OpenAI claim summary for Math-2893

The thermodynamic pressure of every bounded Gaussian Ising perceptron activation

OpenAI's claim

OpenAI presents a proof in the selected manuscript. The Ising Gaussian perceptron is claimed to have an exact pressure formula for every bounded Borel log-potential at positive temperature and density.

Mathematical statement of the claimed result

Result 1.

Fix $\alpha>0$ and any bounded Borel $f:\mathbb R\to\mathbb R$. Let $M=\lfloor\alpha N\rfloor$, and take independent standard Gaussian vectors $g^1,\ldots,g^M\in\mathbb R^N$. With uniform spin probability, \[ p_N(f)=N^{-1}\log\left(2^{-N}\sum_{x\in\{-1,1\}^N} \exp\sum_{a=1}^Mf(g^a\cdot x/\sqrt N)\right). \] For $G\sim N(0,1)$ define \[ \mathcal T_{s,d}U(x)=\begin{cases}d^{-1}\log\mathbb Ee^{dU(x+\sqrt sG)},&d>0,\\ \mathbb EU(x+\sqrt sG),&d=0.\end{cases} \] Let $\mathcal Q$ contain nondecreasing paths $q:(0,1)\to[0,1]$. For a step path with values $q_i$ on $\zeta_i<u<\zeta_{i+1}$, $0=\zeta_0<\cdots<\zeta_k=1$, append $q_{-1}=0,q_k=1$, start $U_k=f$, and recurse $U_{i-1}=\mathcal T_{q_i-q_{i-1},\zeta_i}U_i$, $i=k,\ldots,0$. Define $V_f(q)=U_{-1}(0)$, using its continuous path extension for general $q$. For a finite nonnegative nondecreasing step field $h$, instead start $U_{k-1}(x)=\log\cosh x-h_{k-1}/2$, append $h_{-1}=0$, and recurse $U_{i-1}=\mathcal T_{h_i-h_{i-1},\zeta_i}U_i$. Put $\ell(h)=U_{-1}(0)$ and \[ S_{\mathrm I}(q)=\sup_h\left\{\ell(h)+\frac12\int_0^1h(u)q(u)\,du\right\}. \] The finite limiting value is \[ \lim_N\mathbb Ep_N(f)=\inf_{q\in\mathcal Q}\{\alpha V_f(q)+S_{\mathrm I}(q)\}=:\mathcal P_f(\alpha), \qquad p_N(f)\xrightarrow{\mathbb P}\mathcal P_f(\alpha). \] In particular $f=\beta\phi$ is allowed for every $\beta>0$ and bounded continuous $\phi$.

Outline of the argument

Hierarchical Gaussian enrichment and overlap identities produce a cascade description. Contact-point supersolution estimates for the pattern-density evolution supply the nonconvex upper bound, while cavity calculations give the matching lower bound and approximation treats merely Borel potentials.

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 Ising random perceptron, 2026-09-24; citation key OAI:The-free-energy-of-the-Ising-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.