Only the eigenvalues matter?
A preprint claims the pressure of orthogonally invariant Ising spin glasses at every temperature
Covers result 234 in OpenAI’s catalog: its one paper. See its connections to other results and the work it cites.
4:34. English captions available. The video loads from YouTube when you press play.
Chapters
The claim, in plain words
A spin glass is a collection of magnets, each up or down, coupled by a matrix of interactions. In the Sherrington–Kirkpatrick model the couplings are independent Gaussian numbers, and Parisi’s famous formula, proved by Guerra and Talagrand, gives its free energy. Here the couplings are built differently: choose the eigenvalues, almost any list you like, and rotate the eigenvectors uniformly at random. Semicircle eigenvalues give the Sherrington–Kirkpatrick model; ±1 gives the random orthogonal model of Marinari, Parisi and Ritort. Does the pressure (the log of the average Boltzmann weight per spin) have a limit that depends only on the eigenvalue distribution, at every temperature? Rigorous answers existed only at high temperature (Bhattacharya and Sen; Fan and Wu; Fan, Misiakiewicz, Wang and Wen in 2026, whose explicit condition at the Sherrington–Kirkpatrick scale reaches a quarter of the way to the transition).
Result 234 is one OpenAI preprint, 59 pages, dated September 25, 2026. It claims an exact variational formula for the limiting pressure at every fixed temperature, on average and almost surely, whenever the eigenvalue distribution converges to a compactly supported law and the extreme eigenvalues converge to the edges of its support. The formula is an infimum over Parisi’s overlap profiles, and the spectrum enters only through an edge-extended version of its R-transform from free probability. It adds external fields, random spectra, the ground-state energy as the temperature goes to zero, and the existence of limits for two problems about Gaussian matrices; and it shows compact support matters: with an unbounded spectral law, one enormous eigenvalue can destroy the limit.
What it doesn’t claim
Numerical values, or a separate formula at zero temperature (the ground state is the limit of the formula).
Where it stands
- Posted as a preprint
- OpenAI, 59 pages, dated September 25, 2026
- Peer reviewed
- Not yet
- Checked in Lean
- Yes, main results (docs)
Last checked October 19, 2026. The preprint has not been revised since release. In Lean, a comparator with eleven theorems covers the main results (the field theorem in a slightly different form), with an import closure of about 2,100 files; our scan found no unproved steps, we did not build it, and the item is missing from the formalization list. Not among those OSVerify reviewed; we found no competing claim and no public reaction.
How the proof goes
- Rotational invariance. The couplings are not Gaussian, so the usual interpolation does not apply; what survives is rotational invariance. First, small perturbations make the overlaps ultrametric. Then rotating a little between two eigenspaces gives exact identities, which show that the overlap of two copies inside each eigenspace is fixed by the total overlap alone.
- Bounds. An upper bound by a new minimality argument with Parisi-style trial fields, and a cavity lower bound with a new way of adding spins that keeps the eigenvalue proportions intact.
The numbers in the video
- Ours: four spins with eigenvalues (+1, +1, −1, −1) and one random rotation; the best of the 16 patterns scores 1.51 and the pressure at temperature 1 is 0.126. The expected large-system value at this temperature is 1/(2φ) − (log φ)/4 ≈ 0.189 (φ the golden ratio); exact enumeration up to 22 spins climbs toward it.
- Ours: the paper’s formula, fed the semicircle law, agrees with Parisi’s Sherrington–Kirkpatrick formula on one-step trial profiles at three temperatures, to six decimal places (the paper does not state this). Exact enumeration of all spin states (up to 22 spins, 14,200 samples) extrapolates to the high-temperature value; for the Sherrington–Kirkpatrick spectrum at low temperature such small systems are far from the limit.
Read the source
- All-temperature pressure for orthogonally invariant Ising spin glasses (PDF), 59 pages.
- The Lean formalization notes.
- The collection’s catalog, where this is result 234.
The paper
OpenAI groups related papers into one result; this one has a single paper.
Transcript
Read the full script
0:00The question
A spin glass is a collection of magnets, each pointing up or down, coupled by a matrix of interactions. In the Sherrington–Kirkpatrick model the couplings are independent Gaussian numbers, and Parisi’s famous formula, proved by Guerra and Talagrand, gives its free energy. Here is a different way to build the couplings: choose the eigenvalues yourself, almost any list you like, and rotate the eigenvectors uniformly at random. The Sherrington–Kirkpatrick model is the case where the eigenvalues follow the semicircle law; the random orthogonal model of Marinari, Parisi and Ritort uses plus and minus one. The question is the pressure, the log of the average Boltzmann weight per spin: does it have a limit that depends only on the eigenvalue distribution, and what is it, at every temperature? Physicists proposed replica formulas in the 1990s and 2000s. Rigorous answers existed only at high temperature: Bhattacharya and Sen, Fan and Wu, and in 2026 Fan, Misiakiewicz, Wang and Wen, whose explicit condition at the Sherrington–Kirkpatrick scale reaches only a quarter of the way to the transition. Here is the smallest case our computer worked out: four spins, eigenvalues plus one twice and minus one twice, one random rotation. Of the 16 spin patterns, the best scores 1.51; the pressure at temperature one is 0.126. The expected large-system value at this temperature has a closed form with the golden ratio, 0.189, and our exact enumeration up to 22 spins climbs toward it.
1:41The claim
Result 234 is one OpenAI preprint, 59 pages, dated September 25, 2026. It claims an exact variational formula for the limiting pressure at every fixed temperature, both on average and almost surely, whenever the eigenvalue distribution converges to a compactly supported law and the extreme eigenvalues converge to the edges of its support. The formula is an infimum over Parisi’s overlap profiles; the spectrum enters only through an edge-extended version of its R-transform from free probability. It adds external fields, random spectra, the ground-state energy as the temperature goes to zero, and the existence of limits for two problems about Gaussian matrices. It also shows compact support matters: with an unbounded spectral law, one enormous eigenvalue can destroy the limit. Not claimed: numerical values, or a separate formula at zero temperature.
2:42How the proof goes
For the Gaussian model, interpolation arguments work directly. Here the couplings are not Gaussian; what survives is rotational invariance. First, small perturbations make the overlaps ultrametric. Then rotating a little between two eigenspaces gives exact identities, which show that the overlap of two copies inside each eigenspace is fixed by the total overlap alone. Then come an upper bound, by a new minimality argument with Parisi-style trial fields, and a cavity lower bound, with a new way of adding spins that keeps the eigenvalue proportions intact. As a consistency check, our computer fed the paper’s formula the semicircle law and compared it with Parisi’s formula for the Sherrington–Kirkpatrick model, on one-step trial profiles at three temperatures. They agree to 6 decimal places, which the paper does not state. Our exact enumeration of all spin states, up to 22 spins and 14,200 samples, extrapolates to the high-temperature value, though for the Sherrington–Kirkpatrick spectrum at low temperature such small systems are far from the limit.
3:52Where it stands
The preprint is unrefereed and unchanged since release. In Lean, a comparator with 11 theorems covers the main results, the field theorem in a slightly different form, with an import closure of about 2,100 files, and our scan found no unproved steps; we did not build it, and the item is missing from the formalization list. Not among those OSVerify reviewed, and we found no competing claim and no public reaction. If it holds, the free energy of these spin glasses has an exact formula at every temperature, and once the eigenvectors are random, only the eigenvalue distribution matters.
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.