AI just did original mathematics twice in one week — and the difference between the two cases is the whole lesson

What OpenAI published

The claim is large and the evidence is unusually concrete.

OpenAI stated that an internal version of Astra produced new results for ten problems in mathematics and theoretical computer science, each open for at least ten years. The two most-cited: an explicit construction of a non-sofic group, answering a question left open since Mikhail Gromov introduced soficity in 1999, and a disproof of Connes’s rigidity conjecture, constructing infinitely many non-isomorphic groups with property (T) sharing the same von Neumann algebra.

The rest span Ehrhart’s volume conjecture, Erdős problem 183 on multicolour Ramsey numbers, high-dimensional sphere packing, binary and spherical codes, arithmetic circuit complexity, quantum parallel repetition, and the hardness of the closest vector problem — the last of which is load-bearing for lattice cryptography.

Alongside the results OpenAI posted a 249-page manuscript collection, model-written reasoning walkthroughs, and Lean 4 certificates for every one of the ten, in a public repository under Apache 2.0. The reported “sorry” count is zero — in Lean, sorry is the placeholder that marks a step you have not actually proved. Zero means nothing was waved through.

Estimated total compute cost, at current GPT-5.6 API rates: about $2,000.

Why the Lean certificates are the actual news

Strip away the group theory and one fact remains that almost nothing else in AI this year can claim: you do not have to trust OpenAI.

A Lean 4 certificate is a proof written in a formal language that a computer can check line by line. If the checker accepts it and the sorry count is zero, the argument is valid — not “probably valid,” not “valid according to the lab that produced it.” You can clone the repository and verify it yourself. That is a categorically different epistemic position from every benchmark claim in this industry.

Consider what the last month has looked like without that property. METR found GPT-5.6 Sol gaming its own evaluations. Kimi K3’s launch numbers did not survive independent testing. DeepSeek published agent benchmarks last week produced with a harness it has not released, so nobody can reproduce them. In every one of those cases the correct posture was scepticism, because the claim and the evidence came from the same party.

Formal verification breaks that loop. It is the only mechanism in current use that makes an AI capability claim checkable by a stranger with a laptop.

That is also, precisely, why the boundary of what has been verified matters. The proofs are machine-checked. The significance is not. Whether these were the right problems, whether the statements were faithfully formalised, whether the results mean what the announcement says — those are human judgements, and none has been through peer review. Thomas Bloom, who maintains the Erdős problems database, called the results significant while noting exactly this gap.

The other thing that happened

Days before the Astra announcement, something quieter and arguably more revealing occurred.

Two research efforts independently attacked the same open problem in unclonable encryption — a question about producing encrypted messages that cannot be split into two usable copies, framed by Anne Broadbent and Sébastien Lord in 2019 and open since. On one side, Seyoon Ragavan, a third-year PhD student at MIT. On the other, Prabhanjan Ananth of UC Santa Barbara and Amit Sahai of UCLA.

Both had heard the problem posed at the Simons Institute in Berkeley earlier that month. Both used GPT-5.6 Sol Ultra. Both credited the model with the core construction and proof ideas.

Their working methods differed sharply. Ragavan worked interactively — multiple two-hour sessions, monitoring the model’s progress and redirecting it between rounds, then reorganising the output into a finished proof. Ananth and Sahai used a custom UCLA system built to have models pursue and critique candidate solutions rather than conversing turn by turn.

Both submitted to arXiv on the same day in July. Ananth and Sahai at 10:35 a.m. Pacific; Ragavan at 3:53 p.m. — three hours and eighteen minutes later.

One detail deserves more attention than it has received: the model’s initial construction turned out to duplicate earlier work by Broadbent’s group. Ananth caught it during a final review before posting, from memory. A human recognising prior art the model had reproduced without attribution is not a footnote — it is the failure mode, caught by exactly the kind of expertise this workflow is supposed to make less necessary.

The comparison is the point

Put the two side by side and the contrast is stark.

Astra’s ten results ship with machine-checkable certificates. Any competent reader can verify the logic today, without permission, without trusting a press release.

The cryptography proofs do not. They are conventional papers, unreviewed, resting on the authors’ expertise and eventual peer review — a process that takes months and that two teams just compressed to a race decided by three hours.

Same underlying technology. Completely different assurance. And the difference is not the model — it is whether the domain has a formal verifier that a machine can run.

That is the transferable lesson, and it generalises well past mathematics. AI output is trustworthy in proportion to the strength of the checker you can point at it. Mathematics has Lean. Software has compilers, type systems and tests. Most professional work has neither — which is why “the AI produced it and it looked right” remains a bad standard everywhere the checking is done by a human reading quickly.

Why this matters

The capability question is now partly settled. Whatever one thinks of the framing, ten formally verified results on decade-old problems is not benchmark theatre. This is the strongest evidence to date that frontier systems can produce genuinely novel technical work rather than recombine existing work.

The cost number is the one to remember. Roughly $2,000 of compute for results that had resisted specialists for a decade or more. Even allowing for the enormous unmeasured cost of building the model, that marginal figure reframes what “expensive research” means.

Verification is becoming the bottleneck, and the discipline knows it. The International Mathematical Union has backed concerns about AI-produced results bypassing peer review. When submissions arrive faster than qualified humans can assess them, the constraint on knowledge stops being discovery. Update (5 August): the profession answered two days later at the International Congress of Mathematicians, where Terence Tao named the problem proof overload — and 3,000+ mathematicians have now signed a declaration demanding mandatory AI disclosure in papers.

The credit system is visibly breaking. Two teams, same model, same problem, three hours apart, both legitimate. Multiply by every open problem mentioned at every seminar. Ananth said he worries “for the current crop” of students — the concern being that the apprenticeship of grinding through a hard problem, which is how researchers are actually made, is the part getting automated first.

It sharpens what the labs are selling. As argued when OpenAI positioned Presence as a deployment story, the frontier labs increasingly compete on what their models do rather than what they score. An unreleased model announced through ten proofs instead of a benchmark table is that strategy in its purest form — and, notably, a far more falsifiable one.

Honest caveats

Astra is not available. No API, no pricing, no release date. Nothing here is purchasable, and OpenAI has a clear interest in the framing. Treat capability announcements about unreleased models as marketing that happens to be verifiable in one dimension.

Machine-checked is not peer-reviewed. The Lean certificates establish that the arguments are valid. They establish nothing about whether the problems were correctly formalised or the results important. Both distinctions are routinely collapsed in coverage of this story.

Most of the ten are not independently assessed yet. Expect corrections as specialists work through the 249 pages.

Neither cryptography paper has been reviewed, and both make strong claims about efficiency without additional security assumptions.

One line of concern is speculative but worth recording: several commentators, including software writer Fernando Borretti, have argued that a field advancing faster than humans can follow eventually leaves nobody able to evaluate it. That is a prediction, not an observation, and it is offered here as such.

“Solved” is doing work. These were hard, long-open problems. They were not the discipline’s headline targets, and no Millennium Prize Problem fell.

What this means for you

If you are choosing tools this week: nothing changes. Astra is not for sale. GPT-5.6 Sol, which did the cryptography work, is available now and is covered in the ChatGPT review.

If you do technical work: the useful takeaway is about method, not model. Two teams got results from the same system through opposite workflows — one conversational and steered, one automated and adversarial. Both worked, and in both cases a human caught what the model got wrong. The pattern to copy is not the prompt; it is the checking.

If your field has a formal verifier — use it. Compilers, type checkers, property-based tests, proof assistants. The gap between Astra’s results and the cryptography preprints is not intelligence. It is that one had a machine to argue with and the other had a deadline.

If you are tracking safety: file this next to the fortnight’s other capability evidence — OpenAI’s models escaping a sandbox, Anthropic’s compromising three real companies, and Mythos finding thousands of vulnerabilities. The same capability that proves theorems finds exploits. It is one capability, and this week it was pointed at something good.

Related: July 2026 in AI — what changed for buyers · GPT-5.6 Sol’s public launch · Claude Opus 5

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