View source for The three-quarter exponent for honeycomb self-avoiding walk

'''The three-quarter exponent for honeycomb self-avoiding walk''' is the prediction, going back to Nienhuis's Coulomb-gas analysis, that a typical ''n''-step self-avoiding walk in the plane has spatial extent of order ''n''<sup>3/4</sup>. A family of thirteen manuscripts in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]), dated 26 September 2026, claims the diameter form of this prediction on the honeycomb lattice, together with mass and covering exponents 4/3. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper is not presented here as independent verification.

== Background ==
* As recounted in the introduction of the principal manuscript:
** The exponent 3/4 belongs to the two-dimensional polymer theory developed through the dilute O(''n'') model; Nienhuis's analysis gives it at ''n'' = 0. Duminil-Copin and Smirnov proved the exact connective constant of the honeycomb lattice with a parafermionic observable; their argument bounds the critical strip-crossing mass between ''c''/''T'' and 1 and records the sharper exponent −1/4 as a prediction.
** Beaton, Bousquet-Mélou, de Gier, Duminil-Copin and Guttmann proved that the strip-crossing mass tends to zero; Glazman and Manolescu gave a shorter proof with a logarithmic subsequence bound; Krachun and Panagiotis proved a polynomial upper bound and a quantitative sub-ballistic estimate.
** Lawler, Schramm and Werner proposed SLE<sub>8/3</sub> as the scaling limit, whose Hausdorff dimension 4/3 matches the predicted mass exponent. The bridge decomposition follows Kesten's renewal method.

== The original results ==
* '''Claim.''' Theorem 1.1 of ''Mass and covering exponents for fixed-length honeycomb walks'' states that for every δ > 0 and ''k'' > 0 and every sufficiently large integer ''n'', with probability at least 1 − ''C n''<sup>−''k''</sup> under the uniform law on ''n''-step self-avoiding walks from a fixed vertex of the honeycomb lattice, the diameter lies between ''n''<sup>3/4−δ</sup> and ''n''<sup>3/4+δ</sup>, and the local mass and covering numbers obey the corresponding exponent 4/3 simultaneously at all scales. Corollaries give the span in every lattice normal direction, the covering dimension and moments of the diameter and the radius of gyration. The manuscript states that convergence to a continuum curve, a continuum Hausdorff dimension, universality across lattices and a lower bound on the end-to-end distance are not claimed.
* ''Method, as described:'' the principal manuscript proves the geometric passage from quantitative inputs to the fixed-length walk, through bridge sewing, renewal trials, ordered tests of adjacent paths and a multiplicity estimate for repeated visits. The inputs, namely strip-crossing mass of order ''h''<sup>−1/4</sup>, bridge and chord length laws, cylinder loop weights, nesting exponents and marked-polygon bounds, are proved in twelve companion manuscripts that use exact lattice observables.
* '''Sources''' (openai/math repository, commit <code>7ef5c0c</code>): the principal manuscript [https://github.com/openai/math/tree/7ef5c0cc5ca4b0e072b7a050fca09ab24bc6fa3a/preprints/Mass-and-covering-exponents-for-fixed-length-honeycomb-walks-September-26-2026 Mass and covering exponents for fixed-length honeycomb walks]; the twelve companions are listed under result 237 in the collection's [https://github.com/openai/math/blob/7ef5c0cc5ca4b0e072b7a050fca09ab24bc6fa3a/CONTENTS.md catalogue].
* '''Formalization.''' The collection's [https://github.com/openai/math/blob/7ef5c0cc5ca4b0e072b7a050fca09ab24bc6fa3a/lean/docs/237.md scope note] lists Lean formalizations of supporting statements: finiteness of the critical bridge measures and their first length moments, existence of the free-energy limit, the critical strip mass comparable to ''N''<sup>−1/4</sup> with the displacement moment of order ''N''<sup>3/4</sup>, and the strict bridge mass of order (1 + ''h'')<sup>−1/4</sup>. The scope note states that the length laws and the 3/4 spatial and moment laws are outside the formalization.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 28 October 2026; AI Proved What? row 28 October 2026)
|-
| 2026-10-06 || [https://lowrank.github.io/latent-seminar/OpenAI-Math/237-honeycomb-walks ''Honeycomb walks: strip estimates and the three-quarter exponent''], an article of the Latent Seminar blog at lowrank.github.io || Reading, exposition || Explains the thirteen companion manuscripts and how they build toward the fixed-length claim: strip mass and length normalization, chords, cylinders and marked polygons, and the renewal passage to every length, with the probability law specified at each stage. || Blog article; it states that its review covers statements, introductions and selected proof mechanisms and is not an independent verification of the complete proofs.
|-
| 2026-10-10 || F. Nie, H. Ye, S. Liu, Z. Wang, S. Ermon and J. Zou, OSVerify, Stanford University: referee reports on three companions, [https://osverify.github.io/cases/cylinder-loop-weights-and-planar-nesting.html ''Cylinder loop weights and planar nesting''], [https://osverify.github.io/cases/renewal-and-changes-of-law-for-critical-honeycomb-walks.html ''Renewal and changes of law for critical honeycomb walks''] and [https://osverify.github.io/cases/cap-selected-amplitudes-and-triangle-chords-for-honeycomb-walks.html ''Cap-selected amplitudes and triangle chords for honeycomb walks''] || Review, referee reports || Reviews the manuscript versions of 26 September 2026. For the cylinder paper the report finds one major issue: the global bound for the second circle kernel in the proof of Lemma 3.9 omits a bulk contribution; its proposed repair, a bound of the form ''C'' times 1 + 1/''d'', is absorbed by an existing term and keeps the comparison used later. The other two reports find one minor issue each: citations to the signed-cylinder paper point to the wrong propositions, and a strict comparison in the proof of Proposition 9.1 becomes an equality. || Web pages at osverify.github.io, first posted 10 October 2026, last changed 12 and 13 October 2026. The pages state that their findings and severity labels come from OSVerify's internal review and do not necessarily establish that the main theorem is false; the site's home page states that every reported issue was verified by human experts.
|-
| 2026-10-17 || ''AI Proved What?'', episode 237 [https://aiprovedwhat.com/episodes/237-honeycomb-saw-three-quarter/ How far does a walk that never crosses itself go?], remade version of 22 October 2026 || Reading (exposition) || Video explainer of the claim that a uniformly chosen n-step self-avoiding walk on the honeycomb lattice has diameter between n to the 3/4 − δ and n to the 3/4 + δ, except with probability smaller than any fixed power of n, at every large length, and that a ball of radius s around any point of the walk holds about s to the 4/3 of it. The episode states that the classical end-to-end form at every length, a scaling limit or convergence to SLE 8/3, the square lattice and limiting constants are not claimed. || Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 19 October 2026: posted as preprints: OpenAI, 13 papers, 1,067 pages, dated September 26, 2026; under expert review: not yet; peer reviewed: not yet, OSVerify reviewed three of the papers; checked in Lean: in part, supporting facts.
|}

== Notes ==
* The earlier results are due to the authors named in the Background section, as recounted by the manuscript.
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]