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Uniform marked-polygon estimates and sharp finite bridge moments

OpenAI · September 26, 2026

Lean

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Abstract

We prove a uniform bound for critical honeycomb polygons through two axial marks on a periodic staircase. The bound retains an explicit power of the ratio between the period and the marked separation. We also prove sharp finite strip estimates: bridge mass of order h−1/4, first-length mass at most $Ch^{13/12}$, and mass at least $ch^{-1/4}$ on bridges with length at least $ch^{4/3}$.

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Cite

@misc{OAI:Uniform-marked-polygon-estimates-and-sharp-finite-bridge-moments-September-26-2026,
  author = {{OpenAI}},
  title = {{Uniform marked-polygon estimates and sharp finite bridge moments}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Uniform-marked-polygon-estimates-and-sharp-finite-bridge-moments-September-26-2026/main.pdf}{OAI:Uniform-marked-polygon-estimates-and-sharp-finite-bridge-moments-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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