OpenAI's Astra model solves ten decade-old math problems
OpenAI published ten new results on open problems in mathematics and theoretical computer science, describing them as solved by an internal version of Astra, the company's next major model. The company frames the release as a follow-up to a disproof of the Erdos unit-distance conjecture it shared in May, which it says was discovered while evaluating an unreleased model and has already prompted several follow-up papers by outside researchers.
All ten problems, OpenAI says, had been open with no progress on the main result for at least a decade, and in most cases much longer. They span high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography and extremal combinatorics. The list: new upper bounds on sphere-packing density approaching the Cohn-Elkies threshold; exponentially improved bounds on the maximum size of binary codes at a given minimum distance, with matching results for spherical codes; a construction proving non-sofic groups exist, settling a central open question in group theory; a disproof of Connes's rigidity conjecture, which held that certain groups are uniquely determined by their von Neumann algebras; new lower bounds for computing the permanent with arithmetic circuits and formulas, including an arithmetic-formula lower bound on the order of n4/log n; an exponential parallel repetition theorem for general two-player quantum games; polynomial-factor hardness of approximation for the closest vector problem, a lattice question tied to post-quantum cryptography; a determination, in every dimension, of the maximum volume of a convex body whose centroid is its only interior lattice point (Ehrhart's volume conjecture); a superexponential lower bound for multicolor triangle Ramsey numbers, resolving Erdos problem 183; and results on the compactness and degeneracy conjectures in extremal graph theory, resolving Erdos problems 146 and 180.
According to OpenAI, the total token cost of finding solutions to all ten problems would run to roughly $2,000 at Sol API rates. Humans then turned the model's arguments into manuscripts, working with the same model, after which Astra formalized each argument as a Lean proof certificate. OpenAI is also releasing a narration of the model's thinking process for each solution.
The post ties the release to OpenAI's ChatGPT for Academic Researchers initiative, announced separately, which gives 100,000 scientists and mathematicians free access to OpenAI's best ChatGPT models. On attribution, OpenAI states it takes responsibility for the correctness of the manuscripts and Lean formalizations, while saying the mathematical arguments themselves were generated by its system, and that claiming human authorship for a wholly AI-generated proof would misrepresent both the system's contribution and genuine human intellectual work. It also acknowledges the concerns raised by signers of the Leiden declaration on AI and Mathematics. The source does not name any OpenAI researcher or mathematician involved beyond Astra and unspecified humans, gives no wall-clock time for the work beyond the token cost, and does not say whether outside mathematicians have reviewed any of the ten results beyond OpenAI's own Lean formalization.
Key facts
- OpenAI says an internal version of its unreleased Astra model produced new results on ten math and theoretical computer science problems open for at least a decade, in most cases much longer.
- The ten results include resolving three Erdos problems (183, 146 and 180), disproving Connes's rigidity conjecture, and a new arithmetic-formula lower bound on the order of n4/log n for computing the permanent.
- OpenAI says the total token cost to find the solutions was roughly $2,000 at Sol API rates; humans then wrote the arguments up as manuscripts and Astra formalized each proof as a Lean certificate.
- The release follows a May disproof of the Erdos unit-distance conjecture that OpenAI says has already led to several outside follow-up papers.
- OpenAI states it takes responsibility for the manuscripts' and formalizations' correctness while the mathematical arguments themselves came from its system, and separately notes ChatGPT for Academic Researchers gives 100,000 scientists free access to its best ChatGPT models.
Why it matters
This is a bigger claim than the May Erdos unit-distance disproof: ten results across eight distinct fields, from an internal version of a model OpenAI has not yet released. If the results hold up, it is evidence a frontier model can generate genuinely new mathematical arguments across very different subfields in one pass, not just assist with known techniques in one niche.
Who it affects
Mathematicians and theoretical computer scientists working in high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography and extremal combinatorics, since each of the ten results speaks directly to an open problem in one of those areas. Separately, OpenAI says its ChatGPT for Academic Researchers program gives 100,000 scientists and mathematicians free access to its best ChatGPT models.
How to use it
OpenAI is publishing, for each of the ten results, a Lean proof certificate and a narration of the model's thinking process, so mathematicians can inspect the formal proof and the reasoning trace rather than take the claim on faith. Astra itself remains an internal, unreleased model; OpenAI gives no release date, pricing or access path for it here, only that generating the ten solutions cost about $2,000 in tokens at Sol API rates.
How solid is it
The source is OpenAI's own blog, and every one of the ten results ships with a Lean formalization, a machine-checked proof rather than a prose claim alone. OpenAI also says it takes responsibility for the correctness of the manuscripts and formalizations. Against that: the post does not say whether any outside mathematician has reviewed the results beyond OpenAI's own Lean check, and no OpenAI researcher or mathematician is named as having worked on the manuscripts.
Risks and caveats
OpenAI itself raises the attribution question its post is built around: claiming human authorship for a proof generated entirely by an AI system would misrepresent both the system's contribution and genuine human intellectual work, and it references the Leiden declaration on AI and Mathematics as a marker of community concern on this point. The post gives no exact date for either the ten-result release or the May disproof, no wall-clock time for the work beyond token cost, and does not say whether any of the ten problems had been attempted by other AI systems before.
“We believe attribution should honestly reflect how a result was produced: claiming human authorship for a proof generated entirely by an AI system would misrepresent both the system's contribution and the nature of genuine human intellectual work.”
— OpenAI