How much can one neuron store?

The claim, in plain words

A perceptron is the simplest artificial neuron: N weights and a list of random patterns, each of which must have a positive dot product with the weights. How many patterns can one set of weights satisfy? For weights on a sphere, two per weight: by Cover’s count, the chance of success at that density is exactly one half for every N. For weights that are only ±1, Krauth and Mézard predicted about 0.833 per weight in 1989. In a tiny case, three weights and three patterns, only one of the eight sign choices works; our exhaustive search up to 22 weights falls from 0.92 toward 0.85. Physicists also study a softer version, where each violated pattern costs a factor e^−β, and ask for the pressure: the log of the average Boltzmann factor, per weight.

Result 222 is four OpenAI preprints, 248 pages, dated September 24, 2026. Two claim that pressure exactly, at every density and every positive temperature: for ±1 weights with any bounded reward, and for spherical weights with any bounded continuous reward. Each answer is an infimum over a whole overlap profile, a Parisi-type formula with full replica symmetry breaking; for spherical weights, Györgyi and Reimann predicted it in 2000. A third paper treats patterns from bi-orthogonally invariant random matrices.

The fourth paper, 133 pages, goes to zero temperature in one model: spherical weights, margin −1, and a quadratic penalty. Taking the system size to infinity, then the temperature to zero, then the density down to its critical value from the overcrowded side, it claims a sharp jamming threshold (with no value), one contact per degree of freedom, and cumulative power laws for small gaps and forces with exponents 0.4126930 < γ < 0.4126934 and 0.4231063 < θ < 0.4231088. Physicists predicted these values for hard spheres in 2014 and for this perceptron in 2016.

What it doesn’t claim

The hard capacity itself: the Ising paper says the hard-constraint limit needs extra control of the order of limits. A July 2026 preprint by Shmalo, which says most of its work was done by AI systems, claims to close the remaining gaps in the 0.833 proof by computer. Nor are values of the free energies or of the jamming threshold given. Our evaluation of the Ising paper’s functional at its simplest trial, at zero temperature and outside the theorem, gives a solution entropy that crosses zero at 0.83308.

Where it stands

Posted as preprints

OpenAI, four papers, 248 pages, dated September 24, 2026

Peer reviewed

Not yet

Checked in Lean

In part (docs)

Last checked October 19, 2026. None of the four preprints has been revised since they were added to the repository in its initial commit on October 6, 2026. In Lean, the spherical formula is formalized in full (about 71,000 lines); for the Ising paper, only that the formula is finite, for continuous rewards; the bi-orthogonal paper not at all; the jamming paper only in unlisted supporting lemmas. The linked Lean has no unproved steps; we did not build it. The exponent relation γ = 1/(2 + θ) got a physics-level proof by Parisi and Zamponi in June 2026, credited by the jamming paper. None is among those OSVerify reviewed, and we found no expert reaction or earlier proof.

How the proof goes

1. Upper bound. The free-energy proofs adapt the spin-glass proofs of the Parisi formula. A general reward gives no sign for the usual interpolation, so they use Mourrat’s method: enrich the system with a hierarchy of Gaussian fields, treat the number of patterns as time, and show that if the bound failed, a comparison would have a negative minimum, where overlap identities and Panchenko’s ultrametricity force a contradiction.
2. Lower bound. Remove two patterns, restore them as independent marks, and add a few weights at a time: the cavity method.
3. Jamming. The jamming paper proves its own pressure formula for the unbounded penalty, takes the limits in order, and shows the critical profile scales as a power law near jamming. The exponents come from a number a* where an eigenvalue equals its own parameter: γ = 1 − 2a*, and γ = 1/(2 + θ) follows. a* is pinned down by a 681-line Python certificate printed in the paper; our run matches the printed hash and reaches the same 24 checkpoints, which shows the code runs, not that its error analysis is right.

The numbers in the video

• Ours: Cover’s count; exhaustive enumeration of small Ising perceptrons; the Ising functional at constant paths (0.83308, outside the theorem).
• Ours: gradient-descent minima of the margin −1 model near jamming, up to 800 weights: 1.019, 1.016 and 1.005 contacts per weight; the gap and force slopes have not settled. Local minima at finite size, not the paper’s limit. The Lean line count is ours.

Read the source

• The free energy of the Ising random perceptron (PDF), 45 pages.
• The free energy of the spherical random perceptron (PDF), 30 pages.
• The spherical perceptron with bi-orthogonally invariant disorder (PDF), 40 pages.
• Microscopic jamming in the negative spherical perceptron (PDF), 133 pages.
• The Lean formalization notes.
• The collection’s catalog, where this is result 222.

The 4 papers in this result

OpenAI groups related papers, such as alternative proofs, supporting results and consequences, into one result; we label the first paper it lists the main paper and the others companions.

• The free energy of the Ising random perceptron main paper
• Microscopic jamming in the negative spherical perceptron companion
• The spherical perceptron with bi-orthogonally invariant disorder companion
• The free energy of the spherical random perceptron companion

Transcript

Spot a mistake?

This episode simplifies on purpose, but it shouldn’t be wrong. If something is, open an issue on GitHub or leave a comment on YouTube. Corrections are noted here with a date.