# OpenAI Astra solves ten open maths problems in a single day

> OpenAI Astra cracked ten maths problems open for up to 30 years, with machine-checked Lean proofs, for about 2,000 dollars in tokens.

- Published: 2026-08-06
- Topic: [AI](https://futuretechnologyhq.com/topic/ai/)
- URL: https://futuretechnologyhq.com/article/openai-astra-solves-open-math-problems/
- Publisher: [Future Technology](https://futuretechnologyhq.com/), free to read. Quote as "Future Technology" with a link to the article.

## Key points

- An internal version of OpenAI Astra produced fresh results for ten problems in maths and theoretical computer science, some open for nearly 30 years.
- Every solution ships with a Lean 4 certificate, so the proofs are machine-checked rather than merely plausible.
- OpenAI put the token cost of all ten solutions at roughly 2,000 dollars, and Astra itself is still unreleased.

## Article

OpenAI Astra just did something that is hard to wave away. An internal version of the company's next major model produced new results for ten problems in maths and theoretical computer science that had been open for at least a decade, and a few for close to thirty years. Ten problems that had quietly resisted human effort fell in a single day.

This is not the usual "AI is good at maths homework" story. The problems span serious territory: group theory, von Neumann algebras, high-dimensional geometry, quantum complexity, lattice cryptography and extremal combinatorics. The headline result is an explicit construction of a non-sofic group, a question left hanging since Mikhail Gromov floated the idea of soficity back in 1999. Mathematicians had been circling that one for a quarter of a century.

## Why the Lean proofs matter

Here is the part that separates this from the endless run of benchmark claims. Every one of the ten solutions comes with a Lean 4 certificate. Lean is a proof assistant that forces every step of an argument to be spelled out in machine-readable detail, then checks it. If the proof has a gap, Lean refuses to sign off. So these are not confident-sounding paragraphs that might contain a subtle error. They are verified resolutions, checked by software that does not care how fluent the model sounds.

OpenAI published a 249-page manuscript collection alongside the model's own reasoning walkthroughs, so the working is out in the open for others to poke at. That transparency is doing a lot of heavy lifting for the credibility of the claim.

## The cost is the quiet shock

OpenAI put the token cost for all ten solutions at roughly 2,000 dollars at its current API rates. For context, that is less than many labs spend on a single research trip. Astra itself remains unreleased, so what we are seeing is a teaser rather than a product, but the economics are the thing worth sitting with. If genuinely novel research maths can be produced at that price, the bottleneck stops being raw capability and starts being knowing which questions to point the model at.

It is worth staying level-headed. Ten problems is not the whole of mathematics, and picking tractable open questions is its own skill. Peer review by human mathematicians will still matter, and the Lean certificates make that review faster rather than optional. But the direction of travel is clear, and it fits the wider pattern where the frontier labs keep quietly moving the line on what counts as machine-assisted discovery. It also lands in the same week that AI systems keep [testing their own containment](https://futuretechnologyhq.com/article/huggingface-breach-openai-rogue-agent/), a reminder that capability and control are advancing together.

For anyone tracking where the models are actually heading, this is a bigger marker than another leaderboard win. The maths does not lie, and this time it has been checked. If you want the same kind of watching brief on the [next wave of frontier models](https://futuretechnologyhq.com/article/gemini-3-6-flash-family/) and the [safety alliances forming around them](https://futuretechnologyhq.com/article/open-secure-ai-alliance/), that is exactly what we do here every morning.

## Key takeaways

- OpenAI's Astra model has solved ten open problems in maths and theoretical computer science that had been unresolved for at least a decade.
- The problems span serious territory, including group theory, von Neumann algebras, and quantum complexity.
- Each of the ten solutions comes with a Lean 4 certificate, which verifies the proof through machine-readable detail.
- The token cost for all ten solutions is roughly 2,000 dollars at OpenAI's current API rates.
- The breakthrough highlights the potential for AI to assist in genuine novel research maths, rather than just solving tractable problems.
- The solutions will be subject to peer review by human mathematicians, but the Lean certificates will make the review process faster.

## Sources

- [siliconangle.com](https://siliconangle.com/2026/08/02/openais-astra-solves-10-long-open-math-problems-publishes-proofs/)
- [techtimes.com](https://www.techtimes.com/articles/322710/20260802/openais-astra-solves-ten-decade-old-math-problems-machine-checkable-lean-proofs.htm)