OpenAI's Astra model cracks ten open math problems

OpenAI says an internal, unreleased model called Astra has produced solutions to ten longstanding open problems spanning geometry, cryptography, group theory, and complexity theory. The company disclosed the results in a post titled "Ten advances in mathematics," framing them as a significant leap beyond its earlier math demonstrations.

The list is dense with specialist terrain. It includes new upper bounds on high-dimensional sphere-packing density, exponentially improved bounds on binary and spherical codes, and a construction proving the existence of non-sofic groups. Astra is also credited with disproving Connes's rigidity conjecture and establishing new lower bounds in arithmetic circuit complexity.

Other results touch quantum computing and cryptography directly. The model reportedly proved an exponential theorem for quantum parallel repetition and demonstrated polynomial-factor hardness for the closest vector problem, a result with direct relevance to lattice-based cryptographic schemes. Rounding out the list are progress on Ehrhart's volume conjecture, a superexponential lower bound for multicolor Ramsey numbers, and advances on extremal number conjectures.

How the process worked

OpenAI is careful to describe this as a human-machine collaboration rather than a fully autonomous discovery pipeline. According to the company, human mathematicians first drafted the underlying arguments into manuscripts. Astra's role was to formalize each argument into a verified Lean certificate, the proof-assistant language used to machine-check mathematical logic.

That formalization step is notable mainly for its cost efficiency. OpenAI estimates that generating the tokens behind all ten solutions would run roughly $2,000 at current Sol API pricing, a modest sum for a batch of results touching multiple subfields of pure mathematics.

This is not Astra's first credited breakthrough. OpenAI previously attributed an AI-generated disproof of the Erdős unit-distance conjecture, disclosed in May, to an unreleased model under evaluation, a result widely believed in retrospect to have come from the same system.

Noam Brown, known for his work on OpenAI's reasoning and strategic-play research, is associated with the Astra project, though the company has not detailed the model's architecture or release timeline.

Access for researchers

Alongside the math results, OpenAI is expanding access to its models for academic use. The company recently launched ChatGPT for Academic Researchers, giving roughly 100,000 scientists and mathematicians free access to its top-tier ChatGPT models.

The pairing of the two announcements suggests OpenAI is positioning itself less as a tool vendor for one-off proofs and more as an infrastructure layer for mathematical research broadly, even as Astra itself remains unreleased and its methodology largely undisclosed outside the company's own summary.