Neumann–Zagier 1985 conjecture on tree lattices settled after an AI proof with ChatGPT 5.5 Pro and Opus 4.8
Event confirmedAwaiting review
The takeaway
Joshua Evan Greene, Ahmad Issa and Duncan McCoy prove that a positive definite tree lattice has an irreducible vertex element, which settles a 1985 conjecture of Neumann and Zagier.
Status
- Claim
Event confirmedAwaiting review
- Our reporting
- High confidence
- Verification
- Preprint only
- Importance
- 3 of 5
- Last verified
- 21 October 2026
Your AI and this story
- GPT-6 Astra172 days after its cutoff
- Claude Opus 5.5111 days after its cutoff
- Gemini 4 Argon202 days after its cutoff
- Grok 4.7141 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 111 days before it.
Key facts
- Paper: arXiv 2610.27095 ‘Tree lattices and the Neumann–Zagier conjecture’ (Greene, Issa, McCoy), 19 Oct 2026
- Result: if a tree lattice L(T) is positive definite, some vertex element is irreducible; corollary: if L(T) ≅ Z^n then T has a vertex of weight one (Neumann–Zagier 1985)
- AI methodology: ‘In 2026, JEG and DM submitted this result to the First Proof AI benchmark test. Soon thereafter, JEG and DM were contacted by AI (the second author), who had used AI (artificial intelligence) to prove Theorem 1.1.’
- ‘The AI proof contained two main innovations: the use of the Perron-Frobenius theorem to identify a vertex, and an inductive proof akin to Lemma 3.1 to establish its irreducibility. While the use of Perron-Frobenius appeared to be a stroke of genius, it turned out to be a red herring’; the inductive lemma ‘persisted as an essential linchpin’
- ‘The initial AI proof of Theorem 1.1 used ChatGPT 5.5 Pro and Opus 4.8 … We chucked AI efforts to write this paper in favor of our own, although we engaged it for proofreading.’
- Prior human progress: Greene and McCoy proved the single-bad-vertex case in 2020 and strengthened it in 2023
What happened
A tree with integer weights on its vertices presents an integral “tree lattice”. In 1985 Walter Neumann and Don Zagier conjectured that a tree lattice isometric to the standard lattice Z^n must contain a vertex of weight one. Greene and McCoy proved special cases in 2020 and 2023. In 2026 they submitted the open case to the First Proof benchmark of research problems for AI.
Ahmad Issa then contacted them with a proof he had obtained using ChatGPT 5.5 Pro and Claude Opus 4.8. The authors’ AI methodology section describes how they digested it. The model’s Perron–Frobenius idea “appeared to be a stroke of genius” but was a red herring. Its inductive irreducibility lemma stayed essential, and they strengthened the theorem. The paper itself was written by hand, with AI used only for proofreading.
Why it matters
A problem posted as a benchmark task was solved with frontier models and then confirmed and improved by the experts who posed it. The paper also describes candidly what the AI contributed.
What is disputed or not yet verified
| Verification | Unrefereed preprint by named experts who checked and re-derived the AI argument |
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Sources
1 source from 1 site. Numbers match the chips in the text.
1 source: 1 primary
Primary
Changes
- Filed (PDF read)