Spherical perceptron pressure with orthogonally invariant disorder

Spherical perceptron pressure with orthogonally invariant disorder

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. A spherical perceptron with bi-orthogonally invariant disorder is claimed to have an explicit pressure formula for bounded continuous activation and compact singular-value limits without outliers.

Mathematical statement of the claimed result

Result 1.

Let $M/N\to\alpha>0$, $r_N=\min(M,N)$, and let the law of a real $M\times N$ matrix $A_N$ be invariant under left and right orthogonal multiplication. Suppose its singular values $s_{N,r}$ satisfy, in probability, \[ r_N^{-1}\sum_{r=1}^{r_N}\delta_{s_{N,r}}\Longrightarrow\nu,\qquad \max_{r\le r_N}\operatorname{dist}(s_{N,r},\operatorname{supp}\nu)\to0, \] for a deterministic compactly supported law $\nu$ on $[0,\infty)$. For bounded continuous $f$, with $\sigma_N$ the uniform probability on $\|x\|^2=N$, put \[ F_N=N^{-1}\log\int e^{\sum_{r=1}^Mf((A_Nx)_r)}\,d\sigma_N(x). \] Then $F_N$ converges in probability and mean to a deterministic $F_{\alpha,\nu}(f)$. Its finite spectral-block formula is \[ \sup_{\rho_i>0,\ \sum_i\rho_i=1} \left\{\frac12\sum_iv_i\log(\rho_i/v_i)+ \alpha P_p^f\left((s_j^2\rho_j/\alpha)_{j=1}^h\right)\right\}. \] Here positive singular-value blocks have $n_j/M\to p_j>0$, $v_j=\alpha p_j$, and $\sum_jp_j=\min(1,1/\alpha)$; when $\alpha<1$ an input null block has $v_0=1-\alpha$. The angular functional $P_p^f(a)$ is the deterministic limit, as $M\to\infty$, of \[ M^{-1}\log\int\exp\left(\sum_{r=1}^Mf\left(\sum_jy_{j,r}\right)\right) \prod_jd\sigma_{E_j,Ma_j}(y_j), \] where mutually orthogonal Haar subspaces $E_j\subset\mathbb R^M$ have dimension ratios $p_j$, and $\sigma_{E_j,Ma_j}$ is uniform probability on their spheres of squared radius $Ma_j$ (a point mass for $a_j=0$). Taking increasingly fine operator-norm spectral approximations gives $F_{\alpha,\nu}(f)$. Blocks of vanishing proportion are omitted.

Outline of the argument

Marked probability cascades encode the overlap hierarchy while orthogonal invariance controls coordinate cavity increments. Conditional change-of-measure calculations and spectral data determine the disorder contribution; smooth compactly supported activations are handled first and then approximation removes that regularity.

Formal verification

For family 222, the spherical Gaussian perceptron pressure formula is included. The Ising companion entry only proves finite variational value for bounded continuous log-potentials; it neither gives its pressure convergence nor the bounded-Borel extension. A spherical-field supplement supplies a dual approximation formula. No Comparator is assigned to the selected manuscript here; the family page selects other papers or supporting statements. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

• OpenAI, The spherical perceptron with bi-orthogonally invariant disorder, 2026-09-24; citation key OAI:The-spherical-perceptron-with-bi-orthogonally-invariant-disorder-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main; Statement source: prop:aspect.
• Family 222: Lean scope.
• OpenAI catalogue entry.