Scaling limits of the critical Ashkin–Teller model
From OpenAI Aftermath
Scaling limits of the critical Ashkin–Teller model concern a model that couples two Ising models on the square lattice through a four-spin interaction; along its critical line the large-scale behaviour changes continuously with one parameter, and at one end it becomes the four-state Potts model. Its current representation, due to Lis, gives random clusters on the primal and dual lattices together with a height function. Alcalde López, Heeney and Lis conjectured in 2026 the joint scaling limit of this height and of both complete collections of current clusters along the whole critical line. A manuscript in OpenAI's public mathematics collection (github.com/openai/math), listed as its result 233, claims this conjecture at every point of the critical line, including the Potts endpoint. Its first edition is dated 25 September 2026 and its second edition 6 October 2026. This page summarizes the published claim and records public follow-up work.
Background
- As recounted in the introduction of the manuscript:
- Representations. The model goes back to Ashkin and Teller's four-component lattice model of 1943. The finite spin, percolation and height representation used is due to Lis, whose current expansion gives the edge weights; a noncrossing primal–dual contour representation appears earlier in Rys.
- The double random current point. At zero four-spin coupling the currents are double random currents. Duminil-Copin and Lis identified the finite signed nesting field with a dimer height, and Duminil-Copin, Lis and Qian established conformally invariant limits for the nesting field and recursive cluster-boundary families. Alcalde López, Heeney and Lis stated the joint limit of the height and both complete current-cluster collections at this point and asked, as their Conjecture 1.2, for the extension to the critical line.
- Related work. Alcalde López and Sepúlveda studied a stronger joint decomposition at the Ising point. For the six-vertex model with weights 1, 1 and c, Duminil-Copin, Kozlowski, Lammers and Manolescu proved a height scaling limit for √3 ≤ c ≤ 2 and established inputs on the larger interval 1 ≤ c ≤ 2, which the manuscript uses.
The original result
- Claim. The joint scaling limit of critical Ashkin–Teller currents states that for every fixed point on the critical line, given by sinh 2J = e−2U with U ≤ J and including the four-state Potts endpoint, and in every bounded Jordan domain under admissible polygonal approximations with wired primal and free dual boundaries, the height converges to a Gaussian free field with the predicted coupling constant g = (8/π) arcsin(√(1 + e4U)/2), which ranges over 4/3 < g ≤ 4. Jointly with the height, the wired and free current-cluster collections converge to canonical recursive two-valued local sets of the same field: for the wired current a symmetric two-valued local set at each level, for the free current an asymmetric split inside each hole of a conformal loop ensemble CLE4, repeated in all complementary components. The limit retains every nesting depth and the distinguished wired boundary cluster, in a positive-diameter Hausdorff matching topology. The manuscript states that this is the assertion of Conjecture 1.2 of Alcalde López, Heeney and Lis.
- Method, as described: the manuscript relates the model to the six-vertex model with weights 1, 1 and c = √(1 + e4U), uses the larger-range inputs of Duminil-Copin, Kozlowski, Lammers and Manolescu, and supplies covariance, screening and Gaussianity arguments for the whole interval through reflection positivity. The clusters are identified through thin two-valued local sets, first-passage sets and the canonical excursion decomposition of the Gaussian free field, following the recursive exploration of the double-current case.
- Source. Commit
7ef5c0cof the openai/math repository: The joint scaling limit of critical Ashkin–Teller currents, second edition; the first edition of 25 September 2026 remains in the repository. - Formalization. The repository's contents list gives no Lean scope note or Comparator challenge for this result.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 31 October 2026) |
|---|---|---|---|---|
| 2026-10-06 | The joint Ashkin–Teller height and current-cluster limit, an article of the Latent Seminar blog at lowrank.github.io | Reading (exposition) | Explains family 233: the two coupled planar systems, Gaussian local sets, the claimed joint limit, the difference between joint and marginal convergence, and the structure of the proof. It describes a joint continuum limit for the critical Ashkin–Teller height and two coupled current-cluster collections, with the clusters identified as recursive local sets of the same Gaussian field. | 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_016 in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995) | Review (AI-assisted audit) | Reports no gap and one minor issue: several key inputs, including a regularity and log-moment estimate on the whole parameter range, a convergence criterion and the free-energy correspondence that fixes the normalization, are imported from the companion work of Duminil-Copin, Kozlowski, Lammers and Manolescu, whose literal proofs the manuscript's appendix says contain errors that it repairs; the review notes that these inputs depend on an external, still changing source it could not check. It states that it read the introduction, the model section, the spectral section, part of the Gaussian section and the appendix in full, and checked several parameter ranges and normalizations by hand. | Part of an audit of the collection's manuscripts that the audit describes as complete. The reviews are published under CC BY 4.0. |
| 2026-10-17 | AI Proved What?, episode 233 Two magnets, one free field?, remade version of 22 October 2026 | Reading (exposition) | Video explainer of one OpenAI preprint, 105 pages in its second edition dated 6 October 2026, with a first edition of 25 September: two Ising magnets coupled by a four-spin interaction, with a critical line along which a parameter g runs from just above 4/3 up to 4. The preprint claims the conjecture of Alcalde López, Heeney and Lis at every point of the critical line, including the Potts endpoint: the height converges to the Gaussian free field with the predicted strength, and both complete collections of current clusters converge jointly to the canonical recursive sets of that same field. The episode notes that only the point with zero four-spin coupling had been proved, by Duminil-Copin, Lis and Qian, and states that limits for the spins themselves, magnetization or polarization, spin interfaces, area measures and uniformity as g approaches 4/3 are not claimed, and that the clusters converge as sets, not as curves. | 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, 105 pages, second edition dated October 6, 2026, first edition September 25; under expert review: not yet; peer reviewed: not yet; checked in Lean: no. The page states that the second edition repaired several arguments and replaced one imported convergence step with a new one; that on October 6 a group including Duminil-Copin posted a proof that the loops of the random-cluster model with q = 4 converge to CLE4 and announced an Ashkin–Teller paper in preparation; that the preprint is not among those OSVerify reviewed; and that it found no public reaction. |
Notes
- The background summarizes the manuscript's own account of earlier work.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
