Future TechnologyOpenAI's Astra Solved Ten Open Maths Problems, and the Proofs Are Machine-Checkable
Key takeaways
- Astra produced solutions to ten problems open for a decade or more, for a reported compute bill of around $2,000
- The proofs ship as Lean 4 certificates on GitHub under Apache 2.0, with a
sorrycount of zero - Verification shifts the trust question from OpenAI to the Lean kernel, which is small, open and long-scrutinised
- Astra itself is unreleased, so the process cannot be reproduced outside OpenAI, only the output checked
OpenAI says an internal model called Astra, not yet publicly released, produced solutions to ten problems in mathematics and theoretical computer science that had each been open for ten years or more. The reported compute bill was around $2,000.
What the OpenAI Astra math problems actually cover
The headline result is an explicit construction of a non-sofic group, which settles a question left hanging since Mikhail Gromov defined soficity in 1999. The rest span high-dimensional sphere packing, binary and spherical codes, arithmetic circuit complexity, quantum parallel repetition and the hardness of the closest vector problem. There is also a counterexample in extremal graph theory that knocked over two Erdos problems at once.
None of these are famous open problems in the Riemann hypothesis sense. They are the kind of question a specialist would recognise and a generalist would not, which is roughly the level at which new mathematics actually gets made.
Why the Lean proofs matter more than the results
Every previous claim of this shape ran into the same wall. You had to take the lab's word for it. This one comes with a 249-page manuscript and Lean 4 proof certificates published on GitHub under Apache 2.0. The repository's sorry count is zero, meaning no step is parked as an unproven placeholder.
The workflow ran in four stages: model insight, then human mathematicians writing the argument up readably, then translation into Lean's formal syntax, then the Lean kernel checking every line.
That last stage is the whole point. You do not have to trust OpenAI. You have to trust the Lean kernel, which is small, open source and has been picked over by mathematicians for years. That is a genuinely different position from "the model said so", and it is the bar every future announcement of this kind will now be measured against.
The catch
Astra is not released. Nobody outside OpenAI can reproduce the process, only check the output. A verified proof tells you the theorem is true; it does not tell you how much human steering went into getting there, or whether the same model would do it again.
What to watch next
This lands while the two largest labs circle each other on funding and public markets, which we covered in the Anthropic and OpenAI IPO race. Formal verification is quietly becoming the credibility currency in that fight, in much the same way that Alibaba's Qwen UI agent gets judged on reproducible benchmarks rather than demo reels.
The number to watch is not ten problems. It is the sorry count. If the next announcement ships without certificates attached, treat it as a press release.