OpenAI claim summary for Math-2883

The all-temperature spherical Gaussian perceptron pressure

OpenAI's claim

OpenAI presents a proof in the selected manuscript. The spherical Gaussian perceptron is claimed to have an exact limiting-pressure formula at every positive density and temperature for bounded continuous pattern potentials.

Mathematical statement of the claimed result

Result 1.

Fix $\alpha,\beta>0$, bounded continuous real $\phi$, $M_N=\lfloor\alpha N\rfloor$, and independent standard Gaussian pattern vectors $g^a$. On $S_N=\{x:|x|^2=N\}$, with uniform probability measure, set $H_N(x)=\sum_{a\le M_N}\phi(g^a\cdot x/\sqrt N)$ and $p_N=N^{-1}\log\int e^{\beta H_N}d\sigma_N$. For nondecreasing measurable $m:[0,1]\to[0,1]$, define \[ D_m(t)=\int_t^1m(s)ds,\quad S(m)=\tfrac12\int_0^1\left(D_m(t)^{-1}-(1-t)^{-1}\right)dt, \] \[ V_f(m)=\sup_v\mathbb E\left[f(B_1+\int_0^1m(t)v_tdt)-\tfrac12\int_0^1m(t)v_t^2dt\right], \] where controls are progressively measurable with finite expected cost. The claim is \[ \mathbb E p_N\to\inf_m[\alpha V_{\beta\phi}(m)+S(m)]=\mathcal P, \qquad p_N\to\mathcal P\text{ in probability}. \] Trials of infinite entropy cost have value infinity; $m$ need not equal one near the endpoint.

Outline of the argument

Pattern-addition and coordinate-addition cavity increments are treated separately because the nonlinear Hamiltonian is not Gaussian. Perturbative overlap identities and ultrametricity reduce the array to finite cascades, whose row functional and spherical entropy give the matching variational bounds.

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. The listed Comparator files are IsingFiniteness.lean, PerceptronFreeEnergy.lean, SphericalField.lean. The inventory does not supply a named declaration for these Comparator entries. 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 free energy of the spherical random perceptron, 2026-09-24; citation key OAI:The-free-energy-of-the-spherical-random-perceptron-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• Family 222: Lean scope.
• Comparator IsingFiniteness.json.
• Comparator IsingFiniteness.lean.
• Comparator PerceptronFreeEnergy.json.
• Comparator PerceptronFreeEnergy.lean.
• Comparator SphericalField.json.
• Comparator SphericalField.lean.
• OpenAI catalogue entry.