OpenAI's Astra Solves Ten Open Math Problems for $2,000

OpenAI published a paper this morning claiming that an internal version of Astra, the next major model it has not yet shipped, produced solutions to ten problems in mathematics and theoretical computer science that had seen no progress on the main result for at least a decade, and in several cases far longer. The company puts the token cost of finding all ten at roughly $2,000 at Sol API rates. Every argument also ships as a Lean certificate, so we pulled that repository and counted it: 4,307 Lean files, 838,448 lines, and zero proof holes.

• The results came from an internal version of Astra, OpenAI's unreleased next model, at a stated cost of about $2,000 in tokens at Sol API rates to find all ten solutions.
• The problems span sphere packing, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography and extremal combinatorics. Three are numbered Erdős problems: 146, 180 and 183.
• The formalizations are public at github.com/openai/ten-proofs under Apache 2.0, and they lean on only the three standard Lean axioms with no unproved placeholders.
• The repository's own manifest records its review status as "agent-reviewed". Human mathematical peer review has not happened yet, and OpenAI is openly asking the community to do it.

What did the model actually solve?

These are not competition problems or benchmark items. Each one is a question a research community has been stuck on. The ten, in OpenAI's own framing:

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• High-dimensional sphere packing. New upper bounds on packing density, down to the Cohn-Elkies threshold.
• Binary and spherical codes. Exponentially improved bounds on the maximum size of a binary code at any prescribed minimum distance, with matching results for high-dimensional spherical codes.
• Non-sofic groups. A construction establishing that a finitely presented non-sofic group exists, which has been a central open question in group theory.
• Connes's rigidity conjecture. A disproof of the claim that certain groups are pinned down uniquely by their von Neumann algebras.
• Arithmetic circuit complexity. New lower bounds for computing the permanent, including an arithmetic-formula bound of order n⁴/log n.
• Quantum parallel repetition. An exponential parallel repetition theorem for general two-player quantum games.
• Closest vector problem. Polynomial-factor hardness of approximation for CVP, which sits underneath the lattice assumptions post-quantum cryptography is built on.
• Ehrhart's volume conjecture. The maximum volume of a convex body whose centroid is its only interior lattice point, settled in every dimension.
• Multicolor Ramsey numbers. A superexponential lower bound for multicolor triangle Ramsey numbers, resolving Erdős problem 183.
• Extremal number conjectures. Results on compactness and degeneracy in extremal graph theory, resolving Erdős problems 146 and 180.

The workflow OpenAI describes is a three-stage handoff rather than a single prompt. The model generates the mathematical argument. Humans, working with the same model, turn that argument into a manuscript. The model then formalizes each argument in Lean, and a comparator harness checks the standalone solution against a challenge statement. OpenAI is also releasing the model's own narration of its reasoning for each result.

Does the Lean code actually check out?

This is the part of the announcement that can be inspected rather than taken on trust, so we did. The repository openai/ten-proofs was created at 06:10 UTC on August 1 and last pushed at 07:48 UTC, licensed Apache 2.0. It contains 4,307 Lean files totalling 838,448 lines of proof code. Ten top-level modules carry the headline results, and twelve comparator challenge files hold the formal problem statements.

Two details matter more than the raw size. First, the only sorry keywords in the repository sit inside those twelve challenge files, where they are goal placeholders by design. The proof modules themselves contain none, meaning nothing is stubbed out. Second, the manifest declares the axiom set as propext, Classical.choice and Quot.sound. Those are the three axioms Lean's standard library already assumes, so no extra assumption has been smuggled in to make a proof close.

That is a stronger provenance story than any AI math claim we have covered so far. It is not the same thing as the results being correct, though, and the distinction is worth being precise about. Lean guarantees that the proof follows from the formal statement. It cannot tell you whether the formal statement faithfully expresses the conjecture a mathematician had in mind. Checking that translation is human work, and on twelve statements written in this much machinery it is not a quick read.

Why does "agent-reviewed" matter?

Buried in formalization.yaml is a one-line review field, and it reads agent-reviewed. That is an unusually candid thing to publish. It says the formalizations have been checked by machine, and that the human sign-off a journal would require has not happened. OpenAI's post is consistent with this: it asks the mathematical community to engage with the results, place them in context, and build on them.

The company also stakes out a position on credit that goes against the industry's usual instinct to blur it. It says claiming human authorship for a proof generated entirely by an AI system would misrepresent both the system's contribution and the nature of human intellectual work. OpenAI says it helped prepare the manuscripts and formalize the proofs and takes responsibility for their correctness, while the mathematical arguments themselves came from the system. The post nods directly at the signers of the Leiden declaration on AI and mathematics, the group most publicly uneasy about exactly this.

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1. May 2026OpenAI shares an AI-generated disproof of the Erdős unit-distance conjecture found while evaluating an unreleased model
2. Jul 12, 2026GPT-5.6 Sol produces a proof of the cycle double cover conjecture our earlier coverage
3. Jul 21, 2026Claude Fable 5 returns a Jacobian conjecture counterexample the rival lab's version of the same claim
4. Late Jul 2026Sam Altman previews Astra to US senators reported by The Information; name still tentative
5. Aug 1, 2026Ten results published with Lean certificates and reasoning walkthroughs 06:10 UTC repo creation, 07:37 UTC on Hacker News
6. NextIndependent mathematicians read the formal statements the only check that settles this

What does this say about Astra?

Until this morning, Astra was a name in a scoop. The Information reported that Altman demonstrated the model family to policymakers in Washington, pitching multiple agents that split one hard task and grind on it over long stretches, with advanced mathematics as the showcase. OpenAI has not decided whether it ships as GPT-6, GPT-5.7 or a separate class alongside Sol, Terra and Luna. The timing is not accidental either: the preview landed as an August 1 deadline approached for the federal pre-release model review framework Astra is expected to be the first model submitted under.

Read that way, this paper is the technical exhibit attached to a political argument. Ten decade-old problems and a public Lean repository are a more concrete demonstration of long-horizon reasoning than any benchmark chart, and they arrive precisely when OpenAI needs regulators to see capability rather than risk.

The $2,000 figure deserves its own attention. If a model can convert two thousand dollars of inference into a result a research community could not reach in a decade, the economics of theoretical research change shape, and so does the demand curve for inference compute. For readers tracking the money, the exposed names are the obvious ones: Microsoft through its OpenAI stake, and Nvidia through the inference demand that this kind of workload implies. The signal for investors is not a single announcement, it is whether independent mathematicians confirm the results over the coming weeks. A confirmed ten-for-ten changes the story for research-heavy AI spend. A single serious error in the formal statements does the opposite, faster.

Our take

The claim to be skeptical of is not that a model produced these arguments. It is the word "solved", which is doing a lot of work before anyone outside OpenAI has read the formal statements. What makes this release different from the usual AI-does-math cycle is that OpenAI shipped the artifact that lets you check the work instead of a press release describing it, and then labelled its own review status honestly rather than implying more validation than exists. Nothing here is stubbed out and no exotic axiom is propping anything up, which we verified rather than assumed. That earns the results a serious look. It does not substitute for one.

Original analysis by GenZTech. Primary source: OpenAI. Lean repository statistics counted directly from openai/ten-proofs on August 1, 2026.