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Franz-Parisi jamming exponents in the negative spherical perceptron, at margin -1

In the spherical perceptron with negative margin, x∈NSN−1x\in\sqrt N S^{N-1} must satisfy ga⋅x/N≥σg^a\cdot x/\sqrt N\ge\sigma for M=αNM=\alpha N Gaussian patterns, and the feasible set is nonconvex. Franz and Parisi (2016) identified this model with the jamming universality class of high-dimensional spheres and, from a full replica-symmetry-breaking solution, predicted isostaticity at the SAT-UNSAT threshold and power laws for small gaps g(h)∼h−γg(h)\sim h^{-\gamma} and weak forces p(f)∼fθp(f)\sim f^{\theta} with γ≈0.41269\gamma\approx0.41269, θ≈0.42311\theta\approx0.42311 and γ=1/(2+θ)\gamma=1/(2+\theta), confirmed as universal across negative margins by Franz, Parisi, Sevelev, Urbani and Zamponi (2017). Rigorous work gave only density bounds. Does the negative perceptron have a sharp jamming threshold with these gap and force exponents?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
Spin glasses; jamming; random constraint satisfaction
Posed by
Silvio Franz and Giorgio Parisi, The simplest model of jamming (J. Phys. A, 2016); Franz, Parisi, Sevelev, Urbani and Zamponi (SciPost Phys., 2017)
Year posed
2016
Years open
10y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
22 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for the spherical perceptron with margin −1-1 and quadratic penalty there is a sharp feasibility threshold αc\alpha_c; the ordered limits (size, then zero temperature, then α↓αc\alpha\downarrow\alpha_c) of the contact-removed gap law and mean-one force law exist with GJ(u)=u1−γ+o(1)G_J(u)=u^{1-\gamma+o(1)}, FJ(s)=s1+θ+o(1)F_J(s)=s^{1+\theta+o(1)}, γ=(2+θ)−1\gamma=(2+\theta)^{-1}, 0.4126930<γ<0.41269340.4126930<\gamma<0.4126934, 0.4231063<θ<0.42310880.4231063<\theta<0.4231088, and the critical limit is isostatic. The same holds for a small Gaussian scale mixture of coordinates. Not shown: other negative margins, the unpenalized hard-constraint model directly, other limit orders, or finite-size scaling.

What the AI did

The release README says every result was produced by an unreleased internal OpenAI model with a fixed procedure of roughly three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family has four manuscripts dated September 24, 2026. The jamming paper includes a finite numerical certificate with stated error bounds for the exponent intervals.

Verification

No independent mathematician has checked this yet. Checked here: the background section and Theorem 1.1 of 'Microscopic jamming in the negative spherical perceptron' were read against the Franz-Parisi predictions as the manuscript cites them. The theorem treats margin −1-1 with a quadratic penalty, with limits taken in the order system size, inverse temperature, then density to the threshold from above; the physics prediction is for all negative margins. The exponent intervals rest on a shipped numerical certificate, not run here. The family's Lean formalizations (IsingFiniteness, PerceptronFreeEnergy, SphericalField) concern the free-energy papers, not this theorem, so this entry is unreviewed.

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