OpenAI’s unreleased ‘Astra’ model claims ten advances in maths and theoretical CS, with Lean proofs
Awaiting review
Importance: historic (5 of 5)The takeaway
On 1 Aug 2026 OpenAI published ‘Ten advances in mathematics and theoretical computer science’ by an internal model, Astra (released as GPT-6 Astra on 3 Sep).
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Lean-verified
- Importance
- Historic (5 of 5)
- Last verified
- 19 October 2026
Your AI and this story
- GPT-6 Astra93 days after its cutoff
- Claude Opus 5.532 days after its cutoff
- Gemini 4 Hydrogen123 days after its cutoff
- Grok 4.762 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 32 days before it.
Key facts
- Claims: explicit non-sofic group (Gromov’s question, ~1999); disproof of Connes’s rigidity conjecture; quantum parallel repetition for general two-player entangled games
- Also: Ehrhart volume conjecture (partial per some sources); polynomial-factor NP-hardness of approximating the Closest Vector Problem; permanent circuit lower bound ~n⁴/log n
- Superexponential lower bound for multicolour Ramsey numbers (Erdős #183); Erdős #146 and #180; improved binary and spherical codes
- Sphere-packing upper-bound exponent ~0.5990558 → ~0.6044005, first improvement since Kabatiansky–Levenshtein (1978)
- Evidence: 249-page PDF, Lean 4 proofs (openai/ten-proofs); < $2,000 of tokens per solution at GPT-5.6 Sol prices; prompts not released
- Attribution dispute: Andreas Thom (11 Sep, guest post on Tao’s blog) says the non-sofic proof relies crucially on his 2019 work with Gábor Kun (Prop. 2.3 of OpenAI’s PDF) despite OpenAI’s ‘decade without progress’ framing, and asks whether his own ChatGPT conversations about these techniques reached the model; Mark Sellke replied ‘that did not happen’. Kun and Thom posted a follow-up, arXiv 2608.06222 (6 Aug)
- Independent audit (arXiv 2608.14673): ‘No confirmed substantive mathematical error in a principal result remains’; one chapter needs major revisions, and some stronger results were not reproduced
- Follow-ups: Jihao Liu proved the equality case of Ehrhart’s volume conjecture with GPT-5.6 Sol, Fable 5 and the Danus agent (arXiv 2608.01040); Raphael Steiner extended the superexponential multicolour Ramsey construction to odd cycles with a proof ‘found autonomously by ChatGPT 5.6 Pro/Sol’ (arXiv 2608.02537, later withdrawn and merged into arXiv 2608.02522)
Show 2 more
- The Verge (10 Sep): Thom on Mastodon called Mark Sellke’s categorical answer ‘materially misleading; looking back it was plainly dishonest’ and said ‘De-identification may remove a name; it does not remove the intellectual content of a mathematical idea’. The Verge says OpenAI quietly amended its write-up to credit Kun and Thom without announcing the change
- Scientific American (Joseph Howlett, 6 Aug), ‘OpenAI’s latest math breakthroughs commit research misconduct, experts say’: Stephen Miller (said the sphere-packing proof used his 2016 argument without proper citation): ‘They are running roughshod over the work of others who came before them in a deliberate way. It seems completely systematic to me, and it points to research misconduct.’ Francesco Fournier-Facio (Cambridge) on the soficity result: ‘the big PR machine… does not care about being 100 percent accurate’. OpenAI: ‘We take responsibility for the correctness of these results and are meeting the same standards generally expected of human mathematicians’ and promised small updates
What happened
OpenAI released, in one announcement, ten research results produced by an internal model a month before its launch. Most came with machine-checked proofs.
Why it matters
It moved the frontier from individual AI-assisted results to a lab producing batches of significant theorems. An independent audit largely upheld them.
What is disputed or not yet verified
| Verification | Lean 4 formalizations published for most of the ten results (openai/ten-proofs); results not refereed; a review of 18 chapter reviews (Sienicki & Sienicki, arXiv 2608.14673, unrefereed) found no remaining substantive error in principal results but one chapter needing major revision; experts disputed attribution and citation (Scientific American). Status varies by result |
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Sources
14 sources from 9 sites. Numbers match the chips in the text.
14 sources: 5 primary, 5 press, 4 reactions
Primary
- OpenAI: Ten advances in mathematics and theoretical computer scienceopenai.com, official
- OpenAI: ten proofs manuscript (PDF)cdn.openai.com, paper
- Kun & Thom: Nonsofic wreath products of residually finite groups (arXiv 2608.06222)arxiv.org, paper
- arXiv 2608.01040: The equality case of Ehrhart’s volume conjecture (Liu)arxiv.org, paper
- arXiv 2608.02522: Locally bipartite subgraphs via multicolor Ramsey numbers (Steiner)arxiv.org, paper
Press
- Scientific American: OpenAI’s latest math breakthroughs commit research misconduct, experts say (6 Aug)scientificamerican.com, press
- Quanta: Why the legendary Erdős problems are falling to AIquantamagazine.org, press
- The Verge (11 Aug): The AI takeover of mathematics has beguntheverge.com, press
- The Verge Decoder (20 Aug): Welcome to the AI crisis in math (Robert Hart)theverge.com, press
- The Verge (10 Sep): Mathematicians want proof OpenAI didn’t use their work (Thom)theverge.com, press
Reactions
- A Human Audit of OpenAI’s AI-Generated Mathematical Proofs (arXiv 2608.14673)arxiv.org, discussion
- Simon Willison on the ten advancessimonwillison.net, discussion
- Andreas Thom (guest post on Tao’s blog): On the existence of non-sofic groups (attribution concerns)terrytao.wordpress.com, discussion
- MathOverflow: key new ideas in the non-soficity proofmathoverflow.net, discussion
Changes
- Status confirmed → pending (batch of ten unrefereed results, Lean for most but not all, one chapter needing major revision); the ‘independent audit’ is described as what it is (an unrefereed review of reviews)
- Added video(s) fireship-summer-math-fell-to-the-machines
- Related: added October 2026: the arXiv stream of conjectures settled with disclosed AI help continues (October catalogue, split out of the summer catalogue)
- Added Scientific American’s 6 Aug misconduct article (Stephen Miller, Fournier-Facio, OpenAI response), missed until found while processing October coverage
- Science block: kind: batch (many results in one entry) (not a long-open problem; listed with other AI science results, not in the open-problems count)
- Added The Verge’s Aug 11 feature, Aug 20 Decoder episode and Sept 10 Thom article (found via its Oct 5 storystream)
Show all 9 changes
- Added AI-assisted follow-ups (Ehrhart equality case; odd-cycle Ramsey)
- Added Andreas Thom’s attribution critique of the non-sofic result, Kun–Thom follow-up paper, MathOverflow thread
- Filed
