OpenAI says its next model solved ten open math problems, with Lean proofs attached
A 249-page collection claims ten results on problems stuck for a decade or more, each with a machine-checkable Lean certificate and a token bill under $2,000. Peer review of their significance is still pending.
Ten decade-old problems, $2,000 of tokens each, and a Lean proof attached to every claim.
OpenAI published a 249-page collection on August 1, 2026 claiming that an internal version of its next major model, Astra, produced ten new results on open problems in mathematics and theoretical computer science, each of which had seen no progress for at least a decade. The part that makes this more than a press release: every result ships with a machine-checkable Lean proof certificate and the model's reasoning notes, and OpenAI says each one cost under $2,000 of tokens at GPT-5.6 Sol API rates.
What is confirmed
The proofs are formalized in Lean, which means anyone can run the proof checker and confirm the theorems follow from the stated premises without trusting OpenAI or reading 249 pages of prose. That is a real bar. The results span high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, quantum complexity, lattice cryptography, and extremal combinatorics. Two headline items: a construction of a non-sofic group, answering a question Mikhail Gromov posed in 1999, and a counterexample to Connes's rigidity conjecture about group von Neumann algebras, a structural prediction mathematicians have tested for half a century.
What is not confirmed
Formal verification confirms correctness, not importance. Whether these ten results are landmark advances or competent progress on problems specialists had deprioritized is a judgment only the relevant research communities can make, and that review is still pending. The context that earns the caution: in October 2025 OpenAI drew public criticism from mathematicians over overstated claims about solved open problems. The Lean certificates look like the lesson learned, they move the checkable part out of the marketing department, but the significance claims are still OpenAI's own.
The cost line matters most
Under $2,000 of tokens per result is the detail with consequences. If a frontier model can produce a decade-stuck research result for the price of a laptop, the constraint on this kind of work shifts from talent-hours to compute budget and taste in picking problems. That claim deserves the same scrutiny as the proofs, but unlike most capability marketing, it is denominated in something buyable.
Why a build studio cares
Formal verification is the interesting part for working engineers. A model that can produce output alongside a machine-checkable certificate of correctness is a different tool from one that produces plausible text, and proof-carrying output is a pattern that transfers: to generated code, to security properties, to anything with a checker. The math is the demo; the verification workflow is the product.
Next step: read OpenAI's collection and Simon Willison's notes for a grounded outside read. If verified-output patterns are relevant to something you are building, write to us at hello@gattyworks.com.