OpenAI's Astra Cracks 10 Open Math Problems at Once: When AI Becomes a Real Mathematician

One release, ten problems, breakthroughs across three time horizons (1999-2026)

On August 1, OpenAI published results from its internal next-generation model Astra across mathematics and theoretical computer science, delivering new results on ten long-standing open problems. The problems span eight fields — high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography, and extremal combinatorics. Every one of them had seen no meaningful progress on the main question for at least a decade, and the oldest has been stuck for 46 years.

The motivation is a direct continuation of May's move: OpenAI disproved the Erdős unit-distance conjecture while evaluating an unreleased model. Back then it was a single result. This time it is ten.

Three highlights worth remembering

First: the 27-year-old "non-sofic group" problem. In 1999, Abel Prize laureate Gromov introduced the concept of sofic groups: a countable group is sofic if it can be perfectly approximated by finite permutations. Is every countable group sofic? For 27 years, the world's top mathematicians failed to construct a counterexample. Astra started from the unit group of the Leavitt algebra, combined Kun-Thom extension graphs and Thompson's group V, and pushed the logic to a contradiction — the first-ever proof that non-sofic groups exist.

Second: the 46-year-old high-dimensional sphere packing upper bound. After two Soviet mathematicians set the Cohn-Elkies threshold in 1978, the density upper bound for general dimensions was frozen for 46 years. In 2022, Serhiivna Viazovska won the Fields Medal for solving 8- and 24-dimensional specific cases — but the infinite-dimensional case was still stuck. Astra delivered a fresh proof and computed the exponential decay rate, breaking through the 1978 boundary for the first time.

Third: Connes' rigidity conjecture (proposed by a 1982 Fields medalist) is disproved. Connes claimed that certain groups are uniquely determined by their von Neumann algebras. Astra did not stop at a single counterexample — it constructed a countably infinite family of pairwise non-isomorphic groups that produce exactly the same von Neumann algebra. Fable 5's verdict: "by Fields Medal standards, any one of these would be enough to win a prize."

,000, 10 problems, full Lean 4 verification

The shock is not the results themselves, it is the cost structure. OpenAI reveals that the tokens Astra used to find these solutions cost less than ,000 at Sol API rates — roughly 00 per problem, "about a graduate student's weekend stipend."

The credibility mechanism is what makes this airtight. Every argument was written by Astra itself in Lean 4, with the formal certificate appended to each manuscript. Lean 4's machine-checked verification means: every step was validated by a theorem prover, with no room for hand-waving. Mathematician Elliot Glazer confirmed the news within hours and called it "the most important AI-assisted mathematics result to date."

The openai/ten-proofs GitHub repository is now public, with all ten Lean certificates and the model's own narration of its reasoning chain, every step independently auditable.

100,000 scientists and mathematicians get free Astra access

OpenAI bundled the announcement with a new initiative: free access to its strongest ChatGPT models for 100,000 scientists and mathematicians. The companion reasoning-walkthroughs.pdf is also public — ten papers of the model thinking out loud while solving the problems, essentially a working log of an AI mathematician.

This is consistent with OpenAI's recent pattern: model self-evaluation → unexpected new results during evaluation → publish and open-source. May's Erdős disproof was a "happy accident"; this time the breakthroughs are the deliberate output of Astra's evaluation window. The mathematical community is split: some call it a watershed moment, others remind everyone that independent peer review is the final gate.

What this means: AI mathematicians from "can compute" to "can construct"

If May's Erdős disproof was the first time AI "discovered" something in mathematics, Astra now goes much further. It constructs counterexamples, proves non-approximability, disproves a Fields-medalist conjecture, and tightens a long-standing limit. This is not retrieval or imitation — this is a machine performing genuine mathematical creation, from picking the right construction, to finding the proof path, to writing a self-consistent formal certificate, all done by the model end-to-end.

Test-time compute is nowhere near a ceiling. OpenAI's inference model lead Noam Brown said it publicly: "test-time compute is far from saturated, even million-dollar-scale world problems could be cracked next."

Is mathematics still the glory of the human mind? Starting today, that question probably needs a new answer.

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