Ten advances in mathematics and theoretical computer science

OpenAI's mathematics announcement describes a research process with distinct roles for generating arguments, preparing manuscripts, and checking formal proofs. The company reports ten results produced by an internal version of Astra. Humans then worked with the model to prepare manuscripts, and the model translated the arguments into Lean formalizations.

The accompanying proof repository supplies those formalizations and instructions for building them and for checking them with a separate tool. That gives researchers concrete artifacts to examine alongside the written arguments. OpenAI attributes the mathematical arguments to its system while accepting responsibility for their correctness; the announcement itself does not establish independent acceptance of every result.

“Today, we are sharing a selection of ten results, each of which resolves or makes substantial progress on a long-standing open problem. These problems span high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography and extremal combinatorics.”

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“The results were achieved by an internal version of Astra, our next major model. The total number of tokens needed to find solutions to these problems would cost roughly $2,000 at Sol API rates.”

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“Multicolor Ramsey numbers. A superexponential lower bound for multicolor triangle Ramsey numbers, resolving Erdős problem 183.”

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“We believe attribution should honestly reflect how a result was produced: claiming human authorship for a proof generated entirely by an AI system would misrepresent both the system’s contribution and the nature of genuine human intellectual work.”

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“We helped prepare the manuscripts and formalize the proofs in Lean, and we take responsibility for their correctness, while the mathematical arguments themselves were generated by our system.”

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