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Math-2900 · mathematics · Probability

Critical nonpercolation on locally finite quasi-transitive graphs below one

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every infinite connected locally finite undirected quasi-transitive graph whose independent bond critical probability $p_c$ is less than one, does independent bond percolation at $p_c$ have no infinite open cluster almost surely?

Short English statement

For every infinite connected locally finite undirected quasi-transitive graph whose independent bond critical probability $p_c$ is less than one, does independent bond percolation at $p_c$ have no infinite open cluster almost surely?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2900-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2900

Critical bond percolation is claimed to have no infinite component on every infinite locally finite quasi-transitive graph with $p_c<1$. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

Critical nonpercolation on locally finite quasi-transitive graphs below one

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Critical bond percolation is claimed to have no infinite component on every infinite locally finite quasi-transitive graph with $p_c<1$.

Mathematical statement of the claimed result

Result 1.
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

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

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