OpenAI says an internal Astra model solved ten open problems in mathematics and theoretical computer science

OpenAI published results from an internal version of Astra, described as the company's next major model, solving ten open problems in mathematics and theoretical computer science that had seen no progress on their main result for at least a decade, and in several cases much longer.

What's new

The ten results span high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography, and extremal combinatorics. Highlights include:

• New upper bounds on sphere-packing density down to the Cohn-Elkies threshold
• Exponentially improved bounds on the maximum size of binary and high-dimensional spherical codes
• A construction establishing the existence of non-sofic groups, a central open question in group theory
• A disproof of Connes's rigidity conjecture
• New arithmetic-circuit lower bounds for computing the permanent, including a bound of order n^4/log n
• An exponential parallel repetition theorem for general two-player quantum games
• Polynomial-factor hardness of approximation for the closest vector problem, relevant to post-quantum cryptography
• Resolution of Ehrhart's volume conjecture across all dimensions
• A superexponential lower bound for multicolor triangle Ramsey numbers, resolving Erdos problem 183
• Results resolving Erdos problems 146 and 180 in extremal graph theory

OpenAI says the total token cost to find solutions to all ten problems was roughly $2,000 at Sol API rates. Humans then prepared the arguments into manuscripts with the same model, and the model itself formalized each proof as a Lean certificate, which OpenAI is releasing publicly along with a narration of the model's thinking process for each solution.

Context

This follows OpenAI's May disclosure of an AI-generated disproof of the Erdos unit-distance conjecture, discovered while evaluating an unreleased model, which the company says has already inspired further work in mathematics and theoretical computer science. OpenAI frames both disclosures as part of a broader push to position its models as research collaborators, alongside initiatives like ChatGPT for Academic Researchers, a free-access program for 100,000 scientists and mathematicians announced in July.

Why it matters

OpenAI is explicit that these are not human-assisted proofs in the traditional sense: the mathematical arguments were generated entirely by the system, with humans preparing manuscripts and the model itself handling formalization. The company addresses the attribution question directly, stating that claiming human authorship for an AI-generated proof would misrepresent both the system's contribution and the nature of human intellectual work — a position aligned with the Leiden declaration on AI and mathematics. Coming just as OpenAI previews Astra as its next major model, the release doubles as an early capability signal: an internal, not-yet-released version of the model is already producing results the mathematical community will need to independently verify and contextualize.