Angular pressure and the spherical theorem

Let \(E_1,\ldots,E_h\subset\mathbb R^M\) be mutually orthogonal Haar subspaces with \(\dim E_j=n_j\) and \(n_j/M\to p_j>0\). Thus \(p:=\sum_jp_j\le1\). For \(a_j\ge0\), let \(\sigma_{E_j,Ma_j}\) be uniform probability measure on the sphere in \(E_j\) of squared radius \(Ma_j\), with the point mass at zero when \(a_j=0\). Define the random angular pressure \[ P_M^f(a)=\frac1M\log\int\exp\left\{\sum_{r=1}^M f\left(\sum_jy_{j,r}\right)\right\} \prod_jd\sigma_{E_j,Ma_j}(y_j). \tag{2}\] The subspaces are held fixed in this integral; the vectors \(y_j\) are integrated independently. The subscript \(p\) on its limit records the entire list of proportions \((p_j)_j\).

With this angular limit, the remaining optimization describes how a spherical input divides its norm among the singular-value blocks. For the pressure (1), let \(r_N=\min(M,N)\) and let \(s_{N,1},\ldots,s_{N,r_N}\) be the singular values of \(A\), including zero values in these slots. Structural right or left null coordinates outside these slots are kept separate.

All formulas in this section apply directly to bounded continuous \(f\). They may equivalently be evaluated through a final bounded Lipschitz approximation, uniform on compact intervals; the proof shows that the two prescriptions agree. Conditional deterministic singular values are allowed; the theorem does not require a density, a connected support, or a free-probability regularity condition on \(\nu\).

A Gaussian model with outer and inner columns

The auxiliary Gaussian model distinguishes the vector whose weight we seek from the vectors that witness its projection constraints. For now fix arbitrary column counts \(d_j\ge1\) and a positive exponent \(0<\theta<1\); the application will specify \(d_j\). Here we allow arbitrary positive proportions \(p_j\), without restricting their sum.

Let \(n_j/M\to p_j>0\), and let \[X_j\in\operatorname{St}(n_j,d_j):=\{X\in\mathbb R^{n_j\times d_j}:X^{\mathsf T}X=n_jI_{d_j}\}\] have uniform law. The first columns, jointly across species, are outer variables; the remaining columns have the conditional Stiefel law and are inner variables. For independent standard Gaussian vectors \(g_{r,j}\in\mathbb R^{n_j}\), define the row fields \[\xi_{r,j}(X)=\frac{X_j^{\mathsf T}g_{r,j}}{\sqrt{n_j}},\qquad 1\le r\le M.\] For a Hamiltonian \(H\) set \[ \mathcal Z_{M,\theta}(H)= \left[\int\left(\int e^H\,d\mu_{\rm in}\right)^\theta d\mu_{\rm out}\right]^{1/\theta}. \tag{5}\] The natural tilted path law first samples the outer variables with density proportional to the inner partition function to power \(\theta\), and then samples the tilted conditional inner law.

The first column is therefore drawn once before the inner columns are integrated. In the cascade representation it is shared by visits whose common level is at least \(\theta\); the other columns remain visit variables. The matrix paths below record the covariances of Gaussian row fields seen by such visits. Theorem 3 will identify the limit of the normalized log integral with their variational functional.

Paths with a shared coordinate

Keep the block proportions, column counts, and exponent from Section 2.2. The first coordinate in each block is shared at level \(\theta\).

Choose a common finite grid \[ 0=\zeta_0<\zeta_1<\cdots<\zeta_k=1, \qquad w_i=\zeta_{i+1}-\zeta_i\quad(0\le i<k), \tag{6}\] containing \(\theta\). A trial in block \(j\) consists of symmetric matrices \[0\preceq q_{j,0}\preceq\cdots\preceq q_{j,k-1}\preceq I_{d_j}, \qquad q_{j,-1}=0,\quad q_{j,k}=I_{d_j},\] with \(q_{j,i}e_1=e_1\) whenever \(\zeta_i\ge\theta\). Regard \(q_j(u)=q_{j,i}\) on \([\zeta_i,\zeta_{i+1})\). Let \(F_{j,i}=\mathbb R^{d_j}\) before the sharing level and \(F_{j,i}=e_1^\perp\) thereafter; put \(F_{j,k}=\{0\}\). Define the susceptibility and precision \[ \mathsf K_{j,i}=\sum_{r=i+1}^k\zeta_r(q_{j,r}-q_{j,r-1}), \qquad \mathsf P_{j,i}=(\mathsf K_{j,i}|_{F_{j,i}})^{-1}. \tag{7}\] An eligible trial has strictly positive susceptibility on each nonempty active space. Extend its precision by zero on the orthogonal complement. Let \(q_j^0\) be zero on active coordinates and the identity on shared coordinates, with \(q_{j,k}^0=I\), and define \(\mathsf K_{j,i}^0\) in the same way. The block entropy is \[ 2S_\theta(q_j)=\operatorname{tr}(\mathsf P_{j,0}q_{j,0})+ \sum_{i=1}^k\frac1{\zeta_i} \log\left( \frac{\det\mathsf K_{j,i-1}}{\det\mathsf K_{j,i}} \frac{\det\mathsf K_{j,i}^0}{\det\mathsf K_{j,i-1}^0} \right). \tag{8}\] Each determinant is taken on the active space at its own index. In particular the normalization compensates for the dimension change at \(\theta\).

For a continuous terminal \(D:\prod_j\mathbb R^{d_j}\to\mathbb R\) satisfying \(-C(1+|z|^2)\le D(z)\le C\) for some \(C<\infty\), let \(Z_0,\ldots,Z_k\) be independent centered Gaussian vectors with block-diagonal covariances \(\operatorname{diag}_j(q_{j,i}-q_{j,i-1})\). Set \(U_k=D\) and define backwards, for \(i=k,k-1,\ldots,1\), \[U_{i-1}(z)=\frac1{\zeta_i}\log\mathbb E\exp\{\zeta_i U_i(z+Z_i)\}.\] The single-pattern value is \(V_D(q)=\mathbb EU_0(Z_0)\); this last ordinary expectation averages the root Gaussian data. Define \[ \mathcal G_{\theta,p}(D)=\inf_q\left\{\sum_{j=1}^h p_jS_\theta(q_j)+V_D(q)\right\}, \tag{9}\] where all eligible finite grids and trials are allowed. Refinements always use a common grid for all blocks. The diagonal increment \(I-q_{j,k-1}\) is integrated independently on every visit in the cascade interpretation, including repeated visits to the same leaf.

Explicit evaluation of the angular pressure

Return to \(p=\sum_jp_j<1\) and positive radii \(a_j\). Generate the Haar flag from consecutive column blocks \(G_j\) of a matrix with independent \(N(0,1/M)\) entries, as proved in Section 8. For outer coefficients \(x_j\in\mathbb R^{n_j}\) and inner coefficients \(z_{i,j}\in\mathbb R^{n_j}\), \(j\le i<h\), put \[y=\sum_jG_jx_j,\qquad u_i=\sum_{j\le i}G_jz_{i,j}.\] Thus \(y\) is the physical field and \(u_i\) is a witness in the flag subspace \(E_1\oplus\cdots\oplus E_i\). Requiring \(\lVert y\rVert^2/M=\sum_ja_j\) and \[\lVert y-u_i\rVert^2/M\le\sum_{j>i}a_j\] forces the squared norm in each tail of the orthogonal decomposition of \(y\) to be no larger than the prescribed tail. Equivalently, the initial blocks carry at least their prescribed total mass. After sorting \(a_j/p_j\), Lemma 6 shows that these one-sided tests recover the desired weighted shell. The witnesses are integrated as the inner variables of (5); sending \(\theta\) to zero removes their entropy cost. Section 8 proves the construction and these limits.

Permute the pairs \((p_j,a_j)\) so that \(a_j/p_j\) is nonincreasing; ties may be ordered arbitrarily. Set \[ s_a=\sum_j a_j,\qquad H_p(a)=\frac12\sum_jp_j\log\frac{2\pi e a_j}{p_j},\qquad J_p=\frac12\int_0^p\log(1-u)\,du. \tag{10}\] Here \(H_p\) is the limiting entropy per output coordinate of the product of Lebesgue shells, and \(J_p\) is the limiting logarithmic volume Jacobian of the Gaussian span map. Orthogonalizing the coefficient columns \(x_j,z_{j,j},\ldots,z_{h-1,j}\) after division by \(\sqrt M\) gives \(d_j=h-j+1\) columns in block \(j\). Choose a coefficient radius \(\kappa>0\), penalty strength \(\lambda>0\), and truncation threshold \(B>0\). In each block choose an upper triangular \(d_j\times d_j\) matrix \(R_j\) with positive diagonal and every column norm at most \(\kappa\). For one row \(\xi_j\in\mathbb R^{d_j}\), label the entries of \(\xi_j^{\mathsf T}R_j\) by \(x,j\) and \(z_i,j\), \(j\le i<h\). Write \[y=\sum_j (\xi_j^{\mathsf T}R_j)_{x,j}, \qquad u_i=\sum_{j\le i}(\xi_j^{\mathsf T}R_j)_{z_i,j}.\] The symbols \(y,u_i\) in these row formulas denote single coordinates of the fields above. Let \(g\) be the vector of row functions and \(c\) the corresponding constants in \[ (y^2,s_a),\qquad(-\min\{y^2,B\},-s_a),\qquad \left((y-u_i)^2,\sum_{j>i}a_j\right),\quad 1\le i<h. \tag{11}\] The soft penalties below enforce, in the limit, the row-average inequalities \[M^{-1}\sum_r g_\ell(y_r,u_{1,r},\ldots,u_{h-1,r})\le c_\ell.\] Define \[\begin{align*} \mathcal B_{p,a}(\kappa,\lambda,B,\theta) =\sup_{R_1,\ldots,R_h}\Bigg[& \sum_jp_j\left\{ \log\!\left(\kappa\sqrt{\frac{2\pi e}{p_j}}\right) +\log\frac{R_{j,11}}\kappa +\theta\sum_{\ell=2}^{d_j}\log\frac{R_{j,\ell\ell}}\kappa \right\}\\ &+\theta\min_{0\le t_\ell\le\lambda/\theta} \left\{t\cdot c+\mathcal G_{\theta,p}\big(f(y)/\theta-t\cdot g\big)\right\}\Bigg]. \tag{12}\end{align*}\] The functions \(y,u_i,g\) in the second line depend on the chosen triangular matrices. The terminal is bounded above and has at most quadratic negative growth. The explicit angular formula is \[ P_p^f(a)=J_p-H_p(a)+ \lim_{\lambda\to\infty}\lim_{B\to\infty}\lim_{\theta\downarrow0} \mathcal B_{p,a}(\kappa,\lambda,B,\theta). \tag{13}\] Theorem 1 asserts that these successive limits exist and give the same angular pressure for every sufficiently large fixed \(\kappa\). Values at zero coordinates of \(a\) are obtained by continuity. When \(\sum_jp_j=1\), the evaluation is \[ P_p^f(a)=\lim_{\eta\downarrow0}P_{(1-\eta)p}^f(a), \tag{14}\] with \(a\) fixed. No analytic continuation in \(\theta\) is used: \(\theta\) is a positive exponent throughout.

Marked cascades, zero restrictions, and visit variables

We use Ruelle’s probability cascades [27] and the marked-cascade framework of Panchenko and Talagrand [26]. Their sampling and cavity structure were developed by Bolthausen and Sznitman [4]. The direct calculation below permits nonnegative marks with a positive finite fractional moment; its separate logarithmic estimates cover the restrictions used later. A finite cascade on (6) has Poisson branching exponents \(\zeta_1,\ldots,\zeta_{k-1}\), all strictly below one. At a vertex with exponent \(z\), take independent Poisson points of intensity \(z x^{-1-z}\,dx\). Products along paths are normalized at the leaves. Common-level probabilities for two sampled leaves are \(w_i\), \(0\le i<k\). The step at \(\zeta_k=1\) is an ordinary integral over a fresh visit variable, never a Poisson branching level. Root Gaussian data have coefficient zero and are averaged at the end.

We use the following precise change-of-measure rule. If a positive edge weight \(x\) is multiplied by a nonnegative mark \(F\) satisfying \[ 0<\mathbb E(F^z\mid\text{parent data})<\infty, \tag{17}\] the image of its positive points is again Poisson, with intensity multiplied by that conditional moment, and its marks have density proportional to \(F^z\). This follows by changing variables in the marked Poisson intensity. Points for which \(F=0\) are deleted. Applying this rule backwards with \(F=\exp(U_i-U_{i-1})\) proves both the log-exponential recursion and the tilted child kernel \[ \exp\{\zeta_i(U_i-U_{i-1})\}\,d\mu_i. \tag{18}\] Conditional on root data, the transformed unmarked cascade has its original law and is independent of the path marks with these kernels. Restrictions by an indicator are therefore permitted whenever surviving fractional moments are positive and finite. A subtree with zero normalizer is deleted before the next backward step; the condition is checked at surviving parents, and the final root normalizer must be positive. Restrictions require separate logarithmic integrability estimates; these will be supplied where they are used.

The independence assertion follows through the same backward induction. At each child, include its already transformed unmarked descendant tree in the mark. By induction that tree has its ordinary law independently of the path marks. The parent multiplier depends only on the child and ancestor marks, so its Poisson image intensity factors into the ordinary edge intensity, the tilted child kernel, and the unchanged descendant-tree law. For the normalized multiplier \(F=\exp(U_i-U_{i-1})\), its conditional \(\zeta_i\)-moment is one, including after deletion of zero marks. Thus no ancestor-dependent scale remains, and the transformed unmarked tree has its ordinary law independently of the original root data.

For completeness, the Poisson Laplace formula says that a sum of \(z\)-Poisson points multiplied by independent positive factors with finite \(z\)th moment is a scaled positive stable variable of index \(z\). Such a variable has finite logarithmic second moment and finite positive moments of every order below \(z\). These facts follow by integrating its Laplace transform \(e^{-ct^z}\); its lower tail is bounded using \(\mathbb P(S\le u)\le e^{tu}\mathbb Ee^{-tS}\) and optimizing \(t\). Successive exponents are strictly increasing, so these fractional moments justify all backwards normalizations. For these recursively normalized multipliers, the original and transformed leaf totals have the same law conditional on the root data. Their logarithms are integrable by the preceding stable-moment estimates, so the conditional mean of their difference is zero. Telescoping the multipliers gives the expected logarithm identity.

If the original marks have independent components and their terminal energy is additive, the backward values and tilted kernels factor over those components. The preceding induction shows that conditioning on a finite sampled shape of the transformed unmarked tree preserves this factorization. Averaging the original independent root data therefore leaves the canonical spatial rows independent, with their common averaged row law. This statement concerns the transformed sampled shape; conditioning on the original numerical labels or on realized root fields would be a different assertion.

Merged masses at a branching exponent \(z\) have the law of normalized stable jumps at that exponent. Indeed, after summing descendants, mark the points at that level by the descendant totals; the Poisson mapping and superposition give the assertion. We will use this at the fixed exponent \(\theta\), independently of how finely the other levels are subdivided.

Sampling first-column marks at exponent \(\theta\) and conditional inner variables at each visit implements (5). For the cascade consisting only of the split at \(\theta\), the expected cascade logarithm, conditional on the disorder, is exactly \[\frac1\theta\log\int\left(\int e^H d\mu_{\rm in}\right)^\theta d\mu_{\rm out}.\] Adding redundant cascade levels leaves this representation unchanged when the Hamiltonian has no additional label dependence.

Joint overlap identities and ordered matrix paths

Matrix synchronization is a central feature of vector-spin theory [24]. We follow the joint scalarization and label-overlap formulation of [18], and establish the version needed for the present shared-coordinate construction. For replicas \(a,b\), write \[\mathcal R_j^{ab}=(X_j^a)^{\mathsf T}X_j^b/n_j.\] The array indexed jointly by replica and column is Gram, and the self blocks are fixed. Include also the common-level label overlap. Gaussian perturbations below impose the Ghirlanda–Guerra identities (GG) [8]: for every bounded test \(F\) of the complete arrays on the first \(r\) replicas and every continuous test \(\psi\) of the generated scalarization-and-label vector on one edge, \[ \mathbb E\langle F\psi(T^{1,r+1})\rangle =\frac1r\mathbb E\langle F\rangle\,\mathbb E\langle\psi(T^{12})\rangle +\frac1r\sum_{a=2}^r\mathbb E\langle F\psi(T^{1a})\rangle. \tag{19}\] Here brackets denote Gibbs expectation and \(\mathbb E\) averages all randomness. Initially \(T^{ab}\) denotes the scalarization-and-label vector specified next; after symmetry is proved it determines the full matrix-and-label edge value.

The perturbing scalarizations are \[ v^{\mathsf T}(\mathcal R_j^{ab})^{\circ \ell}v, \qquad \ell=1,2, \tag{20}\] for a countable dense set of vectors \(v\), together with the numerical label overlap. Each is a Gram kernel: use tensor products of the column vectors for the entrywise powers. Products, nonnegative integral powers, and positive sums remain Gram kernels. After individual rescaling their absolute values are at most one and their diagonals are constant. Monomial GG identities give all continuous tests of this generated vector by polynomial approximation on its compact range. At this stage they do not yet identify a nonsymmetric matrix from its scalarizations; the test \(F\) may nevertheless depend on the complete finite matrix arrays.

We recall why the resulting paths have the structure used in (9). A scalar symmetric weakly exchangeable positive semidefinite array with deterministic bounded diagonal and full off-diagonal GG is ultrametric. To check the exact applicability of the established result, normalize a positive diagonal to one (a zero diagonal gives the zero array). The Dovbysh–Sudakov representation [22] writes it as \[T^{ab}=h^a\cdot h^b+a^a\mathbf 1_{\{a=b\}},\qquad a^a\ge0,\] with conditionally iid pairs and \(\lVert h^a\rVert\le1\). The sphere-support theorem [21], whose hypotheses use only off-diagonal GG tests, puts the directing Hilbert-space measure on the sphere of squared radius \(q_*=\sup\operatorname{supp}\mathcal L(T^{12})\). The represented inner-product array therefore has a deterministic diagonal. Its full-overlap GG identities follow from the off-diagonal identities, and [23] applies.

The identities for continuous edge tests extend to bounded Borel tests, since they identify finite Borel measures on the compact edge space. We also need a simple all-pairs extension consequence of GG. On an event requiring every old edge to lie in a fixed marginal bin of mass \(b>0\), GG gives conditional probability \((1-b)/r\) that a new edge from each fixed old vertex misses the bin. A union bound leaves extension probability at least \(b\). Hence positive-mass bins support arbitrarily large all-pairs samples. A bin uniformly below zero is impossible for a Gram array, by testing its large all-ones quadratic form. Thus scalar overlaps are nonnegative. These facts apply to every scalarization, every positive sum of scalarizations, and the label overlap.

For matrix symmetry, we use Panchenko’s argument from the proof of [25], in the scalarized form recalled in [18]. The first- and second-power scalarizations determine each transposed pair of entries up to interchange. If an antisymmetric part persisted with positive probability, choose a small positive-mass joint bin fixing these unordered pairs while keeping a fixed asymmetry gap. The all-pairs extension just used produces arbitrarily large samples in this bin. Finite color thinning gives arbitrarily many ordered vertices whose forward cross matrices choose the same ordering of every pair, to the prescribed tolerance. This elementary thinning follows by successively keeping a color class of the remaining vertices and then taking a common outgoing color. For a homogeneous set of \(2m\) replicas, average each normalized column over its first and second halves, writing these averages as \(a_c,b_c\) for column \(c\). If the bin width is \(\varepsilon\) and \(t_c\) is its same-column overlap center, the fixed self overlaps and the cross overlaps give \[\|a_c-b_c\|^2=\frac{2(1-t_c)}m+O(\varepsilon).\] The averages have norm at most one. Thus \(|a_c\cdot b_d-b_c\cdot a_d|=O(\sqrt\varepsilon)+O(m^{-1/2})\), whereas the homogeneous forward matrices make this difference approach their fixed antisymmetric entry gap. Taking arbitrarily large \(m\) and then arbitrarily small bins contradicts that gap. Thus \(\mathcal R_j^{12}\) is symmetric in the limit. Dense first-power quadratic forms now determine each matrix, so the identities extend to all joint matrix-and-label edge tests.

Finally, the first-power scalarizations and label are comonotone on an edge, by the ultrametric-sum argument in [16]. Otherwise joint GG gives crossed values on edges \(12\) and \(13\). Ultrametricity of each of two crossed coordinates forces edge \(23\) to have the lower value in each coordinate, contradicting ultrametricity of their sum. A positive weighted sum of a countable separating set orders all coordinates simultaneously. Dense quadratic forms therefore give deterministic matrix quantiles \(q_j(u)\) nondecreasing in Loewner order. Gram positivity and self overlap \(I\) imply \(0\preceq q_j\preceq I\). The label coordinate retains its prescribed atom masses, so its sharing threshold lies at quantile \(\theta\). When the first column is shared, \(\mathcal R_j^{ab}e_1=e_1\); hence the matrix quantiles satisfy the pinning condition at that same threshold.

Every finite monotone rounding of this joint order has the cascade law. At any threshold, ultrametricity makes the above-threshold relation an equivalence relation; GG specifies the probability that the next sample joins each existing class. These nested joining probabilities determine the rounded array. To verify that a cascade has them, at a parent of exponent \(z_0\) its child masses of exponent \(z\) are stable jumps tilted by their total to power \(z_0\). The Gamma integral and the Poisson factorial formula show that an existing child with \(n_a\) of \(n\) visits is selected next with probability \((n_a-z)/(n-z_0)\). Multiplying conditional probabilities up the tree gives the GG joining rule. Thus cascade approximations preserve label atoms and matrix pinning.

Fresh fields and the one-pattern variation

Conditionally on finitely many base replicas, independent fresh Gaussian fields have exactly the covariance matrices given by their overlaps. Finite-array convergence therefore passes bounded continuous field tests. Bounded weights bounded away from zero also permit logs and reciprocal partition functions: approximate these uniformly by polynomials on a compact interval, and represent powers by additional replicas. This proves passage of bounded reweightings to finite cascade approximations. Independent residual visit noises are essential when self variance exceeds the largest shared covariance.

For smooth \(D\) the single-pattern functional is Lipschitz in the sum of the \(L^1\) path distances. More precisely, along a step variation, \[ \frac{d}{dt}V_D(q+t\dot q)\Big|_{t=0} =-\frac12\sum_{j,i}w_i\operatorname{tr}\left(\dot q_{j,i}\mathbb E[M_{j,i}M_{j,i}^{\mathsf T}]\right), \tag{21}\] where \(M_{j,i}\) is the conditional mean of the terminal \(j\)-gradient under the successive tilted kernels. Gaussian covariance differentiation inserts half the Hessian and half the step coefficient times the gradient square. Differentiating all increments and propagating Hessians makes these terms telescope; consecutive coefficients differ by \(w_i\), giving (21). The conditional gradients form a martingale, so their second-moment matrices \(G_{j,i}=\mathbb E[M_{j,i}M_{j,i}^{\mathsf T}]\) are nondecreasing and uniformly bounded. To see the resulting order explicitly, put \(T_{j,i}=\sum_{r=i}^{k-1}w_r(\widetilde q_{j,r}-q_{j,r})\) and \(T_{j,k}=0\) on a common grid. Summation by parts gives \[\sum_iw_i\operatorname{tr}((\widetilde q_{j,i}-q_{j,i})G_{j,i}) =\operatorname{tr}(T_{j,0}G_{j,0})+ \sum_{i=1}^{k-1}\operatorname{tr}(T_{j,i}(G_{j,i}-G_{j,i-1})).\] Every term is nonnegative when the tail differences are positive semidefinite. Apply this along the segment between the two paths in (21) to obtain \[ \int_t^1\widetilde q_j(u)\,du\succeq\int_t^1q_j(u)\,du\quad\hbox{for every }j,t \quad\Longrightarrow\quad V_D(\widetilde q)\le V_D(q). \tag{22}\] Approximation by step averages extends these statements to bounded monotone paths. By the fresh-field argument this extended \(V_D\) is the limiting expected log increment of one independent pattern at a GG limit.

A canonical multiplier

Replace each spatial spin row by a standard Gaussian vector \(s\in\mathbb R^d\), with the same sharing rule. Write \(m\) for the index satisfying \(\zeta_m=\theta\). Take independent field increments \[z_0\sim N(0,b_0),\qquad z_i\sim N(0,b_i-b_{i-1})\quad(1\le i<k).\] The first spin coordinate \(s_1\sim N(0,1)\) is sampled at level \(m\); the remaining coordinates \(s_F\sim N(0,I_{d-1})\) are sampled at the terminal visit. All these reference variables are independent.

Here is the one-row recursion defining the canonical integral. For a symmetric multiplier \(C\), start with \[U_k(x,s)=x^{\mathsf T}s+\tfrac12s^{\mathsf T}Cs, \qquad U_{k-1}(x,s_1)=\log\mathbb E_{s_F}e^{U_k(x,s)}.\] For \(i=k-1,\ldots,1\), replace \(U_i\) by \[U_{i-1}(x)=\frac1{\zeta_i}\log \mathbb E_{z_i,\,s_1\text{ if }i=m} \exp\{\zeta_i U_i(x+z_i)\}.\] The argument \(s_1\) is retained until the step \(i=m\) and integrated at that step; its integration and the \(z_m\) integration have the same exponent and may be performed in either order. Finally put \(\Psi(C,b)=\mathbb E_{z_0}U_0(z_0)\). This extended-real function is finite exactly when its successive Gaussian precisions are strictly positive. A zero precision is insufficient: the corresponding Gaussian exponential integral is infinite.

From canonical rows to Stiefel columns

Use \(n\) independent canonical spatial rows. Their unrestricted expected log integral is \(n\Psi(C,b)\), since the recursion factors over rows. Decompose the Gaussian spin columns by Gram–Schmidt, drawing the first column at its prescribed vertex and the remaining columns at visits. Restrict all normalized triangular lengths and cross readings to a fixed small neighborhood of the identity. The first restriction and each conditional inner restriction have probability tending to one, independently of the preceding frame; their recursion normalization costs are \(o(n)\).

On the restriction the directions have the exact Stiefel sharing law. Replacing the quadratic energy by \(n\operatorname{tr}C/2\) costs \(o_\epsilon(n)\). Replacing the linear-field covariance by that of exact Stiefel columns also costs \(o_\epsilon(n)\): its normalized covariance discrepancy is uniformly small, and Gaussian interpolation bounds the expected log change by half the diagonal-minus-pair covariance difference. Let \(n\to\infty\) and then shrink the neighborhood. Restriction gives a lower bound on the canonical integral, so \[\limsup_n l_n(b)\le\Psi(C,b)-\tfrac12\operatorname{tr}(C+b_{k-1})=J(b).\] Lemma 4 proves (23). The same covariance comparison and compact multiplier bound give local uniformity in \(b\). Gaussian differentiation for random reference measures can be justified throughout by finite conditional iid discretizations, followed by truncation: at fixed system size bounded covariances give finite moments of exponential integrals and of Jensen lower bounds for their inverses.

GG at the deterministic contact point

The quadratic penalty in the perturbation parameters implements the contact-point selection used in [16] and [18]. It yields GG at the parameters selected by the comparison, rather than only after averaging over those parameters. The minimizing parameters are deterministic, being selected from an expected pressure. Comparing with all \(u_a=3/2\) shows that their penalty is \(O(e_M^2)\). Every fixed \(u_a\) is consequently interior for large \(M\). Hold all other minimizing coordinates fixed. Convexity in \(u_a\) and minimality give the explicit local remainder bound \[ 0\le\mathbb EP_M(u+s e_a)-\mathbb EP_M(u)-s\partial_{u_a}\mathbb EP_M(u) \le c_a s^2. \tag{34}\] This estimate precedes and does not assume GG.

Differentiation gives \[\partial_{u_a}P_M=e_Mc_a\langle Y_a\rangle/\sqrt M, \qquad \partial_{u_a}^2\mathbb EP_M=e_M^2c_a^2\mathbb E\langle(Y_a-\langle Y_a\rangle)^2\rangle.\] Use convex difference quotients at \(s=M^{-1/4}\), the variance bound (32), and (34). The expected absolute disorder fluctuation of the first derivative is \(O_a(s+1/(s\sqrt M))\). Dividing (34) by \(s^2\) and letting \(s\to0\) also gives \[\mathbb E\langle(Y_a-\langle Y_a\rangle)^2\rangle \le\frac{2}{e_M^2c_a}.\] Consequently the disorder and thermal contributions, respectively, obey \[\frac{\mathbb E\langle|Y_a-\mathbb E\langle Y_a\rangle|\rangle} {\sqrt M e_M} \le C_a\left(\frac{M^{-1/4}}{e_M^2} +\frac1{\sqrt M e_M^2}\right) =O_a(M^{-1/8}+M^{-3/8}).\] In particular, \[ \mathbb E\langle|Y_a-\mathbb E\langle Y_a\rangle|\rangle =o(\sqrt M e_M) \tag{35}\] for every fixed \(a\). Gaussian integration by parts against a bounded test \(F\) on \(r\) replicas yields \[\mathbb E\langle Y_a(1)F\rangle =\sqrt M e_Mc_au_a\,\mathbb E\left\langle F\left(\sum_{b=1}^r\kappa_a(1,b)-r\kappa_a(1,r+1)\right)\right\rangle.\] Replacing \(Y_a\) by its expected mean using (35), and subtracting the one-replica identity, proves GG in every subsequential array limit. Compactness supplies such limits, and Section 3 describes them by jointly ordered paths \(\widetilde q_j\) and the fixed label path.

The contradiction from derivatives

Increase a field increment at index \(i\) by a positive matrix supported on \(F_{j,i}\). The derivative of \(-\mathbb EP_M\) is \(n_j/(2M)\) times the Gibbs mean overlap contracted with that cumulative field increment. At the minimum, the limiting right derivative of [eq:contact] therefore gives \[ \left.\int_{\zeta_i}^1(\widetilde q_j(u)-q_j(u))\,du\right|_{F_{j,i}} \succeq0. \tag{36}\] Outside the active space the pinned-coordinate differences vanish. The tail difference is zero at \(t=1\). On each grid interval, the trial is constant and \(\widetilde q\) is monotone; testing quadratic forms shows the tail inequality between endpoints as well (the tail difference is concave there). Thus (22) implies \(V_D(\widetilde q)\le V_D(q)\).

Poisson differentiation in \(r\) gives minus the expected log integral of one fresh pattern. Its subsequential limit is \(-V_D(\widetilde q)\) by fresh-field convergence. Consequently the limiting \(r\)-derivative of [eq:contact] is at least \(\delta>0\). A minimizing point with \(r>0\) permits a left variation and must have nonpositive left derivative. This contradiction proves \[ \limsup_M\frac1M\mathbb E\log\mathcal Z_{M,\theta}\left(\sum_rD(\xi_r)\right) \le\mathcal G_{\theta,p}(D). \tag{37}\]

Generic parameters and two-pattern identification

There are sets of perturbation parameters of \(\pi\)-measure tending to one on which every subsequential overlap limit of the base system with two patterns omitted satisfies joint GG. To see this, for each fixed perturbation coordinate the pressure variance is \(O(M^{-1})\) as before. Integration by parts bounds its expected first derivative by \(O_a(s_M^2/M)\). Integrating the second derivative controls thermal variance. Integrated convex difference quotients with step \(s\) give the bound \[C_a s\,s_M^2/M+C/(s\sqrt M)\] on the integrated absolute fluctuation of the first derivative. At \(s=M^{-1/8}\) this implies \[\int\mathbb E\langle|Y_a-\mathbb E\langle Y_a\rangle|\rangle/s_M\,d\pi\longrightarrow0.\] The estimates hold on a slightly larger coupling interval, so endpoint difference quotients cause no difficulty. Markov’s inequality and a diagonal selection give the stated good sets for every fixed coordinate. Integration by parts then proves GG exactly as in the upper bound. No separate GG assertion at size \(M+\ell\) is needed.

Choose an arbitrary sequence of good parameters, and pass to joint finite-array subsequential limits of the two-omitted system and the full size-\(M\) system. Write \(\xi^r(X)\) for the base row fields and \(D'_j,D''_{jj}\) for block derivatives. In the full system retain the averages \[\begin{align*} B_j(X,\widetilde X)&= \frac1{n_j}\sum_{r\le M}D'_j(\xi^r(X))D'_j(\xi^r(\widetilde X))^{\mathsf T},\\ c_j(X)&=\frac1{n_j}\sum_{r\le M} \left\{\operatorname{tr}D''_{jj}(\xi^r(X))-(\xi_j^r(X))^{\mathsf T}D'_j(\xi^r(X))\right\}. \tag{38}\end{align*}\] Also retain separately the entries of the one-path Hessian and \(\xi_j^r(D'_j)^{\mathsf T}\) averages. They are bounded because \(D\) is smooth and compactly supported.

Splitting a Stiefel frame

Write both the base and enlarged Stiefel frames with unit-length columns. The enlarged frame admits the decomposition \[ \begin{pmatrix} X_jT_j/\sqrt{n_j}\\ Z_j/\sqrt{n_j+r_j} \end{pmatrix},\qquad T_j^{\mathsf T}T_j=I-\frac{Z_j^{\mathsf T}Z_j}{n_j+r_j}, \tag{40}\] where \(X_j^{\mathsf T}X_j=n_jI\), the bottom block has \(r_j\) rows, and \(T_j\) is upper triangular with positive diagonal. The base frame \(X_j\) has its uniform law and is independent of \(Z_j\). The first column is the outer variable in (5), so its coupling must be fixed before drawing the conditional inner columns. The decomposition has exactly this order: the first column of \(Z_j\) is drawn at the outer vertex independently of the base outer column; its remaining columns are drawn at visits conditionally on that first column and independently of the drawn base frame. These facts follow by splitting the first column and then using rotational invariance and triangular orthogonalization for the conditional remaining columns.

For fixed \(r_j\), the entries of \(Z_j\) converge to independent standard Gaussians. The conditional inner statement holds for every converging bounded outer column: the projection of its orthocomplement onto the last \(r_j\) rows has Gram matrix tending to identity. Representing a uniform frame by orthonormalized standard Gaussian columns proves the assertion and its conditional version.

Fix a small \(\epsilon>0\) and restrict the larger partition function to the thick shell \[ \mathcal I=\left\{\lVert Z_j^{\mathsf T}Z_j/r_j-I\rVert\le\epsilon \text{ for every }j\right\}. \tag{41}\] This only lowers its logarithm. On the restriction replace the larger perturbation by the base perturbation at the same \(u\). For fixed \(\ell\), overlaps differ by \(O_\ell(M^{-1})\). Summable kernel Lipschitz constants, the amplitude \(s_M\), and the exponentially small coefficient tail imply an \(O_\ell(M^{-1/4})+o(1)\) covariance error in unnormalized logarithms. Gaussian interpolation gives a vanishing cost both for expected logs and for bounded expected replica tests. In the latter case differentiation inserts a finite linear combination of covariance differences. The estimates are uniform in \(u\).

There is a second, physical prediction for the new-coordinate overlaps. Under the original larger Gibbs law, \[ (Z_j^1)^{\mathsf T}Z_j^2/r_j\simeq\mathcal R_{j,\mathrm{whole}}^{12}, \qquad (Z_j^1)^{\mathsf T}Z_j^1/r_j\simeq I, \tag{42}\] with exceptional expected probability tending to zero, first as \(M\to\infty\) and then as \(\ell\to\infty\), at each fixed tolerance, uniformly in \(u\). Indeed the disorder-averaged sampled frames are invariant under a common spatial rotation in each species: this is true of the Gaussian patterns, perturbation kernels, and reference marks. Projection of a fixed finite collection of vectors onto \(r_j\) random coordinates approximates all its normalized inner products, by polarization and the spherical squared-projection variance. This uses rotation invariance after disorder averaging, not independence of coordinates under fixed disorder.

The self part of (42) makes the discarded Gibbs mass in (41) small for large \(\ell\). Markov’s inequality transfers (42) to the restricted Gibbs law. On the restriction, whole and base overlaps differ uniformly by \(o(1)\) at fixed \(\ell\). Perturbation replacement therefore preserves the same prediction with the base overlap on the right.

Expansion and fixed-size integrability

Divide the restricted enlarged cascade integral, after the preceding perturbation replacement, by the unrestricted base cascade integral, and call this random ratio \(R_{M,\ell}\). Restriction lowers the numerator and perturbation replacement costs \(o_M(1)\) at fixed \(\ell\), so \(\Delta_{M,\ell}(u)\ge\mathbb E\log R_{M,\ell}-o_M(1)\). After division, the reference measure is the base Gibbs law augmented by fresh outer \(Z\) marks at its labels and the conditional inner \(Z\) variables from (40). It is this ratio that we now expand. In an old pattern the field in block \(j\) is \[T_j^{\mathsf T}\xi_j^r(X)+Z_j^{\mathsf T}h_j^r/\sqrt{n_j+r_j},\] where \(h_j^r\) are independent standard Gaussian vectors of length \(r_j\).

Taylor expansion gives a linear Gaussian field coupled to each row of \(Z_j\). Up to a vanishing error at fixed \(\ell\), it is \(n_j^{-1/2}\sum_{r\le M}h_j^r(D'_j(\xi^r))^{\mathsf T}\); the exact denominator \(\sqrt{n_j+r_j}\) may be replaced by \(\sqrt{n_j}\) because \(r_j\) is fixed. Its conditional cross covariance is therefore \(B_j(X,\widetilde X)\) in the limit. The deterministic second-order correction is a one-path quadratic function \(W(X,Z)\) with coefficients given by the retained one-path averages. Uniformly on the shell, \[ W(X,Z)=\frac12\sum_jr_jc_j(X)+O(\epsilon\ell). \tag{43}\] To check the normalization term, the first-order upper triangular part of \(T_j-I\) has symmetric sum \(-Z_j^{\mathsf T}Z_j/n_j\); at \(Z_j^{\mathsf T}Z_j=r_jI\) it is \(-r_jI/(2n_j)\). The mean fresh quadratic term is the Hessian contracted with \(Z_j^{\mathsf T}Z_j/(2n_j)\). The diagonal specialization of their sum is \(r_jc_j(X)/2\); the retained matrix coefficients are bounded, so moving within the shell costs \(O(\epsilon r_j)\). Summing over \(j\) gives (43). Cross-species fresh quadratic terms are centered. The \(\ell\) added patterns can use their independent base fields at this accuracy.

The expansion requires two estimates: errors in the positive integral must vanish, and logarithms of the resulting ratios must be uniformly integrable. These are separate issues because the restricted insertion may approach zero; its reciprocal also enters normalized Gibbs tests. For fixed \(\ell\), the expected supremum of the dilation and third-order remainders on the shell tends to zero. Compact support controls powers of the base fields occurring in near-identity dilations, while bounded third derivatives and Gaussian moments give an \(O_\ell(M^{-3/2})\) error per old pattern for the fresh shifts. Centered fresh quadratic terms have conditional second moments \(O_\ell(M^{-1})\) at fixed spins and bounded \(Z\). Their exponential absolute-value moments at every fixed power are uniformly bounded for large \(M\), since their supremum is bounded by \[C_\ell\left(1+M^{-1}\sum_{r,j}|h_j^r|^2\right).\] The linear insertion has uniformly bounded conditional exponential moments on the shell. Integration over the independent reference variables thus makes the absolute-integral error from dropping centered terms tend to zero in probability.

Small denominators require a further estimate. At the fixed sharing exponent \(\theta\), merged base Gibbs cluster masses, in transformed order, are normalized \(\theta\)-stable jumps independent of the fresh outer marks. Retain outer \(Z\) norm bands strictly inside the shell. At fixed sufficiently large \(\ell\) their probabilities are positive; conditional inner shell probabilities are uniformly positive on these bands, by the Gaussian frame limit. Hence the reference shell mass dominates a positive constant times an independent thinning of those stable masses. Its negative logarithm has bounded second moment: if \(q>0\) is the retention probability, the retained mass is \[ W_q=\frac{S_q}{S_q+S_{1-q}}, \tag{44}\] with independent positive \(\theta\)-stable totals, and \(S_q\stackrel d=q^{1/\theta}S_1\). Stable logarithmic second moments give the bound. Jensen on the normalized shell handles the Gaussian linear insertion; its conditional variance is bounded for fixed \(\ell\). Other terms are bounded there. These estimates control negative log tails, while the preceding exponential moments control positive tails. Therefore the ratios are tight away from zero in probability and their logs are uniformly integrable.

One can now first truncate exponentials and denominators and then use the finite-array polynomial approximation of Section 3. Shared \(Z\) marks are included through the exact label split; their conditional frame limits hold, and the shell boundary is Gaussian-null. Removing the truncations proves convergence of both logs and bounded Gibbs tests through the expansion. The same estimates give, for each fixed \(\ell\), a uniform lower bound on \(\Delta_{M,\ell}(u)\) even at parameters outside the good set.

Matching physical and canonical overlaps

Lemma 5 and the preceding passage express a lower bound on the limiting increment through arbitrarily fine finite cascade roundings of the joint paths \(q,b\). In such a rounding, draw standard Gaussian \(Z\) rows with the sharing rule, couple them linearly to fields of shared covariances \(b_{j,i}\), retain the shell, add (43) with its deterministic limiting coefficients, and add \(\ell\) independent fresh-pattern factors with covariances \(q\). The linear field also has an independent visit residual with covariance \(B_{j,\mathrm d}-b_{j,k-1}\). Integrating it adds \[\frac12\sum_j\operatorname{tr}\big((B_{j,\mathrm d}-b_{j,k-1})Z_j^{\mathsf T}Z_j\big)\] to the correction. The fresh patterns likewise use their true diagonal covariance. These rounded \(q\) paths describe overlap arrays and need not yet have strict active susceptibility.

We now compare two measures on this same rounded cascade. The first is the restricted cavity measure just obtained from \(R_{M,\ell}\). The second is unrestricted and canonical: keep the new-pattern factors and the Gaussian \(Z\) reference marks, but replace the correction by \(\frac12\sum_j\operatorname{tr}(C_jZ_j^{\mathsf T}Z_j)\), where \(C_j=C(b_j)\) is the multiplier from Lemma 4; do not include a linear visit residual in the canonical comparison. On the shell the difference of log weights is \[ \frac12\sum_jr_j\left\{ c_j+\operatorname{tr}(B_{j,\mathrm d}-b_{j,k-1}-C_j)\right\}+O(\epsilon\ell). \tag{45}\] All constants here are uniform in the rounding grid, because the multipliers and bulk coefficients are uniformly bounded. The constant term in (45) changes the log partition function but cancels when normalizing either measure on the shell. Only the \(O(\epsilon\ell)\) variation affects the comparison of their Gibbs probabilities.

Conditional on the transformed unmarked sampled shape, the canonical spatial rows are iid jointly Gaussian across replicas. Their self covariance is \(I\) and their cross covariance at common level \(i\) is \(q_{j,i}^*\) from Lemma 4. The independent root rows are averaged in this assertion. Additive row energies and new-pattern energies factor the tilted kernels, so new patterns do not bias these row laws. This assertion is used only in the unrestricted canonical measure; the joint shell restriction may couple rows.

Gaussian concentration therefore gives, uniformly over the grids, exponentially small expected canonical Gibbs probability of a shell failure at fixed \(\epsilon\), and of \[ \lVert (Z_j^1)^{\mathsf T}Z_j^2/r_j-q^*_{j,\mathrm{level}}\rVert>\eta \tag{46}\] at fixed \(\eta>0\). Denote these bounds by \(Ce^{-c_\epsilon\ell}\) and \(Ce^{-c_\eta\ell}\). If the canonical shell mass is at least \(1/2\), (45) bounds the actual restricted pair law by \[ 4e^{4C\epsilon\ell} \tag{47}\] times the unrestricted canonical pair law. The probability that this shell mass is below \(1/2\) is at most \(2Ce^{-c_\epsilon\ell}\). Choose \(\epsilon<c_\eta/(8C)\) and then \(\ell\) large. The canonical prediction (46) therefore transfers to the restricted cavity measure with arbitrarily small exceptional probability. This uses row independence only under the unrestricted canonical law; the shell itself can couple the rows.

The restricted cavity measure also retains the physical prediction (42), because the expansion passed bounded Gibbs tests as well as logarithms. It predicts the rounded overlap \(q_{j,i}\) at the sampled common level. After averaging the cascade weights and marks, marked Poisson reweighting, including the shell restriction, keeps the sampled common-level probabilities exactly \(w_i\). Since all matrix overlaps have bounded norm, agreement of both empirical predictions outside small exceptional events gives \[ \sum_{j,i}w_i\lVert q_{j,i}-q^*_{j,i}\rVert\longrightarrow0 \tag{48}\] in the following precise sense. First choose the desired overlap tolerance, then \(\epsilon\) as above, then a sufficiently large fixed \(\ell\). At this fixed \(\ell\), take the finite-array subsequential limit as \(M\to\infty\) and choose a sufficiently fine rounding. Both predictions hold in the resulting joint sampled calculation, and their failure probabilities bound the weighted path distance by twice the chosen tolerance plus a bounded multiple of their sum. No finite-size convergence uniform over all grids is asserted or needed.

For the marking assertion, an outer Gaussian column in a narrower norm band leaves an open set of inner columns satisfying the shell constraint. Both events have positive Gaussian probability, so the surviving root normalizer is positive. Infeasible outer columns simply give zero marks. The canonical fractional moments are bounded by the unrestricted strict-domain recursion; in the actual shell calculation bounded spin norms give finite Gaussian exponential moments. Thus pruning and backward marking apply and preserve the unmarked level law. Independence of canonical spatial rows was used before restriction; after restriction only the comparison (47) is used.

The expected logarithmic cost of the shell

The preceding comparison identifies the overlap path. To bound \(\mathbb E\log R_{M,\ell}\), we also need the log partition function on the canonical shell. Its high Gibbs probability alone does not control its expected logarithmic cost. Write \(Z_c=\int e^{H_c}d\mu\) and \(N_c=\int_{\mathcal I}e^{H_c}d\mu\) for the canonical denominator and shell numerator on the ordinary cascade/spin reference measure.

Let \(A=\mu(\mathcal I)\). At the fixed exponent \(\theta\), choose narrower bands for the outer columns. For all large \(\ell\) their joint probability \(q_\ell\) is bounded below by \(q_0>0\); conditional inner Gaussian Gram bands have probability at least \(c_0>0\). Thus \(A\ge c_0W_{q_\ell}\) with the thinning mass (44). In particular \[ \mathbb E(-\log A)^2\le C(\theta,q_0,c_0), \tag{49}\] independently of every other cascade exponent and of the depth of the grid.

Jensen on the normalized reference shell gives \(\log N_c\ge\log A+\int H_c\,d\mu_{\mathcal I}\). On the shell the quadratic and bounded-pattern terms have magnitude at most \(C\ell\). Conditional on reference spin marks and weights, the averaged linear insertion is centered Gaussian with variance at most \(C\ell\), using the bounded total field covariance and total shell spin norm \(O(\ell)\). Therefore \[ \mathbb E[(\log N_c)_-^2]\le C\ell^2. \tag{50}\] For the denominator, relative entropy gives \(\log Z_c\le\langle H_c\rangle_c\). The canonical spin self covariance is \(I\), its fourth moments are bounded, and the terminal-field fourth moments are uniformly bounded by Lemma 4. Cauchy–Schwarz over \(O(\ell)\) row energies and \(\ell\) bounded patterns yields \[ \mathbb E[(\log Z_c)_+^2]\le\mathbb E\langle H_c^2\rangle_c\le C\ell^2. \tag{51}\] These bounds are uniform in the grid. Since \(\mathbb E(1-N_c/Z_c)\le Ce^{-c_\epsilon\ell}\), the event \(N_c/Z_c<1/2\) has exponentially small probability. On its complement \(-\log(N_c/Z_c)\le\log2\); on the event use (50)–(51) and Cauchy–Schwarz. We obtain \[ \mathbb E[-\log(N_c/Z_c)]=o(\ell). \tag{52}\] This argument depends on the fixed exponent \(\theta\), not on a lower bound for the intermediate grid exponents.

Cancellation and telescoping

We can now evaluate the increment. Additive independent mark recursion gives the unrestricted canonical log value \[\ell V_D(q)+\sum_jr_j\Psi(C_j,b_j).\] Add (45) and subtract the \(o(\ell)\) shell cost. For block \(j\), the coefficient of \(r_j\) is \[\Psi(C_j,b_j)+\tfrac12\{c_j+\operatorname{tr}(B_{j,\mathrm d}-b_{j,k-1}-C_j)\} =J(b_j)+\tfrac12\{c_j+\operatorname{tr}B_{j,\mathrm d}\}.\] Thus the multiplier and the terminal field shift cancel, including the independent visit residual. Lemma 4 and (39) turn the lower bound into \[ \ell V_D(q)+\sum_jr_j\left\{ S_\theta(q_j^*)+\frac12\sum_iw_i \operatorname{tr}((q_{j,i}-q^*_{j,i})b_{j,i})\right\}-o_{\rm tol}(\ell). \tag{53}\] Rounding errors in (39) tend to zero at fixed \(\ell\). The paths \(q_j^*\) are eligible, and \(S_\theta(q_j^*)\) is uniformly bounded: use (24), comparison with \(q^0\), bounded \(b\), and \(S_\theta\ge0\). With (48), the Lipschitz estimate (21), and \(r_j=p_j\ell+O(1)\), division of (53) by \(\ell\) gives a lower bound \(\mathcal G_{\theta,p}(D)-o_{\rm tol}(1)\).

Here is the uniform sequential statement used to sum increments. For every \(\delta>0\), there is \(\ell_0\) such that for each fixed integer \(\ell\ge\ell_0\), every sequence of base sizes and dimensions with \(n_j/M\to p_j\) and increments \(r_j=p_j\ell+O(1)\) satisfies \[ \liminf_{M\to\infty}\frac1\ell\int\Delta_{M,\ell}(u)\,d\pi(u) \ge\mathcal G_{\theta,p}(D)-\delta. \tag{54}\] The constant in \(O(1)\) is fixed across these choices. To prove the statement, begin with any good-parameter sequence and any further compact array limit. The tolerance, shell, and rounding choices above apply to it with uniform canonical constants. Failure of the eventual bound would give such a sequence contradicting (53). The bad parameter sets have vanishing measure and use the fixed-\(\ell\) uniform lower increment bound proved earlier. This establishes (54) after averaging.

For arbitrary target dimensions, telescope along pattern sizes \(M'\) in the residue class of the target \(M\) modulo \(\ell\), with block dimensions \(\lfloor M'n_j/M\rfloor\). The last point is exactly \((M,\boldsymbol n)\). A block increment differs from \(\ell n_j/M\) by at most one; for fixed \(\ell\) and large target \(M\), it therefore differs from \(p_j\ell\) by a fixed bounded amount. The sequential criterion applies uniformly to all large base sizes along these triangular chains; otherwise a failing sequence would contradict (54). Finitely many small initial sizes, including those too small to support the frames, may be skipped with bounded starting cost. The same parameter law \(\pi\) at both endpoints makes the log increments telescope exactly. Divide by the target \(M\), let \(M\to\infty\), then \(\ell\to\infty\) and \(\delta\downarrow0\), and remove the \(o(M)\) perturbation. This proves the matching lower bound in (15). For smooth compactly supported \(D\), convergence in probability follows from bounded differences in the independent patterns, and expectation convergence follows as well.

Extension of the terminal class

The smooth proof extends to continuous potentials satisfying \[ -C(1+|\xi|^2)\le D(\xi)\le C \tag{55}\] by bounded smooth approximation on compact sets. We justify a uniform tail estimate, including its use in the variational infimum.

Omit one row and view its restoration in the cascade representation. Its partition ratio is an integral of \(e^{D(\xi)}\) against an omitted-row Gibbs reference independent of that row. Jensen bounds this ratio from below by \[\exp\{-C(1+\langle|\xi|^2\rangle_{\rm omit})\}.\] If \(R_D\) denotes this ratio, another Jensen inequality gives, for a sufficiently small fixed \(a>0\), \[\mathbb ER_D^{-a}\le e^{aC}\mathbb E\langle e^{aC|\xi|^2}\rangle_{\rm omit}<\infty.\] The bound is uniform because every fresh self field is a standard Gaussian vector of the fixed total dimension \(\sum_jd_j\). The numerator of a tail event is bounded above by \(e^C\) times its omitted-row probability. Split according to whether the denominator is smaller than \(e^{-cR^2}\) and use its negative moment together with Gaussian tails in the numerator. This gives constants \(c,C'>0\), independent of system size and of a trial grid, such that under the expected natural tilted path law \[ \mathbb E\langle\mathbf 1_{\{\lVert \xi\rVert>R\}}\rangle\le C'e^{-cR^2}. \tag{56}\] The same proof applies to one-pattern trial recursions via their cascade representation. It remains valid uniformly over bounded families in (55).

Without perturbations, the expected cascade path law conditional on the common disorder is precisely the natural powered outer/inner law from (5); this follows by the marked change of measure. Interpolation between compact approximations and (56) therefore bounds the \(L^1\) difference of normalized log partition functions by a quantity tending to zero. It also bounds the difference of \(V_D\) uniformly over all eligible paths. Consequently the infima converge and both probability and expectation convergence extend to (55). Bounded continuous functions plus nonpositive quadratic forms are included. Every differentiation used here can first be done with truncated functions; at finite size fields on the compact Stiefel domain are bounded for each fixed disorder.

Differentiability and the powered restriction ratio

Write \(p(t)=\mathcal G_{\theta,p}(D-t\cdot g)\). It is convex as a limit of convex finite pressures. For any bounded continuous row observable \(h\), the limiting pressure with added coupling \(s h\) is differentiable in \(s\). To prove this without assuming uniqueness of a minimizing path, differentiate one finite recursion. Its first derivative is the tilted conditional mean of \(h\). Differentiating the recursion once more sums the conditional variance increments of that mean martingale, each multiplied by its step coefficient \(\zeta_i\in[0,1]\). Without these coefficients the nonnegative increments telescope to at most the terminal variance of \(h\), which is bounded by \(\lVert h\rVert_\infty^2\). Hence \[ 0\le\frac{d^2}{ds^2}V_{D+s h}(q)\le\lVert h\rVert_\infty^2, \tag{57}\] uniformly over grids, paths, and \(s\). Subtract \(\lVert h\rVert_\infty^2s^2/2\) from every entropy-plus-recursion objective. Each becomes concave, and their infimum is concave. The limiting pressure is therefore semiconcave as well as convex, hence differentiable. Infinite-entropy paths do not affect this argument; the baseline trial and the terminal bounds keep the value finite.

Differentiability and exponential tilting imply that the empirical average of \(h\) concentrates at its deterministic derivative under the natural tilted path law, in probability including disorder. The ratio needed for this argument is not an ordinary Gibbs mass. If \(u(x)\) is the conditional inner probability of an event under an additive tilt, then exactly \[ \frac{\mathcal Z(H;\text{event})}{\mathcal Z(H)} =\left(\mathbb E_{\rm out}^{H}u^\theta\right)^{1/\theta}. \tag{58}\] For a one-sided empirical-average event, the usual pointwise exponential bound survives inner integration, the positive outer power, and the final root. Probability convergence of the nearby coupled pressures and differentiability make the ratio exponentially small in probability. More explicitly, if \(v=\mathbb E_{\rm out}^{H}u\) is its natural probability and \(R\) is the restricted ratio, then \(u\le u^\theta\) and Jensen’s inequality give \[v^{1/\theta}\le R\le v.\] Thus an exponentially small \(R\) forces \(v\to0\), while \(v\to1\) forces \(R\to1\) at fixed \(\theta\). These two implications will be used for concentration and for a cost-free restriction, respectively.

Apply this to bounded clippings of every \(g_\ell\). Estimate (56) is uniform for \(t\) in the compact box \([0,d]^s\): all unbounded quadratic coefficients remain nonpositive. Thus \(\bar g\) concentrates at a deterministic vector \(v(t)\), obtained as the limit of the clipped centers. For every direction \(a\) into the box, the one-sided derivative is \[ \partial_a p(t)=-a\cdot v(t). \tag{59}\] To justify passage of slopes through clipping, interpolate only the increment in the coupling. Increasing a nonnegative quadratic coupling adds a nonpositive term. If it decreases, write its contribution, for \(a_\ell<0\) and \(u>0\), as \[-(t_\ell-u|a_\ell|)g_\ell -u|a_\ell|(1-\gamma)(g_\ell-g_{\ell,R}),\qquad 0\le\gamma\le1,\] where \(g_{\ell,R}\) is a nonnegative bounded clipping. Both unbounded coefficients are nonpositive while the direction remains in the box. Uniform tails then make the directional slope error at most \(\sum_\ell|a_\ell|\epsilon_R\), with \(\epsilon_R\to0\). This proves (59), including boundary directions.

Soft duality

Choose a minimizer \(t\in[0,d]^s\) of \(p(t)+t\cdot c\), which exists by continuity. Equation (59) gives the box conditions \[t_\ell=0\Rightarrow v_\ell(t)\le c_\ell,\qquad 0<t_\ell<d\Rightarrow v_\ell(t)=c_\ell,\qquad t_\ell=d\Rightarrow v_\ell(t)\ge c_\ell.\] Equivalently, \[ t_\ell(v_\ell(t)-c_\ell)=d(v_\ell(t)-c_\ell)_+. \tag{60}\] For every configuration the soft Hamiltonian is bounded above by the additive Hamiltonian \[H_t=\sum_rD(\xi_r)-Mt\cdot\bar g+Mt\cdot c,\] because \(d(x-c)_+\ge t(x-c)\) for \(t\in[0,d]\). This proves the upper bound in (16). On \(\lVert \bar g-v(t)\rVert_1\le\delta\), (60) gives the converse inequality up to \(2dM\delta\). This set has natural tilted probability tending to one; by (58) its restriction costs \(o(M)\) in the log partition function. Letting \(\delta\downarrow0\) gives the matching lower bound in probability. Theorem 3 is proved without any interchange of a minimum and an integral.

Gaussian spans and shell entropy

Generate the flag \(E_1\subset E_1\oplus E_2\subset\cdots\) from consecutive blocks of an \(M\times n\) matrix \(G\) with independent \(N(0,1/M)\) entries, \(n=\sum_jn_j\). Gaussian orthogonalization gives its Haar law; see [14] for this construction. On an event of probability at least \(1-e^{-cM}\), its norm and the norm of its inverse on its image are bounded by fixed constants \(C_{\rm op},C_{\rm inv}\). An elementary net proof suffices: an \(r\)-dimensional unit sphere has a \(\delta\)-net of size at most \((1+2/\delta)^r\); fixed-vector Gaussian norms have arbitrarily strong exponential upper tails at a large constant and small-ball probability at most \((C\epsilon)^M\). First obtain the upper bound at a fixed mesh and then take a mesh of order \(\epsilon\) for the lower bound. The strict inequality \(n/M\to p<1\) makes the small-ball exponent dominate the net size for small enough \(\epsilon\).

The volume Jacobian of \(x\mapsto Gx\) is \(\sqrt{\det(G^{\mathsf T}G)}\). Its logarithm divided by \(M\) converges in probability to \(J_p\): successive orthogonal residual lengths are independent with laws \((\chi^2_{M-r+1}/M)^{1/2}\), and chi-square concentration, followed by a Riemann sum, gives \[ \frac1{2M}\log\det(G^{\mathsf T}G)\longrightarrow \frac12\int_0^p\log(1-u)\,du. \tag{62}\]

The angular pressures concentrate. On the good norm and inverse event, the flag projectors are Lipschitz in the Frobenius distance of \(G\), with a fixed operator-norm constant: represent unit vectors by bounded coefficients, compare their distances to the other image, and apply this to each initial block. Differences of consecutive projectors give the individual blocks. Nearby frames may be aligned by projecting and multiplying by the inverse Gram square root; their operator-norm change is bounded by the projector change. Coupling directions and using (61) bounds the pressure change. Boundedness handles pairs farther apart. A Lipschitz extension from the good event has Lipschitz constant \(O(M^{-1/2})\) in standard Gaussian entries, hence exponential concentration. For reference, the Gaussian exponential bound follows by centering with an independent copy, joining the copies by a circular Gaussian interpolation, and applying Jensen to the integrated directional derivative; the independent Gaussian velocity gives \(\mathbb Ee^{t(F-\mathbb EF)}\le e^{Ct^2\operatorname{Lip}(F)^2}\). Smoothing removes differentiability assumptions.

By boundedness, equicontinuity, and concentration, every subsequence has a further subsequence on which \(P_M^f\) converges in probability uniformly on compact sets to a deterministic bounded continuous \(P_\infty\). We identify this function. Radial integration and Stirling’s formula show that the weighted Lebesgue volume in \(E_1\oplus\cdots\oplus E_h\) of a small shell with \(\lVert y_j\rVert^2/M\) near a positive \(a_j\) has logarithm per \(M\), after shrinking the shell, equal to \[ \Phi(a)=H_p(a)+P_\infty(a). \tag{63}\] The same radial calculation gives the upper Laplace bound on compact sets. Set \(\Phi=-\infty\) at a zero coordinate; the divergent radial entropy and bounded \(P_\infty\) justify this convention in upper bounds.

A majorization principle without concavity

The coefficient integral and triangular entropy

The outer coefficient blocks \(x_j\in\mathbb R^{n_j}\) have Lebesgue measure on balls of radius \(\kappa\sqrt M\). For each \(j\le i<h\), let \(z_{i,j}\) independently have the uniform probability law on the same ball. Put \[y=Gx,\qquad u_i=\sum_{j\le i}G_jz_{i,j},\qquad \overline g=\frac1M\sum_{r=1}^M g(y_r,u_{1,r},\ldots,u_{h-1,r}),\] with the tests and constants from (11). Define the soft coefficient pressure \[ Q_M=\frac1M\log\int dx\left[ \int\exp\left\{\frac{\sum_rf(y_r)-M\lambda\sum_\ell(\bar g_\ell-c_\ell)_+}{\theta}\right\}\,d\mu(z) \right]^\theta. \tag{66}\]

In species \(j\), orthogonalize the coefficient columns divided by \(\sqrt M\) in the order \(x_j,z_{j,j},\ldots,z_{h-1,j}\). They are a Haar orthonormal frame times an upper triangular \(R_j\). Conditional on the first column, the remaining frame has precisely the inner Stiefel law, independently of the triangular parameters. Rotational invariance of the independent coefficient balls makes the inner parameters independent of the first length and direction.

For ball probability measure, the normalized log density of these triangular parameters contributes \(p_j\log(R_{j,\ell\ell}/\kappa)\) for each column. Indeed the density of its readings in previous directions and its new perpendicular radius is proportional to that radius to power \(n_j-\ell\) on a fixed-dimensional half ball. Converting the first ball from probability to Lebesgue measure adds \(p_j\log(\kappa\sqrt{2\pi e/p_j})\) by Stirling. Only inner-column entropy is multiplied by \(\theta\) in (66).

For the normalization, write the coefficient columns as \(\sqrt M\,O_jR_j\), where \(O_j^{\mathsf T}O_j=I\), and set \(X_j=\sqrt{n_j}O_j\). Writing \(G_j=M^{-1/2}\widehat G_j\) gives \[G_j(\sqrt M\,O_jR_j) =\frac{\widehat G_jX_j}{\sqrt{n_j}}R_j.\] Thus for fixed \(R_j\) the row readings are exactly \(\xi_j^{\mathsf T}R_j\) in Theorem 3. Apply that theorem with terminal \(f(y)/\theta\), penalty strength \(d=\lambda/\theta\), and the tests in (11). Multiplication by the outer factor \(\theta\) gives the second line of [eq:Bfunctional]. Ordinary finite-mesh Laplace bounds then give \[ Q_M\longrightarrow\mathcal B_{p,a}(\kappa,\lambda,B,\theta) \quad\hbox{in probability}. \tag{67}\] The finite-mesh Laplace bounds remain valid after taking the \(\theta\)th power of the inner integral. On a bounded operator-norm event all normalized energies are uniformly continuous in the triangular parameters, uniformly in frames; the squared and truncated-squared tests use the uniform \(\ell^2\) bounds on readings. Upper and lower finite-mesh sums are therefore valid. For a finite sum and \(0<\theta<1\), its \(\theta\)th power is between the maximum powered summand and the sum of powered summands; the number of cells contributes no extensive cost. Fixed trial neighborhoods give the lower bound. At fixed \(\theta>0\), zero diagonal faces have arbitrarily negative entropy and may be discarded before refining the mesh.

On the bounded-norm event, the normalized log in (66) is Lipschitz in the standard Gaussian entries with constant \(C(\lambda,\kappa)M^{-1/2}\). This constant is independent of \(B\) and \(\theta\). The derivative of a truncated square has the same \(\ell^2\) bound as that of a square, and the inner division by \(\theta\) cancels the outer power in a supremum change bound. Extend from the good event as above. For every fixed deviation, (67) consequently has an exponential concentration rate independent of \(B,\theta\); the size at which its limiting center is accurate may depend on them.

Upper and lower identification of the shell

The upper bound first removes the witnesses. Write \(b_j=\lVert P_{E_j}y\rVert^2/M\) and \(T=\sum_jb_j\). Since \(u_i\) belongs to the \(i\)th flag subspace, \[\frac{\lVert y-u_i\rVert^2}{M}\ge\sum_{j>i}b_j, \qquad \frac1M\sum_r\min\{y_r^2,B\}\le T.\] The first two test penalties therefore sum to at least \((T-s_a)_++(s_a-T)_+=|T-s_a|\), and each remaining penalty is at least the corresponding tail-norm excess. This bound is independent of the witnesses, so it survives their probability integral and the outer power. Change variables from coefficients to the image and use (62) and the radial upper Laplace bound. Along the chosen subsequence, \[ J_p+\mathcal B_{p,a}\le \sup_{\substack{b_j\ge0\\\sum_jb_j\le C(\kappa)}} \left\{\Phi(b)-\lambda\left( \left|\sum_jb_j-s_a\right|+ \sum_{i<h}\left(\sum_{j>i}b_j-\sum_{j>i}a_j\right)_+ \right)\right\}. \tag{68}\] One may take \(C(\kappa)=C_{\rm op}^2h\kappa^2\). Boundary mass splits with a zero coordinate have divergent negative radial entropy. As \(\lambda\to\infty\), compactness forces the total and prefix constraints of Lemma 6; thus the upper limit of (68) is at most \(\Phi(a)\).

For the reverse bound fix \[ \kappa>2C_{\rm inv}\sqrt{s_a+1}. \tag{69}\] Every \(y\) in a sufficiently narrow desired shell and every one of its flag projections then has a coefficient representative strictly inside the appropriate balls, with slack proportional to \(\sqrt M\). A neighborhood of normalized radius \(\delta\) around each witness makes the projection penalty small. Its inner probability is at least \(\exp\{-C(1+|\log\delta|)M\}\), so its cost in (66) is only \(\theta C(1+|\log\delta|)\).

It remains to control loss of squared norm to the bounded lower-norm test. Fix a small \(\eta>0\). Every vector of normalized squared norm at most \(C(\kappa)\) has all coordinates of absolute value larger than \(b_0(\eta)\) within some set of \(\lfloor\eta M\rfloor\) coordinates. The number of such sets is at most \(\exp(Mh(\eta))\), with \(h(\eta)\to0\). One set therefore captures at least that inverse fraction of the weighted desired shell volume. Rotate those coordinates by an independent Haar matrix. The activation energy changes by at most \(2\lVert f\rVert_\infty\eta M\). To quantify the truncation loss, write \(m=\lfloor\eta M\rfloor\) and let \(v\) be the restriction of a captured vector to the selected coordinates. For a Haar rotation \(U\) on these coordinates, \[\mathbb E_U\frac1M\sum_{r=1}^{m}(Uv)_r^4 =\frac{3\lVert v\rVert^4}{M(m+2)} \le\frac{C_1C(\kappa)^2}{\eta}.\] Choose \(B>b_0(\eta)^2\), so the unrotated entries lose no squared norm. Let \(\mu_{\rm cap}\) be the probability measure obtained by normalizing the old activation weight on the captured shell, whose weighted volume is \(W_{\rm cap}\). Extend \(U\) by the identity outside the selected coordinates and put \[L_B(Uy)=\frac1M\sum_r\big((Uy)_r^2-\min\{(Uy)_r^2,B\}\big).\] The inequality \(z^2-\min\{z^2,B\}\le z^4/B\) and weighted Fubini give, for every \(\tau>0\), \[\mathbb E_U\mu_{\rm cap}\{y:L_B(Uy)>\tau\} \le\frac{C_1C(\kappa)^2}{\eta B\tau}.\] Choose \(B\) so the right side is at most \(1/4\). Markov’s inequality shows that with probability at least \(1/2\), the surviving vectors carry at least \(W_{\rm cap}/2\) of the old weighted volume. Their rotated weighted volume is at least \(e^{-2\lVert f\rVert_\infty\eta M}W_{\rm cap}/2\). Rotating the matrix with the vectors preserves the shell norms, flag distances, and Jacobian.

The useful set can depend on the old matrix. To make the probability argument legitimate, supply independent rotations for every candidate coordinate set. Each rotated matrix has the original Gaussian marginal. With probability bounded away from zero, at least one candidate has coefficient pressure satisfying \[ J_p+Q_M\ge\Phi(a)-\epsilon -2\lVert f\rVert_\infty\eta-h(\eta) -\theta C(1+|\log\delta|), \tag{70}\] where at fixed \(\lambda\) the shell, truncation, Jacobian, and witness errors can be made smaller than \(\epsilon\).

The choices respect exactly the order in (13). Fix \(\kappa\) by (69); then fix any finite \(\lambda\) and a desired error \(\epsilon\). The concentration rate for \(Q_M\) at that error is independent of \(B,\theta\). Choose \(\eta\) so that \(h(\eta)\) is below half this rate and \(2\lVert f\rVert_\infty\eta+h(\eta)<\epsilon\). Choose shell widths and \(\delta\) so their errors times \(\lambda\) are small, then choose a sufficiently large \(B\), independently of \(\theta\). Finally take \(\theta\) small enough that the witness entropy is below \(\epsilon\). A union bound over the candidate rotations and the concentration estimate turns (70) into the corresponding deterministic bound for \(J_p+\mathcal B_{p,a}\), with one additional \(\epsilon\) loss. Thresholds may depend on the fixed \(\lambda\); no uniformity in its later limit is required.

For fixed \(M\), the soft integral increases as \(\theta\downarrow0\) because the inner expression is an increasing \(L^{1/\theta}\) norm. It increases with \(B\) and decreases with \(\lambda\). These monotonicities pass to the deterministic limits. The upper bound and the witness lower bound thus establish the successive limits and \[\Phi(a)=J_p+ \lim_{\lambda\to\infty}\lim_{B\to\infty}\lim_{\theta\downarrow0} \mathcal B_{p,a}(\kappa,\lambda,B,\theta).\] Using (63) proves (13) and identifies every subsequential \(P_\infty\). Equicontinuity extends convergence to compact sets and to zero radii.

Full output dimension and bounded continuous activations

If \(\sum_jp_j=1\), retain, independently in each block, a random subspace of dimension \((1-\eta)n_j+o(M)\). Under the untilted angular law, the fraction of squared norm lost in each block is at most \(2\eta\) with probability tending to one. The same is true of the old Gibbs mass with probability tending to one over the chosen retained subspaces: for each fixed old vector the projection failure probability is uniformly small, so Fubini and Markov apply after arbitrary bounded activation reweighting. Ignore zero-radius blocks.

On this good set, rescale each retained projection to its old radius. Orthogonality gives a field change of norm at most \(C\sqrt{\eta M}\), and hence an energy change at most \(C\sqrt\eta M\). Under the untilted angular law the retained directions are uniform and independent of all split lengths. For an upper bound on the old partition function, restrict to its Gibbs-good mass and integrate the transformed weight; this is at most the new partition function times \(e^{C\sqrt\eta M}\). For the reverse bound, restrict the untilted split lengths to the same typical set and use the pointwise opposite energy comparison; its probability tends to one independently of the retained directions. Therefore \[|P_M^f(a)-P_{M,\eta}^f(a)|\le C\sqrt\eta+o_{\mathbb P}(1).\] The retained system has proportions \((1-\eta)p\) and has already been treated. The bound, followed by \(\eta\downarrow0\), proves (14) and convergence, uniformly on compact radius sets. Only the angular dimensions are retained; no input Dirichlet entropy is changed by this prescription.

For \(f\in C_b(\mathbb R)\) choose bounded Lipschitz approximants with a common bound and uniform convergence on compact intervals. Every angular configuration satisfies \(M^{-1}\sum_ry_r^2=\sum_ja_j\). On a compact radius set, the fraction of entries outside \([-R,R]\) is uniformly \(O(R^{-2})\). Thus compact approximation controls the normalized energy uniformly and transfers all angular conclusions. The same estimate applies to (66) on a bounded operator-norm event, independently of \(\lambda,B,\theta\), so its limiting formula has the same continuity. This completes the proof of Theorem 1.

General dimension sequences

Negligible spectral groups inside the edge interval

A block of \(o(N)\) directions can still carry a positive fraction of the input norm under the Gibbs weight. The next lemma records its entropy cost. We will reproduce that cost by reserving one or two directions in macroscopic spectral blocks.

Activation extensions

Examples delimiting the hypotheses