All-temperature pressure for invariant Ising spin glasses

Source and verification

This article explains family 234 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

Overview

A quadratic form is determined by eigenvalues and eigenvectors, while an Ising model samples it only at cube vertices. Family 234 claims an all-temperature pressure formula when the eigenvectors are uniformly random and the spectral law has a compact limit without outliers.

🏷️ Preliminaries: random orientation and pressure

Let ${\Lambda }_N$ be diagonal and $U_N$ a uniformly distributed orthogonal matrix. The interaction is $U_N^{\mathsf{T}}{\Lambda }_N{U}_N$. The probability-normalized Ising pressure is

$$\begin{array}{c}{J}_N=U_N^{\mathsf{T}}{\Lambda }_N{U}_N,\\ Z_N=2^{-N}\sum _{\sigma \in \{-1,1{\}}^N}\exp \left(\frac{1}{2}{\sigma }^{\mathsf{T}}{J}_N\sigma \right),\\ P_N=N^{-1}\log Z_N.\end{array}$$

Temperature is absorbed into ${\Lambda }_N$. Assume the empirical eigenvalue law converges weakly to a compactly supported measure $\mu$, and the extreme eigenvalues converge to the corresponding support edges. A vanishing fraction of spectral outliers could escape weak convergence and still affect the cube optimization, so the edge assumption is additional.

Replicas are independent Gibbs samples conditional on the interaction. Their overlap is $N^{-1}\sigma \cdot \tau$. A trial overlap quantile is a nondecreasing measurable function $p:(0,1)\to [0,1]$.

🏷️ The spectral transform

For $x>0$, solve

$$\int \frac{\mu (d\lambda )}{b(x)-\lambda }=x$$

with $b(x)$ above the upper support edge. If no such solution exists, set $b(x)$ equal to that edge. Define

$$\begin{array}{c}R(x)=b(x)-x^{-1},\\ R(0)=\int \lambda \mu (d\lambda ).\end{array}$$

The edge extension is part of the source’s definition, not an assumed analytic continuation. The final functional needs only $x\in [0,1]$.

For a quantile $p$, put

$$\begin{array}{rl}{D}_p(r)& =1-r-{\int }_0^1(p(s)-r{)}_{+}ds,\\ G(p)& =\frac{1}{2}{\int }_0^{1}R(D_p(r))dr.\end{array}$$

Since $(p(s)-r{)}_{+}\le 1-r$, one has $D_p(r)\ge 0$. This term encodes the interaction’s spectral information.

🏷️ The one-spin entropy term

The entropy contribution is defined through hierarchical Gaussian fields. For a nonnegative nondecreasing step function $h$, partition $(0,1)$ into intervals of lengths $w_i$ with values $h_i$, and let ${\zeta }_i={\sum }_{\ell <i}{w}_{\ell }$, $h_{-1}=0$. The elementary averaging operator is

$$\begin{array}{c}{T}_{a,v}F(z)\\ =\{\begin{array}{ll}{a}^{-1}\log \mathbb{E}{e}^{aF(z+\sqrt{v}G)},& a>0,\\ \mathbb{E}F(z+\sqrt{v}G),& a=0,\end{array}\end{array}$$

where $G$ is standard Gaussian. Start with $F_{k-1}(z)=\log \cosh z$ and recurse downward using $F_{i-1}=T_{{\zeta }_i,h_i-h_{i-1}}{F}_i$. Define

$$\begin{array}{c}\ell (h)=F_{-1}(0)-h_{k-1}/2,\\ S(p)\\ =\underset{h}{\sup }\left\{\ell (h)+\frac{1}{2}{\int }_0^{1}p(s)h(s)ds\right\}.\end{array}$$

This part is independent of $\mu$. Infinite trial values are allowed.

Claimed pressure limit

Under the stated spectral assumptions,

$$\begin{array}{c}{P}_N⟶\mathcal{F}(\mu )\\ =\underset{p}{\inf }\{S(p)+G(p)\}\end{array}$$

almost surely and in expectation, at every fixed temperature. Independence of the orientations across dimensions is not required. The order of this infimum and the entropy supremum is retained. (OpenAI, 2026)

🏷️ Normalization checks and extensions

If every eigenvalue equals $c$, the quadratic energy is $cN/2$ for every spin, so $P_N=c/2$ exactly. The transform is correspondingly $R(x)=c$. This simple case tests the diagonal and probability-prior conventions.

Changing to counting measure adds $\log 2$. Removing the interaction’s diagonal subtracts $\frac{1}{2}\int \lambda \mu (d\lambda )$ in the limit, because ${\sigma }_i^2=1$. Neither adjustment changes the spin-dependent part of the finite Gibbs law.

The paper also treats deterministic external fields whose empirical laws converge in first-moment Wasserstein distance, and random spectra with the specified conditional uniform orientation. These are additional hypotheses, not automatic consequences of weak convergence alone.

For a fixed unscaled energy $E_N$, let $e_N={\max }_{\sigma }{E}_N(\sigma )/N$. Probability normalization gives

$$e_N-\frac{\log 2}{\beta }\le \frac{P_N(\beta )}{\beta }\le e_N.$$

Thus the zero-temperature error is bounded uniformly in $N$. This is the elementary bridge used to extract a ground-state limit from the all-temperature pressure formula.

🏷️ Structure of the variational proof

The source first discretizes the spectrum into finitely many groups. In addition to total overlaps, it tracks overlaps projected onto each spectral group. Small perturbations enforce overlap identities; synchronization and rotation identities express the group information through one common quantile.

The upper comparison enriches the model with hierarchical Gaussian fields. The lower bound adds cavity coordinates and must show that the resulting field supports the entropy dual at the same physical overlap path. These two uses of the quantile cannot be optimized independently without justification. Spectral approximation and tail estimates then remove the finite-alphabet restriction while retaining the no-outlier condition.

🏷️ Key steps and proof outlines

The central issue is to make the spectral energy and the one-spin entropy use the same overlap quantile. An arbitrary Gaussian field gives only one candidate in the entropy supremum. The upper proof obtains an order between quantiles by a constrained minimum; the cavity proof identifies the maximizing field by additional spin tests (OpenAI, 2026).

Resolvent convexity orders the interaction energies

First take a finite spectral law $\mu ={\sum }_a{\rho }_a{\delta }_{{\lambda }_a}$ with all ${\rho }_a>0$. At interpolation time $t$, put $F_t(x)=xR(tx)$. The rotation identities, after synchronization of spectral-group overlaps, identify the limiting interaction energy as

$${\mathcal{E}}_t(p)=\frac{1}{2}{\int }_0^1{F}_t^{\prime }(D_p(r))dr.$$

Why does this energy have a useful order? If $b_t(x)$ solves the resolvent equation for eigenvalues $t{\lambda }_a$, set $M_j={\sum }_a{\rho }_a(b_t-t{\lambda }_a{)}^{-j}$. Implicit differentiation gives

$$b_t^{\prime }=-M_2^{-1},(x{b}_t{)}^{\prime \prime }=\frac{2(M_1{M}_3-M_2^2)}{M_2^3}\ge 0.$$

The last inequality is Cauchy–Schwarz applied to the first and third inverse powers. Since $F_t=(x{b}_t-1)/t$ for $t>0$, it is convex, with continuous endpoint extensions.

For a nondecreasing quantile,

$${\int }_0^1(p(u)-r{)}_{+}du=\underset{0\le s\le 1}{\max }\left\{{\int }_s^{1}p(u)du-(1-s)r\right\}.$$

Thus larger upper-tail integrals of $p$ give smaller $D_p$. Monotonicity of $F_t^{\prime }$ then reverses the energy order: if every upper tail of $p$ dominates that of $q$, ${\mathcal{E}}_t(p)\le {\mathcal{E}}_t(q)$. Also ${\partial }_t[tR(tx)]=F_t^{\prime }(x)$, so the spectral functional is the time primitive of this energy.

A negative minimum would force incompatible derivatives

Fix a finite step trial $q$ whose maximum is below one. Enrich the model with a hierarchical field $h$ on the same partition. Suppose its proposed upper bound fails. The source then minimizes an objective consisting of minus the enriched pressure, minus $\frac{1}{2}\int qh$, plus $S(q)+G_t(q)$, a strict positive term $t\delta$, and small perturbation penalties. Failure supplies a negative minimum bounded away from zero.

At $t=0$, the entropy supremum makes this objective asymptotically nonnegative, so a negative minimum has $t$ bounded away from zero. The field cap is also inactive: the Gaussian recursion and $\log \cosh z\le |z|$ give a lower bound for the objective of the form $c_{q}H-O(\sqrt{H})-O_q(1)$ when the top field level is $H$. Here $c_q>0$ uses $\max q<1$. Choose $H$ after fixing the trial partition.

It is therefore feasible to increase the field in every nonnegative nondecreasing step direction $g$ on the fixed trial partition. The Gaussian derivative, including its diagonal subtraction, gives at the limiting minimum

$${\int }_0^{1}p(s)g(s)ds\ge {\int }_0^{1}q(s)g(s)ds.$$

Taking upper-tail indicators proves the needed order at partition endpoints. The upper-tail integral of $p$ is concave and that of $q$ is affine between endpoints, extending the order to every point. Hence ${\mathcal{E}}_t(p)\le {\mathcal{E}}_t(q)$. But decreasing $t$ is also feasible at the same minimum, whereas its limiting derivative is

$$-{\mathcal{E}}_t(p)+{\mathcal{E}}_t(q)+\delta \ge \delta >0.$$

This contradicts the derivative sign at a minimum. Rounding an arbitrary trial down to finite steps finishes the upper bound without interchanging the field supremum and quantile infimum.

Cavity spin tests identify the entropy maximizer

For rational spectral proportions, add a fixed block of $n$ spins while adding exactly $n{\rho }_a$ directions to spectral group $a$. Fresh directions inside the old eigenspaces couple the two systems and become the Gaussian cavity variables. Resolvent compression and a Schur determinant calculation evaluate the cavity factor at

$$h_p(s)={\int }_0^{p(s)}{R}^{\prime }(D_p(r))dr.$$

For a step quantile, its normalized log factor is

$$\ell (h_p)+\frac{1}{2}{\int }_0^{1}p(s)h_p(s)ds+G(p).$$

This alone falls short of $S(p)+G(p)$, because $S$ is a supremum. The missing assertion comes from testing a cavity spin pair against arbitrary bounded continuous functions of the overlap. Site permutation symmetry identifies the cavity-pair expectation with the total overlap. After the cavity comparison this says that the field’s conditional two-spin product $B_{h_p}$ agrees with $p$.

There is a small measurability point in that conclusion: the tests initially determine only conditional means given $p$. Value roundings of $p$ make the field and its response constant on each plateau of $p$; monotonicity gives compactness. Their limits are functions of $p$, so the test identities imply actual equality almost everywhere. The concavity supporting inequality

$$\ell (v)\le \ell (h_p)-\frac{1}{2}{\int }_0^{1}p(s)(v(s)-h_p(s))ds$$

then proves that $h_p$ attains the entropy supremum in the bounded-path completion. Integrate cavity increments over one common perturbation parameter space so they telescope exactly along the dimension progression. The perturbations cost $o(N)$ in log partition function and disappear after normalization.

Spectral approximation needs more than weak convergence

For spectra of the same size and the same orientation,

$$|P_N(\Lambda ,U)-P_N(\tilde{\Lambda },U)|\le \frac{1}{2}\Vert \Lambda -\tilde{\Lambda }{\Vert }_{\mathrm{op}}.$$

This handles small eigenvalue movements but not changes in multiplicities. For a fixed alphabet with counts $k_a,{\tilde{k}}_a$, Haar symmetry distributes the expected squared spin projection equally among axes within each eigenspace. Convexity of the pressure along the spectral interpolation bounds the difference by endpoint derivatives, yielding

$$|\mathbb{E}{P}_N(k)-\mathbb{E}{P}_N(\tilde{k})|\le \frac{L}{2}\underset{a}{\max }\frac{|k_a-{\tilde{k}}_a|}{\min (k_a,{\tilde{k}}_a)},$$

where $L$ is the alphabet’s spectral diameter. This removes rational-proportion restrictions.

For general bounded spectra, clip the small edge excesses, discretize intervals whose endpoints have zero limiting mass, and bracket eigenvalues in zero-mass intervals using the extreme representatives. The spectral transform converges uniformly, including its capped upper-edge branch. Near zero this uses the uniform expansion $R_{\nu }(x)=\int \lambda d\nu +O(K^{2}x)$ for support in $[-K,K]$; away from zero it follows by monotonicity of the resolvent and convergence of the upper support edge. The resulting uniform bound on $G$ permits taking the variational infimum. This is where the no-outlier assumption enters the passage from finite alphabets to the stated theorem.

See Also

• Family 222 treats nonlinear pattern interactions and a separate invariant-matrix spherical model.
• Family 221 introduces the probabilistic meaning of hierarchical cavity messages.

🏷️ Manuscripts and verification scope

• All-temperature pressure for orthogonally invariant Ising spin glasses — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The scope page describes the pressure formula under its spectral hypotheses, together with specified field, random-spectrum, and ground-state extensions. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. All-temperature pressure for orthogonally invariant Ising spin glasses.