OpenAI's Astra solved 10 decades-old math problems for $2,000

On August 1, 2026, OpenAI announced that its undisclosed internal model, named Astra, had successfully generated fully-verified proofs for ten previously unsolved mathematical problems and a theoretical computer science challenge. The model achieved this feat for a mere $2,000 in computational resources, a fraction of the cost of a weekend's worth of GPU rental.

The impressive results were documented in a 249-page manuscript and the accompanying Lean 4 proof certificates, which were made publicly available under an Apache 2.0 license on GitHub. Lean, a proof assistant, meticulously checked every logical step, ensuring the correctness of the proofs. No gaps were left unchecked, as the repository's "sorry" count (indicating unproven gaps) remained at zero across all ten results.

The problems tackled by Astra span various mathematical domains, including non-sofic groups, sphere-packing bounds, Connes' rigidity conjecture, Ehrhart's volume conjecture, and several problems from Paul Erdős's open-problem catalog, such as problem 183 on multicolor Ramsey numbers. The $2,000 figure underscores the remarkable efficiency of Astra, which essentially condensed years of human effort into an overnight computation using machine-speed proof search.

While initial reactions to the results were positive, particularly from mathematician Timothy Gowers, the proofs have not yet undergone formal peer review, and thus, remain subject to community validation. This breakthrough marks a significant milestone in demonstrating the potential of AI models to independently generate valid mathematical proofs, underscoring the growing influence of formal proof checkers in establishing the veracity of mathematical claims.