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Math-2924-A1 · revision r1 · on Math-2924

OpenAI claim summary for Math-2924

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.

Annealed critical Poisson--Voronoi crossing probabilities are claimed to obey Cardy conformal formula in every bounded Jordan quadrilateral. 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. Cardy crossing probabilities for Poisson–Voronoi 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

Cardy crossing probabilities for Poisson–Voronoi percolation

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Annealed critical Poisson--Voronoi crossing probabilities are claimed to obey Cardy conformal formula in every bounded Jordan quadrilateral.

Mathematical statement of the claimed result

Result 1.
For full-plane Poisson--Voronoi cells at intensity $\varepsilon^{-2}$, independently colour each cell black with probability $1/2$. In a bounded Jordan domain $D$, let $a,b,c,d$ be distinct counterclockwise boundary marks; the crossing event requires a connected black subset of $\overline D$ meeting the closed arcs $ab$ and $cd$. Let $x\in(0,1)$ be the image of $b$ under the conformal map taking $(a,c,d)$ to $(0,1,\infty)$ in $\mathbb H$. Averaging both cell geometry and colours, the claim is \[ \lim_{\varepsilon\downarrow0}\mathbb P(\mathcal C_\varepsilon(D;a,b,c,d)) =\frac{\int_0^x[u(1-u)]^{-2/3}\,du}{\int_0^1[u(1-u)]^{-2/3}\,du} =\frac{\Gamma(2/3)x^{1/3}}{\Gamma(1/3)\Gamma(4/3)} {}_2F_1\left(\frac13,\frac23;\frac43;x\right). \] This is the annealed scalar crossing statement, without extra boundary regularity beyond the Jordan condition.

Outline of the argument

Cluster-boundary circuits are decomposed into fully labeled dual components, including isolated edges, to obtain exact orthogonality. Local plugs open closed field sums, boundary potentials and a finite pin menu control transport along actual arcs, and these identities determine the crossing observable.

Formal verification

Family 224 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