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Buffered comparison and stopping-band resolution in critical Ising

OpenAI · September 23, 2026

Lean

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Abstract

We prove pointwise comparison and finite stopping-band approximation theorems for the critical nearest-neighbor Ising model on the square lattice. Arbitrary common pinned spins are allowed in the comparison: the likelihood ratio is controlled by a zero-field FK connection probability on the graph with those vertices deleted. For an interface exploration across a band of fixed positive width, finitely many regular signed barriers approximate every good conditional target law in total variation, uniformly over bounded observables and sufficiently fine meshes.

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work in progress

Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Formalization

  • OpenAI math release, Lean scope note for family 218 (lean/docs/218.md, October 2026). Selects this paper's finite-graph conditional likelihood comparison (ratio between $\exp(\pm 4\,\mathrm{artanh}\,q_0)$ with common pinned spins and arbitrary fields and boundary mixtures) as the comparator challenge, explicitly excluding the stopping-band approximation theorem.
  • OpenAI, BufferedIsing.lean (OpenAI Math Release, 2026). The comparator defines the finite-graph Ising and FK quantities and states finite_graph_comparison, whose proof in this file is admitted with sorry.

Digests and expositions

Elsewhere

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Cite

@misc{OAI:Buffered-comparison-and-stopping-band-resolution-in-critical-Ising-September-23-2026,
  author = {{OpenAI}},
  title = {{Buffered comparison and stopping-band resolution in critical Ising}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Buffered-comparison-and-stopping-band-resolution-in-critical-Ising-September-23-2026/paper.pdf}{OAI:Buffered-comparison-and-stopping-band-resolution-in-critical-Ising-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI provides Lean files for this family: lean/docs/218.md. The Lean statements have not been checked by people against the paper's claims.

Family 218: Conformal universality for weakly interacting and random-bond Ising models

Weak finite-range square-symmetric even multispin perturbations of the square-lattice Ising model preserve critical bulk spin and energy limits; weak square-symmetric contour interactions also yield chordal SLE3 interface limits. With sufficiently weak iid bond disorder of any fixed bounded nondegenerate mean-zero law, critical spin interfaces converge to the same law in probability over environments.

Other papers in this family: