4. Actual and predictor pieces of b can be treated separately in this first-hit case, since the cancellation below uses the Wick prefix, not a later centering. This completes the stopping rule for every possible first differentiated vertex.

Up to bounded row multipliers at that vertex, and with an unbranched temporal segment before its first label attachment absorbed into A, the incoming matrix is

where P is a positive-length Wick prefix ending in identity. Its Hilbert–Schmidt norm is bounded by \norm{P}_{\op}\norm{A}_{\op}\norm{u}, whereas its diagonal satisfies the exact identity

The diagonal prediction of P is zero, since it is a nonempty Wick string. The Hadamard rule and hiddenness hence show that \E\norm{\diag(M)}^2\to0.

For a continuing term, use Lemma 7.4 on the whole derivative subnetwork of c, including more than its first Jacobian. This gives

in the typical weighted comparison. The exact -2x_j factors in the product and copy rules are bounded labels on the equality forks of this history. An unbranched segment after the last label attachment is a subsequent bounded propagation. Pairing with dq at the stopping boundary now gives zero by Cauchy–Schwarz and its Hilbert–Schmidt budget. Equivalently, later suffix operations may then be opened; their bounded Hilbert–Schmidt maps preserve this zero estimate.

If the derivative terminates at its first differentiated vertex, let E be the matrix of the complete terminating label subnetwork. All its mixed leaves and memory copies have been contracted. By Proposition 5.12 this is a bounded path matrix. Contract its incoming index against M. In indices, the vector left at that vertex is

It tends to zero by the same Hadamard rule, again because the prediction of P is zero. Bounded row or column labels can be included in this identity without changing its conclusion. Pair with the bounded L^2 value of q from Equation (7.19). The terminating and continuing cases together cover every term and correction supplied by the exact product and copy rules.

The finitely many deterministic observation times may be separated within their prescribed spans before taking limits. Integration times coinciding with them have zero Lebesgue measure. The ordinary rough bounds control the limits and all discarded propagation tails. We have proved Equation (7.15); summing Equation (7.6) proves Equation (7.5).□

Completion of the proof of Theorem 7.1. Normalize the limiting endpoint second moments to one, using the rough comparison if an endpoint norm vanishes. For each fixed t, first truncate the hops and elementary likelihoods. Then apply Lemma 7.2, remove the likelihood annuli, and sum with the positive renewal estimate just proved. It gives, for every z'<\lambda_{\rm sp},

The derivative and field-cutoff errors were removed before the large-time limit. Their constants therefore impose no growth condition on Equation (7.1). Choose z'>z, then the predictor cap parameter \epsilon sufficiently small and the tail rate H sufficiently large. The result is bounded by C_ze^{-zt}, enlarging C_z on a bounded interval of times using Proposition 4.2. Restoring the endpoint norms proves Equation (7.2).

Finally, the exceptional starting-state sets in the row estimates have exponentially small integrated mass for the deterministic simplex masks. Fubini and the polynomial endpoint bounds extend the proof to the integrated comparisons in the statement. Every replicated diagram used above is a causal tree; no step calls for a uniform estimate on an adaptively selected bridge or mask.□

Proof. We first establish local boundedness in each norm, and then smoothing in the ordinary norm. Conditional prediction and the path expansion in Theorem 5.1 preserve the visible space. In the ordinary norm, Proposition 4.2 gives the bound and removes likelihood tails in stationary square mean. For the weighted bound, integrate the square evolution against \mu and retain the weights at both edge corners. The principal integration-by-parts term absorbs a fixed multiple of

Here both corners of an edge are retained. The elementary bounds

control the terms differentiating the weight. Cauchy–Schwarz and Young’s inequality absorb the first-order term using \sum_j a_{ij}^2/v_j\le C. The multiplication term is bounded by \norm{aD_v^{-1}}\le C, the comparable edge weights, and v\le1. Consequently

This proves the extension to the weighted completion. Strong continuity holds on smooth recipes by their difference bounds and then on both completions by density and local boundedness.

Analyticity follows by treating the first-order and multiplication terms as a relatively bounded perturbation of componentwise heat flow. Their uniform relative bound is

For example, square the first-order term row by row and use \sum_j a_{ij}^2/v_j\le C; its sum is bounded by C\mathfrak d(p). The remaining term has bounded operator norm. If z is on either ray bounding a sector strictly larger than the negative real half-line, the spectral theorem gives

At sufficiently large |z|, independently of n, the resolvent for \mathcal G^{(n)} is therefore given by a convergent Neumann series. Contour inversion gives uniform analytic bounds. Moreover, Equation (8.3) and the scalar inequality \sqrt u\le\varepsilon u+C_\varepsilon show that the graph norms of L^{(n)} and \mathcal G^{(n)} are uniformly equivalent after adding a fixed multiple of the identity. This proves Equation (8.2). Passing first on the dense recipe class and then by the uniform bounds gives the assertions in the limiting spaces.□

Proof. On the generating span, the path limits make every inner product and fixed-time matrix element deterministic. Self-adjointness comes from finite-dimensional exactness, through the identity

The right side is symmetric in f,g, is nonnegative when f=g, and has operator norm at most e^{-\gamma t} in the gradient norm. These properties pass to the weighted closure. The spectral theorem then gives the generator and its lower bound. The embedding properties follow from Equation (8.1) and the definition of the closures; the final spin-edge bound is Proposition 6.7.□

Proof. It suffices to show that the component orthogonal to \mathcal X_v vanishes. Subtract the weighted orthogonal projection and call the remainder q. It remains weighted-visible and a weighted limit of exact gradients, while its pairing with every generating orbit is zero. Take centered potentials G_n representing this exact limit. The gap gives \norm{G_n}_{L^2(\mu)}^2\le\gamma^{-1}\norm{dG_n}_v^2. For every fixed test potential f=f_{B,\delta},

The exponential gap bound gives a uniform integrable tail; the limit is zero by orthogonality.

The same potential pairing has a local form that detects q as the short time interval shrinks. Reversibility and conditional martingale covariance give the exact identity

Here the derivative matrix has output as row. The factor 2 is the quadratic-variation factor for flip rate w_j/2 and spin increment 2. The derivative matrix dS_rx differs from the identity by O(r) in operator norm: integrate dS_rx-I=\int_0^r dS_uLx\dd u and use \norm{dLx}\le C together with the pointwise rough gradient bound. Also \E_\mu\norm{B(X_s)-B(X_0)}^2=O_B(s).

The passage \delta\downarrow0 is justified in the weighted norm, without comparing the two completions in reverse. For every own-spin independent vector p,

Thus D_w is an isometry from the closure of exact gradients with their weighted norm into ordinary square mean. In Equation (8.8) one may first replace G_n by S_\varepsilon G_n, shrink \delta, and then shrink \varepsilon, using strong continuity on exact gradients. Equivalently one passes in the displayed covariance using Equation (8.9). It follows from the zero pairing that \ip{B}{D_wq}_0=0 for every B.

To conclude from these zero pairings, we must place D_wq in the ordinary visible space. Cut off the coordinates where v_i<\varepsilon. On the retained set a weighted-visible approximation is an ordinary visible approximation, and multiplication by w is an allowed bounded label. On the removed set Equation (8.9) and own-spin independence identify its squared norm with the corresponding weighted mass of q. That mass tends to zero for a fixed element of \mathcal H_v, by density of bounded recipes. Hence D_wq\in\mathcal H. Its orthogonality to all recipes makes it zero, and the isometry makes q=0.□

Proof. We test the claimed transport first on finite recipe projections. Truncate the path expansion of K_{t-s} at fixed order and likelihood level. Between finitely many testing recipes, its matrix elements concentrate at their deterministic stationary values by Theorem 5.1. To remove the endpoint complement, apply the adjoint bounded path word to a testing recipe and project at time t. Conditional prediction on replicated trees makes the remaining squared contraction arbitrarily small. This squared estimate is obtained before multiplying by any scalar test; Cauchy–Schwarz therefore permits arbitrary bounded-L^2 scalar coefficients, including past-dependent tests. Polynomial pointwise bounds and the exponential exceptional-set estimates justify discards. The operator-small path tails then remove the truncations. Increasing the finite basis gives Equation (8.11), since conditional expectation is a contraction. Apply Equation (8.2) to the first tensor factor for the last assertion.□

Proof. We obtain the two budgets from different estimates on the same path. The rough square estimate supplies the time-integrated first line and \norm{p_s(x)}^2\le C\E[\norm{p_b(X_b)}^2\mid X_s=x]. Tilt the entire path by g(X_b)/\E g(X_b), where g=1+\norm{p_b}^2. Its entropy relative to the stationary path is o(n): the original relative log density is o(n) and the extra log density is O(\log n). The tilt depends on the forward terminal state, so it preserves the original reverse transitions. Those transitions have rates w_jr_{s,j}/2, whereas stationary reverse rates are w_j/2. The jump-process entropy chain rule and

give an integrated Hellinger budget o(n) under the tilted path. The preceding conditional bound on p_s converts it into the time integral of Equation (8.13). Markov’s inequality selects a positive-measure set of times where the Hessian budget is bounded and the reverse budget is o(n). Countably many fixed typicality tests can be imposed successively there by their integrated failure bounds and extraction. Auxiliary-time exceptional sets cost their square-mean bounds, rather than a polynomial supremum times their measure.□

Proof. The conclusion concerns the square mean of each random coefficient, so we multiply an adjoint test by a second visible contraction. Write p=p_s, and take e=p^{\mathsf T}B with a fixed smooth scaled recipe B. The flip product rule and Equation (8.12) give \E\sum_jw_j|d_je|^2\le C_B. In Equation (8.14) use

The elementary column estimate needed here is

For the term without r_j, this is the bounded operator estimate for the column-flipped test matrix. For the term with r_j, change x to x^j using detailed balance. Replace p_i(x^j) by p_i(x)-2x_jd_jp_i(x); the first part has the same operator bound, and bounded individual column norms bound the second part by the displayed derivative budget. Local comparability of v at the two corners supplies the same constants for the weighted labels.

Factor 1-r_j=(1-\sqrt{r_j})(1+\sqrt{r_j}) in the remainder. Since |e|\le C_B\norm{p} and |x_j-m_j|\le2, Equations (8.13) and (8.17) make that remainder o(1) by Cauchy–Schwarz: its two squared budgets are o(n) and O(n^{-1}). Derivatives falling on e cost O(n^{-1/2}), by its Dirichlet bound and the same column estimate. Derivatives falling on p cancel exactly, since d_jp_i=d_ip_j and A is antisymmetric. The remaining identity is

Approximate B_A in the visible norm by allowed recipes B and use Bessel’s inequality and the squared contraction comparison before removing that approximation. The result is \norm{\ip{q_s}{B_A}_{\mathcal H}}_{\mathcal S_s}^2=0. The diagonal argument is identical, with the cancellation supplied by d_ip_i=0. It uses the stationary adjoint of the stated diagonal functional and the same n^{-1/2} normalization.□

Proof. The approximation has two stages: smoothing in form norm, followed by finite recipe approximation. By Proposition 8.5 and Lemma 8.1, q lies in the graph domain of componentwise L. First replace it by S_\varepsilon q. The spectral theorem makes this approximation strong in the norm \norm{q}_0^2+\mathfrak d(q). For fixed \varepsilon>0, approximate the input q by finite smooth recipes q_k. Since

the second approximation is also strong in form norm. This norm controls both weighted derivative quantities displayed in the statement. These operations apply to the deterministic vector q; the scalar Hilbert direction used to obtain it has not been differentiated.

For fixed \varepsilon,k, the derivative of the one-state recipe q_k is A/\sqrt n in the smooth word, Hadamard, and column-flip closure. Bounded finite-rank errors disappear in normalized Hilbert–Schmidt norm. Propagate this terminal derivative by the direct single-future identity dS_\varepsilon=K_\varepsilon d, separately for each component. The common columnwise path tail bound approximates this operation in Hilbert–Schmidt square mean, because the fixed input derivative has bounded Hilbert–Schmidt norm.

The remaining conditional expected path matrices lie in the same strong tracial closure. Choose their bounded labels as fixed smooth multitime circuits in the class of Lemma 4.4, with likelihood caps and any finite spin-sign decomposition already fixed. Then simulate the finitely many transitions with the nested accuracies of Theorem 5.1. Conditional pair-trace tests give the squared error. Averaging r independent simulation seeds has conditional Monte Carlo error at most C/r in normalized Hilbert–Schmidt square mean. Each fixed simulated calculation is a one-state word matrix. Only strong Hilbert–Schmidt approximation is required in this step: we neither differentiate a spatial-square-mean approximation of a label nor promote its error to an operator error. When an actual derivative of a fixed multitime circuit is needed elsewhere, its operator truncation and subsequent matrix reduction are instead the separate conclusions of Lemma 4.5 and Proposition 5.12. Thus the derivative approximation here is strong, rather than only a list of converging individual traces. Diagonal extraction is performed before taking closure and is a contraction for the relevant diagonal norm; its convergence also follows directly from the form approximation.

The derivative matrices were strongly approximated by A/\sqrt n with A in the word closure. Multiplication on the two sides by D_{\sqrt v} is a contraction in Hilbert–Schmidt norm. The weighted curl is therefore strongly approximated by D_{\sqrt v}(A-A^{\mathsf T})D_{\sqrt v}/\sqrt n. Since the word closure is linear and transpose-stable, these are precisely weighted antisymmetric test matrices of the type used in Equation (8.16). Equation (8.16) and adjoint annihilation force its inner product with every element of this dense antisymmetric test class to be zero, hence its norm is zero. The diagonal tests do the same for the own-spin derivative. The strong form approximants then give Equation (8.18). Applying the argument to all scalar Hilbert directions, or to a countable orthonormal basis of their separable span, proves the corresponding tensor assertion in square mean.□

Proof. The first assertion is the conditional-variance identity

For the curl estimate, work on one conditional two-spin law before summing over pairs. Condition on the spins outside i,j and write the pair law as p_{ab}, a,b\in\{-1,1\}, with coupling K=J_{ij}. Delete K and normalize to obtain a product law q=q_i\otimes q_j. Put m=\E_q a, m'=\E_q b, A=1-m^2, and B=1-(m')^2. Uniformly in the fields,

Write u_b=d_iQ_i(b) and t_a=d_jQ_j(a). Affinity in one spin gives

Consequently the averaged curl has the exact representation

where

and E_j has the symmetric definition. In particular all three quantities in Equation (8.20) are independent of the two spins. With W=\E_p[v_i(b)v_j(a)], comparison and Jensen give

The log derivative of \operatorname{sech}^2 has absolute value at most 2, so |m_i(\pm)-m|\le|K|e^{2|K|}A. Thus

The last inequality retains the two opposite corners even when one has very small probability. Combining it with W\le e^{6|K|}AB, A^3\le A, and the comparison of p\,v_i with qA, gives

Use the analogous bound for E_j, square Equation (8.20) with factor 3, average over the other spins, and sum ordered pairs. The two error sums together cost 12e^{14\delta_J}R_JO(Q), proving the claim. No lower bound on an individual conditional variance, and no unweighted derivative bound, is used in this removal step.□

Proof. For a full-support posterior P define

Center the minimizer under P to fix its additive constant. On the finite cube the Dirichlet form is positive definite modulo constants, so the minimizer is unique and the following differentiations are legitimate. We estimate its field derivative, then its observation drift, and finally the curl energy at the product endpoint.

The normal equation and its derivative. Since r_i is own-spin independent, stationarity on a single edge gives

Full support therefore implies \sum_i(x_i-m_{P,i})r_i=0 pointwise. Freeze the minimizer and vary the external field in coordinate j. The derivative of the normal functional against \phi is exactly

Indeed the derivative of the measure multiplies the pointwise zero in Equation (8.22); the remaining derivative is \partial_{h_j}m_{P,i}=\1_{i=j}v_{P,j}. Testing Equation (8.22) with \phi=x_j also gives \E_Pv_{P,j}r_j=0. If the heat-bath gap of P is at least \gamma, then the dual norm relative to \mathcal E_P satisfies

This uses Cauchy–Schwarz after centering \phi, followed by the posterior Poincaré inequality.

The drift of the minimum. Let A\succeq0 be the precision per unit time on one segment of the observation. In innovation coordinates the posterior masses satisfy

This is the finite-state filtering identity; it follows by the quotient rule for likelihood weights, as in [21, Equation (2.1)], with A^{1/2} inserted for the stated precision. For a fixed i and rest configuration z, write a=P(z,+), b=P(z,-), and

The edge contribution to the quadratic cost for frozen r is c_i(z)r_i(z)^2. The Hessian of c(a,b)=4ab/(a+b) is

The difference of the innovation vectors of the two corners is 2A^{1/2}e_i. Itô’s formula therefore gives precisely

Summing frozen residual costs bounds this drift below by -\norm{A} M(P).

The drift of the optimized cost also includes motion of the minimizer. This contribution is nonpositive and must be bounded from below. In coordinates on potentials modulo constants, write the cost near its minimum as M+\mathcal E_P(u)-2F(u). For an infinitesimal innovation, minimization subtracts the squared dual norm of the first variation F. Thus this contribution is

by Equation (8.24). This is also obtained directly by Itô differentiating the inverse of the positive Dirichlet matrix. Together with Equation (8.25), it proves

until the posterior-gap stopping time.

The product endpoint. At a product law write \sigma_i^2=v_i and \chi_i=(x_i-m_i)/\sigma_i. Expand the own-independent Q_i in the orthonormal products \chi_T. For each nonempty S\subset[n], put b_{i,S}=\sigma_i\widehat Q_i(S\setminus\{i\}), i\in S. The weighted cost on this support is the squared distance of (b_{i,S})_{i\in S} from the constant vector, since a potential coefficient \widehat g(S) contributes that same value to each b_{i,S}. For |S|=1 this distance is zero. For |S|\ge2,

The ordered-pair sum is exactly the weighted curl energy on S. Consequently

Transporting the curl energy. Fix i\ne j and the remaining spins z. The curl c_{ij}(z)=d_jQ_i-d_iQ_j is independent of both x_i,x_j. For the four joint masses p_{ab}=P(z,a,b) define

This function is homogeneous and concave on the positive cone, since

Extend it continuously to the boundary. If the pair has no coupling, \sum_{a,b}p_{ab}v_i(b)v_j(a)=16H(p). Deleting a coupling \theta_{ij} changes each conditional pair mass and each conditional variance by factors e^{O(|\theta_{ij}|)}, uniformly in the external fields and z. Hence, throughout the observation path where |\theta_{ij}|\le|J_{ij}|,

This concave function of the four joint masses supplies the required transport estimate. Each joint posterior mass is a martingale, so concavity in Equation (8.28) implies \E H(p_t)\le H(p_0). Multiply by c_{ij}(z)^2 and sum over z,i,j. Using Equation (8.29) at the initial point and the product endpoint shows that expected terminal curl energy is at most e^{C\max|J_{ij}|} times its initial value.

Finally stop before the posterior gap fails, using the test-independent stopping rule in Proposition 3.2. The horizon and precision norm are fixed. Equation (8.26) makes e^{Ct}M(P_t) a submartingale up to this stopping time. Its expected terminal value is bounded by Equation (8.27) on good paths. On stopped paths use M(P)\le\E_P\sum_i Q_i^2\le\sup_x\sum_iQ_i(x)^2, a polynomial bound. Their probability is exponentially small. More explicitly, let T be the fixed observation horizon, let \tau be the stopping time, and write

Using the full observation posterior P_T also on the exceptional paths, optional stopping and nonnegativity give

The first inequality uses the product endpoint on paths that reach T without stopping; the second is the concave curl-energy comparison. This proves Equation (8.21). All operations can first be localized where posterior masses are bounded away from zero; bounded finite-cube costs and the displayed inequalities then remove the localization.□

Proof. Apply the preceding results to a deterministic vector obtained by contracting q_s with an arbitrary fixed scalar Hilbert direction. First, Lemma 8.8 gives strong approximants with vanishing curl and own-spin derivative. Lemma 8.9 makes the approximants own-spin independent, preserving their weighted limit and vanishing curl. Apply Lemma 8.10 separately to each fixed polynomial approximant, letting dimension tend to infinity before improving the approximation. Their weighted distance to exact gradients tends to zero. Lemma 8.4 puts the limiting deterministic vector in \mathcal X_v. Orthogonality to \mathcal X_v^\perp after contraction against every scalar direction gives Equation (8.30).□

Proof. The weighted estimate alone cannot be inverted through \iota. Instead, separate the gradient extension into its diagonal differential part

and the multiplication operator ap. Uniformly in x,i,

The first inequality follows by the one-variable mean-value formula for \tanh, with error uniform because \max|J_{ij}|\to0; the second is uniform concentration of row square sums. Integration against stationarity and completion of the square give

Thus its limiting semigroup T_t^{\rm diag} has norm at most e^{-(1-\beta^2)t}. The other part maps weighted to unweighted norm:

Duhamel’s formula consequently gives

Since \norm{q}_v\le\norm{q}_0, the convolution gives Equation (8.31), including equality of the two exponents by taking strict slack. Tensoring and conditional expectation do not increase any norm used here. Finally \lambda\le(1-\beta)^2<1-\beta^2 for 0<\beta<1.□

Proof. There are two steps. First, the hidden estimate bounds the essential norm in the ordinary exact space. Second, the dense intertwining map identifies the resulting finite-dimensional spectral subspaces in the weighted completion.

Essential norm. For every fixed deterministic u\in\mathcal H and bounded recipe B, the finite-dimensional contractions u^{\mathsf T}B concentrate at \ip{u}{B}_{\mathcal H}. This first holds for a fixed finite recipe; for a dense approximation u' use

It need not hold for an arbitrary random scalar times a visible vector.

We turn this concentration into the squared orthogonality required by the hidden theorem. Let u_k be a weakly null unit sequence in \mathcal X, and approximate each u_k within 1/k by a finite exact test combination. At dimension n_k impose concentration for its contractions with the first k recipes, its norm, and the operator pairing currently being tested. Take n_k sufficiently large that the finite pointwise bounds of this approximant are bounded by a fixed power of n_k. A sufficiently slow diagonal sequence k=k(n)\to\infty then has

Thus the representatives are hidden in the squared-contraction sense of Theorem 7.1; weak convergence by itself was not used as a substitute for this property.

To see that this controls the essential norm, take finite-rank orthogonal projections P_m^0\uparrow I on \mathcal X. The Hilbert-space identity

follows because P_m^0K_T is finite rank and every compact operator is uniformly small after applying I-P_m^0. If the claimed essential-rate bound failed, choose unit u_m\in\operatorname{ran}(I-P_m^0) and unit inputs p_m witnessing the excess. The outputs are weakly null, so the previous diagonal construction applies to these arbitrary witnesses; approximate the inputs too by finite exact test combinations while preserving the pairing. Split K_T=K_{T-1}K_1 for T>1. Lemma 8.1 supplies a uniform stationary Hessian budget for the exact inputs K_1p_m. Since their derivatives are independent of the differentiating spin, their w_j-weighted budget equals their v_j-weighted budget. Apply Theorem 7.1 to the hidden outputs and these smoothed inputs. Here the hidden theorem imports the multitime conditional derivative argument: its fixed smooth circuits use the mixed-slot hierarchy, common tensor truncations, and opened network reduction of Lemmas 4.4 and 4.5 and Proposition 5.12. Its path entry comparisons retain the typical integrated-parameter convention of Theorem 5.1; no supremum over auxiliary Duhamel times is added here. The bounded K_1 factor and loss of one time unit disappear after dividing the logarithm by T. This proves Equation (8.33) for all possible norm witnesses.

The dense embedding and its spectral projections. Fix 0<\rho<\lambda_{\rm sp} and choose T so large that the essential norm of T_X:=K_T|_{\mathcal X} is smaller than e^{-\rho T}. Choose r>e^{-\rho T} away from the spectrum, with r<1. The usual resolvent argument for a finite-rank perturbation of an operator of norm smaller than r gives a Riesz decomposition

where E_r is finite dimensional, contains the spectrum outside |z|=r, and the spectral radius of T_X|_{F_r} is at most r. For clarity, a finite-rank approximation with smaller remainder norm makes the resolvent outside that norm the inverse of a finite-dimensional analytic matrix; its zeros are isolated and have finite multiplicity. On a compact annulus |z|\ge r there are finitely many such zeros. This supplies precisely the decomposition just used.

Let T_v=K_T|_{\mathcal X_v}. It is positive self-adjoint and \iota T_X=T_v\iota. Injectivity implies that every generalized eigenvector in E_r embeds as a generalized eigenvector of T_v. Self-adjointness removes all nontrivial Jordan chains and forces these eigenvalues to be real and positive. Write W_r=\iota E_r. For f\in F_r and r'>r, the spectral-radius formula yields

If the T_v-spectral measure of \iota f charged (r',\infty), its kth moment in this inequality would grow faster than (r')^k. The spectral theorem therefore forces that measure into [0,r']; let r'\downarrow r. Thus \iota F_r is orthogonal to W_r. On the dense image \iota\mathcal X the Riesz decomposition is already an orthogonal decomposition into W_r and the low spectral subspace. It follows that

Density now excludes additional weighted spectral mass outside [0,r]. The displayed identity proves agreement of the projections through \iota; no boundedness of an ordinary Riesz projection in the weaker norm was assumed.

Given any rate strictly below \lambda_{\rm sp}, choose \rho above that rate and then r between the essential disk and its corresponding semigroup eigenvalue. The preceding argument accounts for all weighted spectrum below that rate by finitely many eigenvectors in \mathcal X. If the bottom \lambda is strictly below \lambda_{\rm sp}, it is therefore attained and isolated. Its complement has weighted generator spectrum bounded below by \lambda+\eta for some \eta>0. The inverse E_v\to E is bounded because these spaces are finite dimensional, so its composition with \Pi_v\iota gives the asserted projection for vectors of \mathcal H whose image is weighted-exact. Applying Lemma 8.12 to their complementary parts proves the last assertion.□

Proof. The proof has three steps: a limiting contraction with strict slack, uniformity down to an inverse-polynomial norm, and a fixed burn-in. Begin with a block whose initial law has vanishing specific log density. Work first with a fixed D>2, which will be enlarged below. For each possible choice of its origin and terminal horizon, apply Equation (9.1) with \ell=D and terminal function g=S_{T-a-D}f. In the limiting argument we use these restarted objects, denoting the process again by X, so that a=0, T=D, and the endpoint times are prescribed. Normalize N_D=1. The rough square estimate in Proposition 4.2, integrated over the last unit of this interval, supplies a time s\in[D-1,D-1/2] at which

One can impose the reverse Hellinger estimate of Lemma 8.6 at the same time. Both requirements hold on a set of times of positive measure after enlarging their fixed constants. By Proposition 8.11, the visible projection q_s therefore lies in the exact weighted sector, including its scalar Hilbert factor. By Proposition 8.5, Theorem 8.3 and Lemma 8.12, its propagation to the prescribed time 0 has norm at most

Conditional expectation on the scalar factor is a contraction. The upgrade applies to an unweighted visible vector whose weighted image lies in \mathcal X_v; it does not require membership in \mathcal X. The hidden remainder at time 0 satisfies the corresponding estimate with any rate strictly below \lambda_{\rm sp}, by Theorem 7.1, testing against that remainder itself. Finite recipe projections followed by increasing the projection rank justify this test; the norm and Hessian bounds in Equation (9.4) are exactly its input hypotheses. Consequently, for every \rho<\lambda,

Choose \rho\in(\lambda-\xi,\lambda) and then D so large that the right side is strictly smaller than e^{-(\lambda-\xi)D}. This leaves fixed multiplicative slack.

We specify how the comparison time is selected, since the path estimates also contain auxiliary integration variables. First impose finitely many recipe, conditional-pair, and matrix tests at a fixed accuracy. The exponential stationary discard bounds transfer through the upper density bound and every fixed polynomial weight. Bounds for deterministic masks are integrated over their simplex variables before this transfer. Fubini and extraction give convergence in measure on the bounded domains of auxiliary times. The uniform rough L^2 estimates, with the Hessian budget when it occurs, control the resulting integrals. In particular, we do not bound an exceptional time integral by its Lebesgue measure times a polynomial pointwise supremum. Increasing the finite test lists and decreasing their accuracies permits one common choice of s along a subsequence. Strong continuity, or a further fixed positive smoothing time, treats a convergent sequence of selected times. This is the deterministic countable-test content of Theorem 5.1.

To obtain the additive accuracy in Equation (9.3), consider an arbitrary proposed violating sequence of starts, functions, origins a_n, and horizons T_n. Restart each at a_n on [0,D], with g_n=S_{T_n-a_n-D}f_n, as above, and set B=N_D^2 for its restarted endpoint norm. If

the rough estimate already gives N_0^2\le n^{-H}. Otherwise divide p and its potential by \sqrt B. A polynomial pointwise bound remains, since B^{-1/2}\le e^{CD/2}n^{H/2}. If the claimed inequality failed along a sequence of deterministic good disorders, its normalized version would contradict the strict slack in Equation (9.5). Indeed the restarted laws satisfy Equation (9.2), their normalized terminal functions remain polynomially bounded, and every endpoint comparison is now at 0 or D. Only the interior good time is selected after imposing the integrated tests. Equation (9.1) then returns the estimate to the original block, regardless of the sequences a_n,T_n. This proves the required uniformity. The inverse-power remainder comes from normalization and strict slack, rather than a rate assigned to a fixed-time o(1).

We finally prove that one fixed burn-in works with disorder probability tending to one. A deterministic diagonal contradiction will place every required typicality test on the same sequence of disorders; convergence in probability of the limiting block estimate alone would not supply this conclusion. Enumerate the deterministic good-disorder inequalities just used, including the quenched exponential bounds for stationary exceptional sets and the uniform forward superpolynomial discards. For the first k tests, with their fixed tolerances, let their positive exponential rates be c_1,\ldots,c_k. By Proposition 3.5, choose fixed numbers b_k\uparrow\infty so that

Include the density bound at this particular b_k in the finite good-disorder event G_{n,k}.

Suppose every fixed burn-in failed. For each b_k, there would be \varepsilon_k>0 and arbitrarily large dimensions at which its failure event F_{n,b_k} had probability at least \varepsilon_k. Choose one such n_k, increasing sufficiently fast, with

and with all the finitely many dimension thresholds satisfied. Select a deterministic disorder in F_{n_k,b_k}\cap G_{n_k,k}, together with a violating start, function f_k, origin a_k\ge b_k, and horizon T_k\ge a_k+D. Restart on [0,D] with initial law S_{a_k}(x_k,\cdot) and terminal function g_k=S_{T_k-a_k-D}f_k. Its initial density is at most e^{n_k\delta_k} by Equation (9.2) applied after b_k, and \norm{g_k}_\infty\le\norm{f_k}_\infty. Normalize only above the inverse-polynomial cutoff specified above. Every fixed typicality test now holds eventually on this deterministic sequence; its exponential discard survives multiplication by e^{n_k\delta_k} and the fixed polynomial normalization. The limiting argument therefore contradicts the same fixed contraction slack. No lower bound on the sequence \varepsilon_k was used. Thus some fixed b_0 works, as asserted.□

Proof. Iterate the uniform block estimate after the fixed burn-in. Contraction makes the additive remainders summable: for k full blocks their total is at most

The rough square estimate handles the burn-in and the remaining interval of length less than D. Their total length is bounded by b_0+D, so they change only the constant. One may restart this argument at time a, with its arbitrary marginal law; integrating the point-start estimate handles mixtures. Short intervals are covered directly by the same rough estimate. Increase the chosen block accuracy by one fixed power if necessary to absorb constant factors. This proves Equation (9.6). For f(x)=u^tx, the terminal gradient is the constant vector u, giving Equation (9.7).□

Proof. Extend g multilinearly to the cube: let \overline g(z), z\in[-1,1]^n, be its expectation under the product law with coordinate means z_i. Then \partial_i\overline g(z)=\E_z d_ig, since d_ig does not depend on its own spin. Jensen’s inequality gives

Integrate along the Euclidean segment joining two vertices, whose length is at most 2\sqrt n.□

Proof. We will find a late time with bounded gradient energy, propagate that bound back to time zero, and then control oscillation. Relabelling \epsilon, it suffices to work at T=(1+2\epsilon)t_n. Fix |f|\le1 and a starting point. The backward martingale S_{T-s}f(X_s) has jump bracket density 2\sum_iw_i p_{s,i}^2. Hence

On the interval I=[(1+\epsilon)t_n,T], whose length is \epsilon t_n, Equation (9.12) applies. On its good field event,

On the complementary event \norm{p_s}^2\le n, since |f|\le1. There is thus a time s\in I at which

Apply Equation (9.6) between 0 and this time. Uniformly over the starting point and f,

Choose

This choice makes the squared gradient o(n^{-1}), exactly the scale needed to offset the cube diameter in Lemma 9.3. Hence \operatorname{osc}(S_Tf)=o(1) uniformly for |f|\le1. Since \mu S_Tf=\mu f, the variational formula for total variation proves the result.□

Proof. Let \sigma_n be the probability measure characterized by

Its fixed-test weak convergence to the spin spectral law is supplied by Proposition 6.7. For each fixed \delta>0, the definition of its support edge supplies c_\delta>0 such that, with disorder probability tending to one,

One obtains this statement by testing a nonnegative continuous function supported below \lambda+\delta with positive limiting integral. It follows for every t\ge0 that

The fixed spectral-mass bound remains valid for every time in this inequality. It therefore permits logarithmic times without taking a dimension-dependent Laplace-transform limit.

Averaging over the initial equilibrium spin selects a configuration x for which \E_x[x^tX_t] is at least the value in Equation (9.15). The unit linear test f_x(y)=n^{-1/2}x^ty detects this remaining overlap. Its equilibrium mean is zero and its transition mean is at least c_\delta\sqrt n e^{-(\lambda+\delta)t}. Its transition variance is bounded by Equation (9.8); its equilibrium variance is at most the inverse gap, by Theorem 2.1. At t=(1-\epsilon)t_n, choose, for example, 0<\delta<\lambda\epsilon/4. The displayed mean diverges as a positive power of n. Chebyshev’s inequality applied at half this mean gives total variation tending to one.□

Proof. We may choose D>2. First put the block at prescribed relative times using Equation (9.1). In the terminal alternative take \ell=D and g=S_{T-b}f. In the subsequent-block alternative take \ell=2D and g=S_{T-b-D}f. In both cases the restart origin is a=b-D. The new initial law is the old marginal at a, with the same upper density bound and common trajectory Poincaré constant. The gradient and recipe contractions are unchanged at matching times; in particular \widehat A_r=A_{a+r}. The terminal alternative is the same condition on \widehat p_D, while the other alternative gives \widehat N_{2D}\le R\widehat N_D. For the limiting argument denote the restarted process again by X and use a=0,b=D; retain also 2D in the second alternative. Normalize N_D=1.

We first establish exactness at D, then control the mean and fluctuation of the slow coefficient, and finally recover a uniform finite-dimensional estimate. In the terminal alternative, q_b lies in the visible exact sector by assumption. In the subsequent-block alternative, the budget N_{2D}\le R allows Proposition 8.11 at a good time selected in a fixed subinterval compactly contained in (D,2D). Both adjacent time separations thus stay positive after extraction. Transport back to the prescribed time D by Proposition 8.5 gives the same conclusion at b=D. The value of the finite budget R has served this purpose and is not used again.

Retain 0,D and the good time s\in[D-1,D-1/2] in the same scalar extraction, together with 2D and the later good time when that alternative is used. If c_r denotes the random slow coefficient of q_r, projection and Proposition 8.5 give

This transports the mean at the desired rate. To align the vector itself in unweighted square mean, we must also make the centered coefficient small.

Choose a good time s\in[b-1,b-1/2] as in Equation (9.4). Approximate one slow column by a coarse smooth scaled recipe B. The scalar function

approximates the corresponding coefficient c_s in scalar L^2 to error at most C\delta, where \delta is the unweighted approximation accuracy. Indeed multiplication by D_v is a contraction in the ordinary norm, and Equation (8.10) extends the contraction with p_s from fixed recipes to the slow column. The bound uses typicality under the evolving law; stationary mean-square approximation alone would not give it.

The trajectory Poincaré inequality will control this fluctuation once we bound the gradient of F_s. Set c_i=v_iB_i; the exact product rule gives

The columns of B have entries bounded by C_B/\sqrt n, and the matrix (d_jc_i)_{ij} has operator norm at most C_B/\sqrt n. For the first term, mixed-gradient symmetry changes d_jp_{s,i} to d_ip_{s,j}. Also v_i(x^j)\le e^{C|J_{ij}|}v_i(x) and v_i\le2w_i, so

Cauchy–Schwarz, followed by summation over j, proves

The same estimate applies to every one of the finitely many columns.

Before choosing these recipes or D, freeze once and for all a moderate-rate instance of Equation (9.6), say with \rho_0=\lambda/2, and its constant C_{\rho_0}. The Poincaré bound at time a, this instance of propagation applied to the scalar function F_s, and Equation (9.18) give

The coefficient approximation costs C\delta in L^2 before transport. By Equation (9.17), both this error and the square root of Equation (9.19) carry the factor e^{-\lambda(s-a)}. Relative to e^{-\lambda D}, their cost is bounded by

In particular the approximation error does not acquire a factor e^{\lambda D}.

The hidden part and the exact fast complement, propagated from s, cost at most C e^{-(\lambda+\kappa)(D-1)} for some \kappa>0, by Theorem 7.1, Proposition 8.13 and Lemma 8.12. The latter also applies when the visible vector is only in \mathcal H with weighted image in the fast part of \mathcal X_v. After division by e^{-\lambda D}, this fast-sector error is bounded by C e^{\lambda+\kappa}e^{-\kappa D}. First choose \delta sufficiently small in terms of \eta and C_E, and fix the coarse recipes and C_B. Now choose D large so that both normalized exponential errors,

are smaller than the required fixed fraction of \eta. Restoring the common factor e^{-\lambda D} proves the limiting mean and fluctuation estimate with exact slow coefficients and reconstruction.

The block length is now fixed, so we may choose the finer recipes \widetilde E without altering the fluctuation estimate. At this D, an accuracy smaller than c\eta e^{-(\lambda+C)D} makes all changes in the reported A_a, A_b, and the reconstruction smaller than the remaining fraction of \eta e^{-\lambda D}. Here e^{CD} is a rough bound for any endpoint norm needed in this comparison. The finer derivative constants have not replaced C_B in Equation (9.19).

Finally use strict slack, the deterministic good-disorder selection in the proof of Proposition 9.1, and normalization by N_b. The condition N_b\ge n^{-Q} preserves a fixed polynomial envelope; choose every Poincaré and propagation remainder with an exponent larger than 2Q+H+2 before using it. For a violating sequence, perform the one-block or two-block restart above before extraction. The normalized gradients, actual conditional laws, common Poincaré bound, and endpoint ratio meet the same hypotheses at the prescribed relative endpoints; any parameter selected from the family is fixed conditional on the disorder before the path is sampled. The resulting fixed-interval sequence would contradict the limiting estimate. Returning by Equation (9.1) proves Equation (9.16) uniformly at the original block positions, without a rate for any fixed-time o(1). This includes the parameter selected from the preparation family below.□

Proof. Both the added field and its difference matrix are small. Scaled recipe calculus gives

At each refresh attempt, including an attempt that leaves the spin unchanged, the ratio of the perturbed outcome probability to the ordinary outcome probability is at most e^{2\varepsilon_n}. On the common attempt construction the path likelihood is therefore at most e^{2\varepsilon_n N_h}, with N_h\sim\operatorname{Poisson}(nh) under the ordinary process. Started at \mu, the ordinary endpoint has law \mu conditional on N_h=k, because its embedded update kernel preserves \mu. Thus X_h and N_h are independent under this stationary reference law, and for every endpoint y,

Averaging the attempt likelihood therefore proves the endpoint density bound Equation (9.22).

For the variance estimate, we use gradients of the perturbed semigroup. Differentiate Equation (9.21). The exact gradient extension has coefficients a_{ij}^z=d_im_j^z, including the diagonal a_{ii}^z. Its formula is

The diagonal a_{ii}^z p_i is an o(1) zero-order term; the own-spin derivative of an exact gradient is zero. Taylor expansion of \tanh, with Equation (9.24) and the bounded row and column square sums, gives common bounds

For example the leading matrix is the field-difference matrix times the diagonal variance; v_j^z\le2w_j^z controls the displayed weighted row sum. The quadratic Taylor remainders are bounded Schur products of matrices with bounded row and column square sums. Once the right side of Equation (9.24) is at most one, the square-evolution proof of Proposition 4.2 therefore has a common constant C_*. In particular

Thus the gradient estimate remains valid for this possibly nonreversible preparation: it was derived from the stated generator and its exact gradients, without requiring a Gibbs invariant law.

Its jump bracket is at most 4\norm{dT_{h-s}^zf}^2. The conditional martingale variance formula and Equation (9.25) give

The other term in total variance is

Their sum is Equation (9.23).□

Proof. The cone compares the full gradient norm with its mean slow coefficient. Assume N_b\le K|A_b| at the end of a block. Equation (9.16) gives

where C_E' is arbitrarily close to C_E after fine approximation. Choose K\eta<1/4 and then \eta small enough that the last coefficient is at most K, and K\eta/(1-K\eta)\le\epsilon_0. Thus the cone is preserved. At the terminal time it holds by Equation (9.20). At each step, apply the fixed-interval restart in the proof of Proposition 9.6 to the ordinary trajectory with initial block law \nu_zS_a. Its upper density bound follows from Equations (9.22) and (9.2), and its common Poincaré bound is the one fixed above. This restarts only the ordinary part of the evolution, leaving the preparation T_h^z unchanged. At subsequent blocks the required forward norm ratio follows from the preceding cone step:

This ratio is finite for the already chosen block D. Its only role in Proposition 9.6 is to establish exactness at the block endpoint, so its dependence on D does not create a circular choice.

After k=t/D steps,

Take D\ge1 and \eta\le\xi_1/(2K), so that the extra exponent is at most \xi_1. To verify the inverse-power cutoff throughout the iteration, choose

The lower bound for |A|, and |A|\le C_A N, then keep N above n^{-Q} for all the O(\log n) blocks. Choose the inverse-power accuracies used in the supporting Poincaré estimates larger than 2Q+H+2. No new union bound over fixed-time limiting errors is needed: Equation (9.16) holds on one event uniformly over the functions, parameter values, and comparison times in question. The residual estimate in Equation (9.26) follows from the final cone step.□

Proof. We estimate the Jacobian in each unit output direction. For |u|=1, put F=\sum_j u_jF_j and apply Equation (9.26) at the exit from preparation. Differentiation on the finite state space and Duhamel’s formula give the row vector

Indeed \partial_{z_k}L^zf= \sum_i v_i^z\widetilde E_{ik}d_if, also when the target depends on the old spin.

Write p_0=\widetilde E A_0+r under \nu_z. The rough vector extension behind Equation (9.25) propagates r back through preparation at cost at most e^{C_*h/2} in mean square. Its contribution to Equation (9.28) is therefore O(h\epsilon_0|A_0|).

For the main vector, uniformly over 0\le s\le h,

Here the norm is the sum over the fixed number of columns. To verify this estimate, the diagonal likelihood representation has row moments bounded by e^{C h}, and its likelihood difference from one has squared mean O(h), by the likelihood martingale bracket and the bounded row square sums. The entries of a fixed scaled recipe are bounded by C/\sqrt n, so this diagonal error costs O(h) after summation. The part with at least one inserted coefficient matrix costs O(h) in norm by its Duhamel expansion and the common rough bound. Finally the ordinary recipe value changes by o_h(1) in spatial mean square along the short path: its smooth Euclidean extension has Lipschitz constant C_{\widetilde E}/\sqrt n, while the expected number of changed spins is at most nh. These estimates also apply to the perturbed rates, using Equation (9.24); the own-spin zero-order correction is o_n(1). They prove Equation (9.29) without a regularity assertion for the exact eigenvectors.

By Equation (9.22), every marginal in this calculation has upper relative log-density o(n). The one-state typicality comparisons, the chosen fine accuracy, and \norm{v^z-v}_\infty=o(1) therefore give

uniformly in s,z. Combining this with Equations (9.28) and (9.29) yields

Choose \epsilon_0 and the fine accuracy first, and then choose a positive fixed h so that the factor in parentheses is small. Equation (9.26) now proves Equation (9.27), uniformly for every unit output vector.

To pass from the Jacobian estimate to a ball of attainable means, we lift straight line segments by the inverse Jacobian. This argument requires only the proved singular-value bound, not global injectivity. For a target vector v with |v|<a_tR_n, solve the inverse-Jacobian ordinary differential equation

The right side is smooth and |z'(r)|\le |v|/a_t<R_n. The solution stays in a compact subball of the parameter ball and therefore exists to time one. Differentiation gives \mathcal M_t(z(r))=\mathcal M_t(0)+rv. At z=0 the preparation is ordinary stationary evolution, so \mathcal M_t(0)=(\mu F_j)_j. This proves the assertion.□

Proof. Fix 0<\epsilon<1, take 0<\alpha<\epsilon/8 and 0<\xi_1<\lambda\epsilon/4, and round (1-\epsilon)t_n up to an integer multiple t of D. The radius in Lemma 9.9 is at least

The exponent is positive, so the ball of attainable means has diverging radius. In particular, one may choose a preparation for which the mean of F_1 differs from \mu F_1 by a quantity tending to infinity. The variances of F_1 under \nu_zS_t and \mu are bounded: use the common trajectory Poincaré estimate and its bounded pointwise gradient for the first, and the equilibrium gap for the second. Chebyshev’s inequality implies \norm{\nu_zS_t-\mu}_{\TV}\to1. Since \nu_z is a probability law on starting configurations, convexity bounds this distance by \sup_x\norm{S_t(x,\cdot)-\mu}_{\TV}. Monotonicity then gives the same lower bound at the unrounded, earlier time (1-\epsilon)t_n.□

Proof. The upper bound follows from Lemma 9.4. For the lower bound, Lemma 9.5 applies when \lambda=\lambda_{\rm sp}, and Lemma 9.10 applies when \lambda<\lambda_{\rm sp}. These cases exhaust the definition of \lambda.□

Proof. We prove both directions of clock transfer. Write d_c(t) and d_d(k) for the continuous and discrete distances; their kernels satisfy the exact Poissonization identity S_t=e^{nt(P-I)}. For k=\lfloor(1-\epsilon)nt_n\rfloor and T=(1-\epsilon/2)t_n, convexity and discrete-time contraction give

The probability is O((nt_n)^{-1}), while d_c(T)\to1, proving the discrete lower bound.

For the upper bound, a uniform holding probability permits the local Poisson-to-binomial lemma. Restrict to the high-probability event \norm J_\op\le K. At every configuration, n^{-1}\sum_i(Jx)_i^2\le K^2, so at least half of the fields have absolute value at most \sqrt2K. The probability of keeping a selected spin is at least (1+e^{2|(Jx)_i|})^{-1}. Consequently

Choose \vartheta=1-h_0/2\in(0,1) and write

Here R is a Markov kernel preserving \mu. Set M=\lceil(1+\epsilon)nt_n\rceil. For every x and |f|\le1, put u_j=R^jf(x)-\mu f, so |u_j|\le2. For each fixed real y,

The time correction is O_y(\sqrt{\log n/n}), and hence this time exceeds (1+\epsilon/2)t_n for large n. The absolute value of the display is bounded by 2d_c((1+\epsilon/2)t_n)\to0, uniformly over x,f.

We have verified every shifted Poisson average required by Lemma 2.3, uniformly over starts and tests. Apply that uniform transfer: its assumptions are a fixed \vartheta\in(0,1), M\to\infty, bounded arrays, and vanishing Poisson averages for every fixed shift y, all just verified. It gives

uniformly over the test family. Taking half its supremum proves d_d(M)\to0. For convergence in disorder probability, the proof of that lemma uses a fixed finite signed approximation by shifted Gaussian densities; its finitely many Poisson averages are all bounded by the same random quantity 2d_c((1+\epsilon/2)t_n). Thus it applies on the present high-probability events without any additional convergence rate.□

Proof. For the concentration estimate, we use the Maurey–Pisier Gaussian rotation and double-Jensen argument; see Giraud [14, Exercise 1.6.7, equation (1.10), parts A–B]. Rotation between two independent Gaussian vectors expresses the increment as an integral of directional derivatives. First take F smooth with \|\nabla F\|\le D. Let Z' be an independent copy of Z and define the rotation by

For a fixed angle, Z_\theta and V_\theta are independent standard Gaussian vectors. This independence lets us evaluate the exponential moment after applying Jensen’s inequality, first in Z' and then with respect to the uniform measure on [0,\pi/2]:

To pass to a general Lipschitz F, smooth by convolution and use dominated convergence. Linear growth gives the Gaussian exponential integrability required for this passage. Optimization of the exponential Markov bound then gives the tail estimate. The case D=0 follows directly because the function is constant.

For the comparison statement, we interpolate independent realizations of the two processes and show that the expected maximum increases along the interpolation. For \lambda>0, replace the maximum by the smooth function

Gaussian integration by parts gives

The Hessian has zero row sums and nonpositive off-diagonal entries. Consequently the derivative can be expressed using only increment variances. With \Delta_{ab}=B_{aa}+B_{bb}-2B_{ab} denoting their difference, the right side becomes

The limit \lambda\to\infty gives the finite-set comparison. Exhausting a dense set by finite subsets and using continuity and dominated convergence gives the compact-set version. In the applications below, every such process is a linear function of a finite Gaussian vector, which ensures integrability of its supremum.□

Proof. We prove the operator-norm estimate first and then control the row sums and individual entries. The case \beta=0 is deterministic. For \beta>0, add independent diagonal entries of variance 2\beta^2/n to J. The resulting centered Gaussian orthogonal ensemble (GOE) matrix W has a covariance that permits a direct Gaussian comparison: for unit vectors p,q,

With \rho=p^{\mathsf T}q, the increment variance of p^{\mathsf T}Wp is 4\beta^2(1-\rho^2)/n. Bound it by 8\beta^2(1-\rho)/n and compare with the linear process (2\beta/\sqrt n)p^{\mathsf T}z, where z is a standard Gaussian vector. Lemma B.1 yields

In the independent standard Gaussian coordinates defining W, the quadratic form p^{\mathsf T}Wp is Lipschitz with constant \sqrt2\beta/\sqrt n, uniformly over unit vectors. The supremum inherits this constant. Concentration for \lambda_{\max}(W) and \lambda_{\max}(-W) therefore bounds \|W\| by 2\beta+\delta_n/2 with probability tending to one. The largest absolute diagonal entry is at most \delta_n/2 with probability tending to one as well. Removing that diagonal proves the first estimate in Equation (B.2).

For a standard Gaussian g,

Apply the exponential Markov bound to the n-1 independent squares in one row. The probability that their weighted sum exceeds \beta^2+\delta_n is at most \exp(-c_\beta n\delta_n^2) for large n. A union bound over the rows gives the second estimate; Gaussian tails and a union bound over entries give the third. The choice n\delta_n^2=\sqrt n makes all the failure probabilities vanish. The cubic row sums needed for the two-spin remainder are now controlled by the last two estimates together:

□

Proof. The proof has three steps: factor out the two conditional variances, identify the nonnegative part of the expansion, and absorb the cubic remainder into the one-site energies. Condition on all spins outside i,j, and denote the remaining spins by x,y. Their probabilities are

where a,b are the hyperbolic tangents of the external fields left after conditioning. Set Z_0=1+abt. The corresponding conditional means and variances take the form

and the analogous expressions with a,x and b,y interchanged. There are real numbers p,q,r such that

Indeed, expand the restriction of f in the basis 1,x,y,xy and denote the coefficient of xy by r. This isolates the common second-difference coefficient from the two first differences.

Let \E_0 denote uniform expectation over the four pairs (x,y). We now combine the mixed projection energy with the correction term in the claimed inequality. The identities

and their counterparts for y give the exact factorization

where

Only the expansion through second order will contribute to the leading term. Expanding at t=0 and reducing by x^2=y^2=1 gives

The cancellation can be checked directly: before collecting terms, the coefficient of t^2 is

Collecting terms gives the coefficient in Equation (B.4), with the coefficients of 1,x,y cancelling. Thus the displayed expansion contributes through the following single identity:

It remains to control the remainder without excluding nearly deterministic spins; we therefore allow the limiting values |a|=|b|=1 in this estimate. On the complex disk |t|\le1/2, the denominators 1+aty and 1+btx have modulus at least 1/2, while |m_i(y)|,|m_j(x)|\le3. The first term of C is bounded in modulus by 6. In the second term, t+2t^2m_i m_j is bounded by 5; including its remaining factors gives a bound of 600. Hence |C(t;x,y)|\le606 throughout the disk, and Cauchy’s coefficient estimate yields

Since

and 1-a^2-b^2+2a^2b^2=a^2b^2+(1-a^2)(1-b^2)\ge0, we obtain

For |t|\le1/100, the coefficient of r^2 exceeds 1/2.

The leading term is now nonnegative. To finish, the negative cubic term must be charged to the conditional energies while retaining the variance factors. Write A_0=1-a^2 and B_0=1-b^2. Summing over x first gives the exact energy formula

For |t|\le1/4, the denominators lie between 3/4 and 5/4, so

These bounds make P_0p^2 and P_0q^2 at most 20/9 times their respective conditional energies. Multiply the estimate for Q by P_0 and use 9696\cdot20/9<22000 to obtain the conditional inequality. Averaging over the spins on which we conditioned gives Equation (B.3). The estimates are homogeneous in p,q,r, and throughout the argument neither these coefficients nor the external fields were bounded.□

Proof. We use the pair estimate to compare the squared generator with its Dirichlet form. Fix \nu=\nu_{s,\theta} and define the zero-diagonal matrix by T_{ij}=\tanh(sJ_{ij}). On \mathcal E_n, all entries are small enough for Lemma B.3 to apply simultaneously. At each configuration put z_i=v_i d_i f and z'_i=m_i z_i. Summation over ordered pairs gives

where \circ denotes entrywise multiplication. The field-dependent factors remain inside the vectors, so the interaction matrices can be bounded by their operator norms. With M=\max(\|T\|,2\|T\circ T\|), the sum of the two quadratic forms is at most

because v_i^2(1+m_i^2)=v_i(1-m_i^4)\le v_i.

We next control these operator norms using only the disorder event. The Taylor estimate |\tanh u-u|\le C|u|^3, together with the maximum absolute row-sum bound for symmetric matrices, gives

Also \|T\circ T\|\le\max_i\sum_jT_{ij}^2\le\beta^2+o(1). The error bounds depend only on the deterministic tolerances defining the event; they are uniform over s,\theta and J\in\mathcal E_n. The fixed assumption \beta<1/2 therefore gives

Thus, for A=-L_\nu=\sum_i\mathsf D_i and all sufficiently large n,

To read this as a spectral gap, use that A is self-adjoint and nonnegative. Every one-site conditional law has full support, so \mathsf D_i f=0 for all i forces f to be constant along every edge of the cube. Thus the kernel consists exactly of constants. Applied to a nonconstant eigenfunction, the last inequality bounds its eigenvalue below by \kappa_\beta. This proves the gap and variance assertions. The covariance conclusion is the linear-function case f(x)=c^{\mathsf T}x:

□

Proof. We interpolate from the Gibbs measure to product posteriors, compare their expected entropy with the initial entropy, and then remove the observation from the one-site terms. For large n, Equation (B.2) ensures \|J\|<1. Choose K=J+2I; then K is positive definite and satisfies \|K\|<3. Keep it fixed throughout the argument for the realized disorder. Given X\sim\mu and an independent standard Gaussian vector z, use the observation, for 0<u\le1,

The posterior law \nu_{u,y}=\operatorname{Law}(X\mid Y_u=y) is

Discarding the diagonal term, which is constant on the cube, leaves interaction matrix (1-u)J. The observation creates an external field, and a further linear tilt changes only that field. Thus both uniformities in Proposition B.4 apply: every posterior and every such tilt has covariance norm at most C=\kappa_\beta^{-1}. At u=1, the interaction vanishes and the posterior is a product measure.

Derivative of the expected posterior entropy. To quantify the entropy lost to the observation, set H(u)=\E[\Ent_{\nu_{u,Y_u}}(f)]. The derivative identity we need is

The covariance with f here is a vector. We verify the identity using the finite Gaussian mixture defining the observation. Let \phi_u be the centered Gaussian density with covariance uI, and define

These functions give the density of Y_u as q\phi_u and the posterior expectation as p/q=\nu_{u,y}[f]. Their finite exponential sums show directly that both p and q solve the backward heat equation:

Writing r=p/q, the chain rule gives the exact identity

Since \partial_u\phi_u=\tfrac12\Delta_y\phi_u, differentiation and integration by parts yield

Differentiation and integration by parts are legitimate: the spin sums are finite, f is bounded above and below by positive constants, and the remaining dependence on y grows at most exponentially in \|y\|. Gaussian tails therefore control all terms. To identify the derivative as the claimed covariance expression, use

which proves Equation (B.9). At the initial endpoint u\downarrow0, the observations tend to zero in probability and the posterior tends to \mu. Since the entropy functional is bounded on this finite space, H(u)\to\Ent_\mu(f), giving the initial value for the comparison.

Covariance controlled by entropy. We next bound the derivative by the expected entropy itself. Let \nu be any posterior, take a unit vector a, and set Z=a^{\mathsf T}(X-\nu[X]). The second derivative of the log moment-generating function of Z is the variance of a^{\mathsf T}X under a linear tilt of \nu. Applying the uniform covariance estimate to these tilts and integrating twice gives

The change of measure \widehat\nu=f\nu/\nu[f] connects this exponential-moment bound to entropy. Apply Jensen’s inequality to the logarithm under \widehat\nu to obtain

Optimize in \lambda and take a parallel to \Cov_\nu(X,f) when this vector is nonzero. This yields

Equations (B.9) and (B.10) imply H'(u)\ge-C\|K\|H(u). Integrating from zero to one gives

We have now compared the initial entropy with the entropy at the product endpoint. The comparison used covariance control at every interaction scale: controlling linear tilts only at the original interaction J would leave the quadratic change in Equation (B.8) untreated. It remains to express the endpoint entropy in terms of the original one-site energies.

Removing the observation at the product endpoint. For a product probability measure \pi=\bigotimes_i\pi_i, entropy tensorizes:

where in the ith summand the other coordinates are fixed inside the one-coordinate entropy. To prove tensorization, separate one coordinate using the entropy chain rule and apply induction to the remaining product. For a fixed base measure, f\mapsto\Ent(f) is convex; hence averaging the separated coordinate can only decrease the remaining one-coordinate entropies. The required convexity follows from joint convexity of (a,b)\mapsto a\log(a/b) with a=f(x) and b=\pi[f].

Apply tensorization to the product posterior at u=1. We must then average under the joint law of (X,Y_1), retaining the dependence between the observation and the spins. For this step put F=f(X), \varphi(z)=z\log z,

Conditional Jensen and the tower property give

Thus the additional conditioning on the observation decreases the averaged conditional entropy, even though the observation and the complementary spins are dependent. Combining this fact with the posterior tensorization gives

We can now compare each original one-coordinate entropy with the corresponding heat-bath energy. Concavity of the logarithm gives

because the difference of the right and left sides is \E f\,[\log\E f-\E\log f]\ge0. Moreover, summing these conditional covariances gives

The product-endpoint estimate is therefore bounded by the entropy dissipation of the original generator. Substituting into Equation (B.11) proves Equation (B.6), with the permissible choice C_1=e^{3/\kappa_\beta}.

Entropy decay and mixing. For a point start, it remains to bound the initial relative entropy \log(1/\mu(x)) uniformly in the starting point. The identity \|x\|^2=n gives

Reversibility lets us evolve the density f_t=d\rho_{t,x}/d\mu using the same semigroup S_t. For t>0 the density is strictly positive, so the entropy inequality applies to its derivative:

Integrating this differential inequality and using continuity at zero establishes Equation (B.7), with C_2=\log2+1.

To finish, coarsen the measure to an event and its complement; relative entropy then dominates the resulting binary relative entropy. For fixed q\in(0,1), this binary entropy, viewed as a function of p, has value and derivative zero at p=q and second derivative 1/[p(1-p)]\ge4. It is consequently at least 2(p-q)^2. Choosing an event that attains total variation and using the entropy decay just proved yields

which proves the stated mixing upper bound.□

Proof. We first derive the exact evolution identity, then estimate the field differences before completing the squares. The half-difference d_if does not depend on spin i. At the all-positive point, the product rule reads

For j\ne i, the first difference after a flip is d_jf(x^{i,-})=p_j-2r_{ij}. The contribution with j=i to d_iL f is -p_i: both m_i and p_i are independent of spin i. Separating this term gives

Apply the square identity L(g^2)-2gL g=2\Gamma(g) with g=d_if. The equality \Gamma(d_if)=\sum_{j\ne i}w_jr_{ij}^2 then identifies the second-difference energy, proving Equation (C.2).

To use the identity uniformly in the local fields, we must keep the conditional variance in the estimate for a_{ij}. Set z=J_{ij} and m=m_j(\mathbf 1). Flipping spin i changes the j-field by -2z. The hyperbolic-tangent subtraction formula therefore gives

For real arguments with |z|\le1/10 and |m|\le1, the factor 1-m\tanh(2z) is bounded below by 4/5. The bound is uniform as m approaches either endpoint of [-1,1], so the variance factor can be retained without a field cutoff.

To track the constants in the cubic remainder, write T=\tanh(2z) and expand the denominator exactly:

The inequalities |T|\le2|z| and |\tanh u-u|\le |u|^3/3 imply

For the first estimate, the three terms of this decomposition contribute errors bounded by (4/3)|z|^3, (16/3)|z|^4, and 5|z|^3, respectively. Their sum is less than 7|z|^3. The second estimate is a consequence of the first, while the last is immediate from Equation (C.4). The hyperbolic-tangent inequality follows by integrating 1-(\tanh u)'=\tanh^2u\le u^2. Returning to the field difference gives

with the variance factor retained in the remainder.

We can now eliminate the second differences. Since r_{ij}=r_{ji}, the terms for one unordered pair in Equation (C.2) combine as

Completing the square bounds this expression by (a_{ij}p_i+a_{ji}p_j)^2/(w_i+w_j). We compare its numerator with the leading interaction term. For the same pair, write

Equation (C.5) gives

Hence |A^2-B^2|\le7|J_{ij}|^3S^2. The remaining issue is the denominator: it can be small at a nearly saturated field. Weighted Cauchy–Schwarz keeps the variance factors attached to the corresponding weights and gives

This estimate also describes the limiting fields. As m_j\uparrow1, the contribution is at most 4(1-m_j)p_i^2; as m_j\downarrow-1, it is at most 2(1+m_j)^2p_i^2. Both bounds hold without comparing the coupling size with the distance from saturation.

Only the accumulated replacement errors remain. Replacing \sum_{i\ne j}p_i a_{ij}p_j by the first two terms of M_J(p) costs at most

The last inequality uses symmetry and 2|p_i||p_j|\le p_i^2+p_j^2. For the completed squares, Equation (C.6) bounds the total replacement error by

Adding the two errors and using 7+224/27<16 proves Equation (C.3) at \mathbf 1. Sign conjugation then gives the asserted inequality at every x.□

Proof. We now combine the deterministic calculation with the disorder estimates. For the smaller temperatures, the rough bound already contracts at every configuration. For the remaining temperatures, the planted comparison will provide contraction outside a small Gibbs set. Begin on the events of Proposition B.2, where the deterministic tolerances give

Sign conjugation preserves each bound, so they hold for every J^x. The remainder in Lemma C.1 consequently tends to zero uniformly over configurations, test functions, and times.

If \beta\le7/20, it suffices to substitute the disorder estimates into Equation (C.7). Its coefficient is at most

The rational bound follows from 4/(3\sqrt3)+32/27<2 at \beta=7/20, since both terms increase with nonnegative \beta. There is therefore a fixed margin below one. Absorbing the vanishing remainder in Lemma C.1 proves Equation (C.8) with a fixed positive c and \mathcal B_J=\varnothing for all sufficiently large n. The endpoint \beta=0 can also be read directly from Equation (C.2): a_{ij}=0, w_i=1, and (\partial_t-L)U\le-2U. This part of the proof therefore uses no nondegenerate Gaussian field law.

For 7/20<\beta<1/2, we need a sharper estimate on most configurations. The inputs are proved in Sections D and E, independently of Proposition C.2. First, Proposition E.1 bounds the normalized variational expression in Equation (D.5) by 0.9969. Choose \delta=0.0031 and \rho=\delta/2 to retain half this strict margin. Proposition D.1 then gives \kappa>0 for which the event

has planted probability at most e^{-\kappa n} for all sufficiently large n. Under the planted law, the entries above the diagonal are independent with mean \beta^2/n and variance \beta^2/n; Section D defines and analyzes this law. To translate the matrix event into a set of spin configurations, define

Lemma D.4 now yields \mu_J(\mathcal B_J)\le e^{-\kappa n/2} on events whose original disorder probability tends to one. Intersect these with the good disorder events used above. The supremum over all unit vectors in the definition ensures that the set in Equation (C.9) depends only on J, rather than on the evolving function.

The Gibbs mass estimate is established. To obtain the pointwise dissipation bound, take n large enough that 16R_3(J)\le\rho/2. For x\notin\mathcal B_J, homogeneity of M_{J^x} and Lemma C.1 leave the fixed margin

At every configuration, including those in the exceptional set, Equation (C.7) and Lemma C.1 give (\partial_t-L)U\le K U for some nonnegative finite K depending only on \beta. For instance, the eventual estimates \|J\|\le2, R_2(J)\le1, and 16R_3(J)\le1 permit K=8. Choose c=\min\{\rho,\kappa/2\} and C=K+c to combine the Gibbs mass and dissipation bounds on the same event. Since the local calculation applies to every function, this event and these constants work simultaneously for all f_0, t, and x.□

Proof. We first establish the unweighted operator estimate, then insert the continuous kernel G by approximation. The diagonal estimate will require a separate argument that removes one row from the fields. For the operator estimate, truncate X_n=nJ_{ij}^2 at a fixed level T. For a sufficiently small fixed t>0,

For uniform dominated convergence, use |\beta Z+\beta^2/\sqrt n|\le\beta|Z|+\beta^2; the square of the right side has an exponential moment for a sufficiently small parameter. We may therefore choose T so that both the mean tail and the logarithm of the displayed moment are as small as required. Exponential Markov bounds the probability that a row average of (nJ_{ij}^2-T)_+ exceeds the prescribed accuracy by e^{-cn}. Taking the union over the n rows retains a positive exponential rate. Symmetry and entrywise nonnegativity bound the tail matrix’s operator norm by its largest row sum, and the same estimate controls its expectation.

For a fixed unit vector, the quadratic form of the centered truncated matrix is a sum of independent bounded variables with coefficients 2p_ip_j/n, i<j. To bound its moment generating function, recall that a centered variable whose range has length a has logarithmic moment generating function at most a^2s^2/8. The second derivative is a variance under a tilted law and is at most a^2/4; integrating twice gives this bound. Since \sum_{i<j}(2p_ip_j)^2\le2, the present quadratic form has logarithmic moment generating function at most CT^2s^2/n^2, and hence a tail at fixed accuracy bounded by 2e^{-c n^2/T^2}. To obtain an operator estimate, use a 1/4-net of the unit sphere with at most 9^n points. This cardinality follows by comparing volumes of disjoint balls of radius 1/8. For a symmetric matrix A, twice the maximum of |p^{\mathsf T}Ap| on this net bounds \|A\|. The union bound is therefore absorbed by the quadratic exponential rate. It remains to restore the mean and diagonal:

so the resulting mean correction and missing diagonal have norm O_\beta(1/n). Together with the truncation and net bounds, this proves Equation (D.14).

We now insert G. Uniform continuity gives an approximation on a finite rectangular partition of [0,2]^2, of the form \sum_{a,d}g_{ad}\1_{I_a}(w_i)\1_{I_d}(w_j). Multiplication of p by the corresponding coordinate masks reduces each summand to Equation (D.14). The number of summands and their coefficients are fixed. For the remaining kernel error R_{ij}, use

The row-sum bound in Equation (D.10) controls this error, since \sum_j|E_{ij}|\le\sum_jJ_{ij}^2+\beta^2. This proves Equation (D.15). On the operator and row-sum events, the argument is deterministic for every choice of masks and residual coefficients. Thus these weights may depend on the whole interaction matrix.

For the diagonal estimate we must recover independence before applying the weighted tail bound. Fix a row i and condition on

This sigma-field leaves the entries of the off-diagonal row (J_{ij})_{j\ne i} independent. Remove that row from the other fields by defining

The fields so obtained are \mathcal F_i-measurable, with H_j-H_j^{(-i)}=J_{ij}. Keep a\in[0,2] deterministic for now and set c_j(a)=F(a,w(H_j^{(-i)})). Under the conditioning these weights are fixed and lie in [0,32/27], so Equation (D.8) gives a bound e^{-cn} for the event

for all large n, uniformly in \mathcal F_i and a; the mean correction is at most (32/27)\beta^4/n. Take a union first over a fixed finite mesh of a values and then over the n rows. Uniform continuity of F and the bound on \sum_j|E_{ij}| extend the estimate to all a\in[0,2]. Next restore the original fields. On the maximum-entry event, |H_j-H_j^{(-i)}| is uniformly small. The Lipschitz property of w and uniform continuity of F bound the resulting error by a small multiple of the same row sum. Only after these simultaneous estimates do we set a=w_i. Conditioning on w_i at the outset would condition on a row sum and destroy the independence used above. The mesh, continuity tolerances and entry bound are chosen before n; their failures therefore retain positive exponential rates. This completes Equation (D.16).□

Proof. Separate the bin-constant space \mathcal C from the zero-sum space \mathcal U_a in each bin. Then \mathcal C is orthogonal to \bigoplus_a\mathcal U_a, and \overline A preserves each space. Since Qu=u and Q\overline A u=\overline A u, the process expands as

For these fixed bin parameters, C_b is constant in the residual index and has conditional mean zero. The linear residual term uses the \mathcal U–\mathcal C block of the GOE, while the quadratic term uses the \mathcal U–\mathcal U block. Their independence follows from Equation (D.12).

For two indices u,u', set

The linear residual covariance is

Here Q(v_aI+\overline A)p_c lies in \mathcal C and has squared norm V_a. Orthogonality to the residual spaces makes the other covariance contraction vanish. Applying the same covariance identity to the quadratic residual term gives

This follows from

and symmetrization in the bin indices.

Cauchy–Schwarz gives -s_a\le q_a\le s_a, and hence

because the right side minus the left side is (s_a-q_a)(s_d-q_d)\ge0. The inequality therefore holds also for negative overlaps. Combining it with Equations (D.27) and (D.28) bounds the increment variance of the residual process by

This is exactly the increment variance of the linear process

with independent standard Gaussian vectors z_a in \mathcal U_a. Lemma B.1 therefore bounds the expected supremum of the residual process by that of the linear process. We can maximize the latter separately in each residual space: each maximum is the corresponding Gaussian norm multiplied by \sqrt{s_a}. Using \E\|z_a\|\le\sqrt{\dim\mathcal U_a} \le\sqrt{n\widehat\pi_a}, we obtain Equation (D.26). Adding C_b leaves this expectation unchanged because its conditional mean is zero and the bin parameters remain fixed.

Concentration must, however, include C_b. For a unit vector p, the full GOE functional has symmetric coefficient matrix

whose Frobenius norm is at most one. Consequently its coefficient norm in independent standard Gaussian coordinates is at most \sqrt2\beta/\sqrt n. The supremum inherits this Lipschitz constant, proving the assertion for the full process, including C_b.□

Proof of Proposition D.1. We assemble the estimates with five fixed error allowances. Given \delta from the hypothesis, put a=\delta/10. First require an error at most a in Equation (D.17). Next choose the field partition and tolerances to make each error in Equations (D.22) and (D.24) at most a. Take concentration error a in Equation (D.30) and choose \eta so that 2K_{\rm form}\eta\le a. Every choice precedes the limit in n. The rescaling in Equation (D.23) makes the limiting functions admissible for the hypothesis. Functional approximation and concentration therefore bound each net value by 1-\delta+2a. The two form errors and extension from the net add the remaining 3a, giving, for every unit vector,

To verify that this simultaneous assertion retains an exponential rate, collect the failure probabilities after fixing all parameters. Their sum has the form

Each c_i is positive and depends only on \beta and the fixed choices. The initial terms account for the entry, diagonal, norm, empirical-tail and common-shift events; N_w counts the deterministic row-parameter mesh. Only the last term uses an exponentially large sphere net, and its rate quadratic in n absorbs 9^n. The other prefactors are fixed or polynomial in n. Hence the sum is at most e^{-\kappa n} for some \kappa>0 and all sufficiently large n. Finally the event in Equation (D.6) does not involve the auxiliary diagonals, so its probability on the enlarged space equals that on the planted marginal.□

Proof. Let Z_J be the original partition function and

The original disorder law is invariant under sign conjugation. Each summand therefore reduces to the all-positive configuration, and the planted density in Equation (D.1) gives

Here \E Z_J=2^n\exp(\beta^2(n-1)/4).

We next bound the denominator Z_J from below on the exponential scale. Gaussian integration, with an independent uniform sign vector (\varepsilon_i)_{i=1}^n, gives the exact second-moment identity

For the inequality, introduce a standard Gaussian Z to linearize the square. The sign average becomes \cosh(\beta Z/\sqrt n)^n\le e^{\beta^2Z^2/2}, with expectation (1-\beta^2)^{-1/2}. Separate the event \{Z_J\ge\E Z_J/2\} and apply Cauchy–Schwarz to obtain

We upgrade this fixed positive probability using concentration. Each derivative of \log Z_J in an independent standard disorder Gaussian has magnitude at most \beta/\sqrt n, giving Lipschitz constant at most \beta\sqrt{(n-1)/2}. The preceding lower event and Lemma B.1 force

Otherwise \log\E Z_J-\log2 would exceed the mean by a sufficiently large multiple of \sqrt n, and upper-tail concentration would make the preceding event less likely than its fixed lower bound. Lower-tail concentration now gives, for all large n,

Constants here may depend on the fixed \beta and \kappa.

By Equation (D.32) and Markov’s inequality,

On the complement of this event and the event in Equation (D.34), divide the numerator bound by the partition-function lower bound to obtain Equation (D.31). A smaller positive exponent absorbs the sum of the two failure probabilities.□

Proof. We separate one rank-one term from the kernel and bound the remainder by its diagonal. Put q=(w-3/4)/(w+3/4). A finite field has w>0 and |q|<1, so expansion as a geometric series gives

The j=0 term will give \frac32\langle bg\rangle^2. For the remaining terms set \phi_j=\sqrt{3/2}\,gq^j, j\geq1. Their diagonal sum is

The bound (1+m)v=(1-m)(1+m)^2\leq32/27 implies \rho\leq16/27. Moreover \langle\rho\rangle=R, proving that R\geq0. To justify the series for an arbitrary signed b\in L^2, apply Cauchy–Schwarz:

This proves absolute summability of the signed double-integral series, so its sum equals \sum_{j\geq1}\langle b\phi_j\rangle^2. The same estimate bounds that sum by R\langle b^2\rangle. Restoring the j=0 term proves the kernel bound.□