OpenAI · 237-10 · 2026-09-26 · Probability and statistical mechanics
Renewal and changes of law for critical honeycomb walks
preprints/Renewal-and-changes-of-law-for-critical-honeycomb-walks-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 -step self-avoiding walk on the honeycomb lattice has diameter . Its local mass and covering numbers have exponent . These estimates hold at every sufficiently large fixed length, simultaneously across scales, with arbitrarily high polynomial probability.
For critical honeycomb self-avoiding walk, we prove spatial exponent 3/4 for the infinite irreducible-bridge law and for bridges of every sufficiently large compatible even length. We establish the corresponding thermal laws at every large discount scale, including all positive length and spatial moments. For unrestricted uniform walks, the endpoint lower law holds on a common set of lengths of natural density one. We also determine the near-critical exponential correlation scale and small-force free-energy exponent, and prove spatial local lower bounds that permit conditioning on a prescribed terminal vertex.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 237: The three-quarter exponent for honeycomb self-avoiding walk. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.