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Math-1735-A1 · revision r1 · on Math-1735

OpenAI claim summary for Math-1735

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.

For the lattice Z3, whose Bernoulli uniqueness threshold equals its critical threshold by amenable-lattice uniqueness, The cited theorem shows absence of a cluster at that endpoint; it supplies one negative case, not necessary-and-sufficient conditions for all quasi-transitive graphs. The cited theorem rules out bond clusters at pc<1; for amenable quasi-transitive graphs with pc=pu by the usual amenable uniqueness theorem this gives negative endpoint cases, but no full site/bond structural characterization at pu. 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. Uniqueness at the percolation uniqueness threshold
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Result 1.
  5. Result 2.
  6. Outline of the argument
  7. Formal verification
  8. Status caveat
  9. Sources

Uniqueness at the percolation uniqueness threshold

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: Both bond and site percolation on $\mathbb Z^3$ are claimed to have only finite clusters at their respective critical probabilities; Critical bond percolation is claimed to have no infinite component on every infinite locally finite quasi-transitive graph with $p_c<1$. Together they provide the following support within the scopes described below.

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.
Result 2.
Suppose $G$ is infinite, connected, locally finite, undirected, and has finitely many vertex orbits under automorphisms. For Bernoulli bond percolation, define $p_c(G)$ as the infimum of parameters with positive probability of an infinite cluster. The claimed implication is \[ p_c(G)<1\quad\Longrightarrow\quad \mathbb P_{p_c(G)}(\exists\text{ infinite open cluster})=0. \] Equivalently, every fixed vertex $x$ satisfies $\mathbb P_{p_c}(|C_x|=\infty)=0$. The hypothesis $p_c<1$ is essential to the stated scope.

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.

The proof separates growth regimes and uses finitary structural reduction in the polynomial-growth case. Two coordinate moves are glued through fresh corridors using a conditional connection inequality; adaptive exploration would then produce subcritical percolation if an infinite critical cluster existed.

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