OpenAI introduced its next major model on August 1, 2026, and it did so by publishing ten new solutions to open mathematics and theoretical computer science problems, several of which had gone unsolved for decades. The model, called Astra, is unreleased, but the results are already independently verifiable, because OpenAI published them as formal, machine-checked proofs rather than prose claims.
Quick facts
- An internal, unreleased version of Astra, OpenAI’s next major model, produced new results on ten long-open problems spanning geometry, coding theory, group theory, complexity theory, and cryptography.
- OpenAI says the compute needed to find the solutions would cost roughly $2,000 at its Sol API rates.
- Each result was formalized into a Lean certificate, a machine-checkable proof format, and published on GitHub for anyone to verify independently.
- Two of the results directly resolve named open problems: Erdős problem 183 (multicolor Ramsey numbers) and Erdős problems 146 and 180 (extremal graph theory).
- OpenAI explicitly states it takes responsibility for the manuscripts’ correctness while the mathematical arguments themselves were generated by the system, not a human mathematician.
What was actually solved
Per OpenAI’s own publication, the ten results include new upper bounds on sphere-packing density, exponentially improved bounds on binary and spherical error-correcting codes, a construction establishing the existence of non-sofic groups (a central open question in group theory), a disproof of Connes’s rigidity conjecture, new lower bounds on arithmetic circuit complexity for computing the permanent, an exponential parallel repetition theorem for quantum games, polynomial-factor hardness results for the closest vector problem (a foundational post-quantum cryptography question), a resolution of Ehrhart’s volume conjecture, and the two Erdős problems noted above. OpenAI describes all ten as problems that had seen no progress on their main result for at least a decade, and in most cases much longer.
Worth being precise about: this is a separate batch of results from the Erdős unit-distance conjecture disproof OpenAI announced in May 2026, which several secondary outlets have conflated with this release. That earlier result is cited in this announcement as prior work that helped inspire further mathematics, not one of today’s ten.
Why the Lean proofs are the whole point
AI models are well known to produce confident, plausible-sounding claims that turn out to be wrong, which is exactly why the format of this announcement matters as much as its content. A Lean proof is written in a formal language a computer can check mechanically, step by step, with no room for hand-waving. OpenAI published the Lean certificates for all ten results on GitHub, meaning any mathematician, not just OpenAI, can run the verifier and confirm the logic holds. That’s a meaningfully different claim than a benchmark score, which can be gamed, memorized, or cherry-picked. A formally verified proof either checks out or it doesn’t.
How OpenAI is handling attribution
OpenAI’s own writeup addresses a question the mathematical community has been actively debating: who gets credit when an AI system generates a proof. The company points to the Leiden declaration on AI and Mathematics and states plainly that claiming human authorship for a proof generated entirely by an AI system would misrepresent both the system’s contribution and human intellectual work. OpenAI says it helped prepare the manuscripts and formalize the Lean proofs, and takes responsibility for their correctness, while the underlying mathematical arguments came from Astra itself.
Why $2,000 is the detail worth remembering
The cost figure reframes what this milestone actually means. Ten problems that resisted expert mathematicians for a decade or more were solved for roughly the cost of a mid-range laptop in compute. That doesn’t replace mathematicians, choosing which problems matter and interpreting what a result means still requires human judgment, but it does suggest that for a certain class of well-posed, verifiable problems, progress increasingly scales with available compute rather than being bottlenecked purely by the supply of specialists working on them.
Common questions
Is Astra publicly available? No. The math results came from an internal, unreleased version. OpenAI hasn’t given a public release date or a public spec sheet for Astra.
Did AI disprove the Erdős unit-distance conjecture today? No, that result was announced separately in May 2026. Today’s announcement is ten different, new results.
How do I check these proofs myself? OpenAI published the Lean certificates on GitHub; running them through the Lean proof verifier confirms whether each argument holds, independent of OpenAI’s own claims.
Does this mean AI can do all mathematics now? No. These ten problems were selected in areas well suited to systematic search and construction, which is where current AI systems are strongest. Broader mathematical creativity and problem selection still rest with human mathematicians.
Key takeaway
This isn’t evidence of general intelligence, math has clear rules and mechanically checkable answers, which is exactly the kind of problem AI systems are well suited to. But it is a genuine, independently verifiable research contribution, not a benchmark stunt, and it’s a preview of how OpenAI plans to introduce Astra when the model itself eventually ships.
