Result 222, Probability and statistical mechanics

Perceptron free energies and microscopic jamming exponents

Determines finite-temperature variational free energies for Gaussian Ising perceptrons with bounded Borel log-potentials and Gaussian spherical perceptrons with bounded continuous potentials, at every positive pattern density. A spherical extension treats bi-orthogonally invariant disorder with compact limiting singular-value distributions and no outliers. At margin −1, the quadratic-penalty spherical model has a sharp feasibility threshold and limiting gap and force laws, with system size, zero temperature, and critical density taken in that order.

Lean formalization Classification or exact value

The bigger picture

Why it matters

Random constraints can abruptly leave a system unable to satisfy them all. These manuscripts describe both the overall energy balance of perceptron models and the tiny gaps and forces that characterize this transition to jamming.

What changes?

The manuscripts report variational formulas for limiting free energy, the energy-entropy balance, of Gaussian perceptrons: random-constraint models with binary variables (Ising) or vectors on a sphere (spherical). At every fixed positive temperature and pattern density (constraints per variable), they allow bounded Borel log-potentials and bounded continuous potentials, respectively, with convergence in expectation and probability. The spherical extension permits disorder invariant under independent left and right orthogonal transformations, with any compact limiting singular-value distribution and no outliers.

What does that help mathematicians do?

For the margin-minus-one spherical model with quadratic penalty, a reported sharp feasibility threshold locates where constraints become unsatisfiable. Limits proceed in order: infinite system size, zero temperature, then density approaching the threshold from above. Small-value cumulative laws have asymptotic power exponents one minus gamma for gaps excluding contacts and one plus theta for mean-one forces, quantifying near-contact geometry and weak forces. Gamma equals 1/(2+theta), with 0.4126930 < gamma < 0.4126934 and 0.4231063 < theta < 0.4231088.

Are there practical applications?

The immediate value is foundational: the formulas describe macroscopic behavior, while the microscopic laws characterize jamming. Those laws reportedly agree for Gaussian coordinates and an equal mixture of centered Gaussian coordinates with variances one minus epsilon and one plus epsilon, for every sufficiently small fixed epsilon. This identifies a specific change in randomness that leaves the limiting behavior unchanged, not universality for arbitrary disorder.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

4 manuscripts

The free energy of the Ising random perceptron

September 24, 2026 45 pages

We determine the limiting free energy of the Ising perceptron with independent Gaussian patterns for every bounded Borel log-potential, at every fixed positive temperature and pattern density. We give an explicit variational formula for the limit and prove convergence in expectation and probability.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-free-energy-of-the-Ising-random-perceptron-September-24-2026,
  author = {{OpenAI}},
  title = {{The free energy of the Ising random perceptron}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-free-energy-of-the-Ising-random-perceptron-September-24-2026/The-free-energy-of-the-Ising-random-perceptron-September-24-2026.pdf}{OAI:The-free-energy-of-the-Ising-random-perceptron-September-24-2026}},
  year = {2026}
}

Microscopic jamming in the negative spherical perceptron

September 24, 2026 133 pages

We prove a sharp feasibility threshold and limiting gap and force laws for the spherical perceptron with margin −1 and quadratic penalty. The limits are taken successively in system size, inverse temperature, and density approaching the threshold from above. They agree for Gaussian coordinates and for the equal mixture of centered Gaussian coordinates with variances 1−ε1-\varepsilon and 1+ε1+\varepsilon, for every sufficiently small fixed ε. The contact-removed gap cumulative law and the mean-one force cumulative law satisfy

GJ(u)=u1−γ+o(1),FJ(s)=s1+θ+o(1),\displaystyle G_J(u)=u^{1-\gamma+o(1)},\qquad F_J(s)=s^{1+\theta+o(1)},

as u↓0u\downarrow0 and s↓0s\downarrow0, with γ=(2+θ)−1\gamma=(2+\theta)^{-1}, 0.4126930<γ<0.41269340.4126930\lt \gamma\lt 0.4126934, and 0.4231063<θ<0.42310880.4231063\lt \theta\lt 0.4231088. A finite numerical certificate for these exponent intervals, together with its mathematical error bounds, is included.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026,
  author = {{OpenAI}},
  title = {{Microscopic jamming in the negative spherical perceptron}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026/Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026.pdf}{OAI:Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026}},
  year = {2026}
}

The spherical perceptron with bi-orthogonally invariant disorder

September 24, 2026 40 pages

We determine the limiting free energy of a spherical perceptron with a bounded continuous activation and a bi-orthogonally invariant disorder matrix. The singular values may have any compact limiting distribution, provided there are no outliers. We give an explicit variational formula for this limit.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-spherical-perceptron-with-bi-orthogonally-invariant-disorder-September-24-2026,
  author = {{OpenAI}},
  title = {{The spherical perceptron with bi-orthogonally invariant disorder}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-spherical-perceptron-with-bi-orthogonally-invariant-disorder-September-24-2026/The-spherical-perceptron-with-bi-orthogonally-invariant-disorder-September-24-2026.pdf}{OAI:The-spherical-perceptron-with-bi-orthogonally-invariant-disorder-September-24-2026}},
  year = {2026}
}

The free energy of the spherical random perceptron

September 24, 2026 30 pages

We prove an exact variational formula for the limiting pressure of the spherical random perceptron with an arbitrary bounded continuous single-pattern potential. The formula holds at every fixed positive density and inverse temperature, with convergence in expectation and in probability.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-free-energy-of-the-spherical-random-perceptron-September-24-2026,
  author = {{OpenAI}},
  title = {{The free energy of the spherical random perceptron}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-free-energy-of-the-spherical-random-perceptron-September-24-2026/The-free-energy-of-the-spherical-random-perceptron-September-24-2026.pdf}{OAI:The-free-energy-of-the-spherical-random-perceptron-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/222.md.

Perceptron free energies and microscopic jamming exponents

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The linked formalization proves finiteness of the variational value used for the Ising random perceptron. For every nonnegative pattern density and every bounded continuous log-potential, the infimum over admissible overlap paths is a finite real number.

This is a supporting well-definedness result. It does not assert convergence of the finite-system pressure or the paper's extension to all bounded Borel log-potentials.

The formalization proves the variational formula for the spherical random perceptron's limiting pressure for every positive pattern density and inverse temperature and every bounded continuous single-pattern potential. It shows that the variational value is finite and that the pressure converges to it both in expectation and in probability.

The linked spherical-field result supplies a supporting dual formula: finite hierarchy field values converge uniformly on compact parameter sets to an attained dual minimum, and the corresponding bounded stationary-field approximation holds for monotone overlap quantiles bounded away from one.

Comparator links

Result Comparator statement
Finiteness of the Ising-perceptron variational value IsingFiniteness.lean
Variational formula for spherical-perceptron pressure PerceptronFreeEnergy.lean
Spherical linear-field dual formula and approximation SphericalField.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.