Nienhuis's 3/4 exponent for self-avoiding walk on the honeycomb lattice, in diameter form

Let $\mathbb P_n$ be the uniform law on $n$ -step self-avoiding walks from a fixed vertex of a planar lattice. Nienhuis's Coulomb-gas analysis of the dilute $O(n)$ model at $n=0$ (1982) predicts the size exponent $\nu=3/4$ : a typical $n$ -step walk has spatial extent $n^{3/4}$ , equivalently its trace has mass dimension $4/3$ ; Lawler, Schramm and Werner's conjectured $SLE_{8/3}$ scaling limit gives the same value. On the honeycomb lattice Duminil-Copin and Smirnov proved that the connective constant is $\sqrt{2+\sqrt2}$ , but no polynomial size exponent was known; the best fixed-length bound was sub-ballistic (Krachun-Panagiotis: within $Cn/\log n$ ). Does a uniform $n$ -step self-avoiding walk on the honeycomb lattice have spatial extent $n^{3/4+o(1)}$ ?

Result: Proved(see note)
Status: Partial result
AI contribution: AI-discovered
Method: Argument
Field: Self-avoiding walk; two-dimensional critical phenomena
Posed by: Bernard Nienhuis (prediction from the dilute O(n) model)
Year posed: 1982
Years open: 44y
Solved: 2026-09-26
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 58 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: for every $\delta>0$ and $k>0$ there are $C_{\delta,k}$ and $n_0$ such that for every integer $n\ge n_0$ , with $\mathbb P_n$ -probability at least $1-C_{\delta,k}n^{-k}$ , the diameter satisfies $n^{3/4-\delta}\le D(\gamma)\le n^{3/4+\delta}$ , and simultaneously for all visited $z$ and $s\in[1,n]$ the ball mass is within $n^{\pm\delta}$ of $\min\{n,s^{4/3}\}$ and the covering number by $s$ -balls within $n^{\pm\delta}$ of $1+ns^{-4/3}$ . Corollaries: span at least $n^{3/4-\delta}$ in each lattice normal, covering dimension $4/3$ , radius of gyration $n^{3/4+o(1)}$ and $\mathbb E_n[D^p]=n^{3p/4+o(1)}$ . Not shown: a fixed-length lower bound for the end-to-end distance (a companion has it only on a density-one set of lengths), an exponent without $n^{o(1)}$ slack, other lattices, or convergence to $SLE_{8/3}$ .

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family has thirteen manuscripts, all dated September 26, 2026. The principal manuscript proves the fixed-length geometric statement from analytic inputs (strip-crossing mass, nesting around faces, marked-polygon and arc bounds) whose proofs are in the twelve companion manuscripts.

Verification

No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of 'Mass and covering exponents for fixed-length honeycomb walks' were read against Nienhuis's prediction as the manuscript cites it. Theorem 1.1 gives, for every $\delta,k>0$ and every large integer $n$ , an event of probability at least $1-Cn^{-k}$ on which the diameter lies in $[n^{3/4-\delta},n^{3/4+\delta}]$ and local mass and covering numbers have exponent $4/3$ . The proof imports analytic inputs proved in twelve companion manuscripts, which were skimmed, not refereed. The release's Lean formalization for this family (HoneycombBridgeFiniteness, HoneycombFreeEnergy, CriticalStripMass) covers supporting statements only: finiteness of bridge sums, existence of the free energy and the strip-crossing mass law. Its scope note says the mean-length and 3/4 spatial laws are outside it, so this entry is unreviewed, not Lean-checked. The paper states that it gives no lower bound on the end-to-end distance at fixed length, no scaling limit and no universality across lattices.

Nienhuis's 3/4 exponent for self-avoiding walk on the honeycomb lattice, in diameter form

What was actually shown

What the AI did

Verification

Sources

Discussion