OpenAI’s Astra Just Solved 10 Math Problems Nobody Had Cracked in Decades

On August 1, 2026, OpenAI published a report titled Ten advances in mathematics and theoretical computer science. The company said an internal version of Astra — publicly named for the first time today — had solved ten open problems that mathematicians had made no meaningful progress on for at least a decade, and in most cases far longer. Every proof was formalized into a Lean 4 certificate on GitHub so outside researchers can verify the logic line by line.

The cost of generating all ten solutions: roughly $2,000 in tokens at Sol API rates.

Astra AI mathematics: what OpenAI actually solved

The ten results span a wide arc of pure mathematics and theoretical computer science. OpenAI’s own list breaks them down by domain:

• High-dimensional sphere packing — new upper bounds on density down to the Cohn–Elkies threshold, the first improved general sphere-packing exponent since 1978.
• Binary and spherical codes — exponentially improved bounds on the maximum size of binary codes at any prescribed minimum distance.
• Non-sofic groups — a construction establishing that non-sofic groups exist, resolving 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, 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.
• Closest vector problem — polynomial-factor hardness of approximation, a foundational lattice question tied directly to post-quantum cryptography.
• Ehrhart’s volume conjecture — determining, in every dimension, the maximum 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.

The arguments were generated by Astra. Human researchers then prepared them into manuscripts using the same model, and Astra subsequently formalized each proof in Lean 4 to produce machine-checkable certificates. OpenAI has also published reasoning walkthroughs that narrate the model’s thinking for each result.

This is the second time, not the first

In May 2026, OpenAI disclosed that an unreleased reasoning model had disproved the Erdős unit-distance conjecture — an 80-year-old problem in discrete geometry. That result was already triggering follow-on papers from independent mathematicians before today’s announcement. Work citing it has since touched the sum-product conjecture, incidence geometry, and communication complexity.

Today’s release is larger in scope. It spans eight distinct mathematical fields rather than one, and it marks the first time OpenAI has publicly named the model responsible: Astra, described as “our next major model family.” OpenAI researcher Noam Brown confirmed the connection between the models on X, noting the May disproof and today’s ten results came from the same system Sam Altman is currently demoing for Congress.

An internal version of Astra, @OpenAI‘s next major model family, solved 10 major open problems in mathematics, quantum complexity, and theoretical computer science. We believe it will be a major step for scientific reasoning. openai.com/index/ten-adva…

— Noam Brown (@polynoamial) August 1, 2026

What Astra is — and what it isn’t yet

OpenAI describes Astra as a model family built for tasks that run for hours or days with multiple agents working together. That framing is deliberate. The ten proofs were not generated in a single query — they required extended reasoning runs. The $2,000 token cost covers all ten combined, but each individual proof required the model to work through a substantial argument before humans could prepare it into a usable manuscript.

None of the ten problems is a Millennium Prize Problem. Noam Brown acknowledged that directly. OpenAI has also not said whether Astra will attempt them. But the problems it did solve are not minor exercises — the non-sofic group result alone resolves a question that group theorists have held open for decades, and the Connes rigidity disproof touches operator algebras at a foundational level.

The post-quantum cryptography result carries practical weight beyond pure mathematics. The hardness of approximation proof for the closest vector problem strengthens the theoretical foundations that lattice-based encryption schemes depend on. As governments move toward post-quantum standards, that matters.

Attribution and the Leiden question

OpenAI addressed authorship directly in its report. The company stated 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.” Human researchers prepared the manuscripts and take responsibility for correctness — but the mathematical arguments came from Astra.

That framing is a direct response to the Leiden Declaration on AI and Mathematics, signed by a group of mathematicians raising concerns about AI’s role in the discipline. OpenAI cited the declaration by name and said it has “deep respect and understanding” for those concerns. Whether the mathematical community accepts that framing will be one of the more interesting debates in the weeks ahead.

OpenAI has also opened access to its best models for 100,000 scientists and mathematicians through a new initiative called ChatGPT for Academic Researchers — announced separately in the lead-up to today’s release.

GPT-5.6’s role and what comes next

Astra is distinct from GPT-5.6, the model OpenAI currently ships commercially. GPT-5.6 already enabled meaningful work in math and science — Brown noted as much in his X post — but Astra represents the next step up in reasoning capability. OpenAI has not given a release date. The ten proofs are the first public signal of what the model can do.

For context on where GPT-5.6 and the broader Sol API fit into OpenAI’s current stack, see our earlier reporting on GPT-5.6 and the new model family and the 8 million active users milestone on Codex and ChatGPT Work.

The Lean certificates are live on GitHub. The reasoning walkthroughs and the full technical paper are available on OpenAI’s site. The mathematical community now has everything it needs to check the work — and to build on it.

Frequently Asked Questions

What is OpenAI’s Astra model?

Astra is OpenAI’s next major model family, publicly named for the first time on August 1, 2026. It is designed for long-running, multi-agent tasks and demonstrated its capabilities by solving ten open problems in mathematics and theoretical computer science.

Which math problems did Astra solve?

Astra solved ten problems spanning group theory (non-sofic groups, Connes’s rigidity conjecture), high-dimensional geometry (sphere packing, Ehrhart’s conjecture), quantum complexity (parallel repetition), lattice cryptography (closest vector problem), extremal combinatorics (Ramsey numbers, Erdős problems 146, 180, and 183), and arithmetic circuit complexity.

How were the proofs verified?

Every proof was formalized into a Lean 4 certificate and posted publicly on GitHub at openai/ten-proofs, allowing independent mathematicians to verify the logic machine-by-machine without relying on OpenAI’s word.

How much did it cost OpenAI to generate the proofs?

OpenAI states the token cost for all ten solutions combined was approximately $2,000 at Sol API rates — the same API rate available to external developers.