GPT-6 Astra’s Epoch AI run adds more Lean-checked results
Dittert conjecture proved, Ibragimov–Iosifescu and eternal-domination conjectures disproved
Awaiting review
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Lean proof, not yet reviewed
- Importance
- 3 of 5
- Last verified
- 19 October 2026
Your AI and this story
- GPT-6 Astra133 days after its cutoff
- Claude Opus 5.572 days after its cutoff
- Gemini 4 Hydrogen163 days after its cutoff
- Grok 4.7102 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 72 days before it.
Key facts
- Setting: Epoch AI’s LeanOpenProblems harness; pre-release GPT-6 Astra tried each research-open Formal Conjectures statement once, autonomously (see the Köthe entry)
- Dittert conjecture: φ(A) ≤ 2 − n!/n^n for nonnegative n×n matrices with entries summing to n, with equality only for the all-1/n matrix. Lean proof passed the Comparator check (repo tadamcz/dittert); the exposition is not independently reviewed. Humans had earlier proved n ≥ 17 (arXiv 2606.01531) and n = 16 (arXiv 2607.19439, GPT-5.6 Sol-assisted)
- Ibragimov–Iosifescu conjecture (Ibragimov, 1971): disproved with a strictly stationary φ-mixing counterexample; a 13,047-line Lean proof, ‘Lean-checked, statement unaudited’, announced 5 Sep 2026 (repo tadamcz/phi-mixing-clt)
- Strong n-conjecture, n = 4: disproved in Lean with extra SymPy arithmetic checks (repo tadamcz/n-conjecture-strong)
- Eternal domination: 243-vertex graph with γ(G) = γ∞(G) < θ(G), refuting the Gamma–Theta conjecture. Tom Adamczewski & William F. Klostermeyer, arXiv 2609.11500, 10 Sep 2026
- The repositories say they were ‘machine-written by AI assistants at the direction of Tom Adamczewski’
- Same run, Wikipedia collection (‘wikipedia-vega-1000usd’, all 222 research-open statements, $1,000 and 96 h each): disproof of Smale’s mean value conjecture for K = 1 (repo tadamcz/mean-value-problem; own entry GPT-6 Astra autonomously disproves Smale’s mean value conjecture in Lean). Epoch’s LeanOpenProblems README publishes no overall solved count (checked 16 Oct)
What happened
Epoch AI ran pre-release GPT-6 Astra once on each research-open statement in the Formal Conjectures collection. Besides Köthe, several more outputs were packaged as Lean repositories by Tom Adamczewski in the first half of September 2026. For the graph-theory counterexample, domination expert William Klostermeyer co-wrote an arXiv paper.
Why it matters
Autonomous formal proof search now turns out a steady stream of mid-level resolved conjectures, not one-off headlines. The bottleneck is shifting to human auditing of whether the formal statements are the intended ones.
What is disputed or not yet verified
| Verification | Formal proofs in Lean (mechanically checked). Statements and write-ups mostly not independently audited |
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Sources
6 sources from 4 sites. Numbers match the chips in the text.
6 sources: 4 primary, 2 reactions
Primary
- arXiv 2609.11500: A Counterexample to an Eternal Domination Conjecturearxiv.org, paper
- GitHub: tadamcz/dittertgithub.com, code
- GitHub: tadamcz/phi-mixing-clt (Ibragimov–Iosifescu)github.com, code
- GitHub: tadamcz/n-conjecture-stronggithub.com, code
Reactions
- VibeMathed: Ibragimov–Iosifescu conjecture statusvibemathed.com, discussion
- Wikipedia: List of mathematical discoveries by artificial intelligenceen.wikipedia.org, discussion
Changes
- Related: back-links to the Smale mean value and Köthe entries
- Linked the Smale mean-value disproof from the same run; noted that Epoch publishes no overall results count
- Filed (grouped several September 2026 Astra/Epoch results)