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An event occurred in the world of artificial intelligence that mathematicians had been waiting for for decades. On August 1, OpenAI presented proofs for ten mathematical problems that had remained open since 2016 and earlier. This is not just another step in AI development—it is a breakthrough that changes the perception of what large language models can achieve in fundamental science.

The solutions were obtained by an internal version of Astra—the next flagship model that ChatGPT's developer is preparing for launch. According to OpenAI estimates, the computational costs for finding the answers would have been about $2,000 at API rates for Sol, which underscores the efficiency of the new approach. The manuscripts were prepared by researchers together with the model, after which it formalized each proof in Lean—a language for machine-checking theorems. All certificates and reasoning records have been published in open access, allowing the scientific community to verify the results.

Astra is positioned as a separate class of models alongside Sol, Terra, and Luna. OpenAI has not yet decided whether it will be released as GPT-6 or as a version in the GPT-5 line, and no release date has been set. However, on July 26, CEO Sam Altman demonstrated Astra to politicians and regulators in Washington. This is no coincidence: the model may become the first to be reviewed under the new rules of Donald Trump's administration, which require AI developers to submit new systems for evaluation by federal authorities before public launch.

Breakthrough in mathematics: from sofic groups to quantum games

Among the key results is a construction proving the existence of non-sofic groups. This closes a central question that mathematicians had been unable to solve since 1999, when Mikhail Gromov introduced the concept of soficity. The model also disproved Connes's rigidity conjecture on von Neumann algebras, solved Ehrhart's conjecture on volume, and solved Erdős's problem No. 183 on multicolor Ramsey numbers. Astra obtained new lower bounds on the complexity of computing the permanent using arithmetic circuits and proved a theorem on parallel repetition for two-player quantum games.

Of particular note is the improvement of upper bounds on density for sphere packing in high dimensions up to the Cohn–Elkies threshold and the strengthening of bounds for binary codes at any given minimum distance. Mathematician Thomas Bloom of the University of Manchester called these results "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, Astra is not all-powerful. Noam Brown, co-author of the model's reasoning technology, reported that OpenAI also attempted to tackle other major problems, but without success. The model could not solve the "Millennium Problems"—seven questions that the Clay Mathematics Institute named in 2000 as the most important in mathematics, promising $1 million for solving each.

This result marks a new era in the symbiosis of AI and fundamental science. Although a complete solution to the "Millennium Problems" is still far off, Astra's ability to work with formal proofs in Lean is not just a technical achievement but a tool that could accelerate mathematical discoveries by years. In my analysis, this is also a signal for the market: investments in AI research are beginning to yield results that go far beyond commercial applications, strengthening the strategic value of companies like OpenAI.