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Closing the identification step in the GFF limit: Uniqueness of the zero-slope six-vertex phase for −1 ≤ Δ < 1/2

A mix of human-written and AI-generated textHuman understanding: some partsmath.PR — Probabilitymath.MP — Mathematical Physics

Contributed by Yan Ru Pei ↗

Version 1 / Nov 11, 2026 / CC BY 4.0

Abstract

For the six-vertex model with a=b=1a=b=1 and c∈(1,2]c\in(1,2], that is −1≤Δ<12-1\le\Delta<\frac{1}{2}, we prove that the plane measure, the limit of balanced torus measures, is the only translation-ergodic gradient Gibbs measure of slope zero, and hence the unique translation-invariant minimiser of the specific free energy. Both statements had been asserted by reference to Sheffield's uniqueness theory, which is proved for nearest-neighbour potentials; we show that for c≠1c\ne1 the six-vertex weights, which couple faces across the diagonals of each vertex, are not a nearest-neighbour potential in height coordinates. The proof extends Sheffield's cluster swapping to these diagonal interactions by means of residual energies on the diagonals, shows with the Lanford–Ruelle theorem for subshifts of finite type and a Burton–Keane argument that two distinct ergodic zero-slope phases are gradients of Gibbs measures on heights, and shows from the spin representation in the form given by Duminil-Copin, Kozlowski, Lammers and Manolescu that the plane measure is not. This supplies a step that we found missing in the chain of recent results on the Gaussian free field limit of the six-vertex height function. An unrefereed manuscript of OpenAI states the full-plane limit for every c∈(0,2]c\in(0,2]; for 2≤c<2\sqrt{2}\le c<2 it identifies a state built from nearly balanced sectors with the plane measure through the characterisation above. Our result gives that identification, given two properties of the state stated in the manuscript. We also check that the manuscript and the work of Averous, Duminil-Copin, He, Lammers and Manolescu concern the same measure, so that their convergence in domains holds, for c∈[1,2]c\in[1,2] and with constants depending on Δ\Delta, wherever the manuscript's full-plane theorem holds.

Provenance statement

Claude Fable 5.2 and Claude Opus 5.5 (Anthropic), running as agents in Claude Code, were used for proof discovery, checking and writing. Opus agents audited the OpenAI manuscript's proof of the full-plane limit, located the gap in its identification step for √2 ≤ c < √3, and reduced that step to the uniqueness statement of Theorem 1.2; they also proved the results of Section 7. A Fable agent proved Theorem 1.2, through Theorems 1.4 and 1.5. Independent Opus agents audited these proofs blind, with three referees for Theorem 1.2 and a further referee who rechecked its repairs, and an Opus precedence search found the statement of Theorem 1.2 asserted in arXiv:2603.06268 and sketched in arXiv:2608.21306 but found no proof of it. Fable drafted Sections 2 and 7 as an earlier note; Opus wrote the present manuscript, and two Opus reviewers checked it against the audited proofs and the cited sources in one pass. The contributor selected the problem from a list of candidate targets, assigned the models to the steps, directed the structure and presentation of the paper, and made the title and venue decisions. The two properties of the OpenAI state used in Corollary 1.6 were not checked. No formal verification was done.

Tools used

Anthropic
Claude FableVersion 5.2
Anthropic
Claude OpusVersion 5.5

References

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Version history

  1. v1 · 33 pagesSubmitted by Yan Ru PeiInitial depositCurrentNov 10, 2026