Universal gap and force exponents at negative spherical-perceptron jamming

Universal gap and force exponents at negative spherical-perceptron jamming

OpenAI's claim

OpenAI presents a proof in the selected manuscript. At spherical-perceptron margin $-1$, the proposed threshold and microscopic gap and force laws have exponents $\gamma=(2+\theta)^{-1}$, approximately $0.412693$ and $0.423107$.

Mathematical statement of the claimed result

Result 1.

Let $M=\lfloor\alpha N\rfloor$ and let $A$ have independent entries of law \[ \nu_\varepsilon=\tfrac12N(0,1-\varepsilon)+\tfrac12N(0,1+\varepsilon). \] For $\|x\|^2=N$, set $g_\mu=A_\mu x/\sqrt N$, $h_\mu=1+g_\mu$, and $H_N(x)=\frac12\sum_{\mu=1}^M(-h_\mu)_+^2$. Use spherical Gibbs probability proportional to $e^{-\beta H_N}$. SAT means that $H_N$ vanishes somewhere on the sphere. For $0<\delta<u$ and $s>0$, define \[ G_{N,\beta,\alpha}(\delta,u)=\mathbb E\left\langle M^{-1}\sum_\mu\mathbf1_{\{\delta<h_\mu\le u\}}\right\rangle. \] On $C(x)=\{\mu:h_\mu<0\}$ put $\bar r=|C|^{-1}\sum_{\mu\in C}(-h_\mu)$ and $f_\mu=-h_\mu/\bar r$. Set \[ F_{N,\beta,\alpha}(s)=\mathbb E\left\langle |C|^{-1}\sum_{\mu\in C}\mathbf1_{\{f_\mu\le s\}}\right\rangle, \] with value zero when $C$ is empty. There exist $\alpha_c>0$ and $0<\varepsilon_0<1$ such that, for each fixed $0\le\varepsilon<\varepsilon_0$, SAT probability tends to one for $\alpha<\alpha_c$ and zero for $\alpha>\alpha_c$. The ordered limits \[ G_J(u)=\lim_{\delta\downarrow0}\lim_{\alpha\downarrow\alpha_c}\lim_{\beta\to\infty}\lim_{N\to\infty}G_{N,\beta,\alpha}(\delta,u),\quad F_J(s)=\lim_{\alpha\downarrow\alpha_c}\lim_{\beta\to\infty}\lim_{N\to\infty}F_{N,\beta,\alpha}(s) \] exist and are independent of $\varepsilon$. The second is a probability distribution function on $(0,\infty)$, of mean one and with no atom at zero. As $u,s\downarrow0$, \[ G_J(u)=u^{1-\gamma+o(1)},\quad F_J(s)=s^{1+\theta+o(1)},\quad \gamma=(2+\theta)^{-1},\quad 0.4126930<\gamma<0.4126934,\quad 0.4231063<\theta<0.4231088. \]

Outline of the argument

A pressure formula and feasibility repair identify the threshold before the zero-temperature critical layer is analyzed. A convex Brownian martingale problem and contraction characterize all rescaled layer limits; an error-controlled finite numerical certificate brackets the exponents, and coordinate comparison extends the result to the specified Gaussian mixture.

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, Microscopic jamming in the negative spherical perceptron, 2026-09-24; citation key OAI:Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• Family 222: Lean scope.
• OpenAI catalogue entry.