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Disk transfer representations and confined bridge mass

OpenAI · September 26, 2026

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Abstract

We prove that critical honeycomb bridges crossing a strip of width R have mean length $R^{4/3+o(1)}$, with the same exponent after confinement to a fixed multiple of the strip width. The initial boundary port is fixed, the terminal port is summed, and all finite lengths receive their critical weights. The bridge mass is comparable to R−1/4; the half-plane arch kernel at endpoint separation m is comparable to m−5/4.

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Cite

@misc{OAI:Disk-transfer-representations-and-confined-bridge-mass-September-26-2026,
  author = {{OpenAI}},
  title = {{Disk transfer representations and confined bridge mass}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Disk-transfer-representations-and-confined-bridge-mass-September-26-2026/main.pdf}{OAI:Disk-transfer-representations-and-confined-bridge-mass-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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