OpenAI said on August 1 that an internal version of Astra, its next model family, produced solutions to ten open problems in mathematics and theoretical computer science, and released formal proofs that can be checked automatically by software.
The results include new upper bounds in high-dimensional sphere packing, an explicit construction proving the existence of non-sofic groups, and a disproof of Connes’s rigidity conjecture that constructs infinitely many non-isomorphic groups with property (T) sharing the same von Neumann algebra. Three results resolve problems posed by Paul Erdos: a superexponential lower bound for multicolor triangle Ramsey numbers settles Erdos problem 183, and two bipartite constructions disprove the compactness conjecture of Erdos and Simonovits and a degeneracy conjecture, settling problems 146 and 180. The remaining results cover arithmetic circuit lower bounds for computing permanents and an exponential quantum parallel repetition theorem.
Humans remained in the loop. Researchers took Astra’s arguments and wrote them up into full manuscripts, and the model then formalized each proof into a Lean certificate. OpenAI published the resulting 249-page manuscript collection, model-written reasoning walkthroughs and certificates for all ten results in a public repository. Lean is a proof assistant whose kernel either accepts a formal proof or rejects it, which allows the results to be checked without expert review, and the repository manifest reports a sorry_count of zero, meaning the formalizations contain no unproven gaps.
Thomas Bloom, a mathematician at the University of Manchester, called the results big news and more significant than the counterexample to the unit distance conjecture that OpenAI published in May. Noam Brown, the OpenAI researcher behind its test-time reasoning work, noted that no Millennium Prize Problems were among them. OpenAI estimates the tokens needed to find solutions to all ten problems would cost roughly $2,000 at GPT-5.6 Sol API rates, a figure that covers inference and excludes the training run that produced the model.
The verification method sets the announcement apart from most model capability claims. Vendor-published benchmark tables have carried the weight of frontier comparisons this year, including Moonshot’s claims for Kimi K3, where independent reproduction had to wait for the weights and technical report. A Lean certificate can be run by anyone with the repository and the compiler.
That check has limits, because Lean verifies a theorem exactly as formalized, so a formalization that drifts from the informal problem it is meant to answer can compile cleanly while proving something weaker. Independent mathematical review of the ten statements has not been completed, and OpenAI did not claim external peer review for this announcement.
Astra remains unreleased and is still in testing. OpenAI has not said when the system used will be available to customers, or what the results imply for the models it currently sells. By the company’s own account, Astra would be the first model submitted under the Trump administration’s planned federal AI framework, which would require developers to send models to the government before public release.
OpenAI is spending on a different scale elsewhere. The company raised its projected compute budget through 2030 to roughly $750 billion in July, and is in talks with Nvidia over financing for a 10-gigawatt campus in Ohio.
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By the Control Plane Editorial Team