OpenAI Astra Cracks 10 Open Math Problems, Machine-Checked but Not Yet Peer-Reviewed

OpenAI published Ten advances in mathematics and theoretical computer science on August 1, 2026. The claim: an internal version of Astra, an unreleased next-generation model family, produced new proofs for 10 problems where nobody had pushed the main result in over a decade.

There is not a single benchmark chart in the announcement. Instead OpenAI shipped a 249-page paper PDF, a 62-page walkthrough of the model's reasoning, and a Lean 4 formalization repository, openai/ten-proofs, under Apache-2.0. The evidence for "the model is good at this" is a set of files a machine can check, not a score. The repo was created at 06:10 UTC on August 1 and has 327 stars and 34 forks as of writing.

Which of the ten actually matter

Here are the results, with the corresponding Lean file in the repository.

• High-dimensional sphere packing: pins down the exact exponential decay rate of the Cohn-Elkies linear program. The paper's exponent is 0.6044..., the first improvement on the general sphere-packing exponent since the 1978 Kabatianskii-Levenshtein value of 0.59905576.... (SpherePacking.lean)
• Binary and spherical codes: exponentially improves the classical upper bounds at fixed minimum distance, across all parameters. (MetricCodes.lean)
• Non-sofic groups: settles a long-standing question in group theory by explicitly constructing a group that is not approximated by finite permutations. (NonSoficGroup.lean)
• Connes rigidity conjecture, refuted: constructs infinitely many pairwise non-isomorphic property-(T) groups with the same group von Neumann algebra. (ConnesRigidity.lean)
• Permanent lower bounds: division-free circuits need Ω(n² log log n) gates and formulas need Ω(n⁴ / log n) leaves. (Permanent.lean)
• Quantum parallel repetition: an exponential parallel repetition theorem covering all finite two-player entangled games. (QuantumParallelRepetition.lean)
• Closest Vector Problem: a direct reduction from 3SAT yielding hardness of approximation to within an n^(1/400) factor, which touches the hardness assumptions behind lattice-based post-quantum cryptography. (GapCVP.lean)
• Ehrhart volume conjecture: proves in all dimensions that a convex body whose centroid is its only interior lattice point has maximum volume (n+1)ⁿ / n!. (EhrhartVolumeInequality.lean)
• Multicolor Ramsey numbers: a superexponential lower bound giving R_k(3) = k^Θ(k), resolving Erdős problem 183. (MulticolorTriangleRamsey.lean)
• Extremal graph theory: counterexamples to the Erdős-Simonovits compactness conjecture and the Erdős degeneracy conjecture, closing Erdős problems 146 and 180. (CompactnessAndDegeneracy.lean)

The ten do not carry equal weight. Thomas Bloom of the University of Manchester, who maintains the Erdős problems database, called the release "big news" on X and rated it, as a constructive result, above the unit-distance counterexample from May. Bloom has publicly criticized incorrect OpenAI math claims before, which makes the endorsement worth something.

$2,000, one week, and the runs nobody published

The most quoted number is the price. OpenAI wrote that "the total number of tokens needed to find solutions to these problems would cost roughly $2,000 at Sol API rates." Sol is the family that shipped with GPT-5.6's general availability, so that is a rate anyone can pay through the API today.

The production process is recorded in machine-readable form in the repo's formalization.yaml, and that file says more than the announcement does.

automation:
  methods:
    - method: "agent"
      models:
        - "Astra (OpenAI)"
      framework: "Codex"
      cost:
        wall_time: "1 week"
review:
  status: "agent-reviewed"

That is Astra running for one week on top of Codex, OpenAI's own coding-agent framework. The human contribution is spelled out separately in the announcement: the model generated the arguments, people worked with the same model to turn them into manuscripts, and the model then formalized them. Group-theory specialists including Henry Bradford and Francesco Fournier-Facio were consulted, but consultation is not independent certification.

Which is why $2,000 is half a number. The Hacker News thread reached 449 points and 319 comments, and the top-voted objection was that the experimental setup is missing. User aabhay asked four questions: how many problems were handed to the model in total, what fraction it abandoned and at what cost, how many attempts per problem, and what the harness cost if it ran on a job cluster. Report only the token spend on the ten that worked and the figure can be made arbitrarily small.

Agent scores depend on the harness at least as much as on the model, a point already made concrete when swapping two configuration settings moved an ARC-AGI-3 result by an order of magnitude. Here the harness disclosure is two lines: framework: Codex and wall_time: 1 week.

What the Lean kernel certifies, and what it does not

Attaching Lean certificates is not the same as saying verification is finished. This distinction is the whole story.

The machine-checking side is solidly built. The toolchain is pinned to leanprover/lean4:v4.32.0, and the ten Lean files total roughly 21MB, with GapCVP.lean at 5.3MB and MetricCodes.lean at 4.5MB. The ComparatorChallenges/ directory carries 12 per-result JSON configs so the proofs can be rechecked with a kernel OpenAI did not write.

lake exe cache get
lake exe comparator ComparatorChallenges/A_SpherePacking.json

The bottom three rows of the table are the open part. Whether each Lean statement faithfully encodes the problem the field cares about, and whether the passage from informal to formal statement lost anything, is not something the kernel inspects. That is exactly why review.status says agent-reviewed rather than recording human peer review. kingy.ai, which partially reproduced the environment, reported rebuilding the pinned setup and most of the public project, while framing the output as ten formally encoded research claims, not ten settled results in the mathematical canon.

Who is the author, and when does the model ship

OpenAI addressed authorship head-on:

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. We helped prepare the manuscripts and formalize the proofs in Lean, and we take responsibility for their correctness, while the mathematical arguments themselves were generated by our system.

That paragraph is written with June's Leiden Declaration on AI and Mathematics in view. Endorsed by the International Mathematical Union, the declaration criticized AI labs for announcing results via press release instead of peer review, and raised the authorship question directly. OpenAI says it respects the signatories, but did not change how it publishes. On HN, danielrmay noted that taking responsibility for correctness is close to a free promise when the proofs are Lean-verified in the first place.

There is no release date for the model. Astra was first reported by The Information as a family built so multiple agents split a single problem over hours or days, sitting alongside Sol, Terra, and Luna. Whether it ships as GPT-6 or as a GPT-5 variant is undecided, and even the name is provisional.

One more gate stands before release. An executive order signed on June 2 set up a voluntary process giving the federal government up to 30 days of pre-release access to frontier models, with August 1 as the finalization deadline, and Astra is expected to be the first model through it. Sam Altman demonstrated Astra to senators and regulators in Washington shortly beforehand, with Treasury Secretary Scott Bessent and Commerce Secretary Howard Lutnick attending. GPT-5.6 went through something similar, opening to roughly 20 vetted organizations from June 26 to July 9.

What developers can use today is the repository, not the model. It is Apache-2.0, and it shows file by file how far the practice of attaching checkable evidence to agent output has actually come. Automation reached sorry_count: 0. Turning review: agent-reviewed into human review is still the open work.