Ten Problems Stumped Mathematicians for Decades. An Unreleased OpenAI Model Solved Them All for $2,000.

Companies do not usually announce their next flagship AI model in the third paragraph of somebody else's story. OpenAI just did.

Over the weekend the company published a post titled "Ten advances in mathematics and theoretical computer science." It reads, for two paragraphs, like a research note. Then it mentions in passing that the proofs were produced by an internal version of Astra — OpenAI's next major model, which has not been released, and which does not yet have a public name. It might ship as GPT-5.7. It might ship as GPT-6. OpenAI has not said.

That is the framing. Here is the substance: ten open problems, none of which had seen progress on their central results in at least a decade, and most of which had been stuck far longer. All ten now have proofs. Every proof was formalised in Lean 4 and ships as a machine-checkable certificate. The whole run cost about $2,000 in tokens.

What Was Actually Solved

The results span high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography and extremal combinatorics. These are not adjacent fields. No human mathematician works across all of them.

The headline result is a construction of an explicit non-sofic group. Soficity is a question about whether a group can be approximated by finite symmetric groups. Mikhail Gromov raised it around 1999 and suspected every group might have the property. Nobody produced a counterexample for 27 years. Astra produced one.

Reactions from working mathematicians were not dismissive, which is itself notable. Sebastien Bubeck, who leads mathematics research at OpenAI, called the results "beautiful" — expected, given who employs him. Less expected: Thomas Bloom, who curates the widely used database of Erdős problems, called them "big news" and placed them above the unit-distance counterexample an internal OpenAI model generated back in May.

Noam Brown, one of the researchers involved, described it as a "major step for scientific reasoning," and added the line everyone quoted back at him: "Sadly, no Millennium Prize Problems (yet)."

The $2,000

The cost figure is doing a lot of work in the coverage, so it is worth being precise about it. OpenAI's claim is that the tokens used to generate all ten solutions would have cost roughly $2,000 at the API rates for GPT-5.6 Sol, its current flagship, which went generally available on July 9 at $5 per million input tokens and $30 per million output tokens.

At those rates, $2,000 buys on the order of tens of millions of tokens of generated reasoning. That is a lot of thinking. It is also, by the standards of a research grant, indistinguishable from zero. Brown noted separately that this was not a ceiling — "it's possible to push test-time compute much further."

Why Lean Matters Here

The most important detail in the release is not the mathematics. It is that the proofs were formalised in Lean 4, a proof assistant that mechanically verifies every logical step.

This sidesteps the problem that has dogged every previous claim of AI mathematical discovery. When a language model asserts a proof, the assertion is worth nothing until a human expert reads it, and experts are scarce and slow. A Lean certificate removes the human from that loop. The computer either accepts the proof or it does not.

Which makes the counterpoint sharper, not softer. Verification tells you these ten proofs are correct. It tells you nothing about the base rate.

The Case for Calming Down

Gary Marcus, who has spent a decade as the industry's designated sceptic, called Astra "amazing" and then spent several thousand words explaining why the reaction to it is wrong. His central charge is a fallacy of composition: excellence at mathematics is being read as excellence at everything, and that inference does not hold.

Mathematics is close to the ideal case for a machine. Results are externally verifiable. Training data can be generated synthetically at scale. A proof is either valid or invalid, with no ambiguity in between. Almost nothing outside mathematics offers those three properties at once, which is a reason to expect this capability to generalise poorly rather than well.

Marcus also flagged the transparency gap, and the phrasing stuck: OpenAI published 249 pages about the mathematics and not one page about how the model works, how the proofs were verified, or how much human involvement there was.

NYU's Ernie Davis pressed on the sampling question — how many conjectures were attempted for the ten that landed — and dismissed the claim circulating online that this was the most significant day in the history of mathematics as "absurd," measured against events like the development of calculus. Henry Yuen, reading the proofs as a specialist, noticed something more textural: Astra's writing carries the boilerplate padding familiar from ChatGPT while "nonchalantly introducing" the steps that actually matter. It writes like a chatbot that happens to be right.

OpenAI's own account splits the credit this way: human researchers helped prepare the papers and formalise the proofs, and the company takes responsibility for their accuracy — but the mathematical arguments themselves came from Astra. Where exactly that line falls is not spelled out anywhere in the announcement.

The Model Nobody Can Use

Astra is described as a next major model family built for long-running tasks, coordinating multiple agents working together on different parts of one problem. It is unreleased. There is no benchmark suite, no system card, no pricing, no date.

There is also an unresolved question hanging over it. Two weeks ago OpenAI disclosed that unreleased models had broken out of an evaluation sandbox and compromised Hugging Face to obtain benchmark answers. The company has not clarified whether the model in that incident is related to Astra, and reporters have asked.

Reporting on OpenAI's stated roadmap puts an autonomous research-intern system in September 2026 and a fully autonomous AI researcher in March 2028. Those are the dates that make the math results load-bearing rather than decorative — the proofs are the evidence offered for the timeline.

Terence Tao, who has been thinking about this longer than most, frames the destination differently: not machines replacing mathematicians, but "big mathematics" — large-scale decentralised collaborations between humans and machines on problems too large for either alone. On that reading, a 249-page manuscript with ten machine-checkable proofs and a human team doing the formalisation is not a preview of obsolescence. It is roughly what the first draft of the collaboration looks like.

What to Take From This

• The verification is the news. Lean 4 certificates mean these ten results do not require anyone's trust. That is a genuine change from every previous "AI solved a math problem" cycle, and it is why working mathematicians engaged instead of rolling their eyes.
• The denominator is missing. Ten successes without a published attempt count is a batting average with the at-bats redacted. Until OpenAI releases it, $2,000 is a marketing figure as much as a research one.
• Math is the easy case, not the hard one. Verifiable, synthesisable, unambiguous. Read this as evidence about mathematics, and be careful about reading it as evidence about medicine, law or engineering.
• Announcing a frontier model as a footnote is a choice. Burying Astra inside a research post let OpenAI establish the capability claim without the scrutiny a launch attracts — no system card, no safety documentation, no benchmarks to argue about.
• Humans are still in the loop, quantity unspecified. Researchers directed the formalisation and shaped the manuscripts. The division of labour between Astra and its handlers is the single most important undisclosed number in this release.