OpenAI's Unreleased Model Astra Solves Ten Major Open Mathematics Problems

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OpenAI's Unreleased Model Astra Solves Ten Major Open Mathematics Problems

Zvi Mowshowitz<br>Aug 03, 2026

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Math is hard.<br>Math used to be strangely hard for LLMs. People used to gloat about that. Remember?<br>Math is getting easier. AI is getting more capable. Life comes at you fast.<br>Remember this meme?

Why yes. Yes it is.<br>We don’t know the extent to which Astra is a big jump over Fable and Sol in this realm. We do know that Astra can do math. As in real math.<br>OpenAI: We provide new results for the following problems. The results were achieved by an internal version of Astra, our next major model. The total number of tokens needed to find solutions to these problems would cost roughly $2,000 at Sol API rates. These arguments were then prepared into manuscripts by humans with the same model. Afterward, the model formalized each argument in a Lean certificate⁠(opens in a new window). We are also releasing for each solution a model’s narration of its thinking process.<br>High-dimensional sphere packing. New upper bounds on sphere-packing density down to the Cohn–Elkies threshold.

Binary and spherical codes: Exponentially improved bounds on the maximum size of binary codes at any prescribed minimum distance, with analogous results for high-dimensional spherical codes.

Non-sofic groups. A construction establishing the existence of non-sofic groups, addressing a central open question in group theory.

Connes’s rigidity conjecture. Disproof of a longstanding conjecture that certain groups are uniquely determined by their von Neumann algebras.

Arithmetic circuit complexity. New lower bounds for computing the permanent using arithmetic circuits and formulas, including an arithmetic-formula lower bound of order n4/log n.

Quantum parallel repetition. An exponential parallel repetition theorem for general two-player quantum games, extending a foundational principle from classical complexity theory.

Closest vector problem. Polynomial-factor hardness of approximation for the closest vector problem, a foundational lattice question related to post-quantum cryptography.

Ehrhart’s volume conjecture. Determining, in every dimension, the maximum possible volume of a convex body whose centroid is its only interior lattice point.

Multicolor Ramsey numbers. A superexponential lower bound for multicolor triangle Ramsey numbers, resolving Erdős problem 183.

Extremal number conjectures. Results on the compactness and degeneracy conjectures in extremal graph theory, resolving Erdős problems 146 and 180.

Noam Brown (OpenAI): And yes we did try other major problems without success. Sadly no Millennium Prize problems (yet).

But also, we didn’t spend a lot on each problem. It’s possible to push test-time compute much further.<br>Sichu Lu: I do hope they are keeping backlogs of everything that went wrong that's probably more valuable to the future of math than what went right now.

There are Lean proofs. That does not mean that all ten results prove the things they assert that they prove. So far it is looking good.<br>Kevin Roose: almost nobody is pricing in the possibility that the models just keep plowing through every discipline the way they’re plowing through math.<br>James:

Ryan Fedasiuk:

At least some of these were possible before Astra, even without a harness, as both Sol and Fable have now proven the existence of nonsofic groups in their chat interfaces.<br>Ananjan Nandi: Crazy how all of this was started by one guy bullying his Claude during the World Cup final.

A lot of the time the barrier for a particular result is as simple as asking the right question and letting the model cook. With Astra, OpenAI asked it to take a crack at a bunch of open math problems, and it solved 10 of them. Once you know what they are, pointing other models at the same problems, even without ‘hints,’ shows there was a mathematical proofs overhang.<br>We do still have strong statements that Astra is a major step for scientific reasoning.<br>Yu Bai (OpenAI): My jaw dropped 10 times 😅 But really this is going to be an avalanche. Been throwing a few open questions of mine at Astra too, boy is it strong.

Some skepticism is always wise, but I believe that the reason they tried this was that there was a substantial jump in scientific reasoning, at least for some types of problems and probably across the board.<br>Soon we will all have Astra, and other models as good or better than Astra, both at math and at other things, from OpenAI, from Anthropic and soon after that from many other sources.<br>Dean W. Ball: Everyone in the world will soon be able to use the model that made these breakthroughs for every problem they face in life, no matter how mundane, at a cost that will fall dramatically in a matter of months. I still struggle to get my head around this fact.

Approximately zero people have their head around what this means.<br>Table of...