Perceptron free energies and microscopic jamming exponents

1 · The essentials

These manuscripts study mathematical models in which many randomly chosen requirements act on a large collection of adjustable numbers. They ask how the overall behavior settles as the collection grows. The main manuscript gives a formula for that behavior when each adjustable number has two possible values and the random inputs follow a Gaussian distribution. Its result applies at every fixed positive temperature and every fixed positive ratio of requirements to adjustable numbers, under a boundedness condition on how each requirement contributes. Companion manuscripts treat numbers constrained to lie on a sphere and allow a broader class of random inputs under additional conditions. Another companion examines the boundary between possible and impossible requirements in one particular spherical model, describing how small gaps and small forces are distributed near that boundary. The main result has a Lean formalization, but the supplied source does not specify its detailed coverage.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine adjusting many knobs to satisfy a long list of randomly chosen checks, with each check contributing to an overall score. The free-energy results describe a large-system summary of the choices and their scores, while the jamming result examines what happens near the point where satisfying all the checks becomes impossible. This image does not capture the precise randomness, constraints, or order of limits required by the manuscripts.

Why it matters

The free-energy formulas cover every fixed positive pattern density, rather than only a restricted density range, for the stated classes of potentials and disorder. The jamming manuscript adds a microscopic description through limiting gap and force distributions, not just a feasibility threshold. Together, the manuscripts connect large-system formulas with a more detailed analysis of one specific critical regime, without claiming that the latter follows for every model covered by the former.

2 · Going further

The main problem is to determine the limiting free energy of an Ising perceptron with independent Gaussian patterns. For every bounded Borel log-potential, at every fixed positive temperature and pattern density, the main manuscript gives an explicit variational formula and states convergence both in expectation and in probability. The Gaussian spherical companion similarly gives an exact variational formula for limiting pressure with an arbitrary bounded continuous single-pattern potential, at every fixed positive density and inverse temperature, with the same two modes of convergence.

A further spherical companion replaces Gaussian disorder with a bi-orthogonally invariant disorder matrix. Its singular values may have any compact limiting distribution, provided there are no outliers, and the activation must be bounded and continuous. It supplies another explicit variational formula. The abstracts identify variational formulas as the central characterization of these limits, but do not describe the argument used to establish them; a more specific proof strategy cannot be extracted from the supplied text.

The jamming companion concerns the spherical perceptron at margin $- 1$ with a quadratic penalty. A sharp feasibility threshold and limiting gap and force laws are shown in the manuscript, with limits taken successively in system size, inverse temperature tending to infinity, and density approaching the threshold from above. The laws agree for Gaussian coordinates and for an equal mixture of centered Gaussian coordinates with variances $1 - ε$ and $1 + ε$ , for every sufficiently small fixed $ε$ .

After removing contacts from the gap law and normalizing forces to have mean one, the cumulative laws satisfy $G_{J} (u) = u^{1 - γ + o (1)}$ and $F_{J} (s) = s^{1 + θ + o (1)}$ as $u, s ↓ 0$ . The manuscript gives $γ = (2 + θ)^{- 1}$ , with $0.4126930 < γ < 0.4126934$ and $0.4231063 < θ < 0.4231088$ . It includes a finite numerical certificate and mathematical error bounds for these intervals. These microscopic claims concern the specified jamming model and successive limits, rather than the full range of models in the free-energy manuscripts.

Useful notions

Gaussian random inputs
Ising and spherical constraints
Free energy and temperature
Variational formulas
Probability distributions and successive limits

What was known before / what is new

Before

The source provides no account of earlier results or the historical development of these problems. It identifies the model families and their assumptions, but does not say which special cases or exponent predictions were previously known.

New

The main manuscript supplies an explicit limiting free-energy formula for Gaussian Ising perceptrons across all bounded Borel log-potentials and all fixed positive temperatures and densities. The spherical companions supply formulas for bounded continuous potentials, including an extension to bi-orthogonally invariant disorder under spectral restrictions. The jamming manuscript supplies a sharp feasibility threshold, limiting microscopic laws, and certified numerical intervals for their exponents in the specified margin- $- 1$ model.

Limits and assumptions

The Ising result requires independent Gaussian patterns and a bounded Borel log-potential; the Gaussian spherical result requires a bounded continuous single-pattern potential. Their temperature and density parameters are fixed and positive. The invariant-disorder extension requires a compact limiting singular-value distribution with no outliers. The jamming claims require margin $- 1$ , a quadratic penalty, and the stated order of limits; the agreement between coordinate distributions is asserted only for the specified Gaussian mixture with sufficiently small fixed $ε$ . The main result has a Lean formalization, but the supplied source does not identify which individual claims it covers or establish formal verification of the companion results or numerical certificate. No specific remaining open problem is identified in the source.

Mini glossary

Ising constraint: A restriction that each adjustable coordinate takes one of two values.
Spherical constraint: A restriction that the vector of adjustable coordinates lies on a sphere.
Pattern density: The ratio of the number of random patterns to the number of adjustable coordinates.
Variational formula: A formula that characterizes a quantity through an optimization problem.
Feasibility threshold: A boundary separating a regime where the requirements can be satisfied from one where they cannot.
Cumulative law: A function giving the probability that a quantity is at or below a specified value.

Keywords : Perceptron free energy · Variational formulas · Microscopic jamming

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

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.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/222.md

Manuscripts in this family

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The free energy of the Ising random perceptron

24/09/2026

To be verified

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 GitHub folder

Microscopic jamming in the negative spherical perceptron

24/09/2026

To be verified

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 $1 - ε$ and $1 + ε$ , for every sufficiently small fixed ε. The contact-removed gap cumulative law and the mean-one force cumulative law satisfy $G_{J} (u) = u^{1 - γ + o (1)}, F_{J} (s) = s^{1 + θ + o (1)},$ as $u ↓ 0$ and $s ↓ 0$ , with $γ = (2 + θ)^{- 1}$ , $0.4126930 < γ < 0.4126934$ , and $0.4231063 < θ < 0.4231088$ . A finite numerical certificate for these exponent intervals, together with its mathematical error bounds, is included.

PDF GitHub folder

The spherical perceptron with bi-orthogonally invariant disorder

24/09/2026

To be verified

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 GitHub folder

The free energy of the spherical random perceptron

24/09/2026

To be verified

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 GitHub folder