Critical square-lattice FK interfaces below random-cluster weight one
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Definitions and mathematical statement
Statement recorded in the supplied source.
For every fixed $q\in (0,1)$, do square-lattice FK Dobrushin interfaces at the self-dual critical edge parameter converge, as unparametrized ordered curves in smooth marked Jordan domains, to chordal SLE with $\kappa =4\pi /arccos(-\operatorname{sqrt}(q)/2)$?Short English statement
For every fixed $q\in (0,1)$, do square-lattice FK Dobrushin interfaces at the self-dual critical edge parameter converge, as unparametrized ordered curves in smooth marked Jordan domains, to chordal SLE with $\kappa =4\pi /arccos(-\operatorname{sqrt}(q)/2)$?Sources
- S1 Original question source or mathematical context. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Self-dual-random-cluster-interfaces-below-one-September-23-2026/build/sections/00-introduction.tex.
- S2 OpenAI, manuscript citation page. Source of the formulation motivating this local problem. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Self-dual-random-cluster-interfaces-below-one-September-23-2026/README.md.
Statement licence: CC-BY-4.0 · Source file
Contributions 1
OpenAI claim summary for Math-2878
For $0<q<1$, the self-dual square-lattice FK interface is claimed to converge to chordal SLE with $\kappa=4\pi/\arccos(-\sqrt q/2)$ in smooth Jordan domains. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.
Critical square-lattice FK interfaces below random-cluster weight one
OpenAI's claim
OpenAI presents a proof in the selected manuscript. For $0<q<1$, the self-dual square-lattice FK interface is claimed to converge to chordal SLE with $\kappa=4\pi/\arccos(-\sqrt q/2)$ in smooth Jordan domains.Mathematical statement of the claimed result
Result 1.
For fixed $q\in(0,1)$, use square-lattice FK edge-subset weights at $p=\sqrt q/(1+\sqrt q)$, with one boundary arc wired and the other free. Choose a bounded domain $D$ whose Jordan boundary is smooth, and choose different marked boundary points $a,b$. Approximate it by simple closed nearest-neighbor polygons whose boundary parametrizations converge uniformly, and whose marked cyclic parameters converge. The source claims \[ \eta_\delta\Longrightarrow\operatorname{SLE}_\kappa(D;a,b),\qquad \kappa=\frac{4\pi}{\arccos(-\sqrt q/2)}\in(6,8), \] for the Dobrushin medial strand, along the entire mesh sequence, in uniform curve distance modulo increasing reparametrization. The theorem's domain assumption is smooth boundary, rather than every Jordan domain.Outline of the argument
Finite connection and boundary-partition comparisons replace positive-association arguments that fail below one. Sector decay localizes irregular tiled disks, and angular transfer gives a boundary observable determining the Loewner law along the full mesh sequence.Formal verification
Family 223 has no Lean scope page in this release. Its selected result is not presented as formalized. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.Status caveat
OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.Sources
- OpenAI, Self-dual random-cluster interfaces below one, 2026-09-23; citation key
OAI:Self-dual-random-cluster-interfaces-below-one-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main. - OpenAI catalogue entry.