Proof. An empirical mean of bounded exchangeable variables is Cauchy in \(L^2\): in its second-moment calculation all cross terms with distinct indices coincide, and the contribution from equal indices tends to zero. Applying this observation to the powers of \(H_l\) constructs their empirical means simultaneously. The uniformly convergent exponential series then defines \(Z(t)\) and agrees with the empirical mean of \(e^{tH_l}\). In particular, if \(|H_l|\leq C\), then \(e^{-C|t|}\leq Z(t)\leq e^{C|t|}\).

Write \(\varphi(t)\) for the right-hand side of (12). Differentiating a denominator can be represented by averaging over one further replica. Iterating this operation gives \[\varphi^{(d)}(t) =\mathbb E\left[ \frac{\Phi\exp\bigl(t\sum_{l\leq n}H'_l +t\sum_{n<l\leq n+d}H_l\bigr)} {Z(t)^{n+d}} \prod_{j=n}^{n+d-1} \left(\sum_{l\leq n}H'_l+ \sum_{n<l\leq j}H_l-jH_{j+1}\right) \right].\] To justify the replacement by a new replica, average over indices beyond all those already present. The empirical means are unchanged by finite permutations of these indices, and boundedness permits passage to their \(L^2\) limits.

At \(t=0\), condition on the first \(n+d-1\) replicas. For each \(i\leq n\), Equation (11) computes the conditional mean of \(f_i(R_{i,n+d})\). Summing those identities makes the last factor in the product have conditional mean zero. Thus \(\varphi^{(d)}(0)=0\) for \(d\geq1\).

This derivative argument determines the identity for all real \(t\). Indeed, near a fixed real \(t_0\), \[|Z(t_0+z)-Z(t_0)| \leq Z(t_0)(e^{C|z|}-1).\] Consequently the denominator is uniformly nonzero in a complex neighborhood of \(t_0\), and the expected ratio is analytic there by boundedness. The vanishing derivatives prove constancy near zero; overlapping such neighborhoods extends constancy along the real axis. ◻

Proof. Let \(W\) be the empirical future frequency of \(\{R_{il}\in I_i\text{ for all }i<n\}\), and let \(V\) be that of \(\{R_{nl}\geq u\}\). If the asserted probability were zero, then \(V\geq W\) almost surely on \(\mathcal A\). Apply Lemma 2 with \(f_n=\mathbf 1_{[u,\infty)}\), all other \(f_i=0\), and \(\Phi=\mathbf 1_{\mathcal A}\). Put \(q=\mathbb P(R_{12}\geq u)\). On \(\mathcal A\) the numerator contributed by the \(H'_j\) is \(e^{tq}\). For \(t>0\), therefore, \[\mathbb P(\mathcal A) \leq e^{tq}\mathbb E\frac{\mathbf 1_{\mathcal A}} {(1+(e^t-1)W)^n} \leq e^{tq}\mathbb E\frac{W}{(1+(e^t-1)W)^n}.\] For the second inequality, discard the requirements among the first \(n-1\) replicas and exchange replica \(n\) with a future index. The variable \(W\) only tests overlaps against the first \(n-1\) replicas and is unchanged by that exchange, so the empirical mean of the remaining indicator is \(W\).

The positive-probability event \(\mathcal A\) implies \(q<1\). For \(n\geq2\), \[\sup_{0\leq w\leq1}\frac{w}{(1+(e^t-1)w)^n} \leq \frac{(n-1)^{n-1}}{n^n(e^t-1)}.\] The preceding upper bound consequently tends to zero as \(t\to\infty\), contradicting \(\mathbb P(\mathcal A)>0\). ◻

Proof. We first prove the scalar assertions. If \(r=\mathbb P(R_{12}< -\epsilon)>0\), then, on an event where every distinct overlap among \(n\) replicas is below \(-\epsilon\), Equation (11) gives \[\mathbb P(R_{i,n+1}\geq-\epsilon\mid\mathcal F_n) =\frac{1-r}{n}\quad(1\leq i\leq n).\] A union bound leaves conditional probability at least \(r\) that all new overlaps are below \(-\epsilon\). Starting with two replicas, we obtain arbitrarily large such lists with positive probability. If the diagonal is bounded by \(D\), a Gram representation \(R_{ij}=v_i\cdot v_j\) of one list would give \[0\leq\left|\sum_{i=1}^n v_i\right|^2 \leq nD-\epsilon n(n-1),\] which is impossible for large \(n\). Thus the off-diagonal entries are nonnegative.

For ultrametricity, we use the barycenter contradiction of [23]. Suppose ultrametricity fails. By permuting indices there is a support triple with \[R_{12}=a\geq b=R_{13}>c=R_{23}.\] Apply Lemma 3 repeatedly to copies of replicas 1 and 2. For any prescribed finite size and arbitrarily small tolerances, this produces two sets of replicas whose cross overlaps are close to \(a\), whose internal off-diagonal overlaps are at most \(a+\epsilon\), and whose overlaps with replica 3 are close respectively to \(b\) and \(c\). Here is the tolerance choice underlying this iteration. At each step choose a support point in the current positive-probability matching event, and choose the duplication threshold slightly larger than all overlaps of the replica being copied. Allocate a total increase less than \(\epsilon\) among the finitely many steps. The requirements against the other set and replica 3 remain in their original small intervals.

Let \(\bar v_1,\bar v_2\) be the barycenters of Gram vectors for sets of size \(M\). Their squared norms are at most \(a+\epsilon+D/M\), and their mutual inner product is at least \(a-\epsilon\). It follows that \(|\bar v_1-\bar v_2|^2\leq4\epsilon+2D/M\). Their inner products with the vector for replica 3, however, differ by a number arbitrarily close to \(b-c>0\). First take \(M\) large and then the tolerances small to contradict Cauchy–Schwarz. This proves scalar ultrametricity.

For synchronization, we use the coordinate-and-sum argument of [24]. Each component and each sum of two components is again a Gram array satisfying the scalar identities. Suppose two points in the support of the joint pair law have strictly crossed coordinates \(a\) and \(b\). The independent-law term in Equation (11), with \(n=2\), gives positive probability of observing these two points in arbitrarily small neighborhoods on pairs 12 and 13. Ultrametricity in each component forces the corresponding coordinate on pair 23 to equal the smaller of the two coordinates. The sum of coordinates \(a\) and \(b\) on pair 23 is then strictly below its values on both pairs 12 and 13, contrary to ultrametricity for that sum. The support is therefore ordered.

An ordered compact support has at most one point with any prescribed sum of its coordinates. Its coordinates are nondecreasing functions of that sum; in fact each is 1-Lipschitz on the set of possible sums. Applying these functions to the quantile of \(Q_{12}\) supplies the simultaneous quantiles and all the stated conclusions. ◻

Proof. Multiply each point on an edge to level \(i\) by \(e^{X_i-X_{i-1}}\). Conditional on earlier marks, \[\int e^{\zeta_i(X_i-X_{i-1})}\,dK_i=1.\] Poisson mapping therefore preserves the point intensity and changes the conditional mark law to \(\widetilde K_i\). Applying this observation recursively, and ordering the transformed child points at each vertex, gives an ordinary unmarked cascade independent of the root mark. The transformed child marks have the kernels in (14); transformed unmarked descendant processes are independent of their ancestor marks. Positivity and finiteness of the edge-kernel densities ensure that the almost-sure finiteness assumptions remain valid under these kernels.

Let \(T_{\rm new}\) denote the sum of the products of transformed edge points. The multipliers telescope along every path, giving the almost sure identity \[ \log\sum_\alpha\nu_\alpha e^{X_{k-1}(\alpha)} =X_0+\log T_{\rm new}-\log T. \tag{15}\] Conditional on the root, both totals have the original law of \(T\). Thus their log difference has conditional mean zero and conditional second moment at most \(4\mathbb E|\log T|^2\). This proves (13), including integrability. In particular no unconditional first moment of \(e^X\) is required.

The same mapping proves the sampled-mark assertion. Relabeling is performed separately among children of each vertex, so it preserves ancestors and common levels. The residual integration contributes \(e^{X_{k-1}}\) to a leaf mass; conditional residual sampling after the tilt has density \(e^{X-X_{k-1}}\). Independent integration on separate visits gives precisely the last assertion. ◻

Proof. Attach independent standard Gaussian marks to the vertices at level \(i\geq1\), and tilt by \(e^{cg(\alpha)}\), where \(g(\alpha)\) is the mark on the path to \(\alpha\) and \(c\ne0\). By Proposition 5, the unmarked branching law is unchanged and the marks are shifted by \(c\zeta_i\). Their covariance before tilting is \(\mathbf 1_{\{L_{lj}\geq i\}}\). Gaussian integration by parts against a bounded test \(\Phi\) of the common levels of \(n\) replicas gives \[ \zeta_i\mathbb E\Phi =\mathbb E\left[\Phi\left(1+ \sum_{j=2}^n\mathbf 1_{\{L_{1j}\geq i\}} -n\mathbf 1_{\{L_{1,n+1}\geq i\}}\right)\right]. \tag{16}\] For this calculation one can first retain finitely many leaves; Gaussian exponential moments and Jensen’s inequality for inverse partition factors justify passage to the full probability prior. At \(n=1\), Equation (16) gives \(\mathbb P(L_{12}\geq i)=1-\zeta_i\). For general \(n\), it gives the Ghirlanda–Guerra identity for each threshold, and hence for every function of the finite set of levels.

For uniqueness, first consider an array with finitely many possible off-diagonal values. At any one of its thresholds, ultrametricity makes the relation of being above the threshold an equivalence relation on the sampled indices. If its pair probability is \(1-\zeta\), the conditional probability that the next sample joins an existing class \(C\) is \[\frac{1-\zeta+|C|-1}{n}=\frac{|C|-\zeta}{n},\] by applying the identities to a representative of \(C\). The classes at different thresholds are nested. These joining probabilities, and their differences for nested classes, determine the distribution of the whole next row of overlaps. Induction on the sample size gives uniqueness. General bounded off-diagonal distributions follow by nondecreasing finite-valued roundings and convergence on each finite array. Equal rounded values may first be merged into a single level. ◻

Proof. Finite-dimensional centered Gaussian laws depend continuously on their covariance matrices, including at singular matrices. Conditional integration followed by the assumed convergence proves the first claim.

Write \(Z=\langle f\rangle\) for the original Gibbs integral. It belongs to \([\delta,M]\). Uniformly approximate \(\log Z\), or \(Z^{-n}\) for an \(n\)-replica reweighted test, by a polynomial on this interval. Each power of \(Z\) is an integral over additional independent Gibbs replicas. The Gaussian marks on these extra replicas are evaluations of the same field, so their correlations are exactly those of the joint covariance array. Every resulting term is a bounded finite-replica expectation and converges by the first claim. The uniform polynomial errors can then be sent to zero. ◻

Proof. For a finite prior, differentiate the Gibbs ratios and integrate by parts in the Gaussian coefficients. The logarithm gives (17). The covariance part for an \(n\)-replica test is one half the expectation of \(\Phi\) times \[\sum_{l,m\leq n}\dot C_t(l,m) -2n\sum_{l\leq n}\dot C_t(l,n+1) +n(n+1)\dot C_t(n+1,n+2) -n\dot C_t(n+1,n+1).\] The deterministic part inserts \(\sum_{l\leq n}\dot V_t(l)-n\dot V_t(n+1)\). Taking absolute values proves the stated bound.

For a general probability prior, approximate its integral by finite empirical samples, or by finite subsets whose prior mass tends to one. At a fixed system size and on a bounded parameter interval, a uniform variance bound for \(X_t\) and a bound for \(V_t\) give all required polynomial-exponential moments. Inverse partition factors are controlled by Jensen’s inequality: \[\left(\int e^{X_t+V_t}\,d\pi\right)^{-r} \leq\int e^{-r(X_t+V_t)}\,d\pi\qquad(r>0).\] Hölder’s inequality gives the same control for products occurring in the differentiated ratios. These bounds justify passage to the limit in the integrated formulas, and then differentiation wherever the covariance derivatives are continuous. The integrated identities also give the asserted endpoint statements. ◻

Proof. Differentiating \(T_{\zeta_i,v_i}F_i\) in its starting position gives \[m_{i-1}(z)=\mathbb E[m_i(Z_i)\mid Z_{i-1}=z].\] At the terminal level \(m_{k-1}=\tanh\). Thus these conditional means form a martingale, and conditional Jensen proves the monotonicity of their second moments. If two sampled paths separate immediately after level \(i\), their future marks and final spins are conditionally independent given the common history. Their conditional product is \(m_i(Z_i)^2\). Proposition 5 then proves the asserted two-spin interpretation, including equal leaves at the last level.

Apply Lemma 8 to the single-site cascade partition function with energy \(z(\alpha)\sigma\), uniform prior on \(\sigma\in\{-1,1\}\), and covariance \(\mathbb E[z(\alpha)z(\alpha')]=h_{L(\alpha,\alpha')}\). The diagonal is \(h_{k-1}\). Its derivative cancels the derivative of the subtraction \(h_{k-1}/2\). The pair contribution at level \(i\) is \(B_i\), and that level has probability \(w_i\) after the tilt by Proposition 5. This proves (21).

All tilted Gaussian integrals and their conditional means depend continuously on the variances, also when some vanish. This follows by Gaussian coupling and dominated convergence: the payoffs have at most linear growth, their derivatives are bounded, and their exponential weights have locally uniform Gaussian moments. This proves the boundary statement. If an increment is zero, the adjacent positions and backward functions coincide. The corresponding conditional means and their squares therefore coincide as well, proving refinement consistency. ◻

Proof. All derivatives below exist by differentiating smooth Gaussian convolutions; on compact subsets of the strict cone, linear growth of the payoffs gives locally uniform bounds for this differentiation. For \(l<j\), covariance differentiation of the backward recursion gives \[ \partial_{h_j}F_l(z) =c_j-\frac{w_j}{2} \mathbb E[m_j(Z_j)^2\mid Z_l=z], \tag{23}\] where \(c_j\) is independent of \(z\) and vanishes unless \(j=k-1\). To verify the identity, increase the variance entering level \(j\) and decrease the variance entering level \(j+1\). The two Gaussian generators insert respectively \((F_j''+\zeta_j m_j^2)/2\) and \((F_j''+\zeta_{j+1}m_j^2)/2\), propagated by the same conditional expectations. Their difference is \(-w_jm_j^2/2\). At the last level, \(F_{k-1}''+m_{k-1}^2=1\) gives the additional constant \(1/2\). Propagation through the earlier operators gives (23).

The function \(m_j\) is odd and nondecreasing, so \(m_j^2\) is even and nondecreasing in absolute position. By (22), its conditional expectation on the right-hand side of (23) has the same property. Thus \(\partial_{h_j}F_l\) is nonincreasing on \([0,\infty)\), up to its irrelevant constant. In particular, increasing \(h_j\) decreases \(m_i(z)\) for each \(z\geq0\) when \(i<j\).

This change also decreases the sampled absolute positions up to level \(i\). Indeed, their Gaussian variances are unchanged, and the change in every backward tilt \(F_l\), \(l\leq i\), is nonincreasing in absolute position by the identity just proved. The likelihood-ratio comparison following (22), applied successively to the transitions, gives this stochastic decrease. The root transition is unchanged because \(\zeta_0=0\). Combining the pointwise decrease of \(m_i(z)^2\) with the stochastic decrease of \(|Z_i|\) proves the claim. ◻

Proof. Use the original Gaussian increments of variances \(v_i\), and replace the terminal payoff by \(g_c(z)=\log\cosh(\sqrt c\,z)\). Write \(F_i^{(c)}\) for these backward values. This is the same recursion as at covariance \(ch\), in the original position units: \[F_i^{(c)}(z)=F_i^{ch}(\sqrt c\,z).\] For \(c_2\geq c_1>0\), the difference \(g_{c_2}-g_{c_1}\) is even and nondecreasing on \([0,\infty)\). This property propagates through every backward operator. For a positive exponent, the difference of the two backward values can be written \[\frac1\zeta\log\mathbb E_{c_1,z} e^{\zeta(F^{(c_2)}-F^{(c_1)})(Y)},\] where the expectation is in the transition tilted by the lower-\(c\) payoff. For exponent zero it is the ordinary expectation of the difference. In either case (22) shows that the result is nondecreasing in the absolute starting position. It is even by reflection. Hence the tilted absolute-position chain is stochastically increasing with \(c\).

It remains to compare conditional spin means, rather than derivatives in the rescaled position. Define \[a_i^{(c)}(z)= \mathbb E_c[\tanh(\sqrt c\,Z_{k-1})\mid Z_i=z].\] We prove backwards that these functions are odd and, on \(z\geq0\), nonnegative and nondecreasing both in \(z\) and in \(c\). This is true at the terminal level. Conditional on the absolute value \(u\) of a child with parent \(z\geq0\), the mean sign of the child is \(\tanh(zu/v)\): the even tilt cancels in the ratio of its positive and negative densities. The backward conditional mean is thus the integral of \[a_{i+1}^{(c)}(u)\tanh(zu/v)\] against the folded tilted transition. Its factors are nonnegative and nondecreasing in \(u\), while the transition is increasing in \(z\) and \(c\). The induction follows, with zero increments obtained by continuity.

Finally, \(a_i^{(c)}(z)=m_i^{ch}(\sqrt c\,z)\). These are precisely the physical conditional spin means, without an additional scaling factor. Combining their pointwise increase with that of the sampled absolute positions shows that \(\mathbb E_c[a_i^{(c)}(Z_i)^2]=B_i(ch)\) increases with \(c\). Differentiating at \(c=1\) gives (24). ◻

Proof. On the strict cone put \(M_{ij}=w_i\partial_{h_j}B_i\). Equation (21) shows that \(M\) is symmetric and equals \(-2\) times the Hessian of \(\ell\). Lemma 10 and symmetry give \(M_{ij}\leq0\) for \(i\ne j\). Let \(D=\mathop{\mathrm{diag}}(h_0,\ldots,h_{k-1})\). By Lemma 11, every row sum of \(DMD\) is nonnegative: \[\sum_j(DMD)_{ij} =h_iw_i\sum_jh_j\partial_{h_j}B_i\geq0.\] If a symmetric matrix \(A\) has nonpositive off-diagonal entries and row sums \(r_i\geq0\), then \[x^{\mathsf T}Ax =\sum_i r_ix_i^2+ \sum_{i<j}(-A_{ij})(x_i-x_j)^2\geq0.\] Applying this identity to \(DMD\), and using invertibility of \(D\) on the strict cone, proves that \(M\) is positive semidefinite. Hence the Hessian of \(\ell\) is negative semidefinite.

Integrating along line segments gives concavity and the supporting inequality in the strict cone. Approximate boundary covariance vectors by strict ones and use the continuity in Proposition 9 to obtain (25) on the closed cone. If \(h\) and \(v\) initially use different intervals, put both on their common subdivision. Refinement consistency makes this operation harmless. Finally, integrate (21) along the segment from \(h\) to \(v\) and use \(0\leq B_i\leq1\) to prove (26). Monotone step paths are dense in the bounded monotone paths in \(L^1\), proving the extension. ◻

Proof. The zero field gives \(S(p)\geq0\), and monotonicity follows because every admissible field is nonnegative. Suppose \(0<d\leq1\), and define \[g_d(z)=\frac1d\log(2\cosh(dz))-\log2.\] Then \(g_d\geq\log\cosh\): with \(r=1/d\geq1\), the inequality \((a+b)^r\geq a^r+b^r\) applied to \(a=e^{dz}\) and \(b=e^{-dz}\) gives this assertion. For \(0\leq\zeta\leq1\) and \(v\geq0\), \[ T_{\zeta,v}g_d\leq g_d+\frac12\max(\zeta,d)v. \tag{28}\] Indeed, if \(\zeta\geq d\), apply Minkowski’s inequality with exponent \(r=\zeta/d\geq1\) to \(e^{dz+dG}+e^{-dz-dG}\), where \(G\sim\gamma_v\). The \(L^r\) norm of either exponential gains the factor \(e^{rd^2v/2}\), proving the bound \(\zeta v/2\). If \(0<\zeta\leq d\), Jensen’s inequality for the concave power \(\zeta/d\) gives the bound \(dv/2\). At \(\zeta=0\), apply Jensen directly to the logarithm to obtain the same bound.

Iterate (28) through the field recursion. Since \(g_d(0)=(d^{-1}-1)\log2\), this yields \[\ell(h)\leq(d^{-1}-1)\log2 -\frac12\sum_i(1-\max(\zeta_i,d))(h_i-h_{i-1}) =(d^{-1}-1)\log2-\frac12\int_d^1h(s)\,ds.\] For any nonnegative nondecreasing \(h\), \[\int_0^1(1-p(s))h(s)\,ds\geq\int_0^d h(s)\,ds.\] To see this, the density \(1-p\) lies in \([0,1]\) and has mass \(d\); among all such densities the integral against an increasing function is minimized by \(\mathbf 1_{(0,d)}\). For \(d=1\) the assertion is immediate. For \(0<d<1\), subtract this indicator and then subtract the constant \(h(d)\) from the integrand: the resulting product is nonnegative on both sides of \(d\). Combining the last two displays gives (27) after taking the supremum over \(h\).

Finally, for \(p=1\) take the constant field \(h(s)=H\). Its recursion is ordinary expectation at the root, and \[\ell(h)+\frac12\int_0^1h(s)\,ds =\mathbb E\log\cosh(\sqrt H\,G) \geq\sqrt H\,\mathbb E|G|-\log2\longrightarrow\infty,\] where \(G\) is standard Gaussian. This proves the last case. ◻

Proof. We first record the elementary gradient estimates used below. For standard Gaussian data \(g\), \[\mathop{\mathrm{Var}}f(g)\le \mathbb E|\nabla f(g)|^2.\] For completeness, differentiate \(\mathbb Ef(g)f(rg+\sqrt{1-r^2}g')\) in \(0<r<1\); Gaussian integration by parts gives the corresponding scalar product of gradients, bounded by \(\mathbb E|\nabla f(g)|^2\), and integration in \(r\) proves the inequality. Approximation extends it from smooth bounded functions to Lipschitz functions. Applied to \(f(g/|g|)\), radial independence yields the sphere bound \[\mathop{\mathrm{Var}}_{\mathbb S^{d-1}}f \le \frac1{d-2}\mathbb E|\nabla_{\mathbb S^{d-1}}f|^2,\qquad d>2.\] One may cut off at \(g=0\) before passing to this formula.

Here is a corresponding bound on \(SO(d)\). Let its Dirichlet form be the expected sum of squares of the derivatives generated by column-plane rotations, and let \(C_d\) be a Poincare constant for that form. Conditional on a column, the fiber is \(SO(d-1)\). Its conditional variance is bounded using the pairs that do not involve that column. Moving the column in a unit tangent direction transports the entire fiber by a rotation; Jensen’s inequality bounds the squared gradient of the conditional mean by the conditional sum of squares of the derivatives involving that column. Apply the sphere bound to the conditional mean and average over the choice of column. Each pair appears in \(d-2\) fibers and involves two columns, so \[C_d\le\left(1-\frac2d\right)C_{d-1} +\frac2{d(d-2)},\qquad d>2.\] The circle has a finite Poincare constant. Multiplication by \(d(d-1)\) in this recurrence gives \(C_d=O(d^{-1})\). The Dirichlet form of a Frobenius-Lipschitz function is bounded by a fixed multiple of its squared Lipschitz constant.

Conditional on \(U\), apply the backward marked-cascade recursion to \(Z_N=\exp(NP_N^{\mathrm{enr}})\). Proposition 5 gives \[\log Z_N=\mathcal R_N(g_0)+\log S'-\log S,\] where \(g_0\) denotes the common Gaussian marks, \(\mathcal R_N\) is the root recursion value, and \(S,S'\) are unnormalized cascade totals with the same conditional law. For the fixed cascade partition, their log variances are bounded, also conditional on \(U,g_0\). In particular the conditional second moment of \(\log S'-\log S\) is bounded uniformly. The root recursion is \(C\sqrt N\)-Lipschitz in the common standard Gaussian coefficients. Indeed each exponent has that coefficient-norm bound: the spin norm is \(\sqrt N\), the field heights are bounded, and the squared perturbation coefficients sum to \(O(Ne_N^2)\). Integration and log-exponential integration preserve a common Lipschitz bound. The Gaussian inequality thus bounds the conditional variance of the normalized log partition by \(C/N\).

It remains to control its conditional mean as a function of \(U\). For deterministic quadratic terms the normalized exponent is uniformly Lipschitz in Frobenius distance, since \(|\sigma|=\sqrt N\). For the Gaussian terms, compare conditional covariances. Each projected overlap is uniformly Lipschitz in \(U\); the degrees of the monomials grow at most linearly, and \(\sum_j(1+j)4^{-j}<\infty\). Gaussian comparison, divided by \(N\), therefore gives a uniform Lipschitz bound on \(\mathbb E[P_N^{\mathrm{enr}}\mid U]\). The linear enrichment is independent of \(U\). This conditional mean depends only on the spectral projectors, so multiplication of one spectral row by \(-1\) shows that it has the same law on both components of \(O(N)\). The group inequality and conditional variance decomposition prove (32).

Without the added Gaussian fields, \(P_N\) itself is uniformly Lipschitz in \(U\) and depends only on those projectors. Write \(F=P_N-\mathbb EP_N\). Applying the group inequality first to \(F\) and then to \(F^2\) gives \[\mathbb EF^2\le C/N,\qquad \mathop{\mathrm{Var}}(F^2)\le (C/N)\mathbb EF^2\le C/N^2.\] Since \(\mathbb EF^4=\mathop{\mathrm{Var}}(F^2)+(\mathbb EF^2)^2\), this proves (33). Smoothing by small group translates justifies these applications for Lipschitz functions. All estimates used only the common spin norm and normalization of the prior. ◻

Proof. Let \(g_N=\mathbb EP_N^{\mathrm{enr}}\). For each fixed perturbation coordinate \(d\), convexity of the log partition makes \(P_N^{\mathrm{enr}}\) and \(g_N\) convex in \(d\). Expected first derivatives are uniformly bounded on a slightly enlarged interval. More precisely, direct differentiation gives an \(O(e_N)\) bound for a \(v_a\)-derivative, and Gaussian integration by parts gives \(O_j(e_N^2)\) for a fixed \(u_j\)-derivative.

In the first procedure, comparison with the centered choice of one coordinate shows that each fixed coordinate lies in the interior for all large \(N\). At a minimum, its first derivative matches the penalty derivative. Comparison with the choices \(d+s\) and \(d-s\) therefore gives \[0\le g_N(d+s)-g_N(d)-s g_N'(d)\le C_d s^2\] and the analogous negative-increment estimate, whenever the increments remain in the interval. In particular \(0\le g_N''(d)\le C_d\). Convex secants for the random function, followed by Lemma 14, imply \[\mathbb E|\partial_dP_N^{\mathrm{enr}}-\partial_dg_N| \le C_d\left(s+\frac{N^{-1/2}}s\right).\] With \(s=N^{-1/4}\), the right side is \(O_d(N^{-1/4})=o(e_N^2)\). The parameter minimum is deterministic, so the uniform pointwise variance estimate applies without conditioning on the disorder.

For the second procedure, integrate the same convex-secant inequalities in \(d\). The integral of \(g_N'(d+s)-g_N'(d-s)\) is \(O_d(s)\), by the endpoint bounds on the expected first derivatives on the enlarged interval. Thus \[ \int_\Theta \mathbb E|\partial_dP_N^{\mathrm{enr}}-\partial_dg_N|\,d\eta =O_d(N^{-1/4}). \tag{34}\] Convexity and the same endpoint bounds also give \(\int_\Theta g_N''(d)\,d\eta\le C_d\).

These estimates control both thermal and disorder fluctuations. For a group coordinate, \[\partial_{v_a}P_N^{\mathrm{enr}}=e_N\langle A^a_{11}\rangle, \qquad \partial_{v_a}^2g_N =Ne_N^2\mathbb E\mathop{\mathrm{Var}}_{\langle\cdot\rangle}(A^a_{11}).\] Consequently the mean absolute disorder fluctuation of \(\langle A^a_{11}\rangle\) is \(O(N^{-3/16})\), and its mean absolute thermal fluctuation is \(O(N^{-7/16})\). Here and below these bounds hold at a minimum in the first procedure and after integration in the second. At the minima, select a convergent subsequence of the deterministic means \(d_{a,N}=\mathbb E\langle A^a_{11}\rangle\). For the averaged procedure, the means are first kept as functions \(d_{a,N}(\theta)\); select their limits only after choosing a sequence \(\theta_N\in\mathcal G_N\) as below. In either procedure the resulting subsequence satisfies \[ \mathbb E\langle|A^a_{11}-d_a|\rangle\longrightarrow0, \qquad d_a\ge0,\quad \sum_a d_a=1. \tag{35}\] For the Gaussian coordinate, \[\partial_{u_j}P_N^{\mathrm{enr}} =\frac{e_N2^{-j}}{\sqrt N}\langle Y_j\rangle, \qquad \partial_{u_j}^2g_N =e_N^2 4^{-j}\mathbb E\mathop{\mathrm{Var}}_{\langle\cdot\rangle}(Y_j).\] Its mean absolute disorder and thermal fluctuations are respectively \(O_j(N^{5/16})\) and \(O_j(N^{1/16})\). Since \(\sqrt N e_N=N^{7/16}\), this proves \[ \mathbb E\langle|Y_j-\mathbb E\langle Y_j\rangle|\rangle =o_j(\sqrt N e_N). \tag{36}\] In the integrated case, the centers in these estimates are always the expectations at the fixed parameter \(\theta\). Markov’s inequality, followed by a diagonal choice over the finitely many group coordinates and the countably many fixed indices \(j\), produces measurable sets \(\mathcal G_N\) on which all these normalized fluctuation errors tend to zero uniformly. Their probabilities tend to one. This proves the required assertion for every sequence chosen from the sets, rather than only almost every fixed parameter.

Finally, let \(\Phi\) be a bounded continuous overlap test on \(n\) replicas. Conditional Gaussian integration by parts gives \[\begin{align*} \mathbb E\langle Y_j(1)\Phi\rangle ={}&\sqrt N e_N2^{-j}u_j \mathbb E\left\langle\Phi \left(\sum_{l=1}^n\kappa_j(1,l) -n\kappa_j(1,n+1)\right)\right\rangle . \tag{37}\end{align*}\] Compare this with its one-replica version multiplied by \(\mathbb E\langle\Phi\rangle\), use (36), and then use the deterministic diagonal limits (35). Every subsequential limit satisfies GG for each monomial \(\kappa_j\). Polynomials are dense in the continuous functions on the compact overlap state space, so these identities give joint GG for the entire vector. The array structure results of Section 2 now apply. ◻

Proof. Choose row indices \(i\in I_{a,N}\), \(j\in I_{b,N}\), and differentiate under the plane rotation \(x_i'=x_j,\ x_j'=-x_i\). Apply this derivative to \(\langle x_i^1x_j^2\Phi(Q_{12})\rangle\), keeping the Gaussian coefficient tensors and the independent linear enrichment fixed. The Haar-averaged derivative is zero. The total overlap \(Q_{12}\) is rotation invariant, so no derivative hits \(\Phi\). Summing the coordinate derivative over \(i,j\) and dividing by \(N^2\) gives \[\rho_{a,N}A^b_{12}-\rho_{b,N}A^a_{12}.\] The derivative of the main Hamiltonian is \(t_N(\lambda_a-\lambda_b)x_i x_j\). The two-replica Gibbs-ratio derivative thus contributes \[t_N(\lambda_a-\lambda_b) \left(A^a_{11}A^b_{12} +A^a_{12}A^b_{22}-2A^a_{13}A^b_{23}\right).\]

We check that the remaining terms disappear at the required scale. The deterministic perturbation derivative has coefficient \(2e_N(v_a-v_b)\) multiplying \(x_i x_j\), and its normalized contribution is \(O(e_N)\). For a Gaussian perturbation derivative, integrate by parts in its coefficient tensor, conditionally on \(U\) and the cascade. Pairing with the Gibbs exponent supplies coefficient \(Ne_N^2 4^{-j}u_j^2\). Each one-endpoint derivative of its covariance is a sum of at most \(C(1+j)\) terms consisting of bounded overlap factors times \[\frac{x_j^u x_i^v}{N} \quad\hbox{or}\quad \frac{x_i^u x_j^v}{N}.\] Multiplication by \(x_i^1x_j^2\), summation over \(i,j\), and division by \(N^2\) contract these coordinates into products of normalized projected inner products, whose absolute values are at most one. The total error is bounded by \[Ce_N^2\sum_{j\ge1}(1+j)4^{-j}=O(e_N^2).\] The independent linear enrichment contributes nothing. Finite cascade restrictions justify the differentiations as above. Pass to the limit and use the deterministic diagonals to obtain (38).

The same calculation with \(x_i^1x_j^1\) and a one-replica Gibbs integral gives (39). In this calculation the exponent derivative is still \(t_N(\lambda_a-\lambda_b)x_i x_j\), and the Gibbs derivative has one subtraction replica. This accounts for the coefficient without an additional factor of two. ◻

Proof. Conditional on \(Q_{12}=q\), GG gives the marginal of \(Q_{13}\) as one half of the law of \(Q_{12}\) plus one half of \(\delta_q\). Ultrametricity forces \(Q_{13}=Q_{23}\) below \(q\), and forces the other overlap to equal \(q\) when one exceeds \(q\). To include atoms explicitly, suppose the quantile equals \(q\) on \((\alpha,\beta)\). Write \(a_*\) and \(b_*\) for the constant values of \(p_a\) and \(p_b\) on that interval. The preceding conditional laws give \[\begin{split} 2\mathbb E[A^a_{13}A^b_{23}\mid Q_{12}=q] ={}&\int_0^\alpha p_a(u)p_b(u)\,du +b_*\int_\beta^1p_a(u)\,du +a_*\int_\beta^1p_b(u)\,du\\ &+(2\beta-\alpha)a_*b_* . \end{split}\] This is exactly the expression subtracted from \(d_a p_b+p_a d_b\) in (41), evaluated as \(s\uparrow\beta\). Formula (41) is constant across the plateau. The same disintegration with \(\alpha=\beta\) gives the formula at non-atomic values, for almost every conditioning value.

For the algebraic claims, the Stieltjes identity \[d\widehat L=-s\,dl\] follows from (40). If \(c\) denotes (41), its jump at a common discontinuity \(v\) is \[\Delta c=\widehat L_a(v-)\Delta p_b +(\Delta p_a)\widehat L_b(v+).\] Its continuous Stieltjes part obeys the corresponding product rule. Thus \(d\widehat{L_a\star L_b} =d(\widehat L_a\widehat L_b)\). At \(1-\), subtracting the off-diagonal from the diagonal gives \((d_a-p_a(1-))(d_b-p_b(1-))\), the required endpoint product. The symbols consequently agree everywhere up to the choice of versions. Taking \(s\downarrow0\) in (41) gives the root formula. ◻

Proof. By Lemma 17, the two Ward identities combine into \[\rho_bL_a-\rho_aL_b =t(\lambda_a-\lambda_b)(L_a\star L_b).\] Set \(\alpha_a=\widehat L_a(s)\) and \(x=\sum_a\alpha_a\). GG positivity and Gram bounds give \(0\le p_a(s)\le d_a\), so \(\alpha_a\ge0\), and \(x=D_p(p(s))\). If \(x>0\), every \(\alpha_a\) is positive: a zero component paired with a positive component would give \(-\rho_a\alpha_b=0\). Dividing the symbol equations therefore gives a common number \[\frac{\rho_a}{\alpha_a}+t\lambda_a=b \quad\hbox{for every }a.\] The positive sum constraint is the resolvent equation for \(b_t(x)\). Hence \(\alpha_a=f_a(x)\). If \(x=0\), all the symbols are zero, which is the same conclusion.

At the root, put \(r_a=p_a(0+)\), \(p_0=\sum_a r_a\), and \(x=D_p(p_0)\). The root equations are \[\rho_b r_a-\rho_a r_b =t(\lambda_a-\lambda_b) \bigl(f_a(x)r_b+r_a f_b(x)\bigr).\] For \(x>0\), substitution of (42) shows that \[\frac{(b_t(x)-t\lambda_a)^2}{\rho_a}\,r_a\] is independent of \(a\). The sum constraint and (43) give \(r_a=p_0f_a'(x)\). For \(x=0\), the root equations directly give \(\rho_b r_a=\rho_a r_b\), and therefore \(r_a=\rho_a p_0=p_0 f_a'(0)\).

It remains to recover the paths, including their diagonals, from these data. Let \(F_p(r)=|\{s:p(s)\le r\}|\). Then \(D_p'(r)=-F_p(r)\) almost everywhere and \(D_p(1)=0\). For the path in (45), Fubini’s theorem gives \[\begin{split} \widehat L_f(s) &=f(0)+\int_{p(s)}^1 F_p(r)f'(D_p(r))\,dr\\ &=f(D_p(p(s))). \end{split}\] Since \(D_p\) is constant on \([0,p_0]\), its root is \(p_0 f'(D_p(p_0))\). Finally, equal symbols imply equal Stieltjes derivatives of the off-diagonal paths on every \([\epsilon,1)\), because \(d\widehat L=-s\,dl\). The paths differ by a constant, fixed by the root, and the symbol then fixes the diagonal. This also treats jumps accumulating at zero. Applying this calculation to \(f_a\) proves (44).

No step divided by an eigenvalue difference. At \(t=0\), \(f_a(x)=\rho_a x\), so \(p_a=\rho_a p\) and \(d_a=\rho_a\). Tied eigenvalues similarly give proportional paths in the tied groups. If \(p\equiv1\), then \(D_p\equiv0\) and \(p_a=d_a=\rho_a\), also covered by the formulas. ◻

Proof. The mutually orthogonal spaces \(\operatorname{ran}(P_a^+E)\) have orthonormal coordinate maps \(P_a^+E M_a^{-1/2}\). In their direct sum, the cavity coordinate map is the vertical stack \((M_a^{1/2})_a\). Its columns are orthonormal because \(\sum_aM_a=I_n\). Complete these columns to an orthonormal basis, with the complement first, using a rule continuous near the limiting stack \((\sqrt{\rho_a}I_n)_a\). One may project a fixed limiting complementary frame and orthonormalize it. In these coordinates the interaction is the left matrix of (53). Its limit is a conjugate of \(\bigoplus_a\lambda_a I_n\), with cavity column blocks \(\sqrt{\rho_a}I_n\). This proves (53).

The complement to the cavity coordinates in this direct sum lies in the first \(N\) coordinates and has dimension \(d\); its chosen basis gives the special axes. All other eigenvectors also lie in the first \(N\) coordinates, with multiplicities \(k_a(N+n)-n\). Retain them and assign \(A_0\) to the special axes. The resulting base multiplicity is \[k_a(N+n)-n+(n-s_a)=k_a(N+n)-s_a=k_a(N).\] The geometric decomposition is thus \[ \mathbb R^{N+n} =% \BeginAccSupp{method=hex,unicode,ActualText=2A01005F00610020211B005F00610020002800640069006D0020003D0020004E00202212002000640029}% \underbrace{\bigoplus_a\mathcal R_a}_{\dim=N-d}% \EndAccSupp{}% \mathbin{\oplus} % \BeginAccSupp{method=hex,unicode,ActualText=D835DCB1005F004E0020002800640069006D0020003D002000640029}% \underbrace{\mathcal V_N}_{\dim=d}% \EndAccSupp{}% \mathbin{\oplus} % \BeginAccSupp{method=hex,unicode,ActualText=0045211D005E006E0020002800640069006D0020003D0020006E0029}% \underbrace{E\mathbb R^n}_{\dim=n}% \EndAccSupp{}% , \qquad \mathcal R_a=\operatorname{ran}P_a^+\cap\ker E^{\mathsf T}. \tag{56}\] The base interaction agrees with the full interaction on the first summand and uses \(A_0\) on \(\mathcal V_N\). Subtracting the two quadratic forms gives (54).

Conditional on \((M_a)_a\), rotations of the first \(N\) coordinates act transitively on the remaining frame and subspace data. Their conditional law is therefore the invariant probability on this orbit. In particular the base interaction has its Haar law, independently of \((M_a)_a\). After conditioning further on the base projectors, the remaining freedom is the product of the rotations within their eigenspaces. The selected axes consequently have the asserted uniform frame law. Generate the base perturbations independently conditional on the projectors, using their prescribed Gaussian kernels. This preserves that conditional frame law; no full-system Gaussian coefficient realization is inherited by the base.

For a uniform unit vector in a group of dimension \(k_a(N)\), its projections on fixed replicas have covariance \[\frac{(\sigma^l)^{\mathsf T}P_a\sigma^j}{k_a(N)} =\frac{N}{k_a(N)}A^a_{lj}.\] Generate the fixed number of selected vectors by independent Gaussian vectors divided by \(\sqrt{k_a(N)}\), then orthonormalize them. Their Gram matrix tends to the identity. Before orthonormalization the replica projections are already jointly Gaussian with the displayed covariances, which are uniformly bounded. The Gram correction thus does not change the limit. The sphere moment formula, or Gaussian radial independence, gives every fixed moment uniformly, since \(\|P_a\sigma\|\le\sqrt N\) and \(k_a(N)/N\) stays bounded below. This proves (55) for any base Gibbs probability that is independent of the selected frames conditional on the projectors. ◻

Proof. We condition first on all coupled spectral data. Outside the special/cavity coordinates the full and base projectors coincide. Their overlap difference, including the change of dimension divisor, is bounded on two configurations by \[ \frac{C_n}{N}\big(1+|y^1|^2+|y^2|^2\big). \tag{58}\] Multiplying by \(Ne_N\) for the deterministic perturbations or by \(Ne_N^2\) for their Gaussian covariances gives a vanishing coefficient. The changes in these amplitudes for a bounded jump of dimensions also tend to zero. The omitted Gaussian kernels and their derivatives remain summable with the geometric weights. Thus the deterministic perturbation difference is bounded by \(o(1)(1+|y|^2)\), and the covariance difference by \(o(1)(1+|y^1|^2+|y^2|^2)\), uniformly in the parameters.

To see the sign of the comparison, put \(g(\sigma)=1+|y|^2\). Let \(D_0,D_+\) be the deterministic base and full perturbations on \((\sigma,\varepsilon)\), and let \(\mathcal C_0,\mathcal C_+\) be the corresponding Gaussian perturbation covariance kernels, conditional on the coupled spectral data. The preceding estimates give deterministic \(a_N,b_N\to0\), uniform in the parameters and spectral data, such that \[|D_+-D_0|\le a_Ng,\qquad |(\mathcal C_+-\mathcal C_0)(1,2)| \le b_N(g(\sigma^1)+g(\sigma^2)).\] Choose independent centered Gaussian fields \(X_0,X_+\) with these conditional covariances. Keep the full main interaction fixed and add the interpolating perturbation \[(1-\tau)D_0+\tau D_+ +\sqrt{1-\tau}\,X_0+\sqrt\tau\,X_+ -(1-\tau)\delta_Ng,\qquad 0\le\tau\le1.\] Write \(Z_\tau\) and \(\langle\cdot\rangle_\tau^{\mathrm{int}}\) for its partition function and Gibbs probability. The diagonal and pair terms in Lemma 8 are each bounded below by \(-b_N\mathbb E\langle g\rangle_\tau^{\mathrm{int}}\). Consequently \[\frac{d}{d\tau}\mathbb E\log Z_\tau \ge(\delta_N-a_N-2b_N)\mathbb E\langle g\rangle_\tau^{\mathrm{int}}\ge0\] if \(\delta_N\downarrow0\) is chosen to dominate \(a_N+2b_N\). At \(\tau=0\) the partition function is \(Z_N^\theta\langle W_Ne^{-\delta_Ng}\rangle_0\); at \(\tau=1\) it is \(Z_{N+n}^\theta\). Integrating proves (57). The block matrices are uniformly bounded, so Jensen’s inequality bounds its right-hand side below by \(-C\mathbb E\langle1+|y|^2\rangle_0\). Lemma 21 gives the claimed uniform lower bound.

For the test comparison, choose \(c>0\) smaller than every \(\rho_a\). The event \(\min_a\lambda_{\min}(M_a)<c\) has probability tending to zero and can be discarded in bounded tests. On its complement, the orthonormal coordinates of the projection onto \(\bigoplus_a\operatorname{ran}(P_a^+E)\) give \[|y|^2+n =\sum_a\big|M_a^{-1/2}E^{\mathsf T}P_a^+\sigma^+\big|^2 \le c^{-1}\sum_a|E^{\mathsf T}P_a^+\sigma^+|^2, \qquad \sigma^+=(\sigma,\varepsilon).\] At each fixed parameter choice the full disorder law is invariant under site permutations. For each site \(j\), \[\mathbb E\langle(P_a^+\sigma^+)_j^2\rangle_+ =\frac1{N+n}\mathbb E\langle\|P_a^+\sigma^+\|^2\rangle_+ \le1.\] Consequently \[ \limsup_{N\to\infty}\mathbb E\langle\mathbf 1_{\{|y|>B\}}\rangle_+ \le\frac{C}{B^2}. \tag{59}\] This use of symmetry precedes the restriction; no symmetry of the restricted Gibbs probability is needed. Nor is an inverse moment of \(M_a\) required on the discarded event.

For fixed \(B\), the restriction \(|y|\le B\) is a fixed set once the spectral data have been conditioned on. It is nonempty for \(B^2>d\), since uniform base-spin sampling has mean squared projection \(d\). The perturbation differences on that set are bounded by \(O((a_N+b_N)(1+B^2))\). Use the same interpolation without its penalty, normalizing the Gibbs probability on this fixed restriction at every \(\tau\). Lemma 8 bounds the change in the expected bounded Gibbs test by \(O((a_N+b_N)(1+B^2))=o(1)\) for fixed \(B\). At the base endpoint its probability is precisely the base probability tilted by \(W_N\mathbf 1_{\{|y|\le B\}}\). Finally, \(|Q_{12}^+-Q_{12}|\le2n/(N+n)\), so uniform continuity permits replacing the full total overlap by the base one. Equation (59) then removes the restriction in the stated order. ◻

Proof. We first verify the conditional Gaussian integrals needed by the cascade change of measure. We then compute the normalizing constants and the covariances under its tilted kernels. The case \(d=0\) is immediate, so assume \(d\geq1\).

The finite alphabet makes \(b(x)\) well defined above \(\lambda_+\). Writing \(u=1/b\) near \(x=0\) changes its defining equation to \(x=\sum_a\rho_a u/(1-\lambda_a u)\), whose derivative in \(u\) at zero is one. Consequently \(H\) and \(\widetilde H\) extend smoothly to zero, with \(H'(0)=\widetilde H'(0)=\mathop{\mathrm{Id}}\). For positive \(x\), put \(M_2(x)=\sum_a\rho_a/(b(x)-\lambda_a)^2\). Differentiation gives \(b'(x)=-1/M_2(x)\) and \(H'(x)=M_2(x)^{-1}(b(x)\mathop{\mathrm{Id}}-A_0)^{-2}\succeq0\). In particular, all covariance increments in (65) are well defined and positive semidefinite.

Append \(p_k=1\) and put \(p_{-1}=0\). For \(0\leq i\leq k\), define \[ \Sigma_i=\int_{p_{i-1}}^{p_i}H'(D_p(r))\,dr, \qquad H_i=H(D_p(p_i)),\qquad J_i=(\mathop{\mathrm{Id}}-H_iK)^{-1},\qquad K_i=KJ_i. \tag{68}\] The inverse defining \(J_i\) exists: at positive arguments, \(H_i^{-1}-K=b(D_p(p_i))\mathop{\mathrm{Id}}-A\succ0\), and at zero \(H_i=0\). Also \(K_i=(\mathop{\mathrm{Id}}-KH_i)^{-1}K\) is symmetric. The increment \(\Sigma_0\) is common to all leaves, increments \(\Sigma_i\) for \(1\leq i<k\) are attached independently to the edges at level \(i\), and \(\Sigma_k\) is the residual. Since \(D_p\) has slope \(-\zeta_i\) on \((p_{i-1},p_i)\) for \(i>0\), while it is constant on \((0,p_0)\), we have \[ H_k=0,\qquad H_{i-1}-H_i=\zeta_i\Sigma_i\quad(1\leq i\leq k), \qquad \Sigma_0=p_0H'(D_p(p_0)). \tag{69}\] These identities also hold when some adjacent overlap values coincide.

For \(i>0\), write \(\Delta_i=\zeta_i\Sigma_i\). The matrix ordering in the following identity will be important: \[ \mathop{\mathrm{Id}}-\Delta_iK_i=(\mathop{\mathrm{Id}}-H_{i-1}K)J_i, \qquad \det(\mathop{\mathrm{Id}}-\Delta_iK_i) =\frac{\det(\mathop{\mathrm{Id}}-H_{i-1}K)}{\det(\mathop{\mathrm{Id}}-H_iK)}. \tag{70}\] The symmetric Gaussian precision matrix satisfies \[ \mathop{\mathrm{Id}}-\zeta_i\Sigma_i^{1/2}K_i\Sigma_i^{1/2}\succ0. \tag{71}\] To prove this, replace \(\Delta_i\) in (70) by \(t\Delta_i\), \(0\leq t\leq1\), and set \(S_t=H_i+t\Delta_i\). If \(H_{i-1}=0\), then \(\Sigma_i=0\) and the claim is immediate. Otherwise \(H_{i-1}\succ0\), and each nonzero \(S_t\) is positive definite and at most \(H_{i-1}\). Thus \[S_t^{-1}-K\succeq H_{i-1}^{-1}-K\succ0.\] It follows that \(\det(\mathop{\mathrm{Id}}-S_tK)\ne0\); this is also true when \(S_t=0\). By (70) and the identity \(\det(\mathop{\mathrm{Id}}-BC)=\det(\mathop{\mathrm{Id}}-CB)\), the symmetric matrix in (71), with \(\zeta_i\) replaced by \(t\zeta_i\), never has a zero eigenvalue. At \(t=0\) it is the identity, proving (71) by continuity.

For a vector \(u\in\mathbb R^d\), completion of the square now gives the finite integral \[ \frac1{\zeta_i}\log\int \exp\left(\frac{\zeta_i}{2} (u+z)^{\mathsf T}K_i(u+z)\right) \gamma_{\Sigma_i}(dz) =-\frac1{2\zeta_i}\log\det(\mathop{\mathrm{Id}}-\Delta_iK_i) +\frac12u^{\mathsf T}K_{i-1}u. \tag{72}\] Indeed, if \(T_i=(\mathop{\mathrm{Id}}-\Delta_iK_i)^{-1}\), then (70) implies \(J_iT_i=J_{i-1}\) and hence \(K_iT_i=K_{i-1}\). Formula (72) also holds for singular \(\Sigma_i\), by writing \(z=\Sigma_i^{1/2}g\) with \(g\) a standard Gaussian vector.

Starting at \(K_k=K\), backward integration therefore produces the quadratic coefficients \(K_i\) of (68). For \(1\leq i<k\), (72) verifies exactly the conditional fractional exponential moment of exponent \(\zeta_i\) required by the marked-cascade change of measure in Proposition 5. The residual step \(i=k\) uses exponent \(\zeta_k=1\) and verifies its ordinary exponential moment. The root uses exponent zero, namely ordinary expectation of the backward value; its quadratic contribution is \(\mathop{\mathrm{Tr}}(\Sigma_0K_0)/2\). Since a Gaussian quadratic polynomial is integrable, no exponential moment of the root quadratic term is needed. The same recursion and Proposition 5, conditional on the root mark, show that \(Z_K\) is positive and finite almost surely. Their logarithmic integrability assertion applies because the Gaussian integrals above have finite constants and the root value is a quadratic polynomial in a finite-dimensional Gaussian vector. Thus \[ \mathbb E\log Z_K =\frac12\mathop{\mathrm{Tr}}(\Sigma_0K_0) -\sum_{i=1}^k\frac1{2\zeta_i} \log\frac{\det(\mathop{\mathrm{Id}}-H_{i-1}K)}{\det(\mathop{\mathrm{Id}}-H_iK)}. \tag{73}\] Set \(f(x)=\log\det(\mathop{\mathrm{Id}}-H(x)K)\). Its derivative is \(f'(x)=-\mathop{\mathrm{Tr}}(H'(x)K(\mathop{\mathrm{Id}}-H(x)K)^{-1})\). On \((0,p_0)\) this identifies the root term in (73) with \(-\frac12\int_0^{p_0}f'(D_p(r))\,dr\). On every subsequent interval, the change of variable \(D_p'(r)=-\zeta_i\) identifies its determinant term with the corresponding integral. Summing proves (66).

It remains to identify the terminal marks under the quadratic tilt. The marked-cascade rule leaves the root law unchanged and gives, at step \(i>0\), the Gaussian kernel obtained by multiplying the density of \(z\sim\gamma_{\Sigma_i}\) by \(\exp(\zeta_i(u+z)^{\mathsf T}K_i(u+z)/2)\) and normalizing. If \(Y_{i-1}=u\) is the running sum before this step, square completion gives \[ \mathbb E[Y_i\mid Y_{i-1}]=T_iY_{i-1},\qquad \mathop{\mathrm{Cov}}(Y_i\mid Y_{i-1})=T_i\Sigma_i, \qquad T_i=(\mathop{\mathrm{Id}}-\Delta_iK_i)^{-1}. \tag{74}\] The covariance is symmetric and positive semidefinite, since it equals \[\Sigma_i^{1/2} (\mathop{\mathrm{Id}}-\zeta_i\Sigma_i^{1/2}K_i\Sigma_i^{1/2})^{-1} \Sigma_i^{1/2}.\] The conditional covariance is independent of the preceding value, and the kernels on distinct child edges are conditionally independent.

Rescale the running sums by \(U_i=J_iY_i\). Because \(J_iT_i=J_{i-1}\), the process \(U_i\) has centered independent innovations on the tree. Their covariances, and that of the common root, are respectively \[ \mathop{\mathrm{Cov}}(U_i-U_{i-1})=J_{i-1}\Sigma_iJ_i^{\mathsf T}\quad(i>0), \qquad \mathop{\mathrm{Cov}}(U_0)=J_0\Sigma_0J_0^{\mathsf T}. \tag{75}\] For symmetric matrices \(B,C\) for which the indicated inverses exist, the map \(F(B)=(\mathop{\mathrm{Id}}-BK)^{-1}B\) satisfies \[ F(B)-F(C) =(\mathop{\mathrm{Id}}-BK)^{-1}(B-C)(\mathop{\mathrm{Id}}-KC)^{-1}. \tag{76}\] This follows by multiplying on the left by \(\mathop{\mathrm{Id}}-BK\) and on the right by \(\mathop{\mathrm{Id}}-KC\). Applying it to \(B=H_{i-1}\) and \(C=H_i\) in (75) shows that the covariance at step \(i>0\) is \[\frac{\widetilde H(D_p(p_{i-1}))- \widetilde H(D_p(p_i))}{\zeta_i} =\int_{p_{i-1}}^{p_i}\widetilde H'(D_p(r))\,dr.\] Here (67) follows directly from \(H(x)^{-1}-K=b(x)\mathop{\mathrm{Id}}-A\) at positive \(x\). Differentiating (76) along \(H(x)\) also gives \[ \widetilde H'(x)=J(x)H'(x)J(x)^{\mathsf T},\qquad J(x)=(\mathop{\mathrm{Id}}-H(x)K)^{-1}, \tag{77}\] so the root covariance in (75) is \(p_0\widetilde H'(D_p(p_0))\), as required. Since \(J_k=\mathop{\mathrm{Id}}\), the final sum \(U_k\) equals the original terminal mark \(Y_k\). The cascade change of measure gives the original unmarked cascade law independently of these Gaussian innovations, completing the claimed representation.

All the endpoint cases are included. If \(p_{k-1}=1\), then \(\Sigma_k=0\) and the residual step makes no change. If \(p=1\) everywhere, then \(D_p=0\), \(J_i=\mathop{\mathrm{Id}}\), and \(\Sigma_0=\mathop{\mathrm{Id}}\); the sole quadratic contribution is the ordinary root expectation \(\mathop{\mathrm{Tr}}(K)/2\). Equal adjacent overlap values simply give zero increments in the same formulas. ◻

Proof. First perform only the quadratic tilt. By Lemma 23, the terminal marks then have a Gaussian cascade representation associated with \(\widetilde H(x)=(b(x)I_d-A)^{-1}\). Its diagonal covariance is \[\int_0^1\widetilde H'(D_q(r))\,dr,\] whose norm is bounded by \(\sup_{0\le x\le1}\|\widetilde H'(x)\|\). This is finite for the fixed alphabet, independently of the number of steps and their weights. Indeed \(b(x)>\lambda_+\) for \(x>0\), and the inverse has the smooth extension at zero already proved in the lemma.

Let \(\langle\cdot\rangle_{q,K}\) denote the normalized quadratic tilt, still including the uniform cavity-spin probability. The remaining exponent is \[V(y,\varepsilon)=y^{\mathsf T}L\varepsilon +\frac12\varepsilon^{\mathsf T}C\varepsilon.\] For each realization, Jensen’s inequality gives \(\langle e^V\rangle_{q,K}^{-1}\le\langle e^{-V}\rangle_{q,K}\). Taking two replicas therefore gives \[ \mathbb E\langle|y|^r\rangle_{q,W} \le\mathbb E\big\langle |y^1|^r e^{V(y^1,\varepsilon^1) -V(y^2,\varepsilon^2)} \big\rangle_{q,K}. \tag{79}\] Conditional on their branching, the two marks on the right are centered joint Gaussians with uniformly bounded covariance. The spin coefficients range over a fixed finite set. Gaussian polynomial-exponential moments thus bound the right side uniformly, proving (78). This argument includes the residual draws as part of the probability prior. Jensen’s inequality and the same exponential-linear moments also show that the remaining logarithmic normalizer is integrable.

Put \(g_T=(\log W-T)_+\). Since \(\min(W,e^T)=We^{-g_T}\), \[0\le\log\langle W\rangle_q -\log\langle\min(W,e^T)\rangle_q \le\langle g_T\rangle_{q,W}.\] The bound \(|\log W|\le C(1+|y|^2)\) and (78) make its expectation tend uniformly to zero. Restriction of an \(l\)-replica bounded test changes its expectation by at most a constant times \(l\langle\mathbf 1_{\{|y|>B\}}\rangle_{q,W}\). Markov’s inequality and (78) prove the second assertion, including the event where the restricted normalizer is zero. ◻

Proof. Compression of the resolvent of the limiting block matrix onto the cavity coordinates gives, for \(b=b(x)\), \[ \big(bI_n-C-L^{\mathsf T}(bI_d-A)^{-1}L\big)^{-1} =\sum_a\frac{\rho_a I_n}{b-\lambda_a}=xI_n. \tag{83}\] The whole block has eigenvalue \(\lambda_a\) with multiplicity \(n\). Its Schur determinant and the multiplicities in \(A_0\) give \[\begin{align*} C+L^{\mathsf T}\widetilde H(x)L&=R(x)I_n, \tag{84}\\ \log\det(I_d-H(x)K) &=\log\frac{\det(bI_d-A)}{\det(bI_d-A_0)}\\ &=n\left(\log x+\sum_a\rho_a\log(b-\lambda_a)\right). \tag{85}\end{align*}\] To see the multiplicities explicitly, \(\det(bI_d-A)=x^n\prod_a(b-\lambda_a)^n\) whereas \(\det(bI_d-A_0)=\prod_a(b-\lambda_a)^{n-s_a}\). Differentiating (85) and using \(\sum_a\rho_a/(b-\lambda_a)=x\) yields \[ \frac1n\frac{d}{dx}\log\det(I_d-H(x)K) =\frac1x+xb'(x)=xR'(x). \tag{86}\]

Differentiating (84) gives \[L^{\mathsf T}\widetilde H'(x)L=R'(x)I_n.\] Thus, after the quadratic transformation, the field \(L^{\mathsf T}y\) has independent coordinates with common-level covariance path \(h_p\). Its diagonal covariance is \(vI_n\), where \[v=\int_0^1R'(D_p(r))\,dr.\] Writing \(h_*=h_p(1-)\), the independent residual has covariance \((v-h_*)I_n\). Since every cavity spin has squared norm \(n\), integrating this residual and the deterministic cavity term \(C=R(0)I_n\) contributes \(n[R(0)+v-h_*]/2\) to the logarithm. The remaining spin sum is \(\prod_{j=1}^n\cosh z_j\) for the nonresidual field \(z\). Its logarithm is additive over coordinates, so the Gaussian backward recursion and its tilted kernels factor over those coordinates. This recursion contributes \(n[\ell(h_p)+h_*/2]\). Lemma 23 and (86) therefore give \[ \frac1n\mathbb E\log\langle W\rangle_p =\ell(h_p)+\frac12\left[ R(0)+\int_0^1(1-D_p(r))R'(D_p(r))\,dr\right]. \tag{87}\] All three contributions include the root interval; the independent residual was integrated at exponent one.

Fubini’s theorem and the definition of \(D_p\) give \[\int_0^1p(s)h_p(s)\,ds =\int_0^1R'(D_p(r))\int_{\{s:p(s)>r\}}p(s)\,ds\,dr,\] and, almost everywhere in \(r\), \[1-D_p(r)-\int_{\{s:p(s)>r\}}p(s)\,ds=-rD_p'(r).\] Subtracting the pairing with \(h_p\) from the bracket in (87) thus gives \[R(0)-\int_0^1r\,dR(D_p(r)) =\int_0^1R(D_p(r))\,dr=2G_1(p),\] where \(D_p(1)=0\) supplies the endpoint at one. This proves (81).

Both the quadratic transformation and the remaining spin tilt use the marked-cascade rule of Proposition 5. Its tree relabeling preserves all common-level values, and leaves their law unchanged. The scalar residual correction is independent of the spins. Conditional on a common level, the remaining single-coordinate two-spin product is exactly \(B_{h_p}\) from the Ising recursion; the independent coordinates factor through its tilted kernels. This proves (82) and the last assertion. ◻

Proof. For any fixed cavity site \(j\), site permutation symmetry of the full model gives, for every bounded continuous \(\phi\), \[\mathbb E\langle\phi(Q_{12}^+)\varepsilon_j^1\varepsilon_j^2\rangle_+ =\mathbb E\langle\phi(Q_{12}^+)Q_{12}^+\rangle_+.\] Apply the restricted test comparison of Lemma 22, pass through the bounded finite-replica calculations, and then remove the restriction using Lemma 24. Proposition 25 and preservation of the common-level law give \[ \lim_{\bar p\to p}\int_0^1 \phi(\bar p(s))\big(B_{h_{\bar p}}(s)-\bar p(s)\big)\,ds=0. \tag{91}\] No GG property of the full system was used here: the base parameters are good, while site symmetry holds for the full model at every fixed parameter choice.

The functions \(B_{h_{\bar p}}\) are nondecreasing and take values in \([0,1]\), so they are relatively compact in \(L^1\). We explain why any subsequential limit \(B\) is a measurable function of \(p\); this is needed to infer equality from (91). Since each rounding is a function of the values of \(p\), it is constant on every plateau of \(p\). The corresponding field levels are tied there. Their zero increments give identical conditional positions and magnetizations, so \(B_{h_{\bar p}}\) is also constant there, by Proposition 9. An \(L^1\) limit preserves this property on every positive-length plateau. There are only countably many such intervals. Outside their interiors and their null set of endpoints, the generalized inverse of the nondecreasing function \(p\) recovers the quantile coordinate. Thus \(B\) is measurable with respect to \(p\) almost everywhere, including when \(p\) has a singular-continuous part.

Passing to the limit in (91) shows that the signed pushforward of \((B-p)\,ds\) under \(p\) vanishes, since continuous tests determine finite measures on \([0,1]\). The established measurability then gives \(B=p\) almost everywhere. Every subsequential limit is the same, proving (89).

For any fixed \(v\in\mathcal H\), take common refinements of its intervals and those of \(h_{\bar p}\). The supporting inequality from Theorem 12, including tied levels, is \[\ell(v)\le\ell(h_{\bar p}) -\frac12\int_0^1B_{h_{\bar p}}(s) (v(s)-h_{\bar p}(s))\,ds.\] Letting \(\bar p\to p\) proves that the supremum defining \(S(p)\) is at most the right side of (90). Conversely, \(h_{\bar p}\in\mathcal H\), and their values in the definition of \(S(p)\), with \(p\) fixed in the pairing, converge to that same expression. This proves (90). ◻

Proof. Put \(\mathcal V=\inf_{p\in\mathcal P}\{S(p)+G_1(p)\}\). This is finite: \(S\ge0\), \(G_1\) is bounded, and the zero trial has finite \(S\) by Lemma 13. Every sequence of good base parameters has a subsequence with the overlap limits used above. Equations (88) and (90) therefore imply \[\liminf_{N\to\infty}\inf_{\theta\in\mathcal G_N} \big(\mathcal F_{N+n}(\theta)-\mathcal F_N(\theta)\big) \ge n\mathcal V.\] Indeed any failure would select a sequence of parameters whose subsequential overlap limit contradicts the two equations.

Lemma 22 gives a uniform lower bound for the increments also outside \(\mathcal G_N\). Since \(\eta(\mathcal G_N)\to1\), integration over the common parameter space yields \[\liminf_{N\to\infty}\int \big(\mathcal F_{N+n}(\theta)-\mathcal F_N(\theta)\big)\,\eta(d\theta) \ge n\mathcal V.\] The parameter space includes the whole infinite sequence, with the size-\(N\) partition function ignoring later coordinates. Hence these integrated increments telescope exactly along the progression. Dividing their sum by the terminal dimension proves the lower bound for the parameter-averaged perturbed pressure. The perturbations change the expected log partition by \(O(Ne_N)=o(N)\), uniformly in parameters, so the same bound holds for \(\mathbb EP_N\). ◻

Proof. For a measure \(\tau\) with upper support edge \(e_\tau\), put \[H_\tau(z)=\int\frac{\tau(d\lambda)}{z-\lambda}, \qquad z>e_\tau.\] This function is continuous and strictly decreasing. Thus \(b_\tau(x)>e_\tau\) exactly when \(x<H_\tau(e_\tau+)\), and in that case \(H_\tau(b_\tau(x))=x\). In the remaining case \(b_\tau(x)=e_\tau\). In both cases, \[ e_\tau\le b_\tau(x)\le e_\tau+x^{-1}, \qquad -K\le R_\tau(x)\le K. \tag{94}\] The first upper bound follows from \(H_\tau(z)\le(z-e_\tau)^{-1}\). The lower bound on \(R_\tau\) follows from \(H_\tau(z)\ge(z+K)^{-1}\), using the edge limit of this inequality when the transform is capped.

We first work away from zero. Let \(x_j\to x>0\) and pass to a subsequence on which \(b_{\nu_j}(x_j)\to b_*\). The bounds (94) imply that such subsequences exist and that \(b_*\ge e\). If \(b_*>e\), the defining equations are uncapped for large \(j\). Weak convergence, with their arguments uniformly separated from the supports, gives \[H_\nu(b_*)=x.\] Hence \(b_*=b_\nu(x)\). If \(b_*=e\) but \(b_\nu(x)>e\), choose \(c\in(e,b_\nu(x))\). Then \(H_\nu(c)>x\), so \(H_{\nu_j}(c)>x_j\) for large \(j\). Strict monotonicity, including the definition of the capped branch, forces \(b_{\nu_j}(x_j)>c\), a contradiction. These two cases prove \[b_{\nu_j}(x_j)\longrightarrow b_\nu(x).\] The same argument with the measure fixed proves continuity of \(b_\nu\) on \((0,1]\). A compactness argument now gives uniform convergence of \(R_{\nu_j}\) on every \([\varepsilon,1]\).

Near zero the convergence is uniform for a different reason. If \(K=0\) there is nothing to prove. For \(K>0\) and \(x<(4K)^{-1}\), the transform cannot be capped, since \(H_\tau(e_\tau+)\ge(2K)^{-1}\). Write \(b_\tau(x)=x^{-1}+R_\tau(x)\) in its defining equation. The bound \(|R_\tau|\le K\) permits the uniform expansion \[1=\int\frac{\tau(d\lambda)}{1+x(R_\tau(x)-\lambda)} =1-x\left(R_\tau(x)-\int\lambda\,\tau(d\lambda)\right) +O(K^2x^2).\] Therefore \[ R_\tau(x)=\int\lambda\,\tau(d\lambda)+O(K^2x), \tag{95}\] with a constant independent of \(\tau\). Weak convergence on the common bounded interval implies convergence of the means. Combining (95) with the convergence away from zero proves (92).

Finally, \(0\le D_p(r)\le1\) for every admissible \(p\) and \(r\), and hence \[\sup_{p\in\mathcal P}|G_{1,\nu_j}(p)-G_{1,\nu}(p)| \le\frac12\|R_{\nu_j}-R_\nu\|_\infty.\] Taking infima after adding the common functional \(S\) proves (93). These infima are finite: \(S\ge0\) and \(S(0)=0\) by Lemma 13, whereas the transforms are bounded by (94). ◻

Proof. Arrange the two diagonal matrices so that, for each \(a\), \(\min(k_a,\widetilde k_a)\) axes retain the value \(\lambda_a\). Haar invariance makes this rearrangement immaterial. Set \(\Delta=\widetilde\Lambda-\Lambda\) and \[f(s)=\mathbb EP_N(\Lambda+s\Delta,U),\qquad 0\le s\le1.\] Convexity of the log partition function gives \(f'(0)\le f(1)-f(0)\le f'(1)\). At either endpoint, \[f'(s)=\frac1{2N}\mathbb E\left\langle\sum_i\Delta_{ii}x_i^2 \right\rangle_s, \qquad x=U\sigma.\] Within a group of equal endpoint eigenvalues, the expected squared projections are equal on all axes. Indeed, rotating the corresponding rows of \(U\) preserves both its Haar law and the spin Hamiltonian; the spin prior is unaffected. If a group has \(q_a\) axes and \(c_a\) changed axes, its changed axes therefore carry the fraction \(c_a/q_a\) of the group’s expected squared spin norm. Since \(|\Delta_{ii}|\le L\) and the total squared norm is \(N\), \[|f'(s)|\le\frac L2\max_a\frac{c_a}{q_a}, \qquad s\in\{0,1\}.\] At the first endpoint \(c_a=(k_a-\widetilde k_a)_+\), and at the second \(c_a=(\widetilde k_a-k_a)_+\). The endpoint bounds prove (98). ◻

Proof. We first finish the finite-alphabet case. Suppose that \(\mu=\sum_{a=1}^m\rho_a\delta_{\lambda_a}\) with all \(\rho_a>0\) rational. Choose \(n\) so that \(s_a=n\rho_a\) are integers. For each residue \(r\in\{0,\ldots,n-1\}\), choose nonnegative integers \(c_a^r\) with sum \(r\), and at dimension \(N=kn+r\) use multiplicities \[k_a(N)=ks_a+c_a^r.\] They increase by \(s_a\) when \(N\) increases by \(n\), and are positive for all sufficiently large \(N\). Proposition 20 and Proposition 27 give convergence to \(\mathcal F(\mu)\) on each progression. There are finitely many residues, so these canonically chosen multiplicities give convergence on the full sequence of dimensions.

Now allow arbitrary positive proportions \(\rho_a\) and arbitrary multiplicities \(k_a(N)\) with \(k_a(N)/N\to\rho_a\). Approximate \(\rho\) by positive rational probability vectors \(\tau\), and let \(\widetilde k_a(N)/N\to\tau_a\) be the canonical counts just constructed. Lemma 29 gives \[\begin{align*} &\limsup_{N\to\infty} \left|\mathbb EP_N(\Lambda_N,U)- \mathcal F\left(\sum_a\tau_a\delta_{\lambda_a}\right) \right|\\ &\hspace{25mm}\le \frac L2\max_a\frac{|\rho_a-\tau_a|}{\min(\rho_a,\tau_a)}. \end{align*}\] The right side tends to zero as \(\tau\to\rho\). Lemma 28 applies to these laws because their support is the same fixed alphabet. This proves the assertion for every finite alphabet with positive limiting proportions.

We next construct finite alphabets for the general measure. If \(a=b\), then \(R_{\delta_a}=a\) and \(\mathcal F(\delta_a)=a/2\) because \(S\ge0\) and \(S(0)=0\); the assertion follows directly from (96). Hence suppose \(a<b\), and put \[\eta_N=\max\{(a-\lambda_{\min}(\Lambda_N))_+, (\lambda_{\max}(\Lambda_N)-b)_+\}.\] Then \(\eta_N\to0\), and clipping the eigenvalues to \([a,b]\) changes the pressure by at most \(\eta_N/2\). The clipped empirical measures still converge weakly to \(\mu\).

Fix \(\delta>0\). Partition \([a,b]\) into finitely many intervals of length at most \(\delta\), with all interior endpoints of zero \(\mu\)-mass. Use half-open intervals to assign endpoints uniquely, including \(a\) and \(b\) in the first and last intervals. For each positive-mass interval \(I_j\), choose \(r_j\in I_j\cap\mathop{\mathrm{supp}}\mu\), and put \[\nu_\delta=\sum_{j:\,\mu(I_j)>0}\mu(I_j)\delta_{r_j}.\] The smallest and largest representatives satisfy \[a\le r_{\min}\le a+\delta, \qquad b-\delta\le r_{\max}\le b,\] because every neighborhood of a support edge has positive mass.

In a positive-mass interval, replace each clipped eigenvalue by its representative. In all zero-mass intervals together, replace the eigenvalues by \(r_{\min}\) to form \(\Lambda_N^-\), and by \(r_{\max}\) to form \(\Lambda_N^+\). Each zero-mass interval is a continuity set for the clipped measures relative to \([a,b]\), so the fraction of eigenvalues in their finite union tends to zero. Both modified spectra therefore have limiting law \(\nu_\delta\), with only positive limiting groups. Moreover, coordinatewise, \[\Lambda_N^--(\delta+\eta_N)\mathop{\mathrm{Id}} \le\Lambda_N\le \Lambda_N^++(\delta+\eta_N)\mathop{\mathrm{Id}}.\] Monotonicity and (97), followed by the finite-alphabet result, yield \[ \mathcal F(\nu_\delta)-\frac\delta2 \le\liminf_N\mathbb EP_N \le\limsup_N\mathbb EP_N \le\mathcal F(\nu_\delta)+\frac\delta2. \tag{99}\] Thus no estimate involving the fraction changed inside a vanishing group is needed.

As \(\delta\downarrow0\), the laws \(\nu_\delta\) converge weakly to \(\mu\) and their upper support edges converge to \(b\). Lemma 28 gives \(\mathcal F(\nu_\delta)\to\mathcal F(\mu)\), and (99) completes the proof. The thermodynamic limit was taken for each fixed finite alphabet before its refinement. No derivative bound on the finite-alphabet transforms is required uniformly over those refinements. ◻

Completion of the proof of Theorem 1. Proposition 30 supplies convergence of the expected pressures. The spectral hypotheses also imply \(\|\Lambda_N\|_{\mathrm{op}}\le K\) for all sufficiently large \(N\), for some finite \(K\). Comparing the quadratic forms for two orthogonal matrices shows that \(P_N(\Lambda_N,U)\) is \(K\)-Lipschitz in the Frobenius distance, independently of the number of distinct eigenvalues. The fourth-moment argument in Lemma 14 therefore gives \[\mathbb E|P_N-\mathbb EP_N|^4\le \frac{C_K}{N^2}.\] For every \(\varepsilon>0\), Markov’s inequality makes the probabilities of \(|P_N-\mathbb EP_N|>\varepsilon\) summable. Borel–Cantelli, first for positive rational \(\varepsilon\), proves \(P_N-\mathbb EP_N\to0\) almost surely. This argument needs no independence between the orthogonal matrices at different dimensions. ◻

Proof. Let \(f_N(\Lambda)\) be the Haar expectation of the pressure with deterministic eigenvalues \(\Lambda\). The integral against the conditional orbit law is therefore the finite quantity \(f_N(\Lambda_N)\). This is used as a conditional orbit integral, without any assumption of global integrability. Equivalently, all conditional calculations below may first be restricted to \(\{\|\Lambda_N\|_{\mathrm{op}}\le K\}\), on which \(|P_N|\le K/2\). Under the almost-sure spectral hypotheses, Proposition 30 applies to each spectral sequence in the event of convergence and yields \(f_N(\Lambda_N)\to\mathcal F(\mu)\) almost surely. Under the in-probability hypotheses, every subsequence has a further subsequence on which all the spectral hypotheses hold almost surely. Here weak convergence can be expressed by a metric on probability measures, so only finitely many convergence-in-probability assertions are involved. The same deterministic result on this further subsequence proves \(f_N(\Lambda_N)\to\mathcal F(\mu)\) in probability.

For each fixed \(K\), conditional concentration gives \[ \mathbb P\bigl(|P_N-f_N(\Lambda_N)|>\varepsilon, \|\Lambda_N\|_{\mathrm{op}}\le K\bigr) \le\frac{C_K}{\varepsilon^4N^2}. \tag{100}\] The spectral hypotheses in probability imply that the norm bound holds with probability tending to one for a sufficiently large fixed \(K\). This proves convergence of the fluctuations in probability. Under the almost-sure hypotheses the spectra are eventually bounded almost surely. Apply Borel–Cantelli to (100) for integer \(K\) and positive rational \(\varepsilon\) to obtain almost-sure convergence of the fluctuations as well.

Uniform integrability and convergence in probability imply convergence of expectations. Finally, \(|P_N|\le\|J_N\|_{\mathrm{op}}/2\), which proves the sufficient condition. ◻

Proof. Write \(P_N(\beta)\) for the pressure with spectrum \(\beta\Lambda_N\). The uniform probability on the cube gives, for every \(N\) and \(\beta>0\), \[ \frac{P_N(\beta)}\beta \le M_N\le \frac{P_N(\beta)}\beta+\frac{\log2}\beta. \tag{102}\] Theorem 1 applies at each positive integer \(\beta\). Intersecting these countably many probability-one events, the difference between the limit superior and limit inferior of \(M_N\) is at most \(\log2/\beta\) for every such integer. Thus \(M_N\) converges almost surely. Moreover, its limit belongs to each deterministic interval \[\left[\frac{\mathcal F(\mu_\beta)}\beta, \frac{\mathcal F(\mu_\beta)}\beta+\frac{\log2}\beta\right], \qquad \beta\in\mathbb N,\] so it is a deterministic number \(M\). The eventual deterministic bound on \(\|J_N\|_{\mathrm{op}}\) also gives convergence in expectation. For any real \(\beta>0\), take expectations and then \(N\to\infty\) in (102). It follows that \[0\le M-\frac{\mathcal F(\mu_\beta)}\beta \le\frac{\log2}\beta,\] which proves (101) without a uniform positive-temperature estimate. ◻

Proof. Put \(W_N=G_N^{\mathsf T}G_N/N\) and \(\alpha_N=m_N/N\). For every \(Q\in O(N)\), the matrices \(G_NQ\) and \(G_N\) have the same law. Haar averaging this identity against any bounded function of the ordered eigenvalues shows that the conditional law of \(cW_N\) is its orthogonal orbit law, including when the zero eigenvalue is repeated.

The Marchenko–Pastur theorem [18] gives weak convergence in probability of the empirical law of \(W_N\) to \(\mu_\alpha\). To check the edges, let \(s_{\min}^+(G_N)\) be the smallest positive singular value. For \(m_N\ge1\) and \(t>0\), the Gaussian singular-value bounds [10], after scaling and transposing when necessary, give \[\begin{align*} \mathbb P\left(\frac{s_{\max}(G_N)}{\sqrt N}> 1+\sqrt{\alpha_N}+t\right)&\le e^{-Nt^2/2},\\ \mathbb P\left(\frac{s_{\min}^+(G_N)}{\sqrt N}< |1-\sqrt{\alpha_N}|-t\right)&\le e^{-Nt^2/2}. \end{align*}\] Together with positive limiting mass near each continuous spectral edge, these bounds give \(\lambda_{\max}(W_N)\to b_\alpha\) in probability. The lower edge is \(0\) for \(\alpha\le1\) and \(a_\alpha\) for \(\alpha>1\). For \(\alpha<1\), \(W_N\) has \(N-m_N\) zero eigenvalues almost surely for all large \(N\); at \(\alpha=1\), nonnegativity and limiting mass near zero give the lower-edge limit; for \(\alpha>1\), the second bound and limiting mass near \(a_\alpha\) give that limit. Thus both edge excesses vanish. Scaling by a fixed \(c\) preserves this conclusion, reversing the edges when \(c<0\); the case \(c=0\) is immediate.

Since \(\alpha_N\le\alpha\), integrating the first singular-value bound gives \[\sup_N\mathbb E\|W_N\|_{\mathrm{op}}^2 =\sup_N\mathbb E\left(\frac{s_{\max}(G_N)}{\sqrt N}\right)^4<\infty.\] The finitely many possible cases \(m_N=0\) are zero matrices. Hence \(\|cW_N\|_{\mathrm{op}}\) and \(p_N(c)\) are uniformly integrable for each fixed \(c\). Proposition 31 gives convergence in probability, and uniform integrability upgrades it to \(L^1\).

For \(\varepsilon\in\{+1,-1\}\), put \[H_N^\varepsilon=\frac1{2N} \max_{\sigma\in\{-1,1\}^N}\sigma^{\mathsf T}(\varepsilon W_N)\sigma, \qquad A_\beta^\varepsilon= \frac{\mathcal F(\mu_{\alpha,\varepsilon\beta})}{\beta}.\] The same maximum squeeze as in (102) holds pointwise: \[\frac{p_N(\varepsilon\beta)}{\beta} \le H_N^\varepsilon\le \frac{p_N(\varepsilon\beta)}{\beta}+\frac{\log2}{\beta}.\] Also \(|H_N^\varepsilon|\le\|W_N\|_{\mathrm{op}}/2\), so these energies are uniformly integrable. Taking expectations and then \(N\to\infty\) at each fixed \(\beta>0\) gives \[A_\beta^\varepsilon \le\liminf_N\mathbb EH_N^\varepsilon \le\limsup_N\mathbb EH_N^\varepsilon \le A_\beta^\varepsilon+\frac{\log2}{\beta}.\] The expectations are bounded, and the gap between their limit superior and limit inferior is at most \(\log2/\beta\) for every \(\beta>0\). They therefore have a common limit \(e_\varepsilon(\alpha)\), and the same bounds give \[0\le e_\varepsilon(\alpha)-A_\beta^\varepsilon \le\frac{\log2}{\beta},\] which proves (106). Given any error tolerance, choose one sufficiently large fixed \(\beta\) and use (105) in the pointwise squeeze. This proves \(H_N^\varepsilon\to e_\varepsilon(\alpha)\) in probability; uniform integrability gives convergence in \(L^1\).

Dualizing the Euclidean norm and maximizing a linear functional over the cube gives \[\|G_N\|_{\infty\to2} =\|G_N^{\mathsf T}\|_{2\to1} =\max_{\sigma\in\{-1,1\}^N}\|G_N\sigma\|_2.\] Consequently, \[H_N^+=\frac{\|G_N\|_{\infty\to2}^2}{2N^2}, \qquad H_N^-=-\frac1{2N^2}\min_{\sigma\in\{-1,1\}^N}\|G_N\sigma\|_2^2.\] The signs of \(e_+\) and \(e_-\) follow from these identities. Since \(G_N\sigma=\sum_i\sigma_i g_{N,i}\), the \(L^1\) convergence of the energies and the inequality \(|\sqrt{x}-\sqrt{y}|\le\sqrt{|x-y|}\) for \(x,y\ge0\) give (107)–(108). ◻

Proof. Let \(f_h(b)=\ell_b^{\mathrm{can}}(h)+h_{k-1}/2\) be the unshifted initial backward value. Gaussian convolution and differentiation of the log-exponential operators show that \(f_h\) is smooth and strictly convex: its second derivative is a positive tilted average of the terminal second derivative, plus nonnegative conditional-variance terms. Moreover, with \(H=h_{k-1}\), \[ \log\cosh b\le f_h(b)\le\log\cosh b+H/2. \tag{117}\] For the lower bound, compare every backward operator with ordinary expectation and use convexity of \(\log\cosh\). For the upper bound, compare with exponent one throughout and use \(\mathbb E\cosh(b+\sqrt H Z)=e^{H/2}\cosh b\). Convex difference quotients in (117) imply that \(f_h'(b)\to\pm1\) as \(b\to\pm\infty\), uniformly for \(H\) in a bounded set. Thus \(f_h'(b_s(h))=s\) determines a unique bias, bounded uniformly over paths of bounded height for each fixed interior \(s\).

Gaussian differentiation, as in Proposition 9, gives \(\partial_{h_i}\ell_b^{\mathrm{can}}=-w_i\mathcal B_i(h,b)/2\). The envelope rule proves (116). In the remainder of the proof \(s\) is held fixed, so derivatives of \(B_i(h,s)\) include the change in \(b_s(h)\). We use the mean spin as spatial coordinate, so that the varying bias is represented by the fixed initial coordinate \(s\). We will prove that \(\partial_{h_l}B_i\le0\) for \(l>i\) and \(\sum_l h_l\partial_{h_l}B_i\ge0\), then apply the Hessian argument of Theorem 12.

Set \(T=h_{k-1}\) and let \(\zeta(v)=\zeta_i\) for \(h_{i-1}<v<h_i\). The continuous-time version of the unshifted backward recursion solves, on each such interval, \[F_v+\frac12F_{zz}+\frac12\zeta(v)F_z^2=0, \qquad F(T,z)=\log\cosh z.\] For the radial comparison we also need \(\alpha>0\). Let \(F^\alpha\) solve the same recursion with terminal value \(\alpha^{-1}\log\cosh(\alpha z)\) and coefficient \(\alpha\zeta(v)\) in place of \(\zeta(v)\). Write \[u=F_z^\alpha(v,z),\qquad a(v,u)=F_{zz}^\alpha(v,z(v,u)),\qquad -1<u<1.\] Strict convexity gives the inverse \(z(v,u)\). With \(L=\partial_v+\tfrac12\partial_{zz} +\alpha\zeta F_z^\alpha\partial_z\), differentiation gives \(L(F_z^\alpha)=0\) and \(L(F_{zz}^\alpha)=-\alpha\zeta(F_{zz}^\alpha)^2\). The chain rule in inverse coordinates therefore yields \[ a_v+\frac12a^2a_{uu}=-\alpha\zeta(v)a^2, \qquad a(T,u)=\alpha(1-u^2). \tag{118}\] Likewise, continuation of a payoff by the tilted transition kernels satisfies \[ H_v+\frac12a^2H_{uu}=0. \tag{119}\] In particular, if \(H_i\) has terminal condition \(H_i(h_i,u)=u^2\), then \(B_i(h,s)=H_i(0,s)\) at \(\alpha=1\).

We record the boundary regularity that permits comparison on the closed interval \([-1,1]\). For fixed strict level data and \(\alpha>0\), the backward functions have, as \(z\to+\infty\), expansions to every prescribed finite order of the form \[ F^\alpha(v,z) =z+C(v)+c_1(v)e^{-2\alpha z}+\cdots +c_M(v)e^{-2M\alpha z} +O(e^{-2(M+1)\alpha z}),\qquad c_1(v)>0. \tag{120}\] The expansions hold with any fixed finite number of differentiated remainders, uniformly on the closed time intervals between successive levels. To verify this, start from the expansion of the terminal \(\alpha^{-1}\log\cosh(\alpha z)\). An ordinary Gaussian backward step integrates each exponential term. In a positive-exponent step, first factor the linear exponential, expand the remaining exponential, integrate term by term against the Gaussian with the resulting shifted mean, and take the logarithm. The leading correction is multiplied by a strictly positive Gaussian exponential moment. Remainders satisfy the same bounds: on bounded or negative arguments an exponential bound of the indicated order dominates their linear or exponential growth, and Gaussian exponential moments are finite. The argument applies after differentiating any fixed finite number of times. Coefficients are continuous in time, and the positive leading coefficient is bounded away from zero for the fixed data under consideration.

Tilted continuations of \((F_z^\alpha)^2\) obey the same finite expansion property, by taking ratios of the corresponding Gaussian integrals. Since \(1-u=2\alpha c_1(v)e^{-2\alpha z}+O(e^{-4\alpha z})\), inverting this expansion gives bounded spatial derivatives through order four for \(a\) and the functions \(H_i\) on \([-1,1]\). Reflection treats the other endpoint. In particular, \[a(v,u)=2\alpha(1-u)+O((1-u)^2)\quad(u\to1), \qquad a(v,u)=2\alpha(1+u)+O((1+u)^2)\quad(u\to-1).\] Thus \(a\) vanishes at both endpoints, as does \((a^2)_u\). The equations give the required one-sided time derivatives at the finitely many level boundaries.

We next prove two monotonicities. First, every \(H_i\) is convex in \(u\). Indeed \(W=H_{i,uu}\) satisfies \[W_v+\frac12(a^2W)_{uu}=0, \qquad W(h_i,u)=2.\] Expanded as a linear equation, its coefficients are bounded; the diffusion and first-derivative coefficients vanish at the endpoints. Backward comparison preserves nonnegativity. The same comparison in (119) shows that increasing a nonnegative coefficient \(a\) pointwise increases \(H_i\), because its second spatial derivative is nonnegative. These comparison assertions also follow directly by reversing time, multiplying a proposed difference by a sufficiently decaying exponential, and excluding a first negative minimum; the bounded zeroth-order terms are absorbed by the exponential. This argument applies interval by interval in time, including the degenerate endpoints.

At \(\alpha=1\), fix \(l>i\). The covariance derivative calculation for the backward field value, at any \(v\le h_i\), reads \[\partial_{h_l}F(v,z) =\text{a constant independent of }z -\frac{w_l}{2}H_l(v,F_z(v,z)).\] For a variation \(\delta F\), differentiation at fixed inverse coordinate gives \[ \delta a =\delta F_{zz}-\frac{F_{zzz}}{F_{zz}}\delta F_z =a^2\partial_{uu}\bigl[\delta F(v,z(v,u))\bigr]. \tag{121}\] Convexity of \(H_l\) therefore implies \(\partial_{h_l}a\le0\) through time \(h_i\). Applying the preceding continuation comparison at the fixed initial coordinate \(s\) proves \[ \partial_{h_l}B_i(h,s)\le0\qquad(l>i). \tag{122}\]

For the second monotonicity, \(a\) increases with \(\alpha\). To see the sign explicitly, reverse time in (118), write \(A_1,A_2\) for the solutions with \(\alpha_1<\alpha_2\), and set \(D=A_2-A_1\). Then \[D_t=\frac12A_2^2D_{uu}+cD +(\alpha_2-\alpha_1)\zeta A_1^2, \qquad c=(A_2+A_1)\left(\frac12A_{1,uu}+\alpha_2\zeta\right).\] The coefficient \(c\) is bounded, the initial difference is \((\alpha_2-\alpha_1)(1-u^2)\ge0\), and the boundary difference is zero. Comparison proves \(D\ge0\). Position and time rescaling give the identity \[F_h^\alpha(v,z) =\alpha^{-1}F_{\alpha^2h}^{1}(\alpha^2v,\alpha z),\] with the level intervals rescaled on the right. The corresponding initial biases satisfy \(b_s^\alpha(h)=\alpha^{-1}b_s^1(\alpha^2h)\). It follows that increasing \(\alpha\) increases \(B_i(\alpha^2h,s)\), whence \[ \sum_l h_l\partial_{h_l}B_i(h,s)\ge0. \tag{123}\]

Finally, the matrix \(M_{il}=w_i\partial_{h_l}B_i\) is symmetric by (116). Equations (122) and (123) show that \(\operatorname{diag}(h)M\operatorname{diag}(h)\) has nonpositive off-diagonal entries and nonnegative row sums. For a symmetric matrix with these properties its quadratic form is a sum of nonnegative weighted squares of coordinate differences and nonnegative diagonal terms. Thus \(M\) is positive semidefinite on strict level data, proving concavity. Continuity extends the supporting inequalities to the boundary. Tied levels introduce no new conditional mean and simply repeat the same \(B_i\). The bound \(0\le B_i\le1\) in (116), followed by common refinement, gives the \(L^1\) Lipschitz assertion. ◻

Proof. For a given path \(h\), insert the minimizing bias \(b_j=b_{m_j}(h)\) in every coordinate of group \(j\). These biases are bounded uniformly by (117). Consider two unnormalized \(n\)-coordinate recursions: the canonical one with all \(2^n\) spins, and the constrained one with \(\mathcal C_n\). Both use these same biases. Denote the backward values just after level \(i\) by \(U_i\) and \(V_i\), respectively, and put \(\Delta_i=U_i-V_i\ge0\). The canonical values add over coordinates, so its tilted kernels factor over sites. Write \[L_n=U_{-1}(0)-V_{-1}(0) =n\bigl(\ell^{[\mathbf m]}(h)-\ell^{(n)}(h)\bigr).\] We first prove \(L_n=o(n)\) uniformly over the permitted paths, which gives the first estimate. The same loss will bound the relative entropy of two sampled paths; independent-site concentration under the canonical law will then give the second estimate. At the root, \(L_n=\mathbb E\Delta_0\). At a positive-exponent step the canonical transition kernel, denoted by \(K_i^{\mathrm{can}}\), gives \[\Delta_{i-1} =-\frac1{\zeta_i}\log \mathbb E_{K_i^{\mathrm{can}}}e^{-\zeta_i\Delta_i} \le\mathbb E_{K_i^{\mathrm{can}}}\Delta_i.\] Iteration proves \[ 0\le L_n\le\mathbb E_{\mathrm{can}}\Delta_{k-1}. \tag{129}\]

We bound the terminal loss without a factor depending on the number of steps. If a one-coordinate canonical increment \(g\) has Gaussian variance \(v\), its conditional tilted density is proportional to \(\exp\{\zeta F(z+g)\}\), where \(F\) is \(1\)-Lipschitz and \(0\le\zeta\le1\). Gaussian shifting gives, for every \(t\in\mathbb R\), \[ \mathbb E_{\mathrm{tilt}}e^{tg} =e^{t^2v/2} \frac{\mathbb Ee^{\zeta F(z+g+tv)}}{\mathbb Ee^{\zeta F(z+g)}} \le e^{t^2v/2+|t|v}. \tag{130}\] For the root increment the sharper ordinary Gaussian bound holds. Successive conditioning in (130) bounds the exponential moments of each terminal field using only total variance \(H\) and the bounded initial bias. Thus, for the vector \(z\) of terminal fields including their biases, \[ \mathbb E_{\mathrm{can}}\|z\|^2\le C_H n, \tag{131}\] uniformly in the depth and the step weights.

Under the averaged canonical path kernels and final spin draws, the sites are independent and their means are \(m_j\) in group \(j\); this is exactly the bias equation \(f_h'(b_j)=m_j\). Map each spin vector \(\varepsilon\) deterministically to a nearest vector \(\widehat\varepsilon\in\mathcal C_n\). If \(D\) is the number of flipped coordinates, the binomial variance estimate in each group gives \(\mathbb E_{\mathrm{can}}D\le C\sqrt n\). At fixed fields, use the pushforward of the canonical Gibbs spin law as a trial law for the constrained log partition. The finite-space log-sum variational identity then bounds the terminal loss by the expected exponent loss plus the entropy loss. Averaging the former and using (131), \[\mathbb E\bigl|z\cdot(\varepsilon-\widehat\varepsilon)\bigr| \le(\mathbb E\|z\|^2)^{1/2}(4\mathbb ED)^{1/2} \le C_H n^{3/4}.\] The original spin vector is determined by its image and its flip mask. Therefore its entropy loss is at most the entropy of that mask, and subadditivity and concavity of binary entropy give the averaged bound \(n\mathsf h(\mathbb ED/n)\le C\sqrt n\log(n+1)\). Together with (129), this proves \[ 0\le L_n\le C_H\bigl(n^{3/4}+\sqrt n\log(n+1)\bigr)=o(n). \tag{132}\] This establishes the first assertion of (128).

To compare the two-spin products, compare the path-mark and terminal-spin laws under the constrained and canonical kernels. The logarithm of the ratio of their level-\(i\) densities is \(\zeta_i(\Delta_{i-1}-\Delta_i)\) for \(i>0\). The root mark law is unchanged. On allowed terminal spins, the final log-density ratio is \(\Delta_{k-1}\). Hence the chain rule for conditional relative entropies, with weight \(1/\zeta_i\) at level \(i>0\) and weight one for the final spin draw, telescopes to \(L_n\): \[ \sum_{i=1}^{k-1}\frac1{\zeta_i} \mathbb E_{\mathrm{con}}\mathop{\mathrm{KL}}(K_i^{\mathrm{con}}\Vert K_i^{\mathrm{can}}) +\mathbb E_{\mathrm{con}}\mathop{\mathrm{KL}}(K_{\mathrm{spin}}^{\mathrm{con}} \Vert K_{\mathrm{spin}}^{\mathrm{can}}) =L_n. \tag{133}\] Here each conditional entropy is evaluated at the same past, with the outer expectation taken under the constrained past law. Every term is nonnegative. Since \(\zeta_i\le1\), the ordinary relative entropy of the full one-path laws is at most \(L_n\). This bound remains valid for arbitrarily small positive \(\zeta_i\); no lower bound on the step weights or their cumulative sums is used.

Condition now on a specified common level \(i\) of two sampled cascade leaves. The transformed cascade law and the mark representation of Proposition 5 show that each path has the same one-path marginal; its kernels are shared before the split and independent afterwards. The conditional relative entropies of the shared kernels are counted once and those after the split, including terminal spin draws, at most twice. Thus the joint two-path relative entropy is at most \(2L_n\). In the canonical law, sites are independent even conditional on this branching information: the backward values add over sites and each kernel factors over their coordinates. The average two-spin product therefore has mean \(B_i^*(h)\), and the elementary exponential-moment bound for independent variables in \([-1,1]\) yields \[\mathbb P_{\mathrm{can}}\left\{ \left|\frac1n\sum_{r=1}^n\varepsilon_r^1\varepsilon_r^2-B_i^*(h)\right| >\eta\right\} \le2e^{-n\eta^2/2}.\] The constants do not depend on the branching level or the depth. For any event \(A\) and probability laws \(\nu,\pi\), \[\mathop{\mathrm{KL}}(\nu\Vert\pi) \ge\nu(A)\log\frac1{\pi(A)}-\log2,\] as follows by mapping to the two events \(A,A^c\) and expanding binary relative entropy. Apply this with the constrained and canonical two-path laws. Equations (132) and (133) imply that every fixed positive deviation has constrained probability tending to zero uniformly in \(i\) and \(h\). The average spin product is bounded by one, so its expectation converges uniformly to \(B_i^*(h)\). Its constrained expectation is \(B_i^{(n)}(h)\), which proves the second assertion. ◻