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Math-2336-A1 · revision r1 · on Math-2336

OpenAI claim summary for Math-2336

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 at unsolvedmath.com: 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.

At q=1 the source covers isotropic critical square-lattice bond percolation (p_h=p_v=1/2), yielding crossing control in this restricted subcase. It does not cover general anisotropic p_h+p_v=1 or determine the requested general shape class. This entry records partial support; complete coverage of the question is not established by this catalogue.

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

Manuscript

Contents
  1. RSW Theory for Anisotropic Square-Lattice Percolation
  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

RSW Theory for Anisotropic Square-Lattice Percolation

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. The later interface theorem extends the square-lattice FK claim through $q=4$, with the same SLE parameter relation and a separate complete nested-loop limit.

Mathematical statement of the claimed result

Result 1.
Fix $q\in[1,4]$, $p_q=\sqrt q/(1+\sqrt q)$, and \[ \kappa(q)=\frac{4\pi}{\arccos(-\sqrt q/2)}. \] Use the square-lattice random-cluster law with edge-subset weights $p_q^{|A|}(1-p_q)^{|E|-|A|}q^{k_\xi(A)}$, where $\xi$ wires one boundary arc and leaves the other free. For a bounded Jordan domain with two marked boundary points, approximate its boundary uniformly by simple lattice polygons, preserving the marks. The claim gives \[ \eta_n\ \Longrightarrow\ \operatorname{SLE}_{\kappa(q)} \] for the Dobrushin interface in uniform curve distance modulo increasing reparametrization. Under the free infinite-volume law, its full nested loop collection satisfies \[ \Gamma_\delta\ \Longrightarrow\ \text{nested whole-plane }\operatorname{CLE}_{\kappa(q)} \] in the spherical matching topology: all loops above a shrinking diameter threshold are matched with shrinking uniform spherical error. Both limits hold along every mesh sequence, with $q$ fixed.

Outline of the argument

The tiled comparison and angular-transfer method is supplemented by endpoint estimates at $q=4$. Boundary current tests identify the driving process, while full exploration and localization retain macroscopic loops; at $q=1$ this also yields the bond-percolation crossing formula.

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