OpenAI has revealed that an internal version of Astra, its next major model, solved 10 open problems in mathematics and theoretical computer science. Every one of them had resisted proof for at least a decade. Alongside a 249-page manuscript, the company published a machine-checkable Lean 4 certificate for each result on GitHub. That makes this different in kind from an announcement that merely claims a breakthrough.
Not a Claim, but a Proof That Compiles
The first thing worth noting is that all 10 results ship with Lean 4 proofs. Lean is a system for writing mathematical arguments in a formal language and having a computer check them mechanically. A proof either compiles or it does not, and there is no room for hand-waving or convenient rephrasing along the way.
AI-produced proofs have long carried a specific complaint: they take specialists so long to read that independent verification never really happens. With a Lean certificate attached, anyone who has the Lean compiler can confirm the result locally, without having to trust the model or the people running it. It is a formal answer to a criticism the mathematical community has raised repeatedly.
A 27-Year-Old Question in Group Theory Settled
The centerpiece of the 10 is the first explicit construction of a non-sofic group. Mikhail Gromov introduced the notion of soficity in 1999, and whether any group fails to be sofic has stood as a central open question ever since, unresolved for 27 years. No mathematician had been able to prove that such groups exist, or that they do not.
The other 9 results span several fields: a disproof of the Connes rigidity conjecture on von Neumann algebras, a proof of the Ehrhart volume conjecture, solutions to 3 problems from Paul Erdos's catalogue (including problem 183 on multicolour Ramsey numbers), the first improvement since 1978 to the general upper bound on high-dimensional sphere-packing density, a parallel repetition theorem for two-player quantum games, and new lower bounds on the circuit complexity of computing the permanent. Group theory, operator algebras, high-dimensional geometry, quantum complexity, lattice cryptography and extremal combinatorics are all represented.
Total Compute Cost of Roughly 2,000 USD
According to OpenAI, producing all 10 solutions cost roughly 2,000 USD (about 310,000 yen) in compute at Sol API rates. Sebastien Bubeck, who leads mathematics research at the company, confirmed the results on X, called them beautiful, and noted that each one comes with a Lean certificate and a walkthrough of the reasoning.
※1 USD = 156.5 JPY (as of August 3, 2026)
An Uneasy Relationship With the Mathematics Community
The backdrop is far from calm. In June, mathematicians issued the Leiden Declaration, which the International Mathematical Union endorsed. It warns that AI companies are using published research without consent, bypassing peer review, and undermining the integrity of proof and attribution, and it singles out the practice of announcing results through press releases instead of peer-reviewed journals.
OpenAI has been here before. In May it said the same long-horizon model family had disproved the Erdos unit distance conjecture, an 80-year-old problem in discrete geometry. Fields Medalist Tim Gowers said at the time that he would not hesitate to recommend that proof for the Annals of Mathematics. This time, Thomas Bloom, who runs the erdosproblems site, wrote on X that the new results are big news and more significant than the unit distance counterexample. There is praise, then, but how the wider mathematical community will treat findings announced in a blog post remains unsettled.
Astra Itself Is Still Under Wraps
OpenAI has given no release date for Astra, describing it only as its next major model. Some observers, including investor Mark Kretschmann, have speculated that it is the GPT-6 series, though nothing has been confirmed. The company is also giving 100,000 academic researchers free access to its frontier models through 2027, part of a broader push to tie itself more closely to scientific research.
Summary
An unreleased model cleared 10 mathematical problems that had been open for more than a decade, and every result arrived with a machine-checkable Lean 4 proof. The whole effort cost around 2,000 USD in compute. Providing a means of verification is real progress, but the question of announcing results outside peer review is still open, and Astra's own release date is unknown. The debate over how mathematics should use AI looks set to move further from here.
