OpenAI's Astra solved ten maths problems nobody could crack. Here's what actually happened.

A model that nobody can use yet just did something mathematicians couldn't

On 1 August 2026, OpenAI published a research paper with a quiet but extraordinary title: Ten advances in mathematics and theoretical computer science. An internal version of Astra, described as OpenAI's "next major model," had produced solutions to ten problems that had seen no progress for at least a decade, and in most cases much longer. One of those problems had been open since 1999. Another for nearly thirty years.

This is not a benchmark score. It is not a leaderboard ranking. The Astra results are not benchmark scores. They are new mathematical results: constructions, counterexamples, and bounds that professional mathematicians had not been able to produce, released with machine-checkable proofs in the Lean theorem prover.

To understand why that matters, and what the rest of us should take from it, you need to know two things: what a reasoning model actually does differently, and what it means to verify a proof by machine.

What makes a "reasoning model" different from a standard AI

When you ask a typical AI assistant a question, it generates an answer almost immediately. It is doing something like very fast pattern-matching: given the words you typed, what sequence of words most plausibly comes next? That works remarkably well for summarising emails, writing a first draft, or answering questions that have clear answers in its training data. But it has a real weakness on problems where you genuinely need to work through intermediate steps, notice a contradiction, backtrack, and try again.

In September 2024, OpenAI previewed a model that did something strange: it paused, thought, and only then answered. On AIME 2024, a hard high-school maths contest, the previous flagship (GPT-4o) scored about 12%. The thinking model scored about 74%. Nothing about the training data changed that much. What changed was when the compute was spent: at answer time, not just at training time. That shift created a new category: the reasoning model.

Here is the key idea. A reasoning model generates what are sometimes called "thinking tokens" before it commits to a reply. These are internal working notes the model writes to itself. The model generates a long internal thinking trace, a sequence of reasoning steps, exploratory calculations, self-corrections, and backtracking, before producing the final response you see. The thinking trace is usually hidden (though some interfaces show it). The model might explore three approaches to a maths problem, notice that the first two lead to contradictions, and arrive at the correct answer via the third. A standard AI commits to the first plausible path.

The trade-off is real. Where a standard model might respond in two to four seconds, a reasoning model might take 20 to 60 seconds for the same prompt. On hard problems that's a fair trade. On simple ones, it's overkill. Asking a reasoning model to write a thank-you note is like hiring a chess grandmaster to play noughts and crosses.

What Astra actually did

OpenAI describes Astra as a model family built to run long tasks by coordinating multiple agents over extended periods, an extension of the test-time reasoning work associated with research scientist Noam Brown. In plain terms: rather than thinking for a minute, Astra can think for much longer, spinning up multiple sub-processes that check each other's work.

The results run across group theory, high-dimensional geometry, coding theory, quantum complexity, lattice cryptography, and extremal combinatorics. The headline result is probably the most striking. The first-ever explicit construction of a non-sofic group resolved a central question in group theory that had stood since Mikhail Gromov introduced the concept of soficity in 1999. Astra also disproved Connes's rigidity conjecture on von Neumann algebras and resolved three problems from Paul Erdős's famous catalogue.

If those names mean nothing to you, that is fine. What matters is the scale of the ambition. The list spans geometry, group theory, and quantum complexity: territory where progress usually comes from specialists spending years on a single proof. The problems are not obscure curiosities either. They sit at the foundations of how mathematicians think about symmetry, space, and computation.

The thing that makes this claim unusually trustworthy

AI labs make big announcements regularly. Most of them rest on benchmark scores, which are easy to game and hard for outsiders to verify. This one is different because of a tool called Lean.

These solutions are not incremental progress: they are genuine resolutions, verified with Lean. Lean is a programming language and proof assistant that forces every step of a mathematical argument to be spelled out in machine-readable detail. Think of it as a very strict grammar-checker for logical arguments. You cannot hand-wave. If any step is missing or wrong, Lean rejects the proof outright.

OpenAI released a 249-page manuscript and Lean 4 proof certificates on GitHub under an Apache 2.0 licence; the repository's "sorry" count stands at zero, indicating that every step across all ten formalised proofs is fully verified. A "sorry" is what Lean calls a placeholder when a human has written a proof but not yet filled in a step. Zero sorries means zero gaps.

That does not make the claims settled science overnight. The careful conclusion is not "AI has unquestionably settled ten famous problems." It is that OpenAI has published ten concrete mathematical claims with substantially more verification material than a benchmark score or a lab announcement normally provides. Independent mathematicians are still working through them. Researchers are still digesting the claims, with reactions ranging from "big news" to "sadly, no Millennium Prize Problems (yet)."

For what it's worth, Thomas Bloom, who maintains the erdosproblems.com database, called the Astra results "big news" and rated them ahead of the Erdős unit distance counterexample an internal OpenAI model produced in May, a paper he helped verify. That earlier result was considered significant enough that Fields Medalist Tim Gowers said he would recommend it for publication in Annals of Mathematics without hesitation. The bar being cleared here is not trivial.

The $2,000 number is the most interesting part

The cost of producing new results on ten long-standing mathematical problems just fell to $2,000, according to OpenAI, which says its Astra model generated machine-checkable proofs for questions that had resisted human progress for decades.

Pause on that for a moment. A single postdoctoral researcher costs more than $2,000 a week. The kinds of results described here might take a career. The economic implication is not that mathematicians are suddenly obsolete. It is that the cost structure of intellectual discovery has shifted in a way that has no real precedent.

Human researchers turned the model's output into publishable papers, though OpenAI said the mathematical arguments themselves came from Astra. The collaboration looks a lot like what happens when a scientist uses a powerful telescope: the instrument does something no unaided human could do, but you still need someone who knows what they are looking at.

There is also a deeper point about what the cost figure reveals about reasoning models in general. If an internal Astra cracked decades-old problems for roughly $2,000 in tokens, the ceiling for test-time reasoning is nowhere in sight. The old way to make AI more capable was to train a bigger model on more data. That is slow and expensive. The newer approach, more thinking time, more agents checking each other's work, is comparatively cheap and getting cheaper.

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What this does and doesn't mean

It is worth being careful here, because the hype around this story will be loud. A few things worth keeping in mind.

First, Astra is not available to the public yet. OpenAI has not said when Astra will be released publicly, describing it only as its "next major model." Everything announced this week came from an internal test.

Second, the problems solved, while genuine and significant, are not the most famous unsolved problems in mathematics. Solving problems that resisted mathematicians is a major capability signal. It still does not meet the classic broad definition of superintelligence. The useful concept in between is jagged, domain-superhuman intelligence. Astra is extraordinarily good at formal mathematical reasoning. That does not mean it is extraordinarily good at everything.

Third, the Lean certificates make the results verifiable, but verification and independent confirmation by domain experts are different things. The August ten-proof announcement still needs independent review. OpenAI released a 249-page manuscript collection, model-written reasoning walkthroughs, and public Lean certificates for all ten results. The material is there for other mathematicians to scrutinise, and that scrutiny is ongoing.

What does seem clear is that something has changed. The standard picture of AI as a tool for drafting text and summarising documents is already out of date. A system can now sit with a hard problem for an extended period, explore approaches systematically, reject the ones that fail, and arrive at something genuinely new. That is a different kind of capability, and it is reasonable to think carefully about what it means for any field where the core work is solving hard problems.

Mathematics happened to be where the demonstration landed this week. It will not be the last field where the question comes up.