How rough is square ice?
A preprint claims the Gaussian free field limit of the six-vertex model, for every 0 < c ≤ 2
Covers result 225 in OpenAI’s catalog: its one paper. See its connections to other results and the work it cites.
4:26. English captions available. The video loads from YouTube when you press play.
Chapters
The claim, in plain words
Put an arrow on every edge of the square grid so that two arrows point in and two out at every vertex. Of the six kinds of vertex, four get weight 1 and the two saddle-shaped ones weight c; with c = 1 every configuration counts equally, and this is square ice. Read each arrow as a step of one between the faces on either side, and every configuration becomes a random terraced landscape.
Physicists expected that, zoomed out, it looks like the Gaussian free field with roughness 1/arcsin(c/2), for every c up to 2. Kenyon proved it at c = √2, where the model is equivalent to dominoes, and others extended it just around that point. In March 2026, Duminil-Copin, Kozlowski, Lammers and Manolescu proved it for √3 ≤ c ≤ 2, assuming a rotational-invariance result still in preparation, and called c below 1 beyond the reach of current methods.
Result 225 is one OpenAI preprint, 93 pages, dated September 23, 2026. It claims the whole range: for every c above 0 and up to 2, the height field of one specific flat plane state, built as a limit of balanced tori, converges to the Gaussian free field with roughness 1/arcsin(c/2). Below c = 1 the argument is self-contained; from 1 up to 2 it proves convergence itself but takes the identification of the state and the constant from the March preprint; at 2 it follows that preprint’s proof and supplies the missing rotational-invariance input in an appendix.
What it doesn’t claim
Other states or slopes, uniqueness among Gibbs states, domains with boundaries, unequal weights a ≠ b, c above 2, or estimates uniform as c goes to 0.
Where it stands
- Released as a preprint
- OpenAI, 93 pages, dated September 23, 2026
- Peer reviewed
- Not yet
- Checked in Lean
- Not formalized
Last checked October 19, 2026. The preprint has not changed since it was added to the repository in its initial commit on October 6, 2026, and there is no Lean for it at all. A different version of the same paper sits inside item 223’s files, crediting earlier work differently. One day before the release, a human paper (Averous, Duminil-Copin, He, Lammers and Manolescu) still counted only a small part of this range as proved. Release item 233 proves a related result for 1 < c ≤ 2, in domains, by a different route, matching this constant. Not among those OSVerify reviewed; we found no expert reaction.
How the proof goes
- Rows and Bethe roots. The transfer matrix builds the lattice one row at a time; by the Bethe ansatz, every fixed pattern of height marks becomes a sum of determinants over Bethe roots, using exact identities of Slavnov and others. The hard part is controlling the roots uniformly as the rows grow.
- A squeezed constant. In the limit the marks act as currents, and the rows’ angular spectrum gives the coefficient of the logarithm in the height variance. Below c = 1 the paper pins it between a stiffness bound and an overlap bound that meet, with no positive-correlation inequality.
- A free field. Splitting the spectrum into two separated scales gives holomorphic and antiholomorphic currents with Wick-type correlations: the derivatives of a free field.
The numbers in the video
- Ours: exact row variances on cylinders up to 20 sites around, averaged over even and odd sizes, match the prediction within about 1.5% at four values of c from 0.8 to 1.8, and are 6–8% off at c = 0.5 and 2.
- Ours: Monte Carlo on tori up to 128 wide (sampler checked against exact enumeration) gives fitted roughness within 8% for every c tried, mostly within 5%. Variance only, not the Gaussian law.
Read the source
- The Gaussian free field limit of the balanced six-vertex model with variance multiplier 1/arcsin(c/2) (PDF), 93 pages.
- The collection’s catalog, where this is result 225.
The paper
OpenAI groups related papers into one result; this one has a single paper.
Transcript
Read the full script
0:00The question
Put an arrow on every edge of the square grid, so that at every vertex two arrows point in and two point out. That ice rule leaves six kinds of vertex. Give four of them weight one, and the two saddle-shaped ones weight c. With c equal to one, every configuration counts equally: this is square ice. Read each arrow as a step of one between the faces on either side, and every configuration becomes a random terraced landscape. Zoomed out, how rough is the landscape? Physicists predicted in the 1980s that it looks like the Gaussian free field, the standard random surface, with a roughness of 1/arcsin(c/2), expected for every c up to two. Kenyon proved it at c equal to √2, where the model is equivalent to dominoes, and others extended it just around that point. In March 2026, Duminil-Copin, Kozlowski, Lammers and Manolescu proved it for c from √3 to two, assuming a rotational-invariance result still in preparation, and called c below one beyond the reach of current methods.
1:05The claim
Result 225 is one OpenAI preprint, 93 pages, dated September 23, 2026. It claims the whole range: for every c above zero and up to two, the height field of one specific flat plane state, built as a limit of balanced tori, converges to the Gaussian free field, with roughness 1/arcsin(c/2). It makes no claim about other states or slopes, about domains with boundaries, or about uniformity as c goes to zero. How much is new depends on c. Below one, the argument is self-contained. From one up to two, the paper proves convergence itself but takes the identification of the state and the constant from the March preprint. At two, it follows that preprint’s proof and supplies the missing rotational-invariance input in an appendix. One day before the release, a human paper still counted only a small part of this range as proved. And another release paper, 233, proves a related result for c from one to two, in domains, by a different route, matching this constant.
2:15How the proof goes
The proof works with the transfer matrix, which builds the lattice one row at a time. Its rows are solved by the Bethe ansatz: every fixed pattern of height marks becomes a sum of determinants over Bethe roots, using exact identities of Slavnov and others. The hard analytical part is controlling these roots uniformly as the rows grow, including roots that escape to the ends. In the limit, the marks act as currents, and the rows’ angular spectrum gives the coefficient of the logarithm in the height variance. Below c equal to one, the paper pins that coefficient between two bounds that meet: a stiffness bound, from twisting a seam and measuring the curvature of the ground energy, and an overlap bound from a dense array of inhomogeneities, with no positive-correlation inequality needed. Splitting the spectrum into two separated scales gives holomorphic and antiholomorphic currents with Gaussian, Wick-type correlations: the derivatives of a free field. As an illustration, our computer found exact row variances on cylinders up to 20 sites around. Their logarithmic growth, averaged over even and odd sizes, matched the prediction within about 1.5% at four values from 0.8 to 1.8, including square ice, and was 6–8% off at the two ends, where convergence is slow. Monte Carlo on tori up to 128 wide gave fitted roughness within about 8% for every c tried, mostly within 5%. This checks the variance only, not the Gaussian law.
3:51Where it stands
The preprint is unrefereed and unchanged since release. There is no Lean for it at all. A different version of the same paper sits inside another release item, crediting earlier work differently. It is not among those OSVerify reviewed, and we found no expert reaction. If it holds, square ice and its balanced cousins, in this flat plane state, are rough exactly as the Coulomb gas predicted: one formula, 1/arcsin(c/2), from c near zero all the way to two.
Spot a mistake?
This episode simplifies on purpose, but it shouldn’t be wrong. If something is, open an issue on GitHub or leave a comment on YouTube. Corrections are noted here with a date.