OpenAI Astra: Ten Advances in Mathematics and Theoretical Computer Science
OpenAI has announced that an internal version of its next major model, Astra, has solved ten mathematical and theoretical computer science problems that had remained open for at least a decade. These results span diverse fields including high-dimensional geometry, coding theory, and quantum complexity, and are accompanied by formal Lean certificates to ensure correctness.
Mathematical and Theoretical Computer Science Breakthroughs
An internal version of the Astra model generated the mathematical arguments for ten distinct problems. To ensure the results were verifiable, OpenAI researchers worked with the model to prepare manuscripts and formalize each argument into a Lean certificate.
The ten solved problems include:
- High-dimensional sphere packing: The model established new upper bounds on sphere-packing density down to the Cohn–Elkies threshold.
- Binary and spherical codes: The model achieved exponentially improved bounds on the maximum size of binary codes at any prescribed minimum distance, with similar results for high-dimensional spherical codes.
- Non-sofic groups: The model provided a construction establishing the existence of non-sofic groups, solving a central open question in group theory.
- Connes’s rigidity conjecture: The model disproved a longstanding conjecture regarding whether certain groups are uniquely determined by their von Neumann algebras.
- Arithmetic circuit complexity: The model established new lower bounds for computing the permanent using arithmetic circuits and formulas, including an arithmetic-formula lower bound of order $n^{4/\log n}$.
- Quantum parallel repetition: The model developed an exponential parallel repetition theorem for general two-player quantum games, extending classical complexity theory principles.
- Closest vector problem: The model determined polynomial-factor hardness of approximation for the closest vector problem, which is a foundational question for post-quantum cryptography.
- Ehrhart’s volume conjecture: The model determined the maximum possible volume of a convex body whose centroid is its only interior lattice point across every dimension.
- Multicolor Ramsey numbers: The model resolved Erdős problem 183 by providing a superexponential lower bound for multicolor triangle Ramsey numbers.
- Extremal number conjectures: The model resolved Erdős problems 146 and 180 regarding compactness and degeneracy conjectures in extremal graph theory.
Technical Implementation and Cost
The solutions were generated by an internal version of Astra. OpenAI reports that the total number of tokens required to find these solutions would cost approximately $2,000 at Sol API rates. The workflow involved the model generating the arguments, humans collaborating with the model to prepare manuscripts, and the model subsequently formalizing the proofs in Lean.
AI Attribution and Research Responsibility
OpenAI emphasizes that the mathematical arguments were generated by the system, and while humans helped with manuscript preparation and Lean formalization, the system is the primary author of the discovery. The company states that claiming human authorship for AI-generated proofs would misrepresent the nature of intellectual work.
This announcement follows a previous AI-generated disproof of the Erdős unit-distance conjecture in May, which OpenAI notes has already inspired subsequent research in the field, including work on the sum-product conjecture and communication complexity of point-line incidences.