OpenAI's Next-Gen Astra Model Solves Ten Open Math Problems for Just $2,000

OpenAI has announced that an internal version of its next-generation flagship model, Astra, has produced ten new research results in high-dimensional geometry, coding theory, group theory, arithmetic circuit complexity, quantum complexity, lattice cryptography, and extremal combinatorics. These problems share the characteristic that their core conclusions had seen no significant progress for at least a decade, with many awaiting breakthroughs for much longer.

Specific results include: a new upper bound for sphere packing in high dimensions (advancing to the Cohn-Elkies threshold); exponential improvements in the size of binary and spherical codes; construction of a non-sofic group, thereby proving that non-sofic groups indeed exist; a counterexample to Connes's rigidity conjecture; progress on the lower bound for computing the permanent via arithmetic circuits; proof of an exponential parallel repetition theorem for general two-player quantum games; proof of the computational hardness of the closest vector problem under polynomial approximation factors; determination of the maximum volume of a special class of convex bodies (solving Ehrhart's conjecture); establishment of super-exponential lower bounds for multicolor Ramsey numbers (solving Erdős problem #183); and resolution of Erdős problems #146 and #180 in extremal graph theory.

These results were not trained specifically for these problems but came from open-problem evaluations during Astra's development. OpenAI notes that in May of this year, it disclosed a similar attempt: an unreleased model generated a counterexample to the Erdős unit distance conjecture, which spurred further research.

OpenAI states that the tokens consumed to achieve the ten results are worth approximately $2,000 at Sol API prices. The research process involves three steps: the internal version of Astra searches for solutions and generates mathematical arguments; human researchers use the same model to organize the arguments into publicly readable papers; and the model converts each argument into a Lean proof certificate. Lean is an interactive theorem prover that breaks down proofs into a formal process that can be checked step by step by a machine.

OpenAI has also released the model's narrative of the problem-solving process for each result, but these are closer to model-generated reasoning explanations and cannot be simply equated with the complete, raw internal computation that occurred during runtime.

This process indicates a shift in the role of large models in mathematics: they are beginning to venture into true research frontier, proposing arguments that did not previously exist, and subjecting them to step-by-step verification by formal proof systems.

In the past, the application of large models in scientific research was mainly limited to literature search, code writing, data analysis, and paper polishing, rarely touching the core original conclusions of papers. The ten results announced by OpenAI attempt to cross this boundary: the model is no longer just helping to express existing ideas but is searching for possible structures in open problems, forming new mathematical arguments, and using formal systems to check the results.

OpenAI also recently announced ChatGPT for Academic Researchers, planning to offer its most powerful ChatGPT model free to 100,000 scientists and mathematicians. These ten results can be seen as a demonstration of the capabilities of this program.

However, the model providing proofs is only the first step. Whether these ten results will withstand long-term scrutiny from the mathematical community, and whether they can be further simplified, generalized, and integrated into existing theoretical frameworks, still requires verification by professional researchers. OpenAI has placed a sharp question before the mathematical community: when AI can independently produce mathematically valuable arguments, how should authorship, contribution attribution, and the scientific evaluation system be adjusted?