On August 1, OpenAI published solutions to ten mathematical problems that had remained unsolved since at least 2016.
The solutions were generated by an internal version of Astra, the next major model from the developer of ChatGPT. According to OpenAI, the cost of tokens used to find the answers would have been approximately $2000 based on Sol’s API rates.
The manuscripts were prepared by humans using the same model, which then formalized each proof in Lean, a language for machine-verified theorems. The company has released all certificates and reasoning records to the public.
Astra will be a separate class of models alongside Sol, Terra, and Luna, according to The Information. The publication reports that OpenAI has not yet decided whether it will be released as GPT-6 or as a version within the GPT-5 lineup. The company has also not set a release date.
On July 26, the company’s CEO, Sam Altman, demonstrated Astra to politicians and regulators in Washington. The model could be the first to be tested under new rules from the administration of U.S. President Donald Trump, which require AI developers to submit new systems for federal evaluation before public launch.
Problems Solved by Astra
Key results include a construction proving the existence of non-sofic groups, addressing a central question that mathematicians had been unable to solve since 1999 when Mikhail Gromov introduced the concept of soficity.
Other achievements include refuting Connes’ rigidity conjecture on von Neumann algebras, solving Ehrhart’s conjecture on volume, and addressing Erdős problem No. 183 on multicolored Ramsey numbers. Astra also provided new lower bounds on the complexity of computing the permanent with arithmetic circuits and proved a parallel repetition theorem for two-player quantum games.
Additionally, the model improved upper bounds on sphere packing density in high dimensions up to the Cohn-Elkies threshold and strengthened bounds for binary codes at any given minimum distance.
Thomas Bloom, a mathematician at the University of Manchester, described the published results as “big news” and a “significant step” in the field of constructions.
Big news! (And not really my area, but yes, I would rank this as bigger than the unit distance counterexample. Maybe not bigger than a proof of unit distance would have been, but in terms of constructions, this is big.) https://t.co/VDRti1HZ6Z
— Thomas Bloom (@thomasfbloom) August 1, 2026
However, the model did not solve all problems. Noam Brown, co-author of Astra’s reasoning technology, reported that OpenAI attempted to tackle other major issues without success. The model was unable to solve the “Millennium Prize Problems”—seven questions identified by the Clay Mathematics Institute in 2000 as the most important in mathematics, with a $1 million reward for each solution.
In July, Claude from Anthropic disproved a mathematical hypothesis from 1939.
