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Math-2926-A1 · revision r1 · on Math-2926

OpenAI claim summary for Math-2926

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.

Assuming the Voronoi Cardy input without a convergence rate, the expected number of color pivotals is claimed to be asymptotic to $C\varepsilon^{-3/4}$ with $C>0$. 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. The Voronoi pivotal-count amplitude under Cardy’s formula
  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

The Voronoi pivotal-count amplitude under Cardy’s formula

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Assuming the Voronoi Cardy input without a convergence rate, the expected number of color pivotals is claimed to be asymptotic to $C\varepsilon^{-3/4}$ with $C>0$.

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 $\mathbb H$. Assume these scalar crossing probabilities have Cardy's limit for every bounded Jordan quadrilateral. Let $N_\varepsilon$ count nuclei whose colour flip changes the left-to-right crossing of $[0,1]^2$, including outside nuclei whose cells meet that square. There exists $c_V\in(0,\infty)$ with \[ \varepsilon^{3/4}\mathbb EN_\varepsilon\longrightarrow c_V,\qquad (\mathbb EN_\varepsilon)^{-1}\sim c_V^{-1}\varepsilon^{3/4}. \] Expectation averages geometry and colours. The hypothesis does not assume an interface limit or a convergence rate.

Outline of the argument

Four-interface transmission through annuli is decomposed spectrally to separate the conformal information at each end. Uniform signed-state transfer fixes the amplitude, and an intensity derivative connects the transmission calculation to the expected pivotal count.

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