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Square-lattice FK interfaces and nested loops for 1 <= q < 4

OpenAI · September 23, 2026

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Abstract

For every fixed $1\le q\lt 4$, critical square-lattice random-cluster Dobrushin interfaces converge as ordered curves to chordal $\mathop{\mathrm{SLE}}\nolimits _{\kappa(q)}$, where $\kappa(q)=4\pi/\arccos(-\sqrt q/2)$, confirming the Rohde–Schramm prediction in this parameter range. This holds in every bounded Jordan domain under uniform marked boundary approximation. The complete nested plane loop collections converge to whole-plane $\mathop{\mathrm{CLE}}\nolimits _{\kappa(q)}$ in a spherical matching topology retaining multiplicities and traversals. At q = 1 we obtain Cardy's formula for square-lattice bond percolation with free boundary edges.

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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.

Comments and discussion

  • Jim Jardine (Lattice24), openai-math-audit: Independent audit of OpenAI's openai/math release (GitHub repository, audit status dated 2026-10-16). Its "Finding 1" (rated critical) claims a proof gap in this paper's Section 7.3 ("The complete cost of the norm diagrams", estimates hol:strict-budget and hol:baseline), on which Theorem hol:disk, the main theorem and the Cardy corollary depend; it says the finding survived three refutation passes. [AI-generated per source]

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Cite

@misc{OAI:Square-lattice-FK-interfaces-and-nested-loops-for-1-leq-q-lt-4-September-23-2026,
  author = {{OpenAI}},
  title = {{Square-lattice FK interfaces and nested loops for $1\le q<4$}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Square-lattice-FK-interfaces-and-nested-loops-for-1-leq-q-lt-4-September-23-2026/paper.pdf}{OAI:Square-lattice-FK-interfaces-and-nested-loops-for-1-leq-q-lt-4-September-23-2026}},
  year = {2026}
}

Family 223: Random-cluster interfaces: critical, disordered, thermal, and natural-time scaling

Proves chordal SLEκ limits for critical square-lattice random-cluster interfaces for $0\lt q\le4$, with $\kappa=4\pi/\arccos(-\sqrt q/2)$: bounded Jordan domains are allowed for q ≥ 1, and smooth Jordan domains for q < 1, under the stated marked-boundary approximations. For $1\le q\le4$, complete nested plane loops converge to CLEκ.

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