Quick Facts
- OpenAI’s Astra model solved 10 math and theoretical computer science problems, some open for decades, and published Lean 4 machine-verified proofs for all 10.
- The headline result is the first-ever explicit construction of a non-sofic group, resolving a question that stood since 1999.
- OpenAI put the compute cost for all 10 successful solutions at roughly $2,000 at GPT-5.6 Sol API rates, though other attempted problems went unsolved.
OpenAI announced on Aug. 2, 2026, that an internal version of Astra, its next major model family, produced new results for 10 problems in mathematics and theoretical computer science that had been open for at least a decade. The company published machine-checkable proofs alongside every claim.
OpenAI researcher Noam Brown shared the first results on Aug. 1. Brown called the findings “a major step for scientific reasoning.” Sebastien Bubeck, OpenAI’s head of mathematics research, called the results “beautiful.”
The problems span high-dimensional geometry, coding theory, group theory, quantum complexity, lattice cryptography, and extremal combinatorics. The headline result is the first explicit construction of a non-sofic group, resolving a central question in group theory that has stood since Mikhail Gromov introduced the concept of soficity in 1999. No mathematician had proven or disproven whether non-sofic groups exist in the 27 years since.
Other results include a disproof of Connes’s rigidity conjecture, an exponential parallel repetition theorem for general two-player quantum games, and a polynomial-factor hardness of approximation result for the closest vector problem in lattice-based mathematics, which connects directly to post-quantum cryptography.
Astra also produced superexponential lower bounds for multicolor triangle Ramsey numbers, resolved Erdos problems 146, 180, and 183, and achieved the first improved general sphere-packing exponent since 1978. Ehrhart’s volume conjecture also appeared among the resolved results.
OpenAI posted a 249-page manuscript collection, model-written reasoning walkthroughs, and Lean 4 certificates for all 10 results. The certificates are on GitHub under an Apache 2.0 license. The repository reports a “sorry” count of zero, meaning no step in any formalized proof has been left unproven. Lean 4’s trusted kernel gives a binary verdict: the proof either compiles or it does not.
That verification method matters. When OpenAI’s model disproved the Erdos unit-distance conjecture in May 2026, nine external mathematicians read and signed off on the argument. A Lean certificate is different. Anyone can run the check without a PhD or months of peer review. Fields Medalist Tim Gowers said he would recommend the May Erdos proof for publication in Annals of Mathematics without hesitation.
Thomas Bloom of erdosproblems.website described the new results as “big news” and said they are more significant than the unit-distance counterexample OpenAI previously published. Terence Tao said he was “very impressed” and endorsed the Leiden Declaration, a related statement on AI and mathematics, “wholeheartedly.”
Human researchers turned Astra’s output into publishable papers. OpenAI said the mathematical arguments themselves came from Astra. The company describes Astra as a model family built to run long tasks by coordinating multiple agents over extended periods.
Astra sits alongside Sol, Terra, and Luna as a new class of OpenAI models. OpenAI has not decided whether to label it GPT-6 or an additional model in the GPT-5 series. No public release date, model size, context window, API price, or safety card has been announced.
The $2,000 compute figure deserves scrutiny. That cost covers only the 10 attempts that succeeded. Brown confirmed in the same thread, 29 minutes later, that other major problems were attempted without success.
The policy dimension adds weight to the announcement. According to The Information, OpenAI CEO Sam Altman previewed Astra for White House officials, cabinet members, and members of Congress before the public announcement. Astra is also set to be the first model subject to a new U.S. government safety review process requiring official approval before public release.
Read more: OpenAI’s Astra solves 10 long-open math problems and publishes the proofs