Jamming exponents of the negative spherical perceptron
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What gap and force distributions arise at jamming in the spherical perceptron with margin −1 and quadratic penalty? The target takes system size to infinity, then inverse temperature to infinity, then density toward the feasibility threshold from above, in that order. It includes the predicted exponent relation γ=1/(2+θ) and the claimed extension to a sufficiently small fixed Gaussian scale mixture.
Why it matters and what counts as progress
Why it matters: A microscopic derivation would connect the model’s ordered-equilibrium limit to the critical gap and force laws predicted by jamming theory.
What counts as progress: Derive the limiting laws and rigorous exponent bounds under the specified order of limits, or establish the stated mixture extension. A finite certificate for numerical enclosures or analytic input inequalities alone does not establish the full microscopic-limit or universality argument.
How the problem developed
- 13 Jan 2015 · Prediction [1]
Silvio Franz and Giorgio Parisi Derived the jamming exponent relation within a replica-symmetry-breaking analysis for the negative spherical perceptron; the work predicted the exponent regime rather than proving the later microscopic limit laws.
- 24 Sept 2026 · Proof claim [2]
OpenAI Released a manuscript claiming a sharp feasibility threshold and limiting gap and force laws, with narrow certified intervals for the exponents under successive thermodynamic, zero-temperature, and critical-density limits.
AI contribution
Microscopic jamming in the negative spherical perceptron
For the spherical perceptron at margin −1 with quadratic penalty, after successive limits in system size, inverse temperature and density approaching the feasibility threshold from above, the gap and force laws have exponents γ=1/(2+θ), with 0.4126930<γ<0.4126934 and 0.4231063<θ<0.4231088. The same limits are claimed for a sufficiently small fixed Gaussian scale mixture.
Sources
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Publication dates and editorial changes.
OpenAI reports jamming exponents for the negative spherical perceptron