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Critical bond and site percolation on the cubic lattice
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Abstract
We prove that nearest-neighbor Bernoulli bond and site percolation on ℤ3 have no infinite cluster at their respective critical parameters. The proof combines a finite connection inequality for independent hyperedges with a finite-scale extension estimate and an adaptive exploration.
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Notes
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Automatically collected notes (unreviewed)
work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Nutelligence Research, Independent Research Inspired by OpenAI’s Critical Percolation Work (OpenAI Developer Community, post 15, 2026-10-17). This announcement explicitly cites family 213 as inspiration for claimed site-percolation results on periodic, quasi-transitive and specialized lattice graphs, reports Lean verification, and notes an additional assumption for the quasi-transitive result and unresolved novelty assessment. [family-level]
Comments and discussion
- Olle Häggström, What I write about the day after the most earth-shattering event in the history of mathematics (Crunch Time for Humanity, 2026-10-07). Häggström links both manuscripts among percolation problems he had worked on, mentions the role of the van den Berg–Häggström–Kahn inequality, and postpones detailed assessment. [family-level]
Formalization
- OpenAI math release, Critical percolation on every quasi-transitive graph (Lean scope note lean/docs/213.md) (OpenAI math release, 2026-10-06). The release's scope note says the Lean development proves that nearest-neighbour bond and site percolation on $\mathbb Z^3$ have no infinite cluster at their critical parameters, and it states the quasi-transitive bond theorem, linking the two comparator statements. [family-level]
- OpenAI, lean/OAI/Probability/CriticalZ3/Main.lean (openai/math, commit adc7f12). This is the release's Lean main file for this paper; its theorem
critical_no_infinitecombinesbond_critical_no_infiniteandsite_critical_no_infiniteon $\mathbb Z^3$. - Justin Leder (Lean code by Anthropic's Claude models), anthropics/formal-math: percolation (GitHub, commit 795efb8, 2026). This is an independent, earlier Lean formalization of $\theta(p_c)=0$ for nearest-neighbour bond percolation on $\mathbb Z^d$ for all $d\ge2$, which covers the bond half of this paper; the paper cites it as Leder2026. [AI-generated per source]
- Ahmed Bou-Rabee (Lean code by ChatGPT and Claude), percolation-after-anthropic (GitHub, 2026). This is an independent, earlier Lean development of critical site nonpercolation on $\mathbb Z^d$ for $d\ge3$ and of finite bond and hyperedge gluing inequalities, building on Anthropic’s bond-percolation library, which covers the site half of this paper; the paper cites the related exposition as SitePublic2026. [AI-generated per source]
Related prior literature mentioned by others
- Gil Kalai, Amazing: There is no Percolation at the Critical Probability in all Dimensions. (Solved by AI via a conjecture of Gady Kozma and Shahaf Nitzan.) (Combinatorics and more, 2026-09-03). Kalai's 7 October post presents the OpenAI criticality result as extending the setting of this earlier post on the AI-produced $\mathbb Z^d$ bond proof; its comment thread discusses whether the formalization has been verified. [family-level]
- Nikita Gladkov and Aleksandr Zimin, Bond percolation does not simulate site percolation (arXiv:2404.08873, 2024-04-13; revised 2026-02-03). Cited in the earlier site-percolation exposition, its obstruction to approximating site connectivity by independent-bond gadgets provides context for the cubic manuscript’s separate hyperedge treatment of sites.
Possibly relevant literature (found by the agent, not checked)
- Arthur Blanc-Renaudie, Notes on Kozma and Nitzan's proof of θ(p_c)=0 (arXiv:2610.10502, 2026-10-07). This gives a human-written proof of the Kozma–Nitzan conjectures. It is relevant to the paper's finite connection inequality (intro, eq. (1)), which the paper says implies Kozma–Nitzan Conjecture 1.
- Jianping Jiang, Sike Lang, Yuxuan Sun, A connection inequality for the random-cluster model (arXiv:2610.26419, 2026-10-19). This proves a slightly generalized Kozma–Nitzan connection inequality for the random-cluster model with $q\ge1$. It relates to the paper's finite hyperedge comparison, which is stated for independent bits only.
- Lucas Flammant, A finite-graph conjecture related to θ(p_c)=0 for Bernoulli bond percolation on Z^d (arXiv:2609.03492, 2026-09-03). This proposes a finite-graph statement about reaching boundary vertices whose increasing events occur, which implies $\theta(p_c)=0$ on $\mathbb Z^d$. It is close in form to the paper's relay failure bound derived from the comparison inequality.
Digests and expositions
- Perplexity / Orply, Critical Percolation on the Cubic Lattice Has No Infinite Cluster (Orply, 2026-10-11). This AI-assisted exposition summarizes the manuscript’s finite hyperedge comparison, site endpoint convention, and adaptive exploration with conditional failure bounds. [AI-generated per source]
Elsewhere
- Emergent Mind: family 213
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Critical-bond-and-site-percolation-on-the-cubic-lattice-September-24-2026,
author = {{OpenAI}},
title = {{Critical bond and site percolation on the cubic lattice}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Critical-bond-and-site-percolation-on-the-cubic-lattice-September-24-2026/paper.pdf}{OAI:Critical-bond-and-site-percolation-on-the-cubic-lattice-September-24-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/213.md. The Lean statements have not been checked by people against the paper's claims.
Family 213: Critical percolation on every quasi-transitive graph
Resolves the Benjamini–Schramm criticality conjecture for bond percolation on every infinite connected locally finite quasi-transitive graph with $p_c\lt 1$: at the critical probability, there is almost surely no infinite cluster. The family also establishes this conclusion for both nearest-neighbor bond and site percolation on ℤ3.
Other papers in this family:
- No percolation at criticality on quasi-transitive graphs · September 24, 2026