No infinite critical clusters for three-dimensional site and bond percolation
Status OpenLLM status Open unverified
Top Open Problems unknownunknown Not reported cataloguedOpenOpenStatement
Definitions and mathematical statement
Statement recorded in the supplied source.
For independent nearest-neighbor bond percolation and independent site percolation on $Z^3$ at their respective critical parameters, is the probability that the origin belongs to an infinite open cluster zero?Short English statement
For independent nearest-neighbor bond percolation and independent site percolation on $Z^3$ at their respective critical parameters, is the probability that the origin belongs to an infinite open cluster zero?Sources
- S1 Original question source or mathematical context. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Critical-bond-and-site-percolation-on-the-cubic-lattice-September-24-2026/build/introduction.tex.
- S2 OpenAI, manuscript citation page. Source of the formulation motivating this local problem. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Critical-bond-and-site-percolation-on-the-cubic-lattice-September-24-2026/README.md.
Statement licence: CC-BY-4.0 · Source file
Contributions 1
OpenAI claim summary for Math-2938
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.
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 areCriticalPercolation.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
- OpenAI, Critical bond and site percolation on the cubic lattice, 2026-09-24; citation key
OAI:Critical-bond-and-site-percolation-on-the-cubic-lattice-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main. - Family 213: Lean scope.
- Comparator CriticalPercolation.json.
- Comparator CriticalPercolation.lean.
- Comparator CriticalZ3.json.
- Comparator CriticalZ3.lean.
- OpenAI catalogue entry.