OpenAI's Astra solved 10 decades-old math problems for $2,000

Aug 16, 2026 7 min read

A single warmly lit stone tablet among rows of dark unlit tablets — ayraix.com branded hero for OpenAI's Astra math proofs

The headline number: on August 1, 2026, OpenAI said an internal, unreleased model it calls Astra produced fully machine-verified proofs for ten open problems in mathematics and theoretical computer science — several unsolved for decades — for a total inference cost of roughly $2,000. That is less than a weekend of GPU rental, spent on questions professional mathematicians had not cracked in up to 48 years.

Why it's not just a marketing number: OpenAI didn't ask anyone to take its word for it. It published a 249-page manuscript and the underlying Lean 4 proof certificates on GitHub under an Apache 2.0 license. Lean is a proof assistant that checks every logical step by machine — no step gets accepted on vibes. The repository's "sorry" count (Lean's marker for an unproven gap) is zero across all ten results, meaning every formalized step in every proof actually checks out.

What Astra actually solved

Non-sofic groups exist — an explicit construction settling a question open since Mikhail Gromov defined "soficity" in 1999. Astra produced a specific group that cannot be approximated by any finite set, the first concrete counterexample after 27 years.
A better sphere-packing bound — the first improvement to the general upper bound on high-dimensional sphere-packing density since 1978, a 48-year-old ceiling.
Connes's rigidity conjecture, disproved — a long-standing conjecture about von Neumann algebras, overturned with a counterexample.
Ehrhart's volume conjecture, proved, plus three problems pulled directly from Paul Erdős's open-problem catalog — including problem 183, on multicolor Ramsey numbers.

A dense old manuscript of handwritten mathematical notation, close on the worn pages — ayraix.com Lean verification editorial — The proof, not the press release, is the artifact: Lean 4 certificates with a zero "sorry" count across all ten results.

Why the price tag is the real headline

$2,000 for ten proofs that mathematicians couldn't close for decades reframes the cost conversation. It's not that Astra is "smarter" than the mathematicians who worked these problems for years — it's that a model able to search proof space at machine speed, checked step-by-step by Lean, turns a multi-year human research program into an overnight, three-figure compute bill. That ratio — decades of expert effort against $2,000 of inference — is the number worth sitting with, not the model's name.

The reception, and the caveat nobody's skipping

Fields Medalist Timothy Gowers reacted positively, while cautioning the results are still being digested by the field. Thomas Bloom, who curates the erdosproblems.com database, called the ten results "big news" on X — ranking them above an internal OpenAI model's unit-distance counterexample from May 2026, an earlier and smaller signal of the same trend. But the caveat matters as much as the result: none of the ten proofs has been through formal peer review yet. Lean verification confirms the logic is internally consistent; it does not yet carry the community vetting that turns "a machine-checked proof exists" into "the mathematics community has accepted this."

Worn gold-bound archive volumes stacked on library shelving, close up — ayraix.com editorial on unsolved problems finally opened — Ten problems that sat closed for up to 48 years, opened in one run — peer review is still the step nobody gets to skip.

Where this fits the pattern

This isn't an isolated stunt. It follows GPT-5.6 Sol working through a 50-year-old math conjecture in July, and the May 2026 unit-distance counterexample from an earlier internal model. Read together, frontier labs are treating formalized mathematics — where a proof checker gives you an unforgeable pass/fail signal — as the cleanest place to demonstrate that a model can do original research, not just retrieve and remix what's already published.

A full wall of dense library shelving packed with old bound volumes — ayraix.com editorial on formal verification as the trust layer — Formal proof checkers give AI math claims something most AI claims don't have: a pass/fail signal nobody has to trust on reputation alone.

What we'd watch next

Whether any of the ten proofs survive full peer review unchanged, and whether Astra (or its eventual public release) can repeat this on problems chosen by outside mathematicians rather than ones the lab picked itself. A single overnight run is a demo; a track record on adversarially chosen problems is evidence.

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What we're really seeing

The number worth remembering isn't "AI beat mathematicians" — it's "$2,000 bought a machine-checked answer to questions that sat open for up to 48 years." That's only credible because Lean, not OpenAI's word, is doing the checking. Compare that to most AI capability claims, which ask you to trust a benchmark score, a cherry-picked demo, or a company's own eval — exactly the trust problem we've flagged before on unrelated agent-eval work. Formal math is the rare domain where the AI can show its work in a way no one has to take on faith.

If you only remember one thing: the zero "sorry" count is the whole story. Everything else — the $2,000, the decades-old problems, the Fields Medalist's cautious nod — is downstream of the fact that anyone can clone the repo and have Lean re-check every step themselves.

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Question 1 of 3

Roughly how much compute did OpenAI say Astra used to find all ten proofs?