OpenAI's Next Model Solves Ten Open Math Problems

OpenAI says an internal version of Astra, its next major model, has resolved or made substantial progress on ten long-standing open problems spanning high-dimensional geometry, coding theory, group theory, quantum complexity, and extremal combinatorics — and the total compute cost of finding the solutions was roughly $2,000 at Sol API rates.

The company laid out the results in an announcement titled "Ten advances in mathematics and theoretical computer science." The list includes a disproof of Connes's rigidity conjecture, a construction establishing the existence of non-sofic groups, and resolutions of three problems from Erdős's famous problem list: problem 183 on multicolor Ramsey numbers, and problems 146 and 180 on extremal graph theory.

The stakes are significant. Several of the problems addressed sit at the foundations of entire fields. The closest vector problem result — a polynomial-factor hardness-of-approximation proof — bears directly on lattice-based cryptography, the basis of most post-quantum encryption standards. The arithmetic circuit complexity work includes an arithmetic-formula lower bound of order n⁴/log n for computing the permanent, a problem that has stood at the heart of algebraic complexity theory for decades.

How the results were produced

According to OpenAI, the pipeline ran in stages. An internal version of Astra generated the mathematical arguments. Humans then worked with the same model to prepare the arguments into manuscripts. Finally, the model formalized each argument in a Lean certificate, which OpenAI has released on GitHub alongside the ten proofs. For every solution, the company also released a narration of the model's thinking process.

The ten results, as listed by OpenAI:

1. High-dimensional sphere packing. New upper bounds on sphere-packing density down to the Cohn–Elkies threshold.
2. Binary and spherical codes. Exponentially improved bounds on the maximum size of binary codes at any prescribed minimum distance, with analogous results for high-dimensional spherical codes.
3. Non-sofic groups. A construction establishing the existence of non-sofic groups, addressing a central open question in group theory.
4. Connes's rigidity conjecture. Disproof of a longstanding conjecture that certain groups are uniquely determined by their von Neumann algebras.
5. Arithmetic circuit complexity. New lower bounds for computing the permanent using arithmetic circuits and formulas, including an arithmetic-formula lower bound of order n⁴/log n.
6. Quantum parallel repetition. An exponential parallel repetition theorem for general two-player quantum games, extending a foundational principle from classical complexity theory.
7. Closest vector problem. Polynomial-factor hardness of approximation for the closest vector problem, a foundational lattice question related to post-quantum cryptography.
8. Ehrhart's volume conjecture. Determining, in every dimension, the maximum possible volume of a convex body whose centroid is its only interior lattice point.
9. Multicolor Ramsey numbers. A superexponential lower bound for multicolor triangle Ramsey numbers, resolving Erdős problem 183.
10. Extremal number conjectures. Results on the compactness and degeneracy conjectures in extremal graph theory, resolving Erdős problems 146 and 180.

OpenAI states that all ten problems are of substantial interest to their respective mathematical communities, and several are of broad interest across mathematics as a whole.

A follow-up to the Erdős unit-distance disproof

The announcement builds on a milestone from May, when OpenAI shared an AI-generated disproof of the Erdős unit-distance conjecture, discovered while evaluating an unreleased model. The company says that work has already inspired further developments in mathematics and theoretical computer science.

The new results appear to follow the same pattern: problems surfaced during internal model evaluation, then developed into publishable mathematics. The $2,000 compute figure is the detail most likely to draw attention from researchers and rivals alike. It suggests that frontier-level mathematical discovery now costs less than a typical conference travel budget — at least for a lab that already owns the model.

Attribution and the Leiden declaration

OpenAI devoted a substantial section of the announcement to questions of authorship and responsibility, an acknowledgment of growing friction between AI labs and the mathematical community.

"The emergence of systems capable of contributing to mathematical research raises questions that cannot be answered by a technology company alone," the company wrote. "There are many views as to the role of AI in mathematics, and we have deep respect and understanding for those concerned with its impact, including the signers of the Leiden declaration on AI and Mathematics."

The Leiden declaration, signed by mathematicians worried about AI's impact on their field, has become a reference point in the debate over how AI-generated mathematics should be credited and evaluated. OpenAI's position is explicit: AI-generated proofs should not be presented as human work.

"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," the company wrote. "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."

The Lean formalization matters here. Machine-checkable certificates let other mathematicians verify the proofs without trusting either the model or OpenAI's internal evaluation process — a meaningful gesture toward a community that has pushed back on unverifiable AI claims in other domains.

Broader access strategy

The announcement also ties into OpenAI's access strategy for research users. The company recently launched ChatGPT for Academic Researchers, an initiative providing 100,000 scientists and mathematicians with free access to its best ChatGPT models, and says it continues to evaluate its models on open research problems during development.

"As AI systems evolve into more sophisticated research collaborators, ensuring widespread access is fundamental to supporting scientists and mathematicians as they navigate and define the future of their disciplines during this transformative era," OpenAI wrote.

What comes next

The results position Astra as a model with demonstrated capability on problems that have resisted human mathematicians for decades — in one case, Connes's conjecture, for roughly half a century. OpenAI says it hopes the mathematical community will "engage deeply with these results, place them in context, and bring the ideas behind them to life through new research and discovery." The release of Lean certificates and thinking-process narrations means that verification can begin now, and how quickly established mathematicians confirm, contextualize, or contest these proofs will shape the credibility of AI-generated mathematics going forward.