OpenAI · 237-02 · 2026-09-26 · Probability and statistical mechanics
Critical honeycomb chords with prescribed boundary endpoints
preprints/Critical-honeycomb-chords-with-prescribed-boundary-endpoints-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.
Critical self-avoiding walks between prescribed, macroscopically separated boundary ports of a regular honeycomb hexagon have length in probability, where R is the scale of the hexagon. We prove the corresponding statements for half-plane arches, parallel cuts and nonparallel pure cuts. The half-plane law also has mean length . A separate strip argument gives endpoint mean laws on one density-one set of heights and in an aligned, critically weighted mixture of all even heights in a macroscopic interval.
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.