Lipschitz height functions on the triangular lattice

Lipschitz height functions on the triangular lattice are random surfaces whose heights at neighbouring vertices differ by at most a fixed amount. Oded Schramm asked in 2006 whether two such models, one with integer and one with real heights, converge to the Gaussian free field and whether their zero-height interfaces converge to SLE₄. Three manuscripts in OpenAI's public mathematics collection (github.com/openai/math), listed as its result 232, claim Gaussian free field limits for two integer models and for the real-valued model, and the SLE₄ interface limit for the real-valued model at one tuned boundary amplitude. Their first editions are dated 25 September 2026 and their second editions 6 October 2026. This page summarizes the published claims and records public follow-up work.

Background

As recounted in the introductions of the two-arc and real-height manuscripts:
- The problems. Schramm's 2006 problem collection asks for the scaling limits of the uniform odd integer height function with two-arc boundary values and of its zero interface, Problem 2.2, and for the SLE₄ and Gaussian free field limits of the real-valued hard-constraint model, Problem 2.3.
- Integer heights. Glazman and Manolescu developed a two-spin representation with positive association for the uniform integer Lipschitz model and studied its logarithmic fluctuations; Glazman and Lammers studied delocalisation across a larger parameter range, and their 2025 article discusses the Gaussian free field limit as a conjecture.
- Smooth and related models. Schramm and Sheffield established the contour result for the discrete Gaussian free field, and Miller extended contour universality and proved bounded-domain field limits for smooth, symmetric, uniformly convex gradient potentials; the hard constraint considered here lies outside those hypotheses. Miłoś and Peled, and later Cohen-Alloro and Peled, studied the delocalization of hard-core height models on the square lattice, and Duminil-Copin, Kozlowski, Lammers and Manolescu proved a full-plane Gaussian free field limit for a zero-slope six-vertex state.

The original results

Weighted integer heights. Gaussian free-field limits of weighted integer Lipschitz heights states that for integer heights on the triangular lattice with zero boundary values and a factor x for each edge on which the height changes, the field divided by a positive constant depending only on x converges to the zero-Dirichlet Gaussian free field for every fixed x in [1/√2, 1], as a random distribution on every bounded C² Jordan domain under inside lattice approximations. This includes the uniform model, x = 1, and the predicted critical endpoint x = 1/√2.
Two-arc odd integer heights. The Gaussian free field limit of integer Lipschitz heights with two-arc boundary data states that the centered uniform odd integer height function on triangular-lattice approximations of a smooth simply connected domain, with neighbouring differences zero or two and boundary values +1 and −1 on two arcs, converges to a universal multiple of the Dirichlet Gaussian free field, with an absolutely convergent finite-volume formula for the normalization. The manuscript states that this resolves the field part of Schramm's Problem 2.2 and that the interface assertion is a separate question.
Real heights. Uniform real Lipschitz surfaces on the triangular lattice considers real heights sampled uniformly subject to differences at most 1 between neighbours and boundary values +λ and −λ on two arcs. Its main theorem states that there are constants A > 0, σ > 0 and λ in (0, 1/2], independent of the domain, such that for this λ the zero-height interface converges in uniform curve distance to chordal SLE₄ and the centered field divided by σ converges to the Dirichlet Gaussian free field, with σ² = 1/(vA), λ² = π/(8vA) and v = 2/√3; A is a stationary tangent-flux coefficient for which no closed form is proposed. The manuscript states that this resolves Schramm's Problem 2.3.
Method, as described: the two-arc manuscript combines reflection positivity, a spectral sum rule and boundary comparison with Gaussian moment identities, and uses from its weighted companion only an annulus circuit estimate at x = 1. The real-height manuscript builds on Prékopa's log-concavity theorem for the uniform measure on the polytope of admissible heights, the controlled-gradient estimate of Cohen-Alloro and Peled and the harmonic-observable method of Schramm and Sheffield for the interface.
Sources. Commit 7ef5c0c of the openai/math repository, second editions; the first editions of 25 September 2026 remain in the repository:
Formalization. The repository's contents list gives no Lean scope note or Comparator challenge for this result. Its Lean library contains a folder lean/OAI/Probability/LatticeSurface of 243 files at this commit that the contents list does not link; the AI Proved What? status record below describes it as formalizing some building blocks of the real-height manuscript but no limit statement.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 31 October 2026)
2026-10-06	Lipschitz height fields and SLE4 interfaces, an article of the Latent Seminar blog at lowrank.github.io	Reading (exposition)	Explains family 232: hard constraints and generalized fields, the three model-specific statements, stiffness and the boundary amplitude, spectral identification and moment recursion for the integer fields, and conditioned polytopes and interfaces for the real surfaces. It describes Gaussian field limits for two integer models and a real-valued model on the triangular lattice, with the SLE₄ interface limit claimed only for the real-valued model at a tuned boundary amplitude.	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 reviews rank_084 (weighted integer heights), rank_144 (two-arc integer heights) and rank_019 (real heights) in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995)	Review (AI-assisted audit)	All three reviews report no gap and list no issues. The review of the weighted paper states that it read every section, checking the discrete section step by step. The review of the two-arc paper states that it read all ten sections, checking several line by line, and notes that this paper imports only the annulus circuit theorem from the weighted companion. The review of the real-height paper describes itself as a partial audit of a manuscript of about 8,400 lines of LaTeX.	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 232 What does a random landscape look like?, remade version of 22 October 2026	Reading (exposition)	Video explainer of three OpenAI preprints, 261 pages in their second editions dated 6 October 2026, with first editions of 25 September: Gaussian free field limits for integer Lipschitz heights on the triangular lattice with zero boundary values, weighted by x per height change, for every fixed x from 1/√2 to 1; for Schramm's two-arc model of odd integer heights with steps 0 or 2 and boundary values +1 and −1; and for uniform real heights with steps at most one, where at one tuned boundary amplitude, defined only implicitly, the interface converges to SLE₄. The episode recalls Schramm's 2006 questions and the earlier logarithmic-variance results for integer heights, and states that the interface of the integer two-arc model, the values of the scaling constants and the real-height interface at other amplitudes are not claimed.	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 preprints: OpenAI, three papers, 261 pages, second editions dated October 6, 2026, first editions September 25; under expert review: not yet; peer reviewed: not yet; checked in Lean: no. The page states that the papers were revised once, on October 6, with the real-height paper's interface argument nearly doubled; that there is no Lean link for this result, while an uncatalogued Lean folder of 243 files formalizes some building blocks of the real-height paper but no limit statement; that human teams posted proofs of Gaussian limits for the related six-vertex model in 2026; that the papers are not among those OSVerify reviewed; and that it found no expert reaction.

Notes

The background summarizes the manuscripts' 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.