OpenAI says an unreleased model solved ten math problems that stood for decades. None of it has been peer-reviewed yet.

[OpenAI](#/company/openai) said on August 2 that an internal version of Astra, the model family it describes as its next major line, produced new results on 10 problems in mathematics and theoretical computer science that had stood open for at least a decade — in one case, since 1999. The company published a 249-page manuscript collection alongside the claim, along with machine-checkable Lean 4 proof certificates for all ten results, posted to GitHub under an Apache 2.0 license.

Astra is not a product. OpenAI describes it as a model family built to coordinate multiple agents over extended periods on especially hard problems, still in internal testing, with no release date, pricing, or even a settled decision on whether it ships as part of a future GPT line or as something separate. What the company published is a research claim about an unreleased system, not an announcement a reader can go try.

OpenAI researcher Noam Brown, who worked on the results, was candid about the limits alongside the wins: Astra tried and failed to crack other major open problems during the same push, and he described it as "possible to push test-time compute much further" — company shorthand for letting a model spend more inference time reasoning before answering, which is the resource the $2,000 figure is actually measuring. That's a researcher describing a work in progress, not a finished capability.

What the ten problems actually are

OpenAI's headline result is what it describes as the first explicit construction of a non-sofic group, closing a question in group theory it says has stood since Mikhail Gromov introduced the concept of soficity in 1999 — a characterization disputed below. Astra also produced a disproof of Connes's rigidity conjecture, posed in 1980, and a resolution of Ehrhart's volume conjecture. Two more results close specific instances of Erdős problems — the open-problem list maintained by mathematician Thomas Bloom — including Erdős problem 183 on multicolor Ramsey numbers. The remaining results span high-dimensional sphere packing, binary and spherical coding theory, arithmetic circuit complexity, quantum parallel repetition, and the closest vector problem in lattice cryptography.

The verification is the part that doesn't depend on trusting OpenAI. Each argument was written up as a manuscript, then formalized into Lean 4 — a proof assistant that checks every logical step by machine rather than by a human referee. A “sorry” in Lean is a placeholder for a step the author hasn't actually proven; the published certificates carry a sorry count of zero across all ten results, meaning the compiler accepted every step with no gaps left for a human to fill in later. That's a real, independently checkable claim — anyone can clone the repository and run the checker themselves. It is not, on its own, a claim that the results are correct in a deeper mathematical sense the field will find interesting, or that the framing of what each result means is right; that judgment still runs through peer review.

What the $2,000 doesn't cover

That distinction matters because the $2,000 figure is the number every headline repeated, and it describes only the cheapest part of the process. Getting from a model's raw argument to a machine-checked, publishable proof took human mathematicians and engineers formalizing the reasoning in Lean — work OpenAI doesn't put a price on. The marginal token cost of a breakthrough and the total cost of producing one are different numbers, and this story only has the first.

What's actually established, and what isn't

Bloom's read matters because he's the closest thing this story has to an independent referee — he didn't work for OpenAI on this, and his own site is the public ledger several of these problems came from. His verdict wasn't a rebuttal. “Big news,” he called it, ranking the results ahead of a narrower counterexample an OpenAI model found in May to an Erdős problem on unit distances. But he pushed back hard on the framing that this means AI is now doing mathematics unassisted, pointing out that every one of these proofs stands on more than a hundred years of human-built theory, and that turning a model's raw argument into a checkable result still took human mathematicians.

A separate dispute: whose result is this?

Two named mathematicians have gone further than Bloom's framing objection and disputed the results themselves. Steven Miller, at Yeshiva University, says the sphere-packing proof incorporates an argument that originated in his own 2016 paper without proper attribution. “They are running roughshod over the work of others who came before them in a deliberate way,” Miller said. “It seems completely systematic to me, and it points to research misconduct.”

Francesco Fournier-Facio, a group theorist at the University of Cambridge, raised a related but distinct problem with the headline non-sofic-group result: he says it combines ideas from his own 2016 and 2019 papers without adequate framing, and that those papers show mathematicians hadn't actually reached a dead end on the problem — directly undercutting OpenAI's own claim that it had “seen no progress on the main result for at least a decade.” An OpenAI spokesperson responded: “We take responsibility for the correctness of these results and are meeting the same standards generally expected of human mathematicians. We plan to make small updates [to the paper] this week, consistent with standard academic practice.” That statement defends the math as correct; it does not dispute that the citations were incomplete.

This isn't OpenAI's first claimed math result this year — an OpenAI model produced a counterexample to an Erdős problem on unit distances in May, the result Bloom used as this week's point of comparison. What's changed since then is scale and the standard of evidence: ten results instead of one, each backed by a machine-checked certificate rather than a claim resting on the company's word alone. Whether that pattern holds on the next unreleased model, and whether any of the ten survives contact with formal peer review, are the two questions this story leaves open. For a field where a wrong proof has traditionally cost a referee months to catch, a machine-checked certificate changes what "claimed result" even means — it doesn't remove the need for peer review, but it does mean the mechanical part of checking the work no longer has to wait for a human to do it by hand.

Five days later, a different kind of threshold

On August 7, OpenAI disclosed a second, unrelated fact about the same model family: it had paused internal work on parts of Astra after determining the system had reached what the company calls a "critical cybersecurity threshold" under its Preparedness Framework — the ability to independently identify and carry out cyberattacks against well-defended, real-world systems, not a benchmark or a simulation. It is the first OpenAI model to trip that specific bar. The company says it has added isolated testing, restricted access, and real-time monitoring around the affected capabilities, and is working with government agencies and outside safety organizations to stress-test them further.

OpenAI hasn't said whether the math work and the paused cybersecurity work involve the same underlying checkpoint, and this desk could not independently confirm that they do. What's clear is that both disclosures happened inside the same eight-day span about a model OpenAI itself still describes as internal and unreleased — which means the more consequential Astra story this month may not be the math proofs at all, but a frontier lab's own internal safety framework triggering a real capability pause for the first time, rather than staying a paper process.