Gaussian free field limit of the six-vertex height function throughout the regime a = b = 1, 0 < c <= 2

In the square-lattice six-vertex model with weights $a=b=1$ and $c>0$ , the ice rule makes the arrows a height function $h$ on faces. The Coulomb-gas picture predicts that in the critical range $|\Delta|\le1$ , $\Delta=1-c^2/2$ (that is $0<c\le2$ ), the rescaled height converges to a multiple of the Gaussian free field; Di Francesco, Saleur and Zuber (1987) gave the Gaussian action for $1<c<2$ with squared multiplier $1/\arcsin(c/2)$ in unit height steps. Rigorous results covered the free-fermion point $c=\sqrt2$ (Kenyon), perturbations of it, logarithmic delocalization for $1\le c\le2$ , and a GFF theorem for $\sqrt3\le c\le2$ . Does the height function converge to $\sigma(c)$ times the Gaussian free field, with $\sigma(c)^2=1/\arcsin(c/2)$ , for every $c\in(0,2]$ ?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Statistical mechanics; six-vertex model and random surfaces
Posed by: Coulomb-gas prediction; the manuscript cites P. di Francesco, H. Saleur and J.-B. Zuber (J. Stat. Phys. 1987) for the Gaussian action and coupling
Year posed: 1987
Years open: 39y
Solved: 2026-09-23
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 40 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: for each $c\in(0,2]$ the balanced-torus plane law exists and $h(\cdot/\delta)$ converges to $\sigma(c)\Gamma$ , $\sigma(c)^2=2/\arccos(1-c^2/2)=1/\arcsin(c/2)$ , with Green kernel $-(2\pi)^{-1}\log|x-y|$ , jointly on finite-energy mass-zero test measures, in $B^\alpha_{p,q}$ and $W^{\alpha,p}$ restriction spaces for $-1<\alpha<0$ , and with Wick-type limits of increment moments. Priority: Duminil-Copin, Kozlowski, Lammers and Manolescu had proved this for $\sqrt3\le c\le2$ . Not shown: other slopes or Gibbs states, anisotropic $a\ne b$ , or the regime $c>2$ .

What the AI did

The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has a single manuscript.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the prediction. For every $c\in(0,2]$ it constructs the plane law $\mathbb P_c$ as an iterated limit of balanced tori and proves convergence of $h_\delta$ to $\sigma(c)\Gamma$ in finite-dimensional laws, in negative Holder, Besov and Sobolev restriction spaces, and in moments of increments. Scope limits the paper states: one zero-slope plane state only, no uniqueness among Gibbs states or slopes, no uniformity as $c\to0$ , and isotropic weights $a=b$ only. At $c=2$ and for the amplitude on $1\le c<2$ it uses Duminil-Copin-Kozlowski-Lammers-Manolescu (2026), whose rotational-invariance input is cited as in preparation; the manuscript supplies its own endpoint rotation argument. No Lean formalization.

Gaussian free field limit of the six-vertex height function throughout the regime a = b = 1, 0 < c <= 2

What was actually shown

What the AI did

Verification

Sources

Discussion