Claude-written Lean proof claims the dying percolation conjecture θ(p_c)=0 in every dimension
Awaiting review
Importance: historic (5 of 5)The takeaway
In early September 2026 a Lean 4 formalization written by Anthropic’s Claude models (directed by Justin Leder, published in anthropics/formal-math) claimed to prove that critical Bernoulli bond percolation on Z^d has no infinite cluster for every d ≥ 2.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Lean proof, not yet reviewed
- Importance
- Historic (5 of 5)
- Last verified
- 22 October 2026
Your AI and this story
- GPT-6 Astra126 days after its cutoff
- Claude Opus 5.565 days after its cutoff
- Gemini 4 Hydrogen156 days after its cutoff
- Grok 4.795 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 65 days before it.
Key facts
- Problem: θ(p_c)=0 (no percolation at criticality); previously known only for d = 2 and high dimensions (d ≥ 11). Open for 3 ≤ d ≤ 10
- Route: Kozma & Nitzan (arXiv 2401.12397, 2024) showed their Conjecture 3 (‘near-one gluing’) implies θ(p_c)=0 on Z^d for all d ≥ 2
- anthropics/formal-math percolation README: 247 Lean files, ~86,900 lines; axioms only propext, Classical.choice, Quot.sound; ‘no human wrote or edited the Lean code’
- README caveat: ‘has not yet been refereed by human mathematicians or by anyone independent of the author’
- Gil Kalai blog, 3 Sep 2026: ‘If verified, this is a remarkable breakthrough’; he flags missing details in the written proof and the need to check the formalization
- Hugo Duminil-Copin had used θ(p_c)=0 as his main example in an essay on AI and mathematics a few days earlier (‘Care for a little more AI?’, proofsandprompts.com, Aug 30, 2026: AI use ‘does not empower us, it petrifies us’)
- Ahmed Bou-Rabee’s verification page (updated 5 Sep 2026): Kozma–Nitzan Conjectures 1, 2, 4, 6 and Questions 5, 7, 9 proved in stronger form by ‘ChatGPT 5.6 Sol and Claude Fable 5.1, prompted by Ahmed Bou-Rabee’; Question 8 fails under one reading
- Scientific American (Manon Bischoff, Sept 30, 2026) covered it as AI solving a ‘holy grail’ of probability theory: Duminil-Copin had written on Aug 30 that it was ‘only a matter of time before the most famous conjecture in our field... also falls to the bulldozers’; Benedikt Jahnel described ‘ambivalent feelings’
Show 3 more
- Quanta (Konstantin Kakaes, Sept 2026): Justin Leder, an Anthropic member of technical staff on its data science team with no prior percolation papers, posted the proof on GitHub on Aug 28. Gady Kozma: ‘We are still waiting for Anthropic to publish a human readable version.’ Bou-Rabee: ‘clever and technically brilliant, but… does not contain substantial new ideas’
- Related human work: Lucas Flammant, arXiv 2609.03492 (Sept 3, 2026), proposes a finite-graph conjecture that also implies θ(p_c)=0 and proves it for planar graphs; it does not discuss the AI proof
- Human exposition (Jean-Baptiste Gouéré, Université de Tours; arXiv 2610.30063, 21 Oct 2026): ‘we give an exposition of the recent proof of the Kozma-Nitzan conjecture provided by OpenAI’ (Conjecture 1, from OpenAI’s 24 Sep manuscript ‘Critical bond and site percolation on the cubic lattice’); ‘Theorem 1 was proved shortly beforehand in [8]’ (Justin Leder’s Lean 4/Mathlib development in anthropics/formal-math); ‘The proofs in [8] and [9] are very similar and rely on the same ideas’; cites Blanc-Renaudie’s human exposition of the Anthropic proof; no AI disclosure in the note
What happened
Kozma and Nitzan reduced the θ(p_c)=0 problem to an inequality about gluing connection events on finite graphs. In early September 2026 a Claude-written Lean development proved an additive form of that inequality. It went through a “conditioned slack hierarchy” of covariance inequalities, then applied Kozma–Nitzan’s Theorem 6 to get θ(p_c)=0 in all dimensions d ≥ 2. Gil Kalai heard about it from Itai Benjamini and wrote it up on 3 Sep 2026. Separately, Ahmed Bou-Rabee published Lean proofs of several stronger Kozma–Nitzan conjectures, produced with GPT-5.6 Sol and Claude Fable 5.1.
The Wikipedia list credits the result to “Claude + Ahmed Bou-Rabee”. The Anthropic repository itself credits Justin Leder as the director of the Claude run. Anthropic had not put out a press release as of late September 2026.
Why it matters
If the formal statement matches the intended theorem, a famous problem in mathematical physics is settled by machine-written formal mathematics. Commentators stress that Lean confirms the proof is correct but does not confirm the statement is the right one. Human experts still have to check that the formal definitions capture percolation on Z^d.
What is disputed or not yet verified
| Verification | Formal proof in Lean (mechanically checked); the statement’s fidelity and the informal write-up are not yet refereed |
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Sources
12 sources from 9 sites. Numbers match the chips in the text.
12 sources: 6 primary, 2 press, 4 reactions
Primary
- arXiv 2610.30063: Gouéré, On the proof of the Kozma-Nitzan conjecturearxiv.org, paper
- anthropics/formal-math: percolation README (commit 795efb8)github.com, code
- Ahmed Bou-Rabee: Kozma–Nitzan conjectures verification pagenitromannitol.github.io, code
- Kozma & Nitzan: A reduction of the θ(p_c)=0 problem to a conjectured inequality (arXiv 2401.12397)arxiv.org, paper
- Flammant: A finite-graph conjecture related to θ(p_c)=0 (arXiv 2609.03492)arxiv.org, paper
- Anthropic: How Anthropic enables self-service data analytics with Claude (Leder co-author)claude.com, official
Press
- Scientific American: AI solves a ‘holy grail’ problem from probability theoryscientificamerican.com, press
- Quanta Magazine: Transformation (dispatches on AI mathematics, Sept 2026)quantamagazine.org, press
Reactions
- Hugo Duminil-Copin: Care for a little more AI? (Proofs and Prompts, Aug 30, 2026)proofsandprompts.com, discussion
- Gil Kalai: Amazing: There is no Percolation at the Critical Probability in all Dimensionsgilkalai.wordpress.com, discussion
- Proofs and Prompts: Applied mathematics has met the machine before (on verification vs validation)proofsandprompts.com, discussion
- Wikipedia: Dying percolation conjectureen.wikipedia.org, discussion
Changes
- Added Gouéré’s human exposition of the Kozma–Nitzan proof (arXiv 2610.30063)
- Science block: short label for the open-problems table
- Added Quanta details (Aug 28 posting date, Leder background, Kozma and Bou-Rabee reactions) and Flammant’s related arXiv paper
- Linked Duminil-Copin’s Aug 30 essay and added a post file
- Added Scientific American coverage (Sept 30) with Duminil-Copin and Jahnel reactions
- Filed