OpenAI's unreleased Astra AI produced 10 research results, each with a computer-checked proof
The 10 results include three Erdős problems, and anyone can re-check the proofs by computer. OpenAI reported only its successes.
In early August 2026 OpenAI announced ten results in mathematics and theoretical computer science produced by Astra, an in-house model not yet released. Each comes with a proof in Lean 4, a language in which a computer checks every step. The proofs are in the public openai/ten-proofs repository, so anyone can re-check them without having to trust OpenAI.
The ten include better limits on how densely spheres can be packed, stronger bounds for codes, an explicit “non-sofic group” and three problems posed by the mathematician Paul Erdős (#146, #180 and #183). Others concern quantum games and how hard certain computing problems are.
The Next Web reports a 249-page manuscript and about $2,000 of compute at API rates (what OpenAI charges customers). Thomas Bloom, who runs erdosproblems.com, called the release big news, more important than OpenAI’s May disproof of the unit distance conjecture. In reactions gathered by Science Media Centre Spain, Senén Barro noted that only successes were reported, not failures or true costs. Javier Aramayona (ICMAT) said formal verification will let mathematicians take on more ambitious problems, and urged avoiding both fatalism and triumphalism.
What it means
Because the proofs are computer-checked, nobody has to take OpenAI’s word that the formal arguments are valid. What they don’t show is how many attempts failed or what the whole effort cost, so this shows what Astra can do, not how often. Whether each formal statement matches the English claim is still for mathematicians to judge.
Caveats
- Astra was unreleased at announcement, and failure rates and total attempts were not disclosed. The cost figure is OpenAI’s, priced at API rates.
- The results were announced by press release rather than peer review, two months after the IMU-endorsed Leiden Declaration (June 2026) criticised exactly that practice, per The Next Web.
- Lean checks the formal statements; whether they match the English claims is for readers to judge.
- The date is approximate: coverage is dated 1 August, the repository’s only commit 2 August. Epoch AI later referred to the model as GPT-6 Astra (pre-release).