OpenAI's next model, Astra, has solved ten open problems in pure mathematics with formal proofs

OpenAI has published ten new results in pure mathematics and theoretical computer science, each accompanied by a Lean certificate that allows independent computer verification of every logical step. The work came from an internal version of Astra, which OpenAI describes as its "next major model" 1.

Sebastien Bubeck, an OpenAI researcher, announced the results on X. He listed a disproof of Connes' Rigidity Conjecture in von Neumann algebras, improved bounds for high-dimensional sphere packing, new lower bounds for circuit complexity, a result on monochromatic triangles in multicolored graphs, and an existence proof for non-sofic groups 2. Each result ships with what Bubeck described as Lean certificates and chain-of-thought walkthroughs 2.

Roughly ten weeks earlier, OpenAI announced that an internal model had disproved the Erdos unit distance conjecture, an open problem in discrete geometry dating to 1946 3. Fields Medal winner Tim Gowers called that result "a milestone in AI mathematics" 3. University of Toronto professor Daniel Litt described it as "the first example of a result produced autonomously by an AI that I find exciting in itself" 3.

One result made mathematicians take notice. Ten, across unrelated fields, published weeks later, points to a system doing systematic mathematical research rather than landing a single lucky breakthrough.

The credibility question has always been the obstacle. Language models generate text that can look like a proof. Lean is a proof assistant: software that checks mathematical arguments step by step, confirming each logical inference holds. A Lean certificate means any mathematician with a computer can verify the proof without trusting the AI or reading its reasoning 1. The results are not AI text formatted to resemble math. They are machine-checked logical arguments.

OpenAI is specific about the division of labor. According to the company, Astra found the mathematical arguments. Humans then prepared those arguments into publishable manuscripts, using the same model for assistance 1. The Lean certificates verify the underlying mathematics, not just the finished prose.

The cost is where the economics get uncomfortable for anyone funding mathematical research. OpenAI says the total compute required to find these solutions would cost roughly $2,000 at the API rates of GPT-5.6 Sol, one of its commercially available models 1. GPT-5.6 Sol is priced at $5 per million input tokens and $30 per million output tokens 4. One of the ten results disproves Connes' Rigidity Conjecture, a long-standing problem in von Neumann algebras 2. Problems of that caliber typically consume years of expert labor.

A caveat from Ars Technica's analysis of the earlier Erdos result is worth carrying forward. The model "cleverly applied existing ideas drawn from several subfields of mathematics to create a full proof" but "didn't pioneer any genuinely new techniques" 3. Whether Astra's ten new results follow the same pattern, synthesizing known methods across fields rather than inventing new mathematical concepts, is a question the published manuscripts will answer as mathematicians examine them.

The Astra name surfaced in Bubeck's post as an aside. He described it as the company's "next major model," revealed through a math result rather than a product launch 2. He did not specify a release timeline or describe other capabilities. Introducing a model through pure mathematics, the field with the strictest possible standards for correctness, is itself a signal about what OpenAI believes Astra can do.

For anyone building or investing in AI, the structural takeaway is narrow and specific. The bottleneck for machine-generated mathematics was never speed. It was trust. A model that produces a proof in hours instead of years is useless if no one can confirm it is correct. Lean certificates remove that barrier. The same pattern, formal verification replacing human trust as the gatekeeper for machine output, applies to any discipline with a verification layer. Software correctness proofs, cryptographic protocol analysis, and formal logic all share the same structural opening.

The question is not whether AI can do mathematics. On the evidence of these ten results, each machine-verified, it can. The question is what happens to every other field where a machine can do the work and a verifier can prove it.