On August 1, OpenAI dropped an unusual post on its official blog: an internal version of Astra--its next major model--produced new results for ten open problems in mathematics and theoretical computer science, each open for at least a decade (most far longer). Not a tweet claiming a breakthrough--the proofs were formalized in Lean and open-sourced on GitHub for anyone to check.
What happened: ten open problems, roughly $2,000 in compute
The ten problems span high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography, and extremal combinatorics--the hard bones of pure math and theoretical CS. A few examples: new upper bounds on high-dimensional sphere packing down to the Cohn-Elkies threshold; a construction establishing the existence of non-sofic groups (a central open question in group theory); and a disproof of Connes's rigidity conjecture.
OpenAI says the total tokens needed to find these solutions would cost roughly $2,000 at Sol API rates. The arguments were prepared into manuscripts by humans working with the model, then the model formalized each argument into a Lean certificate. Lean is a proof assistant--it does not care whether your argument sounds right, only whether it machine-checks. OpenAI released the ten Lean certificates at github.com/openai/ten-proofs, along with the paper and the model's reasoning walkthroughs.
Why this one is credible: from "just trust us" to machine-checkable
This is the key. Last October, then-OpenAI VP Kevin Weil posted on X that GPT-5 had "solved 10 Erdős problems." Within days, mathematician Thomas Bloom (maintainer of erdosproblems.com) called it a "dramatic misrepresentation"--GPT-5 had merely surfaced existing references, not solved the problems itself. TechCrunch covered it as "OpenAI's embarrassing math."
In May, OpenAI used an unreleased model to genuinely disprove the Erdős unit-distance conjecture (open for nearly 80 years)--this one was real, and Thomas Bloom himself contributed to the follow-up paper. With these ten results in August, OpenAI formalized the proofs in Lean and open-sourced them. You do not have to take the lab's word for it--clone the repo and run Lean yourself. The trust infrastructure for AI doing math has shifted from claim to verification.
Who is Astra: the unreleased next major model
OpenAI calls Astra "our next major model." It is not yet publicly released; this amounts to a flex ahead of launch--using "can solve open math problems" to benchmark the next generation. For everyday users, Astra is out of reach for now; for the industry, it signals that frontier models are starting to treat long-horizon reasoning with verifiable output as the core selling point, not raw benchmark numbers.
Takeaway
The story is not the headline "AI solved math problems"--it is the practice of open-sourcing machine-checkable proofs. It moves the credibility of AI math discoveries from "do you trust the lab" to "does it run." Of course, the pure-math value of these ten results, and whether the model truly "understands" versus searches then formalizes, still needs peer review in each field--Lean verifies logical consistency, not importance or optimality.
But the direction is right. When the output of AI can be independently verified, the debate over "can AI do math" turns from philosophy into engineering.
References
- OpenAI official blog (Ten advances in mathematics): https://openai.com/index/ten-advances-in-mathematics
- Lean proofs open-source repo: https://github.com/openai/ten-proofs
- Paper PDF: https://cdn.openai.com/pdf/ten-proofs-oai.pdf
- 2025 GPT-5 Erdős flop coverage (TechCrunch): https://techcrunch.com/2025/10/19/openais-embarrassing-math