AI Proved What?

Episode 237, October 2026

How far does a walk that never crosses itself go?

Thirteen preprints claim the three-quarter law for self-avoiding walk on the honeycomb lattice

Covers result 237 in OpenAI’s catalog: all 13 papers. See its connections to other results and the work it cites.

Play Episode 237: How far does a walk that never crosses itself go? A long random self-avoiding walk drawn in teal.

4:34. English captions available. The video loads from YouTube when you press play.

Chapters

  1. 0:00The question
  2. 1:28The claim
  3. 2:28How the proof goes
  4. 3:46Where it stands

The claim, in plain words

A self-avoiding walk is a path on a lattice that never visits the same point twice, a simple model of a polymer chain. On the honeycomb lattice every vertex has three neighbours; from one vertex there are 90 walks of six steps (96 paths that never step straight back, minus the 6 that close a hexagon). Pick an n-step walk uniformly at random: how far does it spread? A random walk spreads like n1/2, a straight line like n. Flory guessed in 1949, and Nienhuis derived in 1982, that planar self-avoiding walks spread like n3/4; numerical work confirms it to many digits. Rigorously almost nothing was known: on the honeycomb the best bound, by Krachun and Panagiotis in 2023, was a constant times n/log n. Duminil-Copin and Smirnov proved in 2012 that each extra step multiplies the number of walks by √(2+√2) ≈ 1.85.

Result 237 is thirteen OpenAI preprints, 1,067 pages, dated September 26, 2026. The headline paper claims that a uniformly chosen n-step self-avoiding walk on the honeycomb lattice has diameter between n3/4−δ and n3/4+δ, except with probability smaller than any fixed power of n, at every large length; around every point of the walk a ball of radius s holds about s4/3 of it. If correct, these are the first rigorous three-quarter laws for self-avoiding walk on a planar lattice that we could find.

What it doesn’t claim

The classical end-to-end form at every length (a large diameter does not force the endpoints apart; a sibling paper gets that form only on a set of lengths of density one), a scaling limit or convergence to SLE8/3, anything on the square lattice, or limiting constants.

Where it stands

Posted as preprints
OpenAI, 13 papers, 1,067 pages, dated September 26, 2026
Peer reviewed
Not yet (OSVerify reviewed three of the papers)
Checked in Lean
In part: supporting facts (docs)

Last checked October 19, 2026. The papers were posted on October 6 and have not been revised. OSVerify reviewed three of them: one major issue, a kernel bound in the cylinder nesting paper (which the headline relies on), with a proposed repair it says keeps the rest of the argument intact, and two minor issues in other papers. In Lean, four comparators check supporting facts, including the h−1/4 strip law: about 135,000 lines with no unproved steps (we did not build them). The three-quarter theorem itself is not formalized. We found no public reaction.

How the proof goes

  1. Exact inputs. Three companion papers use Duminil-Copin and Smirnov’s boundary identity, together with integrable loop-model weights from work by Glazman, Manolescu and others. They show that the critical weight of walks crossing a strip of height h decays like h−1/4 up to constants; earlier bounds only pinned it between 1/h and 100·h−10−10. They also get a planar nesting exponent of 1/12 from loops on a cylinder, and a bound on polygon sizes.
  2. All lengths mixed. In the ensemble that weights a walk of length ℓ by 0.5412ℓ, the headline paper shows that bridges of height h have length about h4/3. It also shows that any piece of a walk that is too fast or too slow for its size is rarer than any power of the box size.
  3. One length. The number of n-step walks times 0.5412n is at least 1. That inequality transfers every estimate to one fixed length, with no averaging.

The numbers in the video

  • Ours: every honeycomb self-avoiding walk up to 28 steps, 229,483,600 in all; c28 = 106,794,084, matching the published table (OEIS A001668). The product cn·0.5412n goes from 1.62 at one step to 3.65 at 28 and is always above 1.
  • Ours: a pivot-algorithm sampler up to 10,000 steps, 1,000 samples per length. The mean diameter has log-log slope 0.751 (95 percent interval 0.747 to 0.755). This is numerical evidence for a law no one doubted, not a test of the proof.

Read the source

The 13 papers in this result

OpenAI groups related papers, such as alternative proofs, supporting results and consequences, into one result; we label the first paper it lists the main paper and the others companions.

Transcript

Read the full script

0:00The question

A self-avoiding walk is a path on a lattice that never visits the same point twice: a model for a polymer chain. On the honeycomb lattice each vertex has three neighbors. From one vertex there are 90 walks of 6 steps: 96 paths that never step straight back, minus the 6 that close a hexagon. Pick an n-step walk uniformly at random. How far does it spread? A random walk spreads like the square root of n; a straight line, like n. Flory guessed in 1949, and Nienhuis derived in 1982, that self-avoiding walks in the plane spread like n^(3/4). Numerical work confirms it to many digits. But rigorously, almost nothing was known: on the honeycomb lattice the best bound, by Krachun and Panagiotis in 2023, was spread at most a constant times n/log n. In 2012 Duminil-Copin and Smirnov proved the honeycomb growth rate: each extra step multiplies the number of walks by √(2+√2), about 1.85. Our computer counted every honeycomb walk up to 28 steps, about 230 million in all, matching the published table. Then a pivot algorithm sampled walks of up to 10,000 steps: the diameter grows with slope 0.75 on a log-log plot, the expected three quarters. That is numerical evidence for a law no one doubted; the question is a proof.

1:28The claim

Result 237 is 13 OpenAI preprints, about 1,000 pages, dated September 26, 2026. The headline paper claims that a uniformly chosen n-step self-avoiding walk on the honeycomb lattice has diameter between n^(3/4−δ) and n^(3/4+δ), except with probability smaller than any fixed power of n, at every large length. Around every point of the walk, a ball of radius s holds about s^(4/3) of it: the walk looks four-thirds dimensional across scales. Not claimed: the classical end-to-end form at every length, since a large diameter does not force the endpoints apart; a sibling paper gets that form only for almost all lengths. Also not claimed: a scaling limit, any result on the square lattice, or limiting constants. If correct, these are the first rigorous three-quarter laws for self-avoiding walk on a planar lattice that we could find.

2:28How the proof goes

The headline rests on three companion papers. Using Duminil-Copin and Smirnov’s exact boundary identity, together with integrable loop-model weights from work by Glazman, Manolescu and others, they compute the critical weight of walks crossing a strip of height h: it decays exactly like h^(−1/4), up to constants, where earlier bounds only pinned it between 1/h and a power that barely decays at all. They also work out loops on a cylinder, which gives a nesting exponent of 1/12 for loops around a point, and a bound on polygon sizes. From these, the headline shows that, in the ensemble mixing all lengths, bridges of height h have length about h^(4/3), and that every piece of a walk too fast or too slow for its size is rarer than any power of the box size. A one-line inequality, that the number of n-step walks times the critical weight to the n is at least one, transfers all this to one fixed length, with no averaging. That last step is elementary, and our counts show it: the product is 1.6 at one step and 3.65 at 28, always above one. Everything rests on the companions’ exact computations, and on the long geometric core in between.

3:46Where it stands

The preprints are unrefereed and unchanged since release. OSVerify reviewed three of them: it found one major issue, a kernel bound in the cylinder nesting paper that the headline relies on, and proposed a repair it says keeps the rest of the argument intact; and two minor issues elsewhere. In Lean, four comparators check supporting facts, including the h^(−1/4) strip law, about 135,000 lines with no unproved steps; the three-quarter theorem itself is not formalized. We found no public reaction. If they hold, the planar self-avoiding walk, at least on the honeycomb, spreads like n^(3/4), more than 70 years after Flory’s guess.

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.