OpenAI said on August 1, 2026, that an internal version of its next major model, called Astra, had produced ten new results in mathematics and theoretical computer science, each open for at least a decade, publishing a 249-page manuscript alongside machine-checkable Lean 4 certificates for every result on GitHub. OpenAI says the tokens used to generate all ten solutions would have cost about $2,000 at its Sol API rates. Among the findings, Astra disproved Connes's rigidity conjecture on von Neumann algebras, proved Ehrhart's volume conjecture, and resolved three problems from Paul Erdos's catalog, including problem 183 on multicolor Ramsey numbers. The headline result is the first explicit construction of a non-sofic group, resolving a central question in group theory that has stood since Mikhail Gromov introduced the concept of soficity in 1999; no mathematician had managed to prove or disprove whether non-sofic groups exist in the 27 years since.
Alongside the announcement, OpenAI released the 249-page manuscript and Lean 4 proof certificates on GitHub under an Apache 2.0 license; the repository's "sorry" count stands at zero, indicating that every step across all ten formalized proofs is fully verified. Because Lean's kernel either accepts or rejects a proof outright, independent verification requires nothing more than running the certificates through the compiler. OpenAI's head of mathematics research, Sebastien Bubeck, confirmed the results publicly, describing them as "beautiful"; Thomas Bloom, who maintains the Erdos problems website, called the ten results "big news," stating they are more significant than the unit distance counterexample announced in May 2026. The publication followed a round of closed-door meetings in which OpenAI CEO Sam Altman traveled to Washington, D.C. on July 29 and 30, 2026, to brief senior Trump administration officials and bipartisan senators on the Astra model series, a new family of AI models designed to coordinate multiple agents over long timescales on difficult, multi-step problems.
The announcement lands against a backdrop of escalating tension between AI companies and the mathematics community; in June, mathematicians issued the Leiden Declaration, endorsed by the International Mathematical Union, warning that AI companies are using published research without consent, bypassing peer review, and threatening the integrity of proof and attribution, and specifically citing companies that announce results through press releases rather than peer-reviewed journals. OpenAI cited the Leiden Declaration in its announcement, acknowledging that the mathematical ideas came from Astra rather than from the human researchers who prepared the manuscripts, a disclosure stance the declaration explicitly calls for. The declaration, published June 2, 2026, and endorsed by more than 3,000 signatories including Terence Tao and Peter Scholze, identifies five risks from AI in mathematics: unreliable results, missing citations, dependence on closed commercial systems, exaggerated claims, and loss of scientific independence.
Critics have noted that OpenAI chose which results to publish and that the $2,000 figure covers successful runs rather than every attempt the model made, making it a cost of publication rather than a cost of discovery; outside researchers also noted that OpenAI staff helped prepare the papers and formalize the arguments, and nobody outside the company can run the model that produced the work, so the results cannot be reproduced independently, only checked. Gary Marcus called the release "amazing but vastly oversold," and some specialists expect that once the dust settles, a few of the ten will look genuinely surprising while the rest will be classified as reachable problems that nobody had gotten around to attacking. OpenAI's Noam Brown was candid that none of the Clay Mathematics Institute's seven Millennium Prize Problems, each carrying a $1 million award, fell to Astra, a distinction that matters for calibrating what a $2,000 compute run actually means.
The more useful question is not whether a machine can do mathematics, but whether the mathematical community will treat a Lean certificate from a closed, unreleased model as a credential, and on that point the Leiden Declaration has already drawn a line.