OpenAI published ten new results in mathematics and theoretical computer science on August 1, 2026, crediting the core arguments to an internal version of Astra, its next major model family. Every proof ships with a machine-checkable Lean certificate, and OpenAI puts the inference cost of finding all ten at roughly $2,000 at Sol API rates. Astra itself remains unreleased.
TL;DR
- OpenAI’s post, “Ten advances in mathematics and theoretical computer science”, went live on August 1, 2026, alongside a 249-page manuscript and a public GitHub repo of Lean 4 proof certificates.
- Roughly half the ten are outright resolutions of named conjectures. The other half are improved bounds, which is a real contribution but not the same thing as “solved”.
- The headline result is the construction of an infinite, finitely presented non-sofic group, closing a question open since Gromov introduced soficity in 1999.
- The $2,000 figure covers inference tokens only. It does not include human curation, verification, or the cost of training Astra.
- No Millennium Prize Problem was solved. Noam Brown confirmed it directly: “Sadly no Millennium Prize problems (yet).”
I’ve watched a lot of AI-does-science announcements turn to vapour the moment anyone asked for the working. This one is different in a way that matters: OpenAI released the artifacts. A 249-page manuscript, reconstructed discovery notes, and Lean 4 certificates for all ten claims, sitting in a public repository. Lean is a proof assistant, so any expert can rerun the checker and see for themselves.
That said, the framing around this release is doing some work, and it’s worth pulling apart before you decide how impressed to be. Read on.
What did OpenAI’s Astra actually solve?
Astra contributed results across eight areas: high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography, and extremal combinatorics. OpenAI says each problem had seen no progress on its main result for at least a decade, and in several cases far longer.
Here is the full list, with what each item actually establishes:
| Problem | What Astra produced | Type |
|---|---|---|
| Non-sofic groups (open since 1999) | Construction of an infinite, finitely presented non-sofic group | Resolution |
| Connes’s rigidity conjecture | Disproof: some groups are not uniquely determined by their von Neumann algebras | Resolution |
| Ehrhart’s volume conjecture | Maximum volume determined in every dimension | Resolution |
| Erdős problem 183 (triangle Ramsey) | Superexponential lower bound; growth is k^Θ(k^2) | Resolution of growth scale |
| Erdős problems 146 and 180 | Compactness and degeneracy results in extremal graph theory | Resolution |
| High-dimensional sphere packing | Upper bounds on density down to the Cohn-Elkies threshold | Improved bound |
| Binary and spherical codes | Exponentially improved asymptotic upper bounds on maximum code size | Improved bound |
| Permanent, arithmetic circuits | New lower bounds, including a formula bound of order n^4 / log n | Improved bound |
| Quantum parallel repetition | Exponential parallel repetition for two-player, one-round entangled games below value 1 | New theorem, partial scope |
| Closest Vector Problem | Deterministic reduction: Euclidean GapCVP hard to approximate within any fixed polynomial factor | Improved hardness result |
Thomas Bloom, a mathematician at the University of Manchester, called the results “big news” and a “significant step”. Three separate ErdÅ‘s problems falling in one release is not a small thing.
Were all ten problems really “solved”?
No, and this is the part the headlines flatten. Roughly five of the ten are clean resolutions of named conjectures. The rest are improvements to known bounds, which is genuine research output but a different category of claim.
The sphere-packing result is a good example. It pushes the upper bound on density down to the Cohn-Elkies linear-programming threshold, which tells you the ceiling of that particular method. It does not give exact optimal densities, and it does not produce a new packing construction. An audit by Kingy.ai makes the same point across several items: the permanent result does not prove VP is different from VNP, let alone P against NP, and the quantum repetition theorem does not extend to multiparty or multiround games.
The lattice cryptography item deserves its own warning, because it is the one most likely to be misread. Tightening the hardness of approximating the Closest Vector Problem is not an attack on deployed post-quantum schemes. If you run ML-KEM or ML-DSA, nothing here changes your position. The result refines worst-case hardness, which is the opposite of a break.
What does the $2,000 figure actually cover?
Inference tokens, priced at Sol API rates. That’s it. It works out to about $200 per result, and it is an accounting statement about the compute spent generating proofs, not the cost of the research.
It excludes training Astra, the human researchers who curated and structured the arguments, and the verification effort that will follow over the coming years. Treat it as a demonstration that test-time compute has become cheap enough to let a strong model think for a very long time on one problem. That is the actual claim, and it is interesting enough without the sticker price doing the talking.
How much of this did the AI do on its own?
Less than “AI solves maths” implies, and more than sceptics will want to admit. OpenAI describes a three-stage pipeline: the internal Astra model generated the raw mathematical arguments, human researchers then used the same model to turn those arguments into readable manuscripts, and the manuscripts were fed back through the model to produce the Lean formalizations.
So this is not an autonomous mathematician working alone in a room. It’s a hybrid loop with real human curation in the middle. The novel part is the third stage: because the output is machine-checked, the usual objection to AI-generated proofs, that nobody can tell whether the reasoning is sound, doesn’t apply in the same way. The Lean certificate either compiles or it doesn’t.
Here’s the catch worth keeping in mind. All ten results come from one lab, one unreleased model, and one coordinated manuscript. Formal verification confirms the proofs are internally valid; it does not confirm that the statements proved are the ones the community cares about, or that the framing around them is fair. That takes human specialists reading the paper, and that process is only starting.
What did Astra fail to solve?
The Millennium Prize Problems, all of them. Noam Brown, co-author of Astra’s reasoning technology, said so plainly on X: “Sadly no Millennium Prize problems (yet).” He added that OpenAI tried other major problems without success.
Those seven questions were named by the Clay Mathematics Institute in 2000 and carry a million dollars each. They include the Riemann Hypothesis, P versus NP, and Navier-Stokes existence and smoothness. Only one has ever been settled: Grigori Perelman’s proof of the Poincaré conjecture. The other six are still open, and Astra did not change that.
There’s also a counting quirk. In May 2026, OpenAI shared an AI-generated disproof of the ErdÅ‘s unit-distance conjecture, later confirmed to have come from the same Astra line during evaluation. Count that and the tally is eleven, though OpenAI packaged ten as the official set.
When is Astra being released?
No date has been announced, and OpenAI has not said what it will be called. According to The Information, the company has not decided whether Astra ships as GPT-6 or as a variant inside the GPT-5 lineup.
Sam Altman previewed the model privately to US senators and senior administration officials in Washington on July 29, 2026, with an emphasis on multi-agent collaboration across long-running tasks. Reporting suggests Astra is likely to be the first system submitted under the administration’s voluntary pre-release review framework for frontier models, which allows up to 30 days of government access before broad release and explicitly rules out mandatory licensing.
Who gets credit for an AI-generated proof?
OpenAI’s position is unusually direct, and I think it’s the right one. The company states that attribution should honestly reflect how a result was produced, and that “claiming human authorship for a proof generated entirely by an AI system would misrepresent both the system’s contribution and the nature of genuine human intellectual work.”
That tracks where policy is already heading. US copyright guidance treats fully AI-generated output as lacking human authorship, while protecting human-authored expression layered on top, and journals are moving toward mandatory disclosure of AI involvement. A lab volunteering that its model, not its researchers, produced the argument is a useful precedent.
Should you care if you’re not a mathematician?
It depends on what you took from the headline. If you read it as “AI has solved mathematics”, no, and that reading will keep letting you down. Half these results are bounds, the Millennium problems are untouched, and a human team shaped every manuscript.
The part worth your attention is the pairing of long-horizon reasoning with formal verification. A model that can think for hours on one problem is only useful if you can check the output, and Lean is what makes checking possible here. That combination generalises well beyond pure maths. Anywhere you can express correctness formally, which includes a good deal of software, the same shape applies: let the model run long, then verify mechanically rather than by reading.
That is the story. Not the $2,000.
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