OpenAI has published solutions to 10 long-standing mathematics problems generated by an unreleased model it calls Astra, releasing more than 250 pages of proofs and another 60 pages describing how the ideas came together. Each result was verified in Lean, the proof-checking software mathematicians use to certify formal correctness. OpenAI estimates the whole batch cost around $2,000 in tokens at the current API prices of its Sol model.
The problems span sphere packing in higher dimensions, error-correcting codes, quantum game theory, high-dimensional search relevant to post-quantum cybersecurity, and questions about how large connected networks can grow before structure emerges. Several had gone unresolved for decades. One — the existence of non-sofic groups, infinite mathematical structures that cannot be approximated by finite ones — had been open for the better part of a century, in a lineage stretching back to a Paul Erdős conjecture OpenAI said an internal model cracked in May.
The reaction inside the field has been sharp. Yang-Hui He of the London Institute for Mathematical Sciences told The Verge that he had just returned from a four-week AI and mathematics conference in South Korea where researchers described a phase transition over the past six months, with AI producing genuine and meaningful advances. James Maynard, an Oxford professor and Fields Medal winner, said he has spent the past year soul searching about where the discipline is headed.
“There's a general feeling that [solving] one of these 10 problems would get you a job in academia”— Yang-Hui He, Fellow at the London Institute for Mathematical Sciences
Key facts
- 01OpenAI's unreleased Astra model produced solutions to 10 long-standing math problems, released as 250+ pages of proofs plus 60 pages of methodology.
- 02Each solution was verified with Lean, the mathematical proof-checking software.
- 03OpenAI estimates generating all 10 solutions cost roughly $2,000 in tokens at current Sol-model API prices.
- 04One result addressed the existence of non-sofic groups, a question open for decades — and sparked a credit dispute with researchers Andreas Thom and Gábor Kun.
- 05Mathematicians at a recent four-week AI-and-math conference in South Korea described a 'phase transition' in the past six months.
The non-sofic group result immediately drew scrutiny over credit. OpenAI's initial announcement described the batch as results on problems that had seen no progress for at least a decade, and in most cases much longer. Francesco Fournier-Facio of the University of Cambridge said he and others believed that framing minimized the work of Andreas Thom and Gábor Kun, whose earlier results the proof relies on. OpenAI later rewrote the language to say each result resolves or makes substantial progress on a long-standing open problem, without a correction note.
Kun, a researcher at the Alfréd Rényi Institute of Mathematics in Hungary, called the original framing 'rather comical' and 'rather sloppy,' noting the detailed paper OpenAI attached clearly said the argument builds on his 2016 and 2019 work, the latter co-authored with Thom. He said an OpenAI mathematician emailed him after publication acknowledging the argument relies crucially on his work and that the wording would be revised.
OpenAI confirmed the revision through spokesperson Laurance Fauconnet.
“We updated the language to better reflect the prior research these results build upon. Although the question of whether non-sofic groups exist had remained open for decades, our sofic group proof relies on important mathematical work published more recently.”— Laurance Fauconnet, OpenAI spokesperson
Astra is not the only frontier model producing serious math. In July, Harvard mathematician Levent Alpöge posted that Claude Fable 5, from Anthropic, had disproved the Jacobian conjecture with a small counterexample, ending decades of efforts to prove it true. Together with the Erdős result in May and the Astra batch this month, three of the highest-profile open-problem resolutions of the past year have come from AI systems rather than human research groups.
The economic pressure is what mathematicians are talking about privately. Mathematics is inexpensive to fund — no wet lab, no telescope, no fab — and much of the discipline runs on the salaries of researchers who, as St Andrews professor Colva Roney-Dougal put it, mostly get on with the job without a grant. If AI systems can produce theorems at $2,000 a batch, the funding case for that arrangement changes.
“It's not quite clear whether our universities are going to be willing to pay that much for our theorems”— Colva Roney-Dougal, Professor at the University of St Andrews
There are real caveats. Mathematics has become specialized enough that few researchers can independently evaluate all 10 Astra results, and the Kun episode suggests the framing of what counts as a solved problem is doing a lot of work in OpenAI's presentation. The true cost of generating the proofs was almost certainly higher than the $2,000 token estimate, which reflects API pricing rather than compute and search overhead. And Lean verification confirms a proof is formally correct, not that the problem statement matches what human mathematicians would consider the canonical question.
Still, the trajectory is what matters for the AI market. Astra is described as an internal version of OpenAI's next major model, meaning the capability shown here is upstream of whatever OpenAI ships to paying customers next. If the same reasoning stack that produced 10 open-problem results for $2,000 in tokens is what powers the next generation of ChatGPT and API models, the practical implication is that formal reasoning — the hardest thing for language models to do reliably — has moved from a demo-grade capability to something that produces publishable output. For OpenAI, that is the argument for the next fundraising round. For the rest of the field, including Anthropic and Google, it resets the benchmark that matters.
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