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Signed cylinder propagation and marked polygons on the honeycomb lattice

OpenAI · September 26, 2026

Lean

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Abstract

At the critical activity and loop fugacity two, we prove a 1/6 partition exponent for disjoint honeycomb polygons separating opposite marks on a balanced infinite cylinder, and a 1/12 planar nesting exponent. For ordered first-exit chords in a regular hexagon of side R, with both boundary ports summed and diameter at least $R/10$, the finite critical partition is $R^{3/4+o(1)}$ and the normalized mean length is $R^{4/3+o(1)}$. A separate two-bond estimate on tilted cylinders with controlled site proportions bounds the length-square mass of planar polygons of diameter at most H, modulo translations, by $H^{2/3+o(1)}$.

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Cite

@misc{OAI:Signed-cylinder-propagation-and-marked-polygons-on-the-honeycomb-lattice-September-26-2026,
  author = {{OpenAI}},
  title = {{Signed cylinder propagation and marked polygons on the honeycomb lattice}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Signed-cylinder-propagation-and-marked-polygons-on-the-honeycomb-lattice-September-26-2026/main.pdf}{OAI:Signed-cylinder-propagation-and-marked-polygons-on-the-honeycomb-lattice-September-26-2026}},
  year = {2026}
}

Lean formalization

OpenAI provides Lean files for this family: lean/docs/237.md. The Lean statements have not been checked by people against the paper's claims.

Family 237: The three-quarter exponent for honeycomb self-avoiding walk

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.

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