On August 1, OpenAI released proofs for ten mathematical problems that had remained unresolved since at least 2016.

The solutions were developed using Astra, the internal version of OpenAI's next major model, which was used to explore these complex queries. OpenAI estimates that the cost of tokens used in this research would approximate $2,000 based on Sol API rates.

The proofs were crafted by humans using the same model, which then formalized each proof in Lean, a language designed for machine verification of theorems. All certificates and reasoning records have been made publicly available.

Astra is set to be a distinct class of models alongside Sol, Terra, and Luna, according to The Information. The publication noted that OpenAI has not yet determined whether Astra will be released as GPT-6 or as a version within the GPT-5 series, and no release date has been announced.

On July 26, CEO Sam Altman showcased Astra to politicians and regulators in Washington. This model may be the first to undergo evaluation under the new regulations set by the Trump administration, which require AI developers to submit new systems for federal assessment before public launch.

Problems Solved by Astra

Among the significant achievements is a construction proving the existence of non-sophic groups, effectively resolving a central question that mathematicians have struggled with since 1999, when Mikhail Gromov introduced the concept of sophicity.

Other milestones include the refutation of Connes' rigidity hypothesis regarding von Neumann algebras, the resolution of Erhart's volume hypothesis, and the solution to Erdős problem No. 183 concerning Ramsey multicolor numbers. Astra also provided new lower bounds for the complexity of computing permanents using arithmetic circuits and proved a theorem on parallel repetition for two-player quantum games.

Additionally, the model enhanced upper density estimates for sphere packing in high dimensions up to the Kohn–Elkies threshold and strengthened bounds for binary codes at any specified minimum distance.

Thomas Bloom, a mathematician from the University of Manchester, referred to 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 succeed with every challenge. Noam Brown, a co-author of Astra's reasoning technology, noted that OpenAI attempted to tackle other major problems but was unsuccessful. The model could not resolve the Millennium Prize Problems, seven questions designated in 2000 by the Clay Mathematics Institute, which promised $1 million for the solution of each.

It is worth mentioning that in July, Anthropic's Claude disproved a mathematical hypothesis from 1939.