OpenAI’s Astra Solves Ten Long‑Standing Open Problems for About $2,000

OpenAI revealed that its internal‑version Astra model tackled ten long‑standing open problems across high‑dimensional geometry, coding theory, group theory, arithmetic circuit complexity, quantum complexity, lattice‑based cryptography and extremal combinatorics. The model produced the mathematical arguments, which human researchers then edited into readable papers and finally formalized as Lean certificates.

According to OpenAI, the total token usage for finding all ten solutions, priced at the Sol API rate, amounts to about $2,000. This figure demonstrates that large‑scale mathematical discovery can be pursued at a modest computational cost.

The ten results include:

High‑dimensional sphere packing : Astra derived a new upper bound on packing density that reaches the Cohn–Elkies threshold.

Binary codes and spherical codes : For any prescribed minimum distance, the model exponentially improved the known upper bound on the size of binary codes and achieved comparable gains for spherical codes.

Non‑sofic group : Astra constructed an explicit non‑sofic group, confirming the long‑open question of whether such groups exist.

Connes rigidity conjecture : The model produced a counterexample, showing that the conjectured uniqueness of von Neumann algebras for certain groups does not always hold.

Arithmetic circuit complexity of the permanent : Astra advanced lower‑bound results for arithmetic circuits/formulas computing the permanent, presenting a new quantitative bound (illustrated in the original figure).

Quantum parallel repetition : The model proved an exponential‑parallel‑repetition theorem for general two‑player quantum games.

Closest‑vector problem : Astra demonstrated polynomial‑approximation hardness for the closest‑vector problem, a core lattice problem tied to post‑quantum cryptography.

Ehrhart volume conjecture : It identified the maximal volume achievable by a special class of convex bodies in any dimension, resolving a conjecture posed by Ehrhart.

Multicolor Ramsey numbers : Astra established a super‑exponential lower bound for multicolor triangle Ramsey numbers, thereby solving Erdős problem 183.

Extremal graph theory conjectures : The model settled compactness and degeneracy conjectures in extremal graph theory, addressing Erdős problems 146 and 180.

The workflow described by OpenAI consists of three stages: (1) Astra searches for solutions to open problems and generates raw mathematical arguments; (2) human researchers refine these arguments into publishable papers; (3) the model translates each argument into a Lean proof certificate, enabling step‑by‑step machine verification.

OpenAI also released narrative explanations of each solution, though these are not the full internal computation trace. The authors note that while the first‑step proofs are impressive, the mathematical community must still scrutinize the results, simplify, generalize, and integrate them into existing theory.

Beyond the technical achievements, the announcement raises broader questions about authorship, credit, and evaluation in a future where AI can independently produce research‑level mathematics. OpenAI’s parallel launch of ChatGPT for Academic Researchers, offering free access to its strongest models for 100,000 scientists, underscores the company’s intent to accelerate AI‑driven scientific discovery.