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Math-2878-A1 · revision r1 · on Math-2878

OpenAI claim summary for Math-2878

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

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.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Critical square-lattice FK interfaces below random-cluster weight one
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Result 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

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