OpenAI presents ten mathematical advances from Astra

OpenAI says an internal version of its upcoming model, Astra, has found new results for ten open problems in mathematics and theoretical computer science. Several had seen no progress on their main result for at least a decade, while some had remained unresolved for much longer.

The problems span very different areas: high-dimensional geometry, coding theory, group theory, quantum complexity, lattice-based cryptography and extremal combinatorics. This is not a single conjecture, but a collection of results that specialists will now need to review and place in context.

The announced advances include:

• New upper bounds for the density at which spheres can be packed in high-dimensional spaces.
• Exponential improvements to bounds on the maximum size of certain binary and spherical codes.
• A construction proving the existence of non-sofic groups, an important open question in group theory.
• A refutation of Connes' rigidity conjecture, related to the information that can be recovered from certain algebraic structures.
• New lower bounds for computing the permanent, a mathematically difficult operation, using arithmetic circuits and formulas.
• A parallel repetition result for two-player quantum games.
• A proof of approximation hardness for the closest vector problem, which is relevant to post-quantum cryptography.
• An answer to the Ehrhart volume conjecture in all dimensions.
• A superexponential lower bound for multicolor triangle Ramsey numbers.
• Results resolving Erdős problems 146 and 180 in extremal graph theory.

The company says the model generated the mathematical arguments. People then prepared the manuscripts with help from the same system and formalized each proof in Lean, a tool that can automatically verify mathematical proofs. OpenAI also says it takes responsibility for the correctness of the results, while attributing the generation of the central ideas and arguments to the model.

“Claiming that a result was created by human authors when a proof was generated entirely by an AI system would misrepresent both the system's contribution and the nature of human intellectual work.”

The cost of generating the solutions was reportedly around $2,000, calculated using its API rates. The company also announced an initiative to provide 100,000 scientists and mathematicians with free access to its best models through ChatGPT for academic researchers.

What changes for you

These results do not automatically make AI an autonomous mathematician. A proof can contain errors, rely on hidden assumptions or solve a different formulation from the one that interests the community. Independent review is therefore still necessary, even when the argument has been formalized in Lean.

What matters is that models are beginning to take part in a part of scientific work that once seemed almost entirely reserved for specialists: proposing conjectures, finding useful structures and constructing proofs. In practice, a researcher could use AI to explore thousands of paths and spend their time checking, interpreting and extending the ones that look promising.

OpenAI had already reported in May an AI-generated refutation of the Erdős conjecture on unit distances. It now presents ten more results, but the next step does not depend on the model alone. It depends on other mathematicians verifying the arguments, identifying their limits and discovering which new questions they open.