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Math-2938-A1 · revision r1 · on Math-2938

OpenAI claim summary for Math-2938

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.

Both bond and site percolation on $\mathbb Z^3$ are claimed to have only finite clusters at their respective critical probabilities. 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. No infinite critical clusters for three-dimensional site and bond 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

No infinite critical clusters for three-dimensional site and bond percolation

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Both bond and site percolation on $\mathbb Z^3$ are claimed to have only finite clusters at their respective critical probabilities.

Mathematical statement of the claimed result

Result 1.
For independent nearest-neighbor bond percolation on $\mathbb Z^3$, write $p_c^{\mathrm{bond}}$ for the first parameter permitting an infinite cluster. The source asserts \[ \mathbb P_{p_c^{\mathrm{bond}}}(\exists\text{ infinite open cluster})=0. \] For independent nearest-neighbor site percolation, it separately claims \[ \mathbb P_{p_c^{\mathrm{site}}}(\exists\text{ infinite open cluster})=0. \] These are two cubic-lattice statements at their respective critical parameters.

Outline of the argument

Independent hyperedges provide a finite connection inequality that accommodates the two models. Finite-scale extension bounds allow an adaptive exploration to maintain successful connections; assuming critical percolation would make this construction survive below the threshold, a contradiction.

Formal verification

For family 213, the Lean scope covers cubic-lattice bond and site critical nonpercolation, and quasi-transitive bond graphs when $p_c<1$. The listed Comparator files are CriticalPercolation.lean, CriticalZ3.lean. Named declarations include OAI.CriticalZ3.critical_no_infinite. 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