Gaussian free field limits for the balanced six-vertex model
From OpenAI Aftermath
The six-vertex model assigns an arrow to each edge of the square lattice so that two arrows enter and two leave every vertex, and its configurations define a random height function on the faces. In the balanced case a = b = 1, the two vertex types with both horizontal arrows pointing inward or both outward carry a weight c, and for 0 < c ≤ 2 the height function is expected to converge, after spatial rescaling, to a multiple of the Gaussian free field. A manuscript in OpenAI's public mathematics collection (github.com/openai/math), dated 23 September 2026 and listed as its result 225, claims this convergence for every 0 < c ≤ 2 for one specified plane state, with the variance multiplier 1/arcsin(c/2). The family has no Lean formalization in the repository's catalogue. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper is not presented here as independent verification.
Background
- As recounted in the introduction of the manuscript:
- Lieb computed the exact square-ice entropy at c = 1. In the Coulomb-gas approach, Di Francesco, Saleur and Zuber gave a Gaussian-action prediction for 1 < c < 2 that agrees with the multiplier claimed here.
- At the free-fermion point c = √2 the model is equivalent to dimers, where Kenyon proved conformal invariance and Gaussian free field convergence for domino tilings. Near this point, Falco obtained arrow-correlation asymptotics and logarithmic height variance, and Giuliani, Mastropietro and Toninelli proved Gaussian free field convergence for a weakly interacting dimer model and later height-covariance identities that include the six-vertex interaction.
- Duminil-Copin, Kozlowski, Krachun, Manolescu and Tikhonovskaia established the Bethe-root analysis of the free energy; Duminil-Copin, Karrila, Manolescu and Oulamara proved logarithmic delocalization for 1 ≤ c ≤ 2; Lis and, later, Glazman and Lammers obtained further delocalization results.
- Duminil-Copin, Kozlowski, Lammers and Manolescu gave a Gaussian free field theorem for √3 ≤ c ≤ 2 with the same multiplier, using a rotational-invariance input listed as in preparation in the version cited.
The original result
- Claim. The main theorem of The Gaussian free field limit of the balanced six-vertex model with variance multiplier 1/arcsin(c/2) states that, for every fixed 0 < c ≤ 2, the plane state obtained from tori with balanced columns, first letting the length tend to infinity and then the even width, exists, and that the rescaled height function converges to σ(c) times the plane Gaussian free field with Green kernel −log|x − y|/(2π), where σ(c)2 = 2/arccos(1 − c2/2) = 1/arcsin(c/2). The convergence is stated for finite-energy measure tests, in negative-regularity Hölder, Besov and Sobolev restriction spaces, and for moments of separated height increments. The manuscript states that it makes no uniqueness assertion among Gibbs states or slopes and no estimate uniform as c tends to 0.
- Method, as described: for 0 < c < 2 the manuscript builds two Gaussian currents from finite row identities of the algebraic Bethe ansatz, Slavnov's scalar products and the determinant formulas of Kitanine, Maillet, Slavnov and Terras, controlled by uniform thermodynamic estimates; joint spectral values of commuting angular row operators are treated as inner functions; for c < 1 a stiffness lower bound and a vacuum-overlap upper bound identify the covariance coefficient directly, while for c ≥ 1 the normalization is taken from the conditional theorem of Duminil-Copin, Kozlowski, Lammers and Manolescu. At c = 2 an appendix supplies the rotational-invariance input from fair orientations of random-cluster loops and the rotation theorem for the random-cluster model.
- Source (openai/math repository, commit
7ef5c0c): The Gaussian free field limit of the balanced six-vertex model with variance multiplier 1/arcsin(c/2), dated 23 September 2026.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 30 October 2026) |
|---|---|---|---|---|
| 2026-10-06 | The balanced six-vertex height and its Gaussian limit, an article of the Latent Seminar blog at lowrank.github.io | Reading (exposition) | Explains the height convention, the balanced-torus prescription of the plane state and the order of its limits, the meaning of convergence of a distribution-valued field tested against averages, and the claimed multiplier 1/arcsin(c/2), stressing that an arbitrary boundary condition or slope is not asserted to give the same state. | Blog article; it states that its review covers statements, introductions and selected proof mechanisms and is not an independent verification of the complete proofs. |
| 2026-10-16 | J. Jardine, Lattice24, referee review rank_168 in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995) | Review (AI-assisted audit) | Reports no gap. It has two minor notes: several analytic steps of the angular-space and normalization sections are given as compressed sketches that a referee would ask to see written out, although the reviewer found no failing step; and at c = 2 the manuscript does not spell out that the plane law it takes from earlier work is obtained with the same order of limits as its own definition. The reviewer lists its own checks, including the multiplier identity, kernel transforms and a numerical check of a Toeplitz symbol. | Part of an audit that its README describes as complete for the collection's manuscripts without a Lean formalization. The review states which sections were read in full. It is published under CC BY 4.0. |
| 2026-10-17 | AI Proved What?, episode 225 How rough is square ice?, remade version of 22 October 2026 | Reading (exposition) | Video explainer of the claim and its history. The episode states that the argument is self-contained below c = 1, that from 1 to 2 it proves convergence but takes the identification of the state and the constant from the March 2026 preprint of Duminil-Copin, Kozlowski, Lammers and Manolescu, and that at c = 2 it supplies the missing rotational-invariance input. It lists other states or slopes, uniqueness among Gibbs states, domains with boundaries, unequal weights, c above 2 and estimates uniform as c tends to 0 as not claimed. It notes that a different version of the same paper appears among the files of the collection's result 223, that a paper by Averous, Duminil-Copin, He, Lammers and Manolescu posted one day before the release still counted only part of this range as proved, and that the collection's result 233 claims a related result for 1 < c ≤ 2 in domains. | Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 27 October 2026: posted as a preprint: OpenAI, 93 pages, dated September 23, 2026; under expert review: not yet; peer reviewed: not yet; checked in Lean: not formalized. The page states that it found no expert reaction. |
Notes
- The background summarizes the manuscript's own account of earlier work.
- For the collection's random-cluster claims, which include the case of cluster weight 4, see Conformal invariance of square-lattice random-cluster interfaces; for the free-fermion point and dimer loops, see also The double-dimer CLE4 conjecture in the half-plane.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
