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

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 $SLE_{8/3}$ .

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 26, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Nienhuis's 3/4 exponent for self-avoiding walk on the honeycomb lattice, in diameter form · Honeycomb SAW 3/4 exponent (diameter)

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

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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, and HoneycombBridgeMassSupport added on 7 October 2026: strict bridge mass of order $(1+h)^{-1/4}$ from a fixed port) covers supporting statements only. 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.

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