Two magnets, one free field?
A preprint claims the joint scaling limit of critical Ashkin–Teller heights and currents, at every point of the critical line
Covers result 233 in OpenAI’s catalog: its one paper. See its connections to other results and the work it cites.
4:32. English captions available. The video loads from YouTube when you press play.
Chapters
The claim, in plain words
Take two Ising magnets on the same square grid and couple them with a four-spin interaction of strength U. This is the Ashkin–Teller model, first posed in 1943 as a four-component lattice. With U = 0 the two magnets are independent; when U reaches the strength of the ordinary couplings it becomes the four-state Potts model. Through both runs a line of critical points, along which the large-scale behaviour changes continuously through one parameter g, from just above 4/3 up to 4.
The paper looks not at the spins but at random edge sets built from them, called currents (after Marcin Lis), one with wired and one with free boundary, plus a height function that steps by ½ across corners. In February 2026 Alcalde López, Heeney and Lis conjectured the large-scale picture for the height and the currents: the height becomes a Gaussian free field, and every cluster of both currents, at every nesting depth, becomes an explicit random set cut out of that same field. Only the point U = 0 had been proved, by Duminil-Copin, Lis and Qian.
Result 233 is one OpenAI preprint, 105 pages in its second edition, dated October 6, 2026 (first edition September 25). It claims that conjecture at every point of the critical line, including the Potts endpoint, in every bounded Jordan domain with suitable lattice approximations: 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.
What it doesn’t claim
Limits for the spins themselves, their magnetization or polarization, the spin interfaces, area measures, or uniformity as g comes down toward 4/3. Clusters converge as sets, not as curves.
Where it stands
- Posted as a preprint
- OpenAI, 105 pages, second edition dated October 6, 2026 (first edition September 25)
- Peer reviewed
- Not yet
- Checked in Lean
- No
Last checked October 19, 2026. The second edition repaired several arguments and replaced one imported convergence step with a new one. There is no Lean for this result. On October 6, the same day the repository went public, a group including Duminil-Copin posted a proof that the q = 4 random-cluster loops converge to CLE4, and announced an Ashkin–Teller paper in preparation. Not among those OSVerify reviewed; we found no public reaction.
How the proof goes
- Gaussian without rotational symmetry. The main outside input is a 2026 six-vertex paper by Duminil-Copin, Kozlowski, Lammers and Manolescu, itself unrefereed, whose plane Gaussian limit rests on a rotational-invariance input still in preparation that covers only part of the line. The new paper avoids it: reflection positivity in frames set at 45 degrees to each other cancels the square lattice’s angular wobble, so the covariance has the free field’s shape; its strength is fixed afterwards.
- Rooms and explorations. The picture is cut into flat rooms using exact loops; the height is Gaussian in each room; the currents are explored inward from the boundary, and each exploration converges to a canonical local set of the limiting field.
- One match per cluster. Every lattice cluster is matched to exactly one canonical set, using a classical theorem of Sierpiński: a continuum cannot be split into countably many, but at least two, disjoint closed pieces.
The numbers in the video
- Ours, exact: the paper’s current law against the spin construction on tiny boxes, 18 comparisons, all matching; at the Potts end the currents become the critical q = 4 random-cluster model. One site joined by four wires is cut off with probability 1/9 for two independent Ising magnets and 1/21 at the Potts end.
- Ours, Monte Carlo of twice the height on boxes up to 64: the centre variance grows with the log of the size, with slopes 0.28 ± 0.03 (predicted 0.30) at U = −1 and 0.22 ± 0.03 (predicted 0.20) at U = 0. At the Potts end the slope is still about twice the prediction at this size; slow corrections are expected there, so this run confirms nothing at that end.
Read the source
- The joint scaling limit of critical Ashkin–Teller currents (PDF), 105 pages.
- The collection’s catalog, where this is result 233.
The paper
OpenAI groups related papers into one result; this one has a single paper.
Transcript
Read the full script
0:00The question
Take two Ising magnets on the same square grid, each spin plus or minus one, and couple them with a four-spin interaction of strength U. This is the Ashkin–Teller model, first posed in 1943 as a four-component lattice. With U equal to zero the two magnets are independent; when U reaches the strength of the ordinary couplings it becomes the four-state Potts model. Through both runs a line of critical points, along which the large-scale behavior changes continuously, through one parameter g, from just above 4/3 up to four. The paper looks not at the spins, but at random edge sets built from them, called currents, following Marcin Lis: one with wired boundary and one with free boundary, plus a height function that steps by one half across corners. In February 2026, Alcalde López, Heeney and Lis conjectured the large-scale picture for the height and the currents: the height becomes a Gaussian free field, and every cluster of both currents, at every nesting depth, becomes an explicit random set cut out of that same field. Only the point U equals zero had been proved, by Duminil-Copin, Lis and Qian. Our computer checked the current law exactly on tiny boxes: 18 comparisons between the paper’s formula and the spin construction, all matching. At the Potts end the currents become the critical random-cluster model with four colors. In the smallest case, one site joined by four wires, the chance that the site is cut off is 1/9 for two independent Ising magnets, and 1/21 at the Potts end.
1:38The claim
Result 233 is one OpenAI preprint, 105 pages in its second edition, dated October 6, 2026; the first edition is dated September 25. It claims that conjecture at every point of the critical line, including the Potts endpoint, in every bounded Jordan domain with suitable lattice approximations: 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. Not claimed: limits for the spins themselves, their magnetization or polarization, the spin interfaces, area measures, or uniformity as g comes down toward 4/3. Clusters converge as sets, not as curves.
2:29How the proof goes
The main outside input is a 2026 six-vertex paper by Duminil-Copin, Kozlowski, Lammers and Manolescu, itself unrefereed. Its plane Gaussian limit rests on a rotational-invariance input that is still in preparation and covers only part of the line. The new paper avoids it: reflection positivity in frames set at 45 degrees to each other cancels the square lattice’s angular wobble, so the covariance has the free field’s shape; its strength is fixed afterwards. From there the paper cuts the picture into flat rooms using exact loops, proves the height is Gaussian in each room, and explores the currents inward from the boundary. Each exploration converges to a canonical local set of the limiting field. Finally it matches every lattice cluster to exactly one canonical set, using a classical theorem of Sierpiński: a continuum cannot be split into countably many, but at least two, disjoint closed pieces. As an illustration, our computer simulated twice the height on boxes up to 64. Its variance at the center grows with the log of the size, with slopes close to the predicted ones at two points of the line. At the Potts end, the slope is still about twice the prediction at this size; slow corrections are expected there, so this run confirms nothing at that end.
3:51Where it stands
The preprint is unrefereed; the second edition repaired several arguments and replaced one imported convergence step with a new one. There is no Lean for this result. The same day the repository went public, a group including Duminil-Copin posted a proof that the q-equals-four random-cluster loops converge to CLE₄, and announced an Ashkin–Teller paper in preparation. Not among those OSVerify reviewed, and we found no public reaction. If it holds, at every point of the critical Ashkin–Teller line, the height and both currents are described by one Gaussian free field, at every depth of nesting.
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.