OpenAI · 237-06 · 2026-09-26 · Probability and statistical mechanics

Cylinder amplitudes and logarithmic bridge-length windows on the honeycomb lattice

preprints/Cylinder-amplitudes-and-logarithmic-bridge-length-windows-on-the-honeycomb-lattice-September-26-2026

Family 237. The three-quarter diameter exponent for honeycomb walks.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

Proves the diameter form of Nienhuis's predicted three-quarter exponent: a uniformly chosen $n$ -step self-avoiding walk on the honeycomb lattice has diameter $n^{3/4 + o (1)}$ . Its local mass and covering numbers have exponent $4/3$ . These estimates hold at every sufficiently large fixed length, simultaneously across scales, with arbitrarily high polynomial probability.

We prove a logarithmic window of critical honeycomb bridge lengths: summed through height $2 h (lo g h)^{1/64}$ , bridges of lengths between $h^{4/3} (lo g h)^{- 1/8}$ and $2 h^{4/3} (lo g h)^{1/2}$ have mass at least $h^{3/4} (lo g h)^{- C}$ .

Preprint source and Lean scope

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A Lean scope note is stored for family 237, but it does not name this preprint. It names 237-09, 237-10, 237-11, 237-02. This page still has the full preprint source. No compilation is asserted. O.1 stays NOT_CLOSED.