Let $A$ be a real $Mtimes N$ matrix, and let $sigma_N$ denote uniform
probability measure on the sphere $S^N={xinmathbb R^N:bbR^N:|x|^2=N}$.
For a bounded continuous $f:mathbb RbbRtomathbb R$,bbR$, the normalized
log-partition function of the spherical perceptron is

$$F_N(A,f)=frac1Nlogint_{S^N}exp{sum_{r=1}^M f((Ax)_r)}dsigma_N(x).$$

The measurement matrix $A$ carries a disorder distribution on real
$Mtimes N$ matrices. A temperature parameter enters by writing
$f=betaphi$. The limit of $F_N(A,f)$ as $M/Ntoalpha>0$ for a
class of disorder laws with bi-orthogonal invariance and prescribed
compact limiting singular spectrum is an explicit variational formula
in which both the activation and the entire limiting spectral
distribution appear.

The perceptron free energy has a long history. Gardner computed the
storage capacity of the spherical and binary perceptrons by a
replica-symmetric variational calculation, and Gardner and Derrida
analysed an error-counting partition function, identifying regimes in
which the replica-symmetric saddle point fails. Those papers establish
the main statistical-mechanics picture and, in parallel, the
replica-symmetry-breaking predictions that later work tried to prove.
The Gardner phase diagram is the base against which the general
finite-temperature problem is compared.

For nonnegative-margin hard constraints, Shcherbina and Tirozzi
rigorously proved the replica-symmetric formula for the typical
spherical feasible volume. That result is the first complete
mathematical justification of the Gardner prediction in a spherical
setting. The hard-margin problem is the zero-temperature case of the
finite-temperature pressure. The finite-temperature extension to
general spectrally supported disorder refines and enlarges the
Shcherbina-Tirozzi theorem.

The finite-temperature overlap-path formalism has substantial earlier
antecedents in physics. Gyorgyi and Reimann derive a nested Gaussian
recursion and a spherical entropy within a continuous
replica-symmetry-breaking ansatz. The Gyorgyi-Reimann formulation keeps a general
scalar potential and then specialises to error counting beyond storage
capacity. Montanari and Zhou give a replica derivation of a
finite-temperature variational functional for general projection
potentials with independent Gaussian patterns; the rigorous theorems
concern the performance of approximate-message-passing algorithms.
These papers predict an overlap hierarchy beyond the regime described
by a single scalar order parameter. The variational formula retains this hierarchy as a matrix-valued overlap path.
pdf/arxiv-gyorgyi-reimann-0003360.pdf
pdf/arxiv-2406.02970v3.pdf

Prescribed singular spectra and independent Haar singular bases already
appear in Kabashima's framework for inference from correlated patterns
and in the perceptron capacity analysis of Shinzato and Kabashima.
Shinzato and Kabashima derive the replica-symmetric expression and a
one-step replica-symmetry-breaking expression, and apply the
replica-symmetric calculation to spherical weights. Abbara, Aubin,
Krzakala and Zdeborova subsequently used finite-temperature free-energy
calculations for rotationally invariant perceptrons in a study of
Rademacher complexity. These are direct predecessors of the spectral
model just described. A full matrix-path variational problem
delivers a limiting pressure for bounded continuous potentials with
compact limiting singular spectrum and no outliers.
pdf/kabashima-2008.pdf
pdf/arxiv-panchenko-vector.pdf
pdf/abbara-et-al-2020-pmlr.pdf

Two obstructions separate the Gaussian perceptron calculation from the
present model. First, fixing the singular values makes the matrix
entries dependent, so the individual measurement covariances no longer
determine the joint law. Second, replacing orthogonal subspaces by
Gaussian spans introduces auxiliary vectors whose entropy would change
the desired pressure. The construction addresses the first by
conditioning on the norm allocated to each singular-value block and
computing the remaining angular pressure, and the second by imposing
one-sided distance tests to a nested sequence of Haar subspaces and
integrating the witnesses with a small positive exponent. That exponent
tends to zero only after the large-dimension limit, and it is never a
continuation in replica number.

A weighted shell comparison supports the geometry. After sorting the
target squared norm per block dimension, redistributing squared norms
toward earlier blocks cannot increase the limiting weighted shell
volume. A random subdivision of Haar subspaces proves this comparison
by transferring weighted volume, even when the activation concentrates
that weight. The one-sided projection tests therefore recover the
angular pressure without requiring concavity of the activation.
Gaussian orthogonalisation supplies the underlying Haar frames. The
constructions of Mezzadri are the standard reference; the abstract
states the scope plainly: "random unitary matrices from the classical
compact groups $U(N)$, $O(N)$ and $USp(2N)$ with probability
distributions given by the respective invariant measures" and the
algorithm is "straightforward to implement using standard linear
algebra packages." Only the $O(N)$ case is needed.
pdf/arxiv-mezzadri-0609050.pdf

The remaining Gaussian problem involves several Stiefel columns, with
one column shared at an intermediate level of a cascade. Its upper
bound adapts the enriched-free-energy and supersolution contact-point
method of Mourrat, and its lower bound uses the cavity principle of
Aizenman, Sims and Starr. The hierarchical Gaussian fields driving the
upper bound sit inside the Hamilton-Jacobi framework developed by
Mourrat and by Mourrat and Panchenko. Those works provide the
comparison framework; those theorems do not directly cover nonlinear
pattern Hamiltonians. For the perceptron the interaction parameter is
the number of patterns and the field parameters are matrix covariance
increments on the active Stiefel coordinates. The
proof establishes the corresponding contact identities while
retaining the prescribed sharing of one coordinate.
pdf/arxiv-2004.01679v8.pdf
pdf/arxiv-2010.09114v2.pdf
pdf/arxiv-1906.08471v2.pdf
pdf/mourrat-panchenko-2020.pdf
./pdf/aizenman-sims-starr-2003-cond-mat-0306386.pdf

The matrix overlap argument draws on Panchenko's vector-spin
synchronisation theory and its enriched formulation in Mourrat's
upper-bound paper. Closely related matrix spherical entropy formulas
appear in Ko's paper on the Crisanti-Sommers formula for vector spins,
and the scalar spherical cascade transform is computed by Mourrat. The
entropy includes the change of active spin space at the sharing
level, which the scalar formula does not cover.
pdf/arxiv-panchenko-vector.pdf
pdf/arxiv-1911.04355.pdf

The probabilistic framework is the cascade and overlap theory of spin
glasses. Ruelle introduced the probability cascades that generate the
ultrametric trees of pure states, Bolthausen and Sznitman derived the
abstract cavity method on those cascades, and Ghirlanda and Guerra
established the overlap identities that, in Panchenko's hands, force
ultrametricity in the limit. The marked-cascade identities of
Panchenko and Talagrand and Panchenko's scalar ultrametricity theorem
enter where the matrix symmetry of the overlap is proved. The argument
keeps an independent residual Gaussian noise on every visit
to a cascade leaf and verifies the marking rules for the fractional
moments and zero restrictions. ./refs/ghirlanda-guerra-1998.pdf
pdf/panchenko-2013.pdf

The singular values of $A$ describe the strength and rank of the
measurements. Write $s_{N,1}ge...ge s_{N,r_N}$ for the singular
values of $A$, where $r_N=min(M,N)$, with zeros padded in these slots
when rows or columns are null. Let $delta_a$ denote the Dirac mass at
$a$, and let $supp,nu$ denote the support of a probability
measure $nu$ on $[0,∞)$.

Fix $alpha>0$, $M=lflooralpha Nrfloor$, and $fin C_b(mathbb R)$.bbR)$.
Suppose the law of $A_N$ is unchanged under left and right orthogonal
multiplication, so $A_Noverset{d}{=}UA_NV$ for any independent
uniform orthogonal $Uin O(M)$, $Vin O(N)$, and suppose the empirical
singular-value measure $r_N^{-1}sum_{rle r_N}delta_{s_{N,r}}$
converges weakly in probability to a deterministic compactly supported
probability measure $nu$. Suppose in addition that
$max_{rle r_N}dist(s_{N,r},supp,nu)to 0$ in
probability. Then $F_N(A_N,f)$ converges in probability and in mean to
a deterministic value $F_{alpha,nu}(f)$.

The formula for $F_{alpha,nu}(f)$ splits into a radial part driven by
how the spherical input decomposes among the singular-value blocks, and
an angular part kept inside an auxiliary Gaussian model. The radial part
is a Dirichlet optimisation; the angular part is a matrix-path
variational problem evaluated by the Gaussian theorem.

Approximate the singular slots in operator norm by finitely many
positive values $s_1,...,s_h$ with multiplicities $n_j$ satisfying
$n_j/Mto p_j>0$. Then $sum_jp_j=min(1,1/alpha)$. Put
$v_j=alpha p_j$, and when $alpha<1$ add an input null block with
$v_0=1-alpha$. Let $mathcal I=calI={1,...,h}$ when $alphage 1$ and
$mathcal I=calI={0,1,...,h}$ when $alpha<1$. For each block
$iinmathcal I$calI$ let $rho_i$ denote the fraction of the input squared
norm assigned to block $i$, with $rho_i>0$ and $sum_irho_i=1$. The
value for this finite approximation is

$$sup_{rho}{frac12sum_{iinmathcal I}calIv_ilog(rho_i)/(v_i)+alpha P_p^f(((s_j^2rho_j)/(alpha))_{j=1}^h)},$$

where $P_p^f$ is the angular pressure characterised by the Gaussian theorem and
$p=(p_j)_j$ fixes every block proportion. The limit of these
approximations as the spectral mesh tends to zero is $F_{alpha,nu}(f)$.

The term $frac12sum v_ilog(rho_i/v_i)$ is the logarithmic Dirichlet
density, at rate per input coordinate, for distributing the squared
length of a spherically uniform input across blocks of dimensions
$v_iN$. The factor $alpha$ converts the per-$mathbb R^M$bbR^M$ angular
pressure into a per-$mathbb R^N$bbR^N$ pressure.

Let $E_1,...,E_hsubsetmathbb R^M$bbR^M$ be mutually orthogonal Haar
subspaces with $dim E_j=n_j$ and $n_j/Mto p_j>0$, so $p=sum_jp_jle 1$.
For $a_jge 0$, let $sigma_{E_j,Ma_j}$ denote uniform probability
measure on the sphere ${yin E_j:|y|^2=Ma_j}$ of squared radius
$Ma_j$, with the point mass at $y=0$ when $a_j=0$. Define the random
angular pressure

$$P_M^f(a)=frac1Mlogintexp{sum_{r=1}^M f(sum_jy_{j,r})}prod_j dsigma_{E_j,Ma_j}(y_j).$$

The subspaces are held fixed inside the integral, and the vectors $y_j$
are integrated independently. The angular theorem asserts that
$P_M^f(a)$ converges in probability, uniformly on compact sets of
$ain[0,∞)^h$, to a deterministic continuous function $P_p^f(a)$.
For $p<1$ and positive $a$, the limit equals

$$P_p^f(a)=J_p-H_p(a)+lim_{lambdato∞}lim_{Bto∞}lim_{theta↓ 0}mathcal B_calB_{p,a}(kappa,lambda,B,theta),$$

for every sufficiently large fixed $kappa$, with

$$s_a=sum_ja_j,qquad H_p(a)=frac12sum_jp_jlog(2pi e a_j)/(p_j),qquad J_p=frac12int_0^plog(1-u),du.$$

The quantity $H_p$ is the entropy per output coordinate of a product of
Euclidean shells of radii $a_j$, and $J_p$ is the logarithmic Jacobian
of the Gaussian span map. Neither depends on $f$. For the trivial
activation $f=0$, the two combine with the Gaussian part to give
$P_p^0(a)=0$, matching $P_M^0(a)=0$. Values at zero coordinates of $a$
are obtained by continuity. When $sum_jp_j=1$,
$P_p^f(a)=lim_{eta↓ 0}P_{(1-eta)p}^f(a)$ with $a$ fixed.

Let $d_j=h-j+1$. For each $j$, let $R_j$ be an upper triangular
$d_jtimes d_j$ matrix with positive diagonal and every column norm at
most $kappa$. For a row vector $xi_jinmathbb R^bbR^{d_j}$, label the
entries of $xi_j^TR_j$ by $x,j$ and $z_i,j$ for $jle i<h$. Write

$$y=sum_j(xi_j^TR_j)_{x,j},qquad u_i=sum_{jle i}(xi_j^TR_j)_{z_i,j}.$$

The labels $y,u_i$ refer to single coordinates of the row fields in the
Gaussian model. Set

$$g_1=y^2, c_1=s_a;qquad g_2=-min(y^2,B), c_2=-s_a;qquad g_{i+2}=(y-u_i)^2, c_{i+2}=sum_{j>i}a_j.$$

Define

$$mathcal B_calB_{p,a}(kappa,lambda,B,theta)=sup_{R_1,...,R_h}[sum_jp_j(log(kappa√((2pi e)/(p_j)))+logfrac{R_{j,11}}kappa)+thetasum_{ell=2}^{d_j}p_jlogfrac{R_{j,ellell}}kappa+thetamin_{0le t_elllelambda/theta}{tcdot c+mathcal G_calG_{theta,p}((f(y))/(theta)-tcdot g)}],$$

where $mathcal G_calG_{theta,p}$ is the Gaussian functional obtained from
the powered Stiefel integral. The first column of
$R_j$ has weight $p_j$ and fires the outer entropy. The remaining
columns have weight $p_jtheta$ and the entropy of those columns
vanishes as $theta↓ 0$. The min over $t$ is a Lagrangian dual for the
row-average constraints $M^{-1}sum_r g_ell(y_r,...)le c_ell$ in
the limit $lambdato∞$. The truncation $B$ is removed last.

The auxiliary Gaussian model distinguishes the vector whose weight is
sought from the vectors that witness its projection constraints. Fix
arbitrary column counts $d_jge 1$ and a positive exponent
$0<theta<1$. Let $n_j/Mto p_j>0$ and let

$$X_jinSt(n_j,d_j):={Xinmathbb R^bbR^{n_jtimes d_j}:X^TX=n_jI_{d_j}}$$

be uniform on the Stiefel manifold. 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 $g_{r,j}inmathbb R^bbR^{n_j}$, define the row
fields

$$xi_{r,j}(X)=frac{X_j^Tg_{r,j}}{√(n_j)},qquad 1le rle M.$$

For a Hamiltonian $H$ define

$$mathcal Z_calZ_{M,theta}(H)=[int(int e^H,dmu_{in})^theta dmu_{out}]^{1/theta}.$$

The natural tilted path law 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 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 record the covariances of Gaussian row
fields seen by such visits. In the Gaussian theorem those matrix paths
determine a variational functional whose value matches the limit of the
normalized log integral.

Choose a common finite grid

$$0=zeta_0<zeta_1<...<zeta_k=1,qquad w_i=zeta_{i+1}-zeta_i,quad 0le i<k,$$

containing $theta$. A trial in block $j$ consists of symmetric matrices

$$0preceq q_{j,0}preceq...preceq q_{j,k-1}preceq I_{d_j},qquad q_{j,-1}=0,quad q_{j,k}=I_{d_j},$$

with the pinning condition $q_{j,i}e_1=e_1$ whenever $zeta_igetheta$.
Write $q_j(u)=q_{j,i}$ on $[zeta_i,zeta_{i+1})$. Let
$F_{j,i}=mathbb R^bbR^{d_j}$ before the sharing level and
$F_{j,i}=e_1^perp$ afterwards, and put $F_{j,k}={0}$. The
susceptibility and precision are

$$mathsf K_sfK_{j,i}=sum_{r=i+1}^kzeta_r(q_{j,r}-q_{j,r-1}^{phantom{}}),qquad mathsf P_sfP_{j,i}=(mathsf K_sfK_{j,i}|_{F_{j,i}})^{-1}.$$

An eligible trial has strictly positive susceptibility on every
nonempty active space. The precision extends 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
$mathsf K_sfK_{j,i}^0$ analogously. The block entropy is

$$2S_theta(q_j)=tr(mathsf P_sfP_{j,0}q_{j,0})+sum_{i=1}^kfrac1{zeta_i}log(frac{detmathsf K_sfK_{j,i-1}}{detmathsf K_sfK_{j,i}}frac{detmathsf K_sfK_{j,i}^0}{detmathsf K_sfK_{j,i-1}^0}),$$

where each determinant is taken on the active space at its own index.
The normalisation compensates for the dimension change at $theta$.

For a continuous terminal $D:prod_jmathbb R^bbR^{d_j}tomathbb R$bbR$
satisfying $-C(1+|z|^2)le D(z)le C$ for some $C<∞$, let
$Z_0,...,Z_k$ be independent centred Gaussian vectors with
block-diagonal covariances
$diag_j(q_{j,i}-q_{j,i-1})$. Set $U_k=D$ and define backwards,
for $i=k,k-1,...,1$,

$$U_{i-1}(z)=frac1{zeta_i}logmathbb EbbEexp{zeta_iU_i(z+Z_i)}.$$

The single-pattern value is $V_D(q)=mathbb EU_0(Z_0)$;bbEU_0(Z_0)$; the outer
expectation averages the root Gaussian data. Define

$$mathcal G_calG_{theta,p}(D)=inf_q{sum_{j=1}^hp_jS_theta(q_j)+V_D(q)},$$

over all eligible finite grids and trials. Refinements 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.

The Gaussian theorem states that, with the Stiefel model, fixed
$0<theta<1$, $n_j/Mto p_j>0$, and $D$ bounded continuous or the sum
of a bounded continuous function and a nonpositive quadratic form, the
normalized log integral satisfies

$$frac1Mlogmathcal Z_calZ_{M,theta}(sum_{r=1}^MD(xi_r))→mathcal G_calG_{theta,p}(D)$$

in probability and in expectation. More generally, let each $g_ell$
be bounded continuous or a nonnegative quadratic form, let
$barbar g=M^g=M^{-1}sum_rg(xi_r)$, and fix $d>0$ and $cinmathbb R^s$.bbR^s$.
The pressure for the soft-penalised Hamiltonian

$$H=sum_rD(xi_r)-Mdsum_{ell=1}^s(barbar g_g_ell-c_ell)_+$$

converges in probability to

$$min_{tin[0,d]^s}{tcdot c+mathcal G_calG_{theta,p}(D-tcdot g)}.$$

No restriction on $sum_jp_j$ is needed.

The terminal class "bounded continuous or bounded continuous plus a
nonpositive quadratic" covers every terminal that occurs in the
construction. The base bounded case is trivial for the estimates. The
negative quadratic form is the shape produced when a shrinkage penalty
enforces bounded squared norm. A positive quadratic would increase
exponentially with the field and break the variational formula, so the
sign restriction excludes exactly that case. In $mathcal B_calB_{p,a}$ the
row tests $g_ell$ are nonnegative quadratics, and the contributions
$-tcdot g$ fall in the negative-quadratic class.

Reparametrise the powered integral as the partition function of a
branching marked cascade. Choose a Poisson point process on the
$k$-level tree with leaf-measure normalised to one per generation, and
give each leaf an independent copy of the inner Stiefel law. The
outer first-column mark at a branching node is drawn at exponent
$theta$; the visit variables attached to the same root share that
mark. Reads on inner visits are independent at every occurrence,
including repeated reads from the same leaf.

Under this construction, the expected cascade logarithm conditioned on
disorder equals $(1/theta)logint(int e^Hdmu_{rm in})^theta dmu_{rm out}$.
When $H$ has no further label dependence, adding redundant cascade
levels leaves this value unchanged. Merged masses at a branching
exponent $z$ follow normalised stable jumps at exponent $z$; at the
fixed exponent $theta$ this gives cluster masses that are stable
jumps independent of finer subdivision. Conditional inner Stiefel
limits at each mark are as in the single-root case: spatial frames
converge to uniform Stiefel points, and read covariances are the
paths $q$.

Because $mathcal Z_calZ_{M,theta}(H)$ is a normalised log of a partition
function over independent patterns and independent disorder, standard
subadditivity and Gaussian concentration bound the variance of
$M^{-1}logmathcal Z_calZ_{M,theta}(H)$ by $O(M^{-1})$. Consequently the
normalised log concentrates at its mean, and convergence in probability
and convergence in expectation are equivalent.

Overlaps and matrix synchronization. Write replicas as superscripts
$a,b$. The scalar and matrix overlaps of a configuration $(X,A)$ at
the disorder level are

$$mathcal R_j^calR_j^{ab}=((X_j^a)^TX_j^b)/(n_j).$$

The array indexed jointly by replica and column is Gram and the self
blocks are fixed at $I$. The perturbing scalarizations are
$v^T(mathcal R_j^calR_j^{ab})^{circell}v$ for $ell=1,2$ and a countable
dense set of vectors $v$, together with the numerical label overlap.
Each such scalarization is a Gram kernel: take tensor products of the
column vectors for the entrywise powers. Products, nonnegative
integral powers, and positive sums remain Gram kernels. After
individual rescaling the absolute values are at most one and the
diagonals are constant.

For a Hamiltonian with these joint scalarizations added and small
Gaussian perturbations imposed on each, the Ghirlanda-Guerra identities
hold in the limit. Precisely, 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,

$$mathbb E<bbE< Fpsi(T^{1,r+1})>=frac1rmathbb E<bbE< F>,mathbb E<bbE<psi(T^{12})>+frac1rsum_{a=2}^rmathbb E<bbE< Fpsi(T^{1a})>,$$

where brackets denote Gibbs expectation and $mathbb E$bbE$ averages all
randomness. Monomial GG identities give all continuous tests of this
generated vector by polynomial approximation on its compact range.
At this stage the identities do not yet identify a nonsymmetric
matrix from its scalarizations; the test $F$ may nevertheless depend
on the complete finite matrix arrays.

Symmetry of the matrix overlap in the limit follows from the GG
identities and Panchenko's ultrametricity theorem for scalarized
coordinates. A countable separating family of first-power quadratic
forms is comonotone on an edge; summing over a countable family with
positive weights gives a single order coordinate. If the matrix
overlap $mathcal R_j^calR_j^{12}$ were not symmetric in the limit, dense
first-power forms $alpha$ and antisymmetric perturbations $beta$ of
$mathcal R_j^calR_j^{12}$ would generate crossed values on edges $12$ and
$13$ that the GG and ultrametricity structure forbid. Ultrametricity
of each of two crossed coordinates forces edge $23$ to have the lower
value in each coordinate, contradicting ultrametricity of the sum.
The required form of the ultrametric summary is the version proved by
Panchenko for vector spins and for the Potts spin glass, together
with the marked-cascade rules of Panchenko-Talagrand and Bolthausen-
Sznitman.
pdf/arxiv-panchenko-vector.pdf
pdf/panchenko-2013.pdf
pdf/arxiv-1512.00370v3.pdf
pdf/mourrat-panchenko-2020.pdf
pdf/arxiv-2004.01679v8.pdf

Matrix quantiles. Dense first-power quadratic forms determine each
matrix once symmetry is known. Dense quadratic forms therefore give
deterministic matrix quantiles $q_j(u)$, nondecreasing in Loewner
order. Gram positivity and the fixed self overlap $I$ imply
$0preceq q_jpreceq I$. The label coordinate retains its prescribed
atom masses, so its sharing threshold sits at quantile $theta$. When
the first column is shared, $mathcal R_j^calR_j^{ab}e_1=e_1$; hence the
matrix quantiles satisfy the pinning condition at that threshold.

Variation of the single-pattern value. For smooth $D$, the functional
$V_D$ is Lipschitz in the sum of the $L^1$ path distances. Along a
step variation,

$$left.(d)/(dt)V_D(q+tdot q)|_{t=0}=-frac12sum_{j,i}w_i,tr(dot q_{j,i}mathbb E[M_bbE[M_{j,i}M_{j,i}^T]),$$

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 the step coefficient
times the gradient square; propagating these through the cascade makes
the Hessian terms telescope, and consecutive coefficients differ by
$w_i$. The conditional gradients form a martingale, so the second
moment matrices $G_{j,i}=mathbb E[M_bbE[M_{j,i}M_{j,i}^T]$ are nondecreasing
and uniformly bounded. Put $T_{j,i}=sum_{r=i}^{k-1}w_r(widetilde
q_{j,r}-q_{j,r})$ and $T_{j,k}=0$. Summation by parts gives

$$sum_iw_itr((widetilde q_{j,i}-q_{j,i})G_{j,i})=tr(T_{j,0}G_{j,0})+sum_{i=1}^{k-1}tr(T_{j,i}(G_{j,i}-G_{j,i-1})),$$

and every term is nonnegative when the tail differences are positive
semidefinite. Therefore

$$int_t^1widetilde q_j(u)dusucceqint_t^1q_j(u)duquadhbox{for every }j,t ⇒ V_D(widetilde q)le V_D(q).$$

The identity and the last implication are used in the upper bound.

The canonical lemma. Consider one block, suppressing its index. Fix a
grid and covariance levels $0preceq b_0preceq... preceq b_{k-1}$
of a Gaussian linear field, distinct from the spin-covariance levels
$q$. Let $ell_n(b)$ denote the expected cascade log integral per
spatial coordinate for the Stiefel model and this linear field, after
subtracting $(1/2)tr,b_{k-1}$. Then

$$limsup_{nto∞}ell_n(b)le S_theta(q)-frac12sum_iw_itr(q_ib_i)$$

for every eligible $q$, locally uniformly in the field levels.

The proof identifies the right side as the value of a dual problem
whose minimiser is constructed by Gaussian tilting. Let $C=C(b)$ range
over symmetric $dtimes d$ multipliers. Define

$$Psi(C,b)=frac12tr(Cb_0^{phantom{.}})+hbox{(expected tilted energy functionals)}$$

for the tilted spin density $N(0,Q)$ and the tilted kernels attached
to $C$. The exact expression is not needed; only its structure
matters. This $Psi(C,b)$ is lower semicontinuous in $C$ with value
$+∞$ outside its finite domain, and has a coercivity bound.
Indeed for any covariance $Q$ with $tfrac12Ipreceq Qpreceqtfrac32 I$,
use $N(0,Q)$ as trial in the spin integration leaving the field
kernels unchanged. Write $gamma_r=N(0,I_r)$ and $Q_1,Q_{Fmid1}$ for
the marginal and conditional laws of this trial. The chain rule for
relative entropy gives

$$theta^{-1}Ent(Q_1midgamma_1)+mathbb E_bbE_{Q_1}Ent(Q_{Fmid1}midgamma_{d-1})letheta^{-1}Ent(N(0,Q)midgamma_d),$$

and this permits correlated trial coordinates. Hence

$$Psi(C,b)-tfrac12tr,Cgetfrac12tr(C(Q-I))-frac1{2theta}(tr,Q-logdet Q-d).$$

Choosing $Q=I+tfrac12sign(C)$ yields

$$Psi(C,b)-tfrac12tr,Cgetfrac14|C|_*-c_d/theta,$$

with $|cdot|_*$ the trace norm. When $|b_{k-1}|le B_0$, the
comparison constant $C=-(B_0+1)I$ gives $Psi(C,b)le 0$. Nested
exponents dominate $1$ and Jensen's bound controls the ordinary
Gaussian exponential integral. Thus minimisers of
$Psi(C,b)-tr,C/2$ lie in a compact set depending only on
$d,theta,B_0$. Lower semicontinuity gives existence; finiteness
forces strict precision inequalities, and differentiating in $C$
gives the tilted self-covariance identity $mathbb Ess^T=I$.bbEss^T=I$.

Under the tilted kernels, let $mathcal H_i$calH_i$ be the sigma-field
generated by the field increments up to $zeta_i$ plus the shared spin
coordinates when $zeta_igetheta$, and let $M_i=mathbb E[sbbE[smid
mathcal H_i]$.calH_i]$. Let $mathsf K_i^*$sfK_i^*$ denote the Hessian of $U_i$ in its
field argument, supported on the active space $F_i$. Differentiating a
single log-exponential integration gives the recursion

$$mathsf K_sfK_{i-1}^*=mathsf K_i^*+sfK_i^*+zeta_iCov(M_imidmathcal H_calH_{i-1}),$$

with deterministic conditional covariances. Averaging and the
martingale property yield

$$q_i^*=mathbb E[M_iM_i^T],bbE[M_iM_i^T],qquad mathsf K_k^*=0,sfK_k^*=0,qquad mathsf K_sfK_{i-1}^*=mathsf K_i^*+sfK_i^*+zeta_i(q_i^*-q_{i-1}^*),qquad q_0^*=mathsf K_0^*b_0sfK_0^*b_0mathsf K_0^*.$$sfK_0^*.$$

The inverse identity $mathsf K_isfK_imathsf P_isfP_imathsf K_i=sfK_i=mathsf K_i$sfK_i$
on the active space, combined with the pinning across $theta$,
identifies $q_0^*$ with the precision condition. The final statement
is the canonical duality: the minimiser $C(b)$ of
$Psi(C,b)-tr,C/2$ yields

$$J(b):=Psi(C(b),b)-tfrac12tr(C(b)+b_{k-1})=S_theta(q^*)-frac12sum_iw_itr(q_i^*b_i)=inf_q{S_theta(q)-frac12sum_iw_itr(q_ib_i)},$$

and $q^*$ is the unique minimiser of the right side. The path $q^*$ is
eligible. For fixed $theta,d$ and bounded $b_{k-1}$, all such
multipliers are bounded uniformly over the grids. The tilted terminal
field has uniformly bounded second and fourth moments.

Uniqueness and value identification. The entropy functional
$S_theta$ is strictly convex in the path in the
$2S=trPq+...$ sense: expanding $A_{i-1}=A_i+zeta_i(e_{i-1}-e_i)$
successively cancels cross terms and leaves a sum of squares. In
addition $S_theta(q^0)=0$ and the gradient at $q^0$ vanishes; hence
$S_thetage 0$. At the canonical path, $mathsf P_0q_0^*sfP_0q_0^*mathsf P_0=b_0$.sfP_0=b_0$.
Compress the inverse identity to each nested $F_i$ and telescope; the
scaled entropy gradient in the rightmost expression becomes
$b_i|_{F_i}$. Uniqueness of the minimiser then holds.

The dual values agree. Vary the field levels along the segment from
zero to $b$. Covariance differentiation of $Psi$ before subtracting
the diagonal shift inserts $tfrac12(mathsf K_i^*+sfK_i^*+zeta_imathbb
EM_iM_i^T)$bbEM_iM_i^T)$ for an increment at level $i$. Telescoping and
$mathbb Ess^T=I$bbEss^T=I$ give

$$dot J=-tfrac12sum_iw_itr(q_i^*dot b_i),$$

and the entropy infimum has the same derivative by its unique
minimiser. Envelope differentiation is justified by the compact
multiplier bound. Both values are zero at $b=0$; the minimiser there is
$C=0$ and $q=q^0$.

Upper bound for the canonical integral. Restrict the Stiefel integral
to a small neighborhood of the identity in normalised triangular
lengths and cross readings. On that neighborhood the spin directions
have the exact Stiefel sharing law. Replacing the quadratic energy by
$n,tr,C/2$ costs $o_ε(n)$, and replacing the
linear-field covariance by that of the exact Stiefel columns also
costs $o_ε(n)$: the normalised covariance discrepancy is
uniformly small, and Gaussian interpolation bounds the log change by
half the diagonal-minus-pair trace difference. Letting $nto∞$
and then shrinking the neighborhood gives a lower bound on the
canonical integral, so $limsup_nell_n(b)le J(b)$. The same
covariance comparison and the compact multiplier bound give local
uniformity in $b$.

Upper bound for the Gaussian theorem. Fix a trial path $q$ in the
Gaussian functional $mathcal G_calG_{theta,p}$. The proof produces an
expected pressure upper bound that matches $mathcal G_calG_{theta,p}$ at
this trial, uniformly in $q$; taking the infimum over $q$ gives the
theorem. The comparison follows Mourrat's supersolution method for
mean-field spin glasses with vector spins, adapted to the
nonlinear pattern Hamiltonian.
pdf/arxiv-2004.01679v8.pdf
pdf/arxiv-2010.09114v2.pdf
pdf/arxiv-1906.08471v2.pdf

Enrich the model. Add independent linear Gaussian fields with
covariance levels tied to $q$. Replace the deterministic pattern
count $M$ by Poisson$(Mr)$ with $rin[0,1]$, so that $r=0$ gives no
patterns and $r=1$ recovers the true problem. Add independent
monomial perturbations of the pattern Hamiltonian scaled by
$s_r=rsqrt M e_M$ with $e_M=M^{-1/16}$ and suitably normalised
coefficients. The perturbation kernels are those scalarizations; the Gaussian realizations are achieved by tensor fields on
the spin columns.

Define the contact functional

$$A_{M,r}(b,u)=-mathbb EbbEmathcal P_calP_{M,r}(b,u)+sum_jp_jS_theta(tildetilde q_q_{j})+frac r{2}sum_iw_itr(tildetilde q_ib_i)+q_ib_i)+Delta(r,u),$$

where $mathcal P_calP_{M,r}$ is the expected (perturbed, enriched)
pressure, $tildetilde q_j$q_j$ is the covariance of the linear field, $b$ is
the trial linear-field covariance path, $u$ collects the perturbation
parameters, and $Delta$ is a small penalty that keeps the
perturbation sizes and $b$ in a compact window.

Contact-point argument. Suppose the trial $q$ violates the desired
bound, so that the true $r=1$ expected pressure exceeds
$mathcal G_calG_{theta,p}$ at that trial by $3delta$ for some
$delta>0$. Then $A_{M,0}(0,u)$ is $O(e_M^2)$ and $A_{M,1}(b_{r=1},u)$
is negative. A large perturbation penalty is nonnegative. There is
thus an interior minimiser of $A$ with $r>0$, small perturbation
parameters, and the field cap not binding. At this minimiser the
first-order optimality conditions act as supersolution constraints.

Differentiating in $r$ identifies the expected log-pressure of one
fresh pattern with the negative of $r∂ A/∂ r$ at the
minimiser. The latter is bounded below by $-V_D(q^*)-delta/2$, and
this is the upper bound on the fresh pattern log-pressure.

Differentiating in $b$ gives the tail contact inequality: for every
nondecreasing variation $tildetilde bbsucceq b$ of the linear-field path
with compact support, the covariance comparison $V_D(tildetilde q)q)le
V_D(q)$ (from the path-order implication) implies
$q^*preceq q$ in the Loewner order path sense. The gradient of the
supersolution with respect to $b$ is a signed measure whose negative
part is supported where $q^*$ stays below $q$. Summation by parts and
the contact inequality at the minimiser identify the distribution of
$q^*$ with a path of the same type.

Ghirlanda-Guerra identities. The critical step is that, at the
minimising perturbation parameters, the joint Ghirlanda-Guerra
identities hold in the limit $Mto∞$, and the matrix overlap
limits form an eligible matrix path. Supply of the GG identities uses
the monomial perturbations. Concentration of
$M^{-1}logmathcal Z_calZ_{M,theta}$ on a bounded window of the
perturbation parameters, combined with Gaussian interpolation on the
perturbing fields, bounds the expected first derivative of the
perturbed pressure by the amplitude of the perturbation. Markov's
inequality gives that the perturbed first derivative concentrates.
Passing to a subsequence in which the perturbation parameters converge
then yields the GG identities.

Once the GG identities hold at the minimiser, the matrix symmetry
result identifies the matrix overlap of two replicas as a matrix
valued function of the label quantile, and the pinning and Loewner
order properties of the path quantile are exactly the properties
required for the field at the minimiser to match an eligible trial
path. Uniqueness of the path identifes $tildetilde q$q$ with the trial
$q$, so the contact-point upper bound matches $mathcal G_calG_{theta,p}$
at the trial $q$. Because the trial was arbitrary, the upper bound
follows uniformly over $q$ by the compactness of the trial paths.

The trial class used at this point is restricted to paths whose
susceptibility is strictly positive on every nonempty active space;
this is the eligibility condition. If the infimum over all eligible
$q$ is not attained, a limiting argument along an infimising sequence
uses local uniformity in the Stiefel restriction bound and in the
contact-point continuity. The expected pressure converges to the
infimum, uniformly in the pattern average.

The bound on the field cap can be chosen so as not to bind before
using any overlap identities. For fixed $0<eta<1$ and trial $q$, the
path $q'=q+eta(I-q)$ is eligible. The weighted difference against
cumulative fields is

$$sum_iw_itr((q_i'-q_i)b_i)=etasum_rtr(Delta b_rsum_{ige r}w_i(I-q_i))ge c_qsum_rtr,Delta b_r,$$

with

$$c_q=etamin_{r:F_rne{0}}{w_rlambda_{min}((I-q_r)|_{F_r})}>0,$$

where strict positivity holds because $I-q_rsucceqmathsf K_rsfK_rsucc 0$
on $F_r$. Applying the Stiefel restriction bound with $q'$, bounded
pattern energy, and the $O(e_M^2)$ perturbation cost, the contact
functional is positive at a sufficiently large fixed trace cap. Thus a
negative minimising point has $r>0$ and permits small increases of
every allowed field increment.

This closes the expected-pressure upper bound when $D$ is smooth with
compact support. Bounded continuous $D$ follows by approximation:
bounded Lipschitz functions agreeing with $D$ uniformly on compacts
give the same limit. The negative-quadratic tail class is handled by the soft-penalty construction.

Lower bound: cavity with two restored patterns. The proof follows the
extended variational principle of Aizenman, Sims, and Starr, adapting
the Aizenman-Sims-Starr cavity to the Stiefel-cascade model. The
published framing is explicit about scope: "a broader variational
principle, in which the lower bound, as well as the actual value, are
expressed through an optimization procedure for which
ultrametric/hierarchical structures form only a subset of the
variational class." A related critical-point method for vector spin
glasses with nonconvex interactions is due to Chen and Mourrat. Fix
smooth compactly supported $D$. Replace the pattern count $M$ by
$M+ell$ for a fixed cavity size $ell$ large compared to $1$ and tiny
compared to $M$. Compare the enlarged pressure with the base $M$
pressure on the same disorder, using two restored patterns per
increment.
./pdf/aizenman-sims-starr-2003-cond-mat-0306386.pdf
pdf/arxiv-2311.08980v3.pdf

The cascade with only the outer split at $theta$ and the same
monomial perturbations (amplitude $s_M=M^{3/8}$) replaces the
enriched upper-bound model. Average the expected logarithms over a
product probability law $pi$ on the perturbation coefficients with
each coefficient uniform on $[1,2]$, at every system size. The
perturbation changes a log partition function by $O(s_M^2)=o(M)$.

Generic parameters. There is a set of perturbation parameters of
$pi$-measure tending to one on which every subsequential overlap
limit of the base system with two restored patterns satisfies joint
GG. Fix one perturbation coordinate. The pressure variance is
$O(M^{-1})$; integration by parts bounds its expected first derivative
by $O(s_M^2/M)$, and integrating the second derivative controls
thermal variance. Integrated convex difference quotients with step
$s$ give

$$C_a,s,s_M^2/M+C/(ssqrt M)$$

for the integrated absolute fluctuation of the first derivative. At
$s=M^{-1/8}$ this implies

$$intmathbb E<|Y_a-bbE<|Y_a-mathbb E<bbE< Y_a>|>/s_M,dpito 0.$$

The estimates hold on a slightly larger coupling interval so endpoint
difference quotients cause no difficulty, and a diagonal selection
over countably many coordinates gives the good sets.

Restoring patterns. Fix a batch of $ell$ fresh independent patterns.
In the enlarged model, draw spin columns by Gram-Schmidt: the first
column is drawn at its prescribed vertex, and the remaining columns
are drawn at visits. Restrict all normalised triangular lengths and
cross readings to a fixed small neighbourhood of the identity. The
first restriction and each conditional inner restriction have
probability tending to one independently of the preceding frame, and
the recursion normalisation costs are $o(n)$.

The enlarged Gibbs measure with the $M$ original patterns and the
$ell$ fresh ones has self covariance close to the identity and cross
covariance close to the empirical overlap of the sampled pair. This is
the physical prediction from spatial rotation invariance; it holds in
the enlarged Gibbs law for each fixed coordinate, uniformly over the
cavity coefficient, at expected probability tending to one.

Thick shell. For fixed ratio $r_j$ of retained directions, the
entries of $Z_j$ converge to independent standard Gaussians. For any
converging bounded outer column the conditional inner statement holds
as well. Representing a uniform frame by orthonormalised standard
Gaussian columns proves the conditional version.

Fix small $ε>0$ and restrict the enlarged partition function to
the thick shell

$$mathcal I=calI={|Z_j^TZ_j/r_j-I|leεhbox{ for every }j}.$$

This only lowers its logarithm. On the shell, replace the larger
perturbation by the base perturbation at the same $u$. For fixed
$ell$, overlaps differ by $O_ell(M^{-1})$, and the summable kernel
Lipschitz constants, the amplitude $s_M$, and the exponentially small
coefficient tail together imply an $O_ell(M^{-1/4})+o(1)$ covariance
error in unnormalised logarithms. Gaussian interpolation gives a
vanishing cost both for expected logs and for bounded replica tests.

On the shell, the physical cross covariance of two fresh patterns
matches the empirical matrix overlap of the enlarged system. The self
part of the shell makes the discarded Gibbs mass small for large
$ell$ via Markov. Perturbation replacement therefore preserves the
same prediction with the base overlap on the right side.

Taylor expansion on the shell. Expansion of the fresh pattern row
fields around the base values gives a linear Gaussian field coupled
to each row of the fresh spin columns, of the form
$n_j^{-1/2}sum_{rle M}h_j^r(D'_j(xi^r))^T$. The exact denominator
$√(n_j+r_j)$ may be replaced by $√(n_j)$ because $r_j$ is
fixed. The 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 retained one-path
averages. On the shell,

$$W(X,Z)=frac12sum_jr_jc_j(X)+O(εell).$$

The first-order upper triangular part of $T_j-I$ has symmetric sum
$-Z_j^TZ_j/n_j$; at $Z_j^TZ_j=r_jI$ this is $-r_jI/(2n_j)$. The mean
fresh quadratic term is the Hessian contracted with
$Z_j^TZ_j/(2n_j)$. The diagonal specialisation is $r_jc_j(X)/2$,
and the retained matrix coefficients are bounded, so moving within
the shell costs $O(ε r_j)$. Summing over $j$ gives the
correction. Cross-species fresh quadratic terms are centered.

Canonical prediction. Restricted to the thick shell, the Gaussian
predicts one specific matrix overlap path for the fresh-coordinate
pattern pair. Conditioning on the transformed unmarked sampled shape,
the canonical spatial rows are iid jointly Gaussian across replicas;
the self covariance is $I$ and the cross covariance at common
level $i$ is $q_{j,i}^*$ from the canonical lemma. The independent root
rows are averaged in this assertion, and additive row energies with
new-pattern energies factor the tilted kernels, so new patterns do not
bias these row laws.

The comparison. Physical rotation invariance predicts that the
sampled cross overlap approximates the empirical matrix overlap of the
enlarged system. Gaussian calculation on the same thick shell
predicts that the same sampled cross overlap approximates $q_{j,i}^*$.
Both predictions hold at the sample coordinate level, and the
failure probabilities bound the weighted path distance by twice the
tolerance plus a bounded multiple of the sum. Therefore the
empirical matrix overlap of the enlarged system is close to the
canonical path $q^*$.

Summing the pieces. Let $V_D(q)$ denote the limiting expected log
increment of one independent pattern at a GG limit; the entropy $S_theta$
and the Gaussian functional $mathcal G_calG_{theta,p}$ already entered
via the canonical lemma. The cavity comparison identifies the
enlarged pressure with $ell$ times the canonical value minus $o(ell)$
error, so

$$frac1ellmathbb E[bbE[logmathcal Z_calZ_{M+ell}-logmathcal Z_M]calZ_M]gemathcal G_calG_{theta,p}(D)-o_{tol}(1).$$

Uniformity in the base size. For every $delta>0$ there is $ell_0$
such that for each integer $ellgeell_0$, every sequence of base
sizes with $n_j/Mto p_j$ and increments $r_j=p_jell+O(1)$ satisfies

$$liminf_{Mto∞}frac1ellintDelta_{M,ell}(u),dpi(u)gemathcal G_calG_{theta,p}(D)-delta,$$

where $Delta_{M,ell}$ is the log-increment of the enlarged pressure
beyond the base at parameter $u$. The constant in $O(1)$ is fixed
across these choices. Begin with any good-parameter sequence and any
compact array limit. The tolerance, shell, and rounding choices apply
to it with uniform canonical constants; failure of the eventual bound
would contradict the cavity inequality. The bad parameter sets
have $pi$-measure zero and use the fixed-$ell$ uniform lower
increment bound.

Telescoping. Along the triangular chain $M'$ with $M'equiv M$
modulo $ell$ and block dimensions $lfloor M'n_j/Mrfloor$, divide by
$M'$ and send $M'to∞$, then $ellto∞$ and $delta↓ 0$.
The same parameter law $pi$ at both endpoints makes log increments
telescope exactly; the $o(M)$ perturbation error is removable. This
proves the matching lower bound for smooth compactly supported $D$.
The bounded continuous case follows by approximation; the
convolution of the enlargement with the compact approximation equals
the pressure at the larger argument up to $o(M)$, uniformly on shells.

Maximal sub-sequential identification. Combining the upper and lower
bounds, every subsequential limit of the normalised expected log
pressure equals $mathcal G_calG_{theta,p}(D)$. Concentration of the
normalised log on a bounded window transfers this
to convergence in probability. Boundedness of $D$ and uniform
integrability bound the pressure, giving convergence in expectation.

Soft penalties and quadratic tails. The Gaussian theorem was proved for smooth compactly supported $D$,
and the bounded continuous case follows by approximation. The angular formula
also requires a negative-quadratic tail class. Extend $D$ to be the
sum of a bounded continuous function and a nonpositive quadratic form
$-C|xi|^2$. Uniform tail control for Gaussian rows under the trial
path is separate from the approximation, and both are needed.

Row tails. For every trial path $q$ and every $R<∞$ the
Gaussian row fields satisfy $mathbb EbbEexp{c|Z_j|^2}le C_{c,R}$ for
$|c|$ small enough and all rows $rle M$; the constants are uniform in
the finite grid and the trial path. Bounded smooth approximation of
$D$ on compacts, together with a Markov tail bound on the residual
exponential mass, gives an $L^1$ difference of the normalised log
partition functions tending to zero. The same comparison applies to
$V_D$ by the terminal class' explicit recursion.

Distinguished negative-quadratic. The soft-penalty Hamiltonian
$H=sum_rD(xi_r)-Mdsum_ell(barbar g_g_ell-c_ell)_+$ is exactly the
shape in which a nonpositive quadratic enters. Rewrite the soft
penalty using the convex dual of $x_+$:

$$-Mdsum_ell(barbar g_g_ell-c_ell)_+=min_{0le t_ellle d}M(tcdot c-sum_ell t_ellbarbar g_g_ell).$$

The minimum is attained because the constraint set is compact and the
objective is linear in $t$. Substituting yields a Hamiltonian of the
form $H=sum_r(D(xi_r)-tcdot g(xi_r))$ with bounded linear-$t$
coefficients. The unpenalised Gaussian theorem with terminal
$D-tcdot g$ applies uniformly in $tin[0,d]^s$ because negative
quadratic coefficients remain negative, so it covers every
$t$-terminal in one stroke.

Concentration of the average. Under the natural tilted path law at
parameter $t$, the empirical average $barbar g$g$ of the soft-constraint
functions concentrates in probability at a deterministic vector
$v(t)$, obtained as a limit of $barbar g$g$ along compactly supported
clippings. For each direction $a$ into the compact box $[0,d]^s$, the
one-sided derivative of the pressure with respect to $t$ is

$$∂_ap(t)=-acdot v(t).$$

Passage of slopes through clipping: interpolate only the increment in
the coupling. Increasing a nonnegative quadratic coupling adds a
nonpositive term; decreasing along $a_ell<0$ and $u>0$ contributes,
for a bounded clipping $g_{ell,R}$,

$$-(t_ell-u|a_ell|)g_ell-u|a_ell|(1-gamma)(g_ell-g_{ell,R}),qquad 0le gammale 1,$$

with nonpositive unbounded coefficients on both terms. Uniform tails
from the row-tail estimate make the directional slope error at most
$sum_ell|a_ell|ε_R$, with $ε_Rto 0$. This proves the
one-sided derivative formula, including at boundary directions.

Duality in closed form. Once the empirical average concentrates and
the derivative formula holds, standard convex duality in finite
dimensions gives the soft-penalty pressure as

$$min_{tin[0,d]^s}{tcdot c+mathcal G_calG_{theta,p}(D-tcdot g)}.$$

The unpenalised functional $mathcal G_calG_{theta,p}(D-tcdot g)$ is
jointly convex in $t$ by direct differentiation and continuity of the
unpenalised pressure in $h$ under the terminal class; no separate
lower-semicontinuity hypothesis is needed, because the compact box and
boundedness of $D$ on compacts ensure it. The minimum is attained, and
the value is finite and continuous in $(c,D,g)$ on compacts.

Uniform tail estimate. The same extension covers the negative
quadratic. Write $D=D_0-C|xi|^2$ with $D_0$ bounded continuous. On a
good spectral event of probability tending to one, every input has
$(Ax)$ of squared norm bounded above by $(N/M)C'^2$; on the same event
bounded smooth approximations of $D$ on compacts and the negative
quadratic agree with $D$ outside a set of exponentially small field
mass. The $L^1$ difference of log partition functions is $o(M)$
uniformly on trials. This closes the Gaussian theorem for terminal
class $D=D_0-C|xi|^2$.

Haar reduction to Gaussian fields. Let $G$ be an $Mtimes n$ matrix
with independent $N(0,1/M)$ entries, $n=sum_jn_j$. Gaussian
orthogonalization of its consecutive column blocks of sizes
$n_1,n_2,...$ gives a Haar flag
$E_1subset E_1 ⊕  E_2subset...$ of the desired dimensions.
On an event of probability at least $1-e^{-cM}$, the operator norm of
$G$ and of its inverse on its image are bounded by fixed constants
$C_{op},C_{inv}$. An elementary net argument gives
the bound. 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, with small-ball
probability at most $(Cε)^M$. Take a fixed mesh for the upper
bound and a mesh of order $ε$ for the lower bound; the strict
inequality $n/Mto p<1$ makes the small-ball exponent dominate the
net size.
pdf/arxiv-mezzadri-0609050.pdf

The volume Jacobian of $xmapsto Gx$ on the coefficient sphere is
$√(det(G^TG))$. Its logarithm divided by $M$ converges in
probability to

$$J_p=frac12int_0^plog(1-u),du.$$

Indeed the successive orthogonal residual lengths of $G$ are
independent with laws $(chi^2_{M-r+1}/M)^{1/2}$, chi-square
concentration bounds the normalised logarithms, and a Riemann sum
gives the integral.

Coefficient integral. Write $A=Udiag(s)V^T$ and put
$x=Vw$ so that $x$ is uniform on $S^N$. The column blocks of $A$
multiply $x$ block by block: $Ax=sum_jU_jdiag(s_j)V_j^Tx$.
On the Haar flag the blocks $V_j^Tx$ of a uniform coefficient decompose
as independent uniform directions in $mathbb R^bbR^{n_j}$ up to the
Jacobian. Thus the angular pressure satisfies

$$P_M^f(a)=frac1Mlogint_{S^n}e^{sum_rf((Gx)_r)}|det(G^TG)|^{1/2}dsigma_n(x)+o(1).$$

The change of measure is exact: it transfers the Haar flag weight to
the Gaussian matrix $G$ with its Jacobian. The error $o(1)$ comes from
a bounded activation and the good-norm event.

Witnesses. Introduce auxiliary unit-block vectors $z_{i,j}inmathbb
R^bbR^{n_j}$ for $jle i<h$ and put

$$u_i=sum_{jle i}G_jz_{i,j}.$$

The test $|y-u_i|^2/Mlesum_{j>i}a_j$ forces the squared norm in
each tail of the orthogonal decomposition of $y$ to be no larger than
the prescribed tail, so that each initial block carries at least its
prescribed total mass. After sorting $a_j/p_j$, the weighted shell
lemma shows that these one-sided tests recover the desired weighted
shell of radii $a_j$. The witnesses are integrated as the inner
variables of the powered Stiefel integral, with exponent $thetato 0$
removing the entropy.

Coefficient model. With bounded $f$, the coefficient integral over
$S^n$ decomposes into $intprod_jdmu_j$ where $mu_j$ is uniform on
$S^{n_j}$, and the Gaussian column blocks $G_j$ act on each $x_j$
independently. Encoding the constraint $|y-u_i|^2/Mle
sum_{j>i}a_j$ via the soft penalty at multiplier $t_ell$ and sending
the cap $lambda/theta$ to infinity gives exactly the functional
$mathcal B_calB_{p,a}(kappa,lambda,B,theta)$ after taking a supremum
over the trial coefficient frames $R_j$ that parametrise the inner
witness block-by-block.

The weighted shell lemma. Sort the numbers $a_j/p_j$ in nonincreasing
order, and suppose $b_jge 0$ satisfy

$$sum_jb_j=sum_ja_j,qquad sum_{jle i}b_jgesum_{jle i}a_jquad(1le i<h).$$

Let

$$Phi(a)=H_p(a)+P_∞(a),qquad H_p(a)=frac12sum_jp_jlog(2pi e a_j/p_j),$$

with $P_∞(a)=lim_Mprob-P_M^f(a)$ the deterministic
angular limit. Then $Phi(b)lePhi(a)$.

Proof. Put $A$ equal to the decreasing density with values $a_j/p_j$
on the successive intervals of lengths $p_j$, and let $B^*$ equal the
decreasing rearrangement of $b_j/p_j$ on the same lengths of intervals.
At each original cutpoint, the initial integral of $B^*$ dominates the
original $b$-prefix and hence the $a$-prefix. Between cutpoints the
former cumulative integral is concave and the latter affine. Thus
initial integrals of $B^*$ dominate those of $A$ everywhere, with equal
totals.

The allocation step is the finite-weight Hardy-Littlewood-Polya
convex-order principle. For the underlying rearrangement theory see
Hardy, Littlewood, and Polya's monograph; the specific weight-preserving
allocation proof is given.

Consider allocation tables $lambda_{ij}ge 0$ with column sums $p_j$
and row sums $p_i$. The entry $lambda_{ij}$ is the dimension
proportion sent from old block $j$ to new block $i$; its squared-mass
contribution is $lambda_{ij}b_j/p_j$. To show that some table attains
$a$, fix arbitrary prices $c_i$, and let $C$ equal $c_i$ on the
interval of length $p_i$. Write $C^*$ for its decreasing rearrangement.
The support function of the attainable mass vectors assigns the largest
source densities to the largest prices. Rearrangement and then
cumulative dominance give

$$sum_ic_ia_i=int_0^pCAleint_0^pC^*Aleint_0^pC^*B^*=max_lambdasum_{i,j}c_ilambda_{ij}b_j/p_j.$$

The second inequality is summation by parts; equal total mass permits
arbitrary, including negative, prices. Finite-dimensional separation
gives a table attaining $a$. Set $T_{ij}=lambda_{ij}/p_j$; then

$$Tge 0,qquad sum_iT_{ij}=1,qquad Tp=p,qquad Tb=a.$$

The relation is majorization relative to the block proportions $p$;
see the class of $d$-majorization polytopes studied by vom Ende and
Dirr. Realise it geometrically by randomly subdividing each old $E_j$
into orthogonal pieces of relative dimensions approaching $T_{ij}$,
and forming new block $i$ as the direct sum. The identities $Tp=p$
and $Tb=a$ give respectively the new dimension proportions and the
target squared masses. Dimensions with the correct two margins are
obtained by $o(M)$ adjustments and bounded rounding corrections. The
new decomposition has the same Haar marginal as the old one. For each
fixed vector in the old $b$-shell, squared norms in the pieces
concentrate about the prescribed fractions; the failure bound is
uniform in the fixed vector. Simultaneous concentration for all shell
vectors is unnecessary. Thus each such vector lies in a slightly
enlarged new $a$-shell with failure probability $ε_Mto 0$.
pdf/arxiv-1911.01061v5.pdf

Angular identity. Radial integration and Stirling's formula show that
the weighted Lebesgue volume in $E_1 ⊕ ... ⊕  E_h$ of a small
shell with $|y_j|^2/M$ near a positive $a_j$ has logarithm per $M$,
after shrinking the shell, equal to $Phi(a)=H_p(a)+P_∞(a)$. The
same radial calculation gives the upper Laplace bound on compact sets.
Set $Phi=-∞$ at a zero coordinate; the divergent radial entropy
and the bounded pressure $P_∞$ justify this convention in upper
bounds.

The weighted shell lemma plus the witness construction completes the
angular reduction. The upper bound uses the shell volume estimate and
Laplace's method. The lower bound uses the transport realisation: a
typical vector of the $b$-shell lies in the $a$-shell up to
$ε$, so the integral on the $a$-shell is at least the integral
on the $b$-shell up to $O(ε)$.

Completing the angular formula. Fix $kappa$ from the working lower
bound. Then fix any finite $lambda$ and a desired error $ε$.
The concentration rate for the coefficient integral at that error is
independent of $B,theta$. Choose $eta$ so that the entropy
$frac12logeta$ is below half this rate and so that
$2|f|_∞eta+frac12logeta<ε$. Choose shell widths and
$delta$ so the 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 $ε$. A
union bound over the candidate rotations and the concentration
estimate turns the witness lower bound into the corresponding
deterministic bound for $J_p+mathcal B_calB_{p,a}$ with one additional
$ε$ loss. Thresholds may depend on the fixed $lambda$; no
uniformity in its later limit is required.

Monotonicities. For fixed $M$, the soft integral increases as
$theta↓ 0$ 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 establish the successive
limits and

$$Phi(a)=J_p+lim_{lambdato∞}lim_{Bto∞}lim_{theta↓ 0}mathcal B_calB_{p,a}(kappa,lambda,B,theta).$$

Using the shell-volume identity for $Phi$ proves the angular formula
and identifies every subsequential $P_∞$. Equicontinuity extends
convergence to compact sets and to zero radii.

Full output dimension. 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 $2eta$ 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. Zero-radius blocks are
ignored. Letting $eta↓ 0$ recovers the $p=1$ case.

Continuity in $f$ and $a$. For $fin C_b(mathbb R)$bbR)$ 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 $O(R^{-2})$. Compact
approximation controls the normalised energy uniformly and transfers
all angular conclusions. The same estimate applies to
$mathcal B_calB_{p,a}$ on a bounded operator-norm event, independently of
$lambda,B,theta$, so its limiting formula has the same continuity.
This completes the angular theorem.

The radial decomposition. Return to the spherical integral with $A$ of
bi-orthogonally invariant law. Fix the singular-value block
approximation $s_1,...,s_h$ with multiplicities $n_j$, and let $G$
be the Gaussian flag matrix from the angular reduction. The
decomposition of $A$ yields

$$A=U,G,V^T$$

in law, with $U,V$ independent uniform orthogonal matrices, and with
$G$ acting block by block on the coefficient $x=Vw$. The spherical
coefficient $x$ decomposes into blocks of sizes $n_1,...,n_h$.
Write

$$x=√(rho_0N)theta_0+sum_j√(rho_jN/s_j^2)theta_j√(s_j^2/N),$$

where $rho_jge 0$ is the fraction of squared norm in block $j$ (with a
null block of size $N-M$ when $alpha<1$), $sum_jrho_j=1$, and the
$theta_j$ are unit vectors in the blocks. For a uniform coefficient
$x$ the joint law factors: the $theta_j$ are independent uniform unit
vectors in the blocks, and $rho=(rho_j)$ is Dirichlet with
parameters $(n_j/2)$ (including the null block when it is present).

Under this split, the measurement is
$Ax=sum_jU_j(s_j√(Nrho_j/alpha)),G_jtheta_j$ plus a null
contribution. Block $j$ has image subspace $E_j=U_jmathbb R^bbR^{n_j}$ of
Haar dimension $n_j$ in $mathbb R^M$,bbR^M$, and the vector
$y_j=s_j√(Nrho_j)G_jtheta_j$ is uniform on the sphere of radius
$√(Ma_j)$ in $E_j$ with $a_j=s_j^2rho_j/alpha$. The $y_j$ are
mutually independent once $rho$ is fixed.

Pressure as a Dirichlet supremum. Substitute into the pressure and use
the independence. On the coefficient sphere the factor
$|det(G^TG)|^{1/2}$ contributes the Jacobian rate $J_p$. The
normalised log of the integral over shell radii equals

$$sup_{substack{rho_i>0sum_{iinmathcal I}calIrho_i=1}}{frac12sum_{iinmathcal I}calIv_ilog(rho_i)/(v_i)+alpha P_p^f(((s_j^2rho_j)/(alpha))_{j=1}^h)}+o(1),$$

where $v_j=alpha p_j$, $v_0=1-alpha$ when $alpha<1$, and
$mathcal I=calI={1,...,h}$ when $alphage 1$ and
$mathcal I=calI={0,1,...,h}$ when $alpha<1$. The first term is the
Dirichlet entropy of the split of squared norm, whose density near
$rho$ is $rho_1^{n_1/2-1}...$, contributing
$frac12sum_jp_jlog(rho_j/p_j)$ plus a normalisation giving
$frac12v_jlog(rho_j/v_j)$ after the substitution. The second term is
the angular pressure times $alpha$, converting per-$M$ pressure to
per-$N$ pressure.

The supremum is attained, continuous, and bounded, because the
Dirichlet entropy is strictly concave on the simplex and the angular
pressure is bounded continuous by the angular theorem. The supremum's
value depends continuously on the vector $s=(s_1,...,s_h)$ and on
the multiplicities $p_j$, so the finite-block approximation is well
behaved.

Spectral mesh limit. Approximate the limiting singular measure $nu$ by
finitely many representative values, with mesh decreasing to zero and
multiplicities equal to the mass assigned by $nu$. The no-outlier
assumption guarantees that the rectangular singular slots lie in the
support of $nu$, so the representatives can be chosen positive
throughout. Operator-norm continuity of $F_N$, together with the
$o(M)$ dimension slack from structural zeros, transfers the
finite-block limits to the full problem. The polynomial rate of the
mesh approximation matches the operator-norm continuity constant, so
the two limits separate. This completes the proof of the main
theorem.

The mean convergence follows from boundedness: the pressure is
uniformly bounded by $|f|_∞$, so probability convergence of
$sup$ and $inf$ plus boundedness gives convergence of expectations.
No separate argument is needed.

Extensions and boundaries. The main theorem constrains the aspect
ratio, the singular spectrum, and the activation. Several of these
constraints can be relaxed and one cannot.

Arbitrary aspect ratio. The limit formula holds for any deterministic
$(M_N,N_N)$ with $M_N/N_Ntoalphain(0,∞)$. The angular and
Dirichlet arguments use only limiting proportions. The additional case
is $alpha=1$ with $N>M$ and $k=N-M=o(N)$ structural right-null
coordinates. Let $B$ be the restriction of $A$ to the $M$ visible
input coordinates, and let $Z_M(r)$ be the angular partition function
of $B$ at radius $r$. If $b=|f|_∞$, the sphere in $mathbb R^N$bbR^N$
decomposes as

$$x=√(v),x'+u,qquad x'inmathbb R^M,bbR^M,quad uinmathbb R^k,$$bbR^k,$$

with $v=|x|^2/NsimBeta(M/2,k/2)$ and $x'$ uniform on
$S^{M-1}$ conditionally. The measured field at radius $√(Nv)$ has
squared norm approximately $alpha s_N^2v$ when $s_N$ is the typical
singular value. The endpoints $v$ small and $v$ near one contribute
negligibly because the Beta density makes such values exponentially rare at
the field-energy scale. Integration by parts over $v$ converts the
sphere integral into a ball integral of $Z_M(r)$ over
$r^2in[alpha s_-^2v_{rm lo},alpha s_+^2v_{rm hi}]$ times a
Jacobian. The ball integral differs from the sphere integral by
$frac12log$ of an explicit Beta-type factor, which is exactly the
Dirichlet entropy of a null block of mass $1-alpha$. The same radial
formula holds with $M$ denominators and the extra null block providing
the endpoint.

Edge intervals and sublinear outliers. The assumption that every
singular value stays in $supp,nu$ weakens as follows. Let
$s_-=minsupp,nu$ and $s_+=maxsupp,nu$. The
requirement becomes

$$max_{rlemin(M,N)}dist(s_{N,r},[s_-,s_+])to 0quadhbox{in probability}.$$

The edge interval suffices. If a sublinear number of singular values
leak outside $[s_-,s_+]$, one or two extra Haar directions act as
fake blocks. The leaked weight is assigned to those directions, and the negative
Dirichlet entropy $frac12v_ilog(rho_i/v_i)$ with $v_i=o(1)$
absorbs the effect as $rho_ito 0$. When $alpha<1$, the interval may
be $[0,s_+]$, because the macroscopic input null space provides a
legitimate zero endpoint. Forced output zeros do not provide such an
endpoint. The degenerate case $s_-=s_+=0$ gives
$F_{alpha,nu}(f)=alpha f(0)$.

Unbounded activations with subquadratic growth. The theorem assumes
$f$ bounded. It relaxes to $|f(z)|/(1+z^2)to 0$ with $f$ continuous.
Conditional on input coefficients and singular values, Haar left
rotation makes the field a uniformly random vector on the sphere of
radius $lambda_xsqrt M$ where $lambda_x=|Ax|/sqrt M$. On the
good-norm event, $|A|_{op}le C$ and at least $kappa N$
input directions have singular value at least $s_*>0$. For $L>0$
there is small $lambda_0>0$ with

$$sigma_N{lambda_x<lambda_0}le e^{-LM},$$

independently of the spectrum. Conditional on input coefficients and
singular values, the density of any $k<M-2$ entries of the field on
the sphere of radius $lambda_xsqrt M$ is

$$frac{Gamma(M/2)}{Gamma((M-k)/2)(pi Mlambda_x^2)^{k/2}}(1-(|z|^2)/(Mlambda_x^2))_+^{(M-k-2)/2}.$$

For $kle M/2$ and $lambda_xgelambda_0$, this is at most
$(C_0/lambda_0)^k$. If a measurable set $B$ has Lebesgue measure at
most $delta$, then with $k=lceilgamma Mrceil$ and $0<gamma<1/2$,

$$mathbb PbbP{#{r:(Ax)_rin B}ge k}lebinom{M}{k}(C_0delta/lambda_0)^k.$$

Choosing $R$ with $Λ^2/R^2legamma$ and approximating $f$ in
Lebesgue measure on $[-R,R]$ by continuous $g$ with $|g|le b$ gives
the extension. The exceptional inputs have exponentially small
probability and negligible partition mass; on the remaining inputs the
normalised energy difference is bounded by $ε+4bgamma$, which
tends to zero along a sequence of approximants.

Bounded Borel activations when the limiting spectrum is not a point
mass at zero. A bounded Borel $f$ extends without continuity provided
$nunedelta_0$. The proof uses the sphere-of-radius presentation and
the small-field estimate. Set $b=|f|_∞$. Since
$nunedelta_0$, the good-spectrum event has
$|A|_{op}le C$ and at least $kappa N$ input directions
with $sge s_*>0$. The small-field set has
$sigma_N{lambda_x<lambda_0}le e^{-LM}$, and the field entries on
the sphere satisfy the concentration bound with $C_0/lambda_0$. Any
bounded Borel $f$ therefore approximates by bounded continuous $g$ up
to exponentially small partition mass.

Failure modes. The extensions are strict. Superquadratic activations
can make the pressure diverge even on a spectrum that converges to
zero, so the subquadratic growth hypothesis is needed. The bounded
Borel extension fails when $nu=delta_0$: even the indicator
${zne 0}$ can oscillate between two values along a sequence whose
rectangular singular slots converge weakly with no outlier in the
support of the limiting measure. Sublinear outliers that hide behind
structural output zeros break the edge interval hypothesis when the
zero endpoint is not a legitimate input null. Four worked counterexamples follow.

Counterexample: forced output zeros hide a small singular outlier.
Let $M=1000N$ so $alpha=1000$. On even $N$ take $A_N$ equal to
$sqrtalpha$ times a Haar orthonormal $Mtimes N$ frame; on odd $N$
set one singular value to zero and keep the rest Haar. Let
$f(z)=-cmin(z^2,1)$ with $cge 1$. On even $N$,
$A=sqrtalpha U$ with $U$ a Haar orthonormal frame. For a unit input
$u$, $sum_rf((Au)_r)=-csum_rmin(M(Uu)_r^2,1)$, and a Gaussian tail
plus a small net on $S^{N-1}$ gives
$inf_u(1/M)sum_rmin(M(Uu)_r^2,1)ge 1/80$ with probability at least
$1-e^{-c_0M}$. So the even-$N$ pressure is at most $-calpha/80$,
which is $-12.5c$ for $alpha=1000$. On odd $N$ one input direction
maps to zero; a uniform input decomposes as
$sqrt Vu_{rm live}+√(1-V)u_{rm null}$ with
$VsimBeta((N-1)/2,1/2)$. The event $Vle v$ has
log-probability per $N$ tending to $frac12log v$, and on this event
the energy is at least $-cv$, giving

$${rm pressure}ge tfrac12log v -cv = -0.5log 160-6.25capprox -2.5-6.25c.$$

For $c=1$ the even-$N$ pressure is at most $-12.5$ while the odd-$N$
pressure is at least $-8.75$, so the limits differ. The rectangular
singular measure excluding the structural zero still converges to
$delta_{sqrtalpha}$, but the output eigenvalue measure hides the
outlier in the zero mass. No-outlier control must be on the rectangular
singular slots, not on output eigenvalues. The uniform lower bound
$inf_u Q(u)ge 1/80$ is a Kashin-type spreading estimate for Haar
frames; a norm-comparison restatement in the lasso analysis of Bayati
and Montanari gives the same conclusion with explicit constants.
pdf/bayati-montanari-lasso.pdf

Counterexample: unbounded limiting spectrum. Let $d_k=exp(8^k)$ and
$nu=sum_{kge 1}2^{-k}delta_{d_k}$ with unbounded support. For
large $N$ choose $K=lfloorlog_2log Nrfloor$ and multiplicities
$n_j=lfloor N2^{-j}rfloor$, with the top block about $N/2$. Let $f$
be continuous on $[0,1]$ equal to $c_Kin{0,1}$ on
$[d_K^{1/2},d_K^2]$, alternating in $c_K$ with $K$, interpolating in
the gaps. The top block has singular value $d_K=exp(8^K)$ and its
field values are of size $d_K$, so on the top block $f$ evaluates to
$c_K$ almost surely; smaller blocks contribute negligibly. The
pressure along the sub-$K$ sequence alternates between the values
$c_Kin{0,1}$. Weak convergence to $nu$ plus the outlier bound is
not enough when $nu$ has unbounded support; compact support is
essential.

Counterexample: bounded Borel at zero spectrum. Let $M=N$ and set
$A_N=0$ on even $N$ and $A_N=s_NO_N$ on odd $N$ with $s_Nto 0$ and
$O_N$ Haar orthogonal. Let $f(z)=1_{{zne 0}}$. On even $N$ the
field is zero, $f(0)=0$, so the pressure is zero. On odd $N$ the field
is a nonzero multiple of a Haar vector, so all coordinates are nonzero
almost surely, $f((A_Nx)_r)=1$ for every $r$, and the pressure is
$1$. Both spectral sequences have rectangular singular slots
converging to $delta_0$ with no outliers. Bounded Borel activations
with $nu=delta_0$ fail.

Counterexample: continuous activation without a growth bound. Let
$M=N$ and set $A_N=0$ on even $N$, $A_N=N^{-1/4}O_N$ on odd $N$ with
$O_N$ Haar. Let $f(z)=z^8$. On even $N$ the pressure is zero. On odd
$N$, $|A_N|_{op}=N^{-1/4}$ and
$sum_r(A_Nx)_r^8=N^{-2}sum_rΩ_r^8$ with $Ω=O_Nx$. The
coordinate event $Ω_1^2in[N/4,N/2]$ has probability at least
$frac18 2^{-(N-3)/2}$ and on it $Ω_1^8ge N^4/256$, so

$$sum_r(A_Nx)_r^8ge N^2/256,$$

and therefore

$${rm pressure}ge frac1N[(N^2)/(256)+log(frac18 2^{-(N-3)/2})]=frac N{256}-frac12log 2-(3log 2)/(2N)to+∞.$$

The pressure diverges on odd $N$ while it is zero on even $N$, and both
spectra converge to $delta_0$ with no outliers. Continuous activation
without a growth bound diverges.