OpenAI's Astra Cracks 10 Math Problems That Stumped Humans for Decades

OpenAI announced its next major model by doing something unusual: dropping a batch of solved mathematical problems that had stumped the world's best mathematicians for decades. The company revealed Astra, an unreleased model designed to tackle complex, long-running tasks, after an internal version produced ten significant advances in mathematics and theoretical computer science. These are new mathematical results, not benchmark scores or leaderboard climbs.

What Astra actually produced

The ten results span eight fields and arrive with machine-checkable proofs in the Lean 4 theorem prover. OpenAI published a 249-page manuscript alongside Lean 4 certificates for every result on GitHub.

1. High-dimensional sphere packing — the first improvement to the general upper bound on sphere-packing density since 1978.
2. Binary and spherical codes — exponentially improved bounds on the maximum size of binary codes at any prescribed minimum distance.
3. Non-sofic groups — the first explicit construction of a non-sofic group, resolving a central question in group theory open since Mikhail Gromov introduced soficity in 1999.
4. Connes's rigidity conjecture — a disproof, constructing infinitely many non-isomorphic groups with property (T) that share the same von Neumann algebra.
5. Arithmetic circuit complexity — new lower bounds for computing the permanent, including an arithmetic-formula lower bound of order n4/log n.
6. Quantum parallel repetition — an exponential parallel repetition theorem for general two-player quantum games.
7. Closest vector problem — a polynomial-factor hardness of approximation result with direct connections to post-quantum cryptography.
8. Ehrhart's volume conjecture — determining the maximum possible volume of a convex body whose centroid is its only interior lattice point, in every dimension.
9. Multicolor Ramsey numbers — a superexponential lower bound resolving Erdős problem 183.
10. Extremal number conjectures — resolving Erdős problems 146 and 180 in extremal graph theory.

The headline result: non-sofic groups

The non-sofic group result is drawing the most attention. A group is the mathematical description of a set of symmetries. A sofic group is one whose structure can be approximated by shuffling a finite deck of cards. Every group anyone had examined turned out to be sofic, and no one could prove that all of them are. Astra built the exception.

The second major result is a disproof of the Connes Rigidity Conjecture, posed by Fields Medalist Alain Connes in 1980. The conjecture asks whether the von Neumann algebra L(G) constructed from a property (T) group G retains enough information to reconstruct G. Astra showed it does not, by constructing a family of groups that all produce the same algebra while remaining structurally distinct.

How Astra works

OpenAI describes Astra as a model family built to run long tasks by coordinating multiple agents over extended periods, extending the test-time reasoning work associated with research scientist Noam Brown. The key capability is long-horizon reasoning: maintaining a coherent chain of thought across a lengthy problem-solving session, backtracking from dead ends, and verifying intermediate steps rather than producing a single-shot answer.

The workflow for these proofs followed four stages:

1. Astra autonomously generated the core mathematical arguments.
2. Human researchers prepared those arguments into manuscripts using the same model.
3. Astra formalized every argument as a Lean certificate, enabling machine verification.
4. For each solution, the model also produced a narration of its own reasoning process.

Why Lean certificates matter

Every one of the ten results ships with a machine-checkable certificate in OpenAI's GitHub repository, under an Apache 2.0 license. The repository reports a "sorry" count of zero. In Lean, a sorry is a placeholder marking an unproven gap. Zero of them means the entire proof chain is complete and machine-verified.

The mathematical community has long objected that AI-generated proofs are difficult to verify independently. Machine-checkable proofs sidestep that problem: anyone with the Lean compiler can validate them without trusting the model or its operators.

The cost: surprisingly modest

The total inference tokens needed to find all ten solutions would cost roughly $2,000 at current API rates. That figure excludes model training costs and the labor involved in writing the papers, but for ten results that collectively stumped the global mathematics community for decades, the inference cost is strikingly low.

What the math community is saying

Thomas Bloom, a University of Manchester mathematician who runs erdosproblems.com, called the results "big news" on X, rating them more significant than the counterexample to the unit distance conjecture published in May. He also pushed back on the idea that AI is replacing mathematicians, noting that the model draws on more than a century of mathematical theory, was built by mathematicians, and was trained on everything mathematicians have ever written.

Noam Brown said on X that OpenAI had also tried and failed to crack other major problems. "Sadly, no Millennium Prize Problems (yet)," he wrote. The broader academic response has mixed excitement with caution, with many researchers emphasizing the need for external verification. The proofs are public, so that scrutiny is already underway.

What this signals about Astra

Chief Scientist Jakub Pachocki said on OpenAI's official podcast that the company wants to build AI systems capable of working on a problem for hours or days. Current systems are largely limited to short tasks. These ten proofs are the first public evidence that sustained, autonomous mathematical reasoning at this level is achievable.

Astra continues OpenAI's celestial naming scheme introduced with GPT-5.6, which assigned names like "Sol," "Terra," and "Luna" based on capability tiers. OpenAI has not yet decided whether to release Astra as an extension of the GPT-5 family or as GPT-6. It is also expected to be the first model subject to the AI safety review framework the Trump administration is pushing forward, which would require companies to submit new models to the federal government before public release.

OpenAI was explicit about attribution: claiming human authorship for a proof generated entirely by an AI system misrepresents both the system's contribution and the nature of human intellectual work. That stance will force journals and universities to develop clearer policies, and faster than most had planned.

The Lean certificates are on GitHub and the reasoning walkthroughs are publicly available. If the proofs hold up to community scrutiny, the assumption that open mathematical problems require human insight to crack will need to be revised.