OpenAI said on August 1, 2026, that an early, unreleased build of Astra, the model the company plans to ship next, had generated original solutions to ten open problems spanning mathematics and theoretical computer science. The announcement doubles as the first public confirmation that Astra exists at all. OpenAI introduced its next flagship model not with a demo, but with a folder of proofs.
The price tag is almost as notable as the results. OpenAI estimates that the compute burned searching for all ten proofs would have run about $2,000 based on current Sol API pricing, the token cost of asking a frontier model to work like a research mathematician across a batch of decades-old problems.
A third detail matters most for judging whether any of this is real. OpenAI says the system converted each argument into a Lean certificate, output from a proof assistant that checks a chain of logical steps one by one and rejects anything that does not follow strictly from established axioms. That is a categorically different claim than “the model says it solved this.” A Lean certificate can be run by anyone, including a skeptic inside OpenAI’s own walls, and either it checks out or it does not. It turns an assertion into a verifiable object, which is the same standard mathematics itself uses to separate a proof from a claim.
Three of the ten results show what is actually being claimed. The system produced a construction proving that non-sofic groups exist, settling a question group theorists have puzzled over for years about whether every group can be approximated using finite permutations. A group that fails this test forces mathematicians to rethink assumptions used across the field. The model also produced a disproof of Connes’s rigidity conjecture, which held that certain groups can always be recovered from the structure of an associated von Neumann algebra, an idea operator algebraists have leaned on for decades. And it showed that the closest vector problem, a lattice question underpinning several cryptographic schemes now being adopted as defenses against future quantum computers, resists efficient approximation. That kind of hardness result is exactly what cryptographers point to when arguing a scheme will hold up against attackers.
OpenAI also credits the system with resolving three items from the long list of unsolved questions posed by the mathematician Paul Erdos, numbered 146, 180 and 183 in the community’s informal registry. That extends a line of results that started in May, when an earlier internal model produced a disproof of Erdos’s unit-distance conjecture, a result OpenAI says has already spurred follow-on papers from outside researchers.
OpenAI’s account of how credit should work is the least technical and most consequential part of this release. The company argues that describing a machine-generated proof as human work would misstate what the system contributed and diminish what human mathematicians actually do, and it says it will not make that move. Its own role, OpenAI says, was assembling the written manuscripts and turning the arguments into checkable Lean proofs. The company stands behind the correctness of that packaging work, while the underlying mathematical reasoning came from the model itself. OpenAI also nods to the researchers behind the Leiden declaration, a statement addressing how AI is changing mathematics, whose signers have pressed labs to be transparent about authorship and credit, framing its disclosure as a response to that pressure rather than a dismissal of it. The distinction matters because mathematical prestige has always tracked to a named author. Once a lab publishes results where no person did the reasoning, journals, universities and prize committees need a convention for saying so, instead of letting the humans who operated the software collect credit by default.
None of this should be read as settled science. A Lean certificate proves that an argument is logically valid. It says nothing about whether the problem was important, whether the method offers real insight to working mathematicians, or whether a simpler human proof already exists somewhere unpublished. No outside mathematician has yet reviewed these ten results, and everything here, including the $2,000 figure and the description of Astra itself, comes from OpenAI’s own account of a model it has not released. If the results survive scrutiny from mathematicians outside the company, they mark a genuine jump in what AI systems can contribute to research-level mathematics. Operators evaluating frontier-model capability claims should treat Astra as a name to watch closely over the next few months, not as a benchmark already confirmed.
OpenAI published these findings on August 1, 2026, in a post titled “Ten advances in mathematics and theoretical computer science.”
