On August 1, 2026, OpenAI announced that Astra — an internal version of its next major model — generated fully machine-verified solutions to 10 problems in mathematics and theoretical computer science, each unsolved for at least a decade. The headline result is an explicit construction of a non-sofic group, open since 1999. OpenAI published a 249-page manuscript collection and Lean 4 proof certificates with a 'sorry' count of zero, and put the total compute cost at roughly $2,000 using GPT-5.6 Sol API rates.
OpenAI's own framing is that this marks the moment AI crosses from doing tasks to doing original research. Astra isn't public yet, but the company released everything needed to check the claim independently: a 249-page collection of manuscripts, model-written reasoning walkthroughs, and Lean 4 certificates for all 10 results, posted to GitHub under an Apache 2.0 license.
What Astra Actually Solved
| Result | Field | Open Since |
|---|---|---|
| Explicit construction of a non-sofic group | Group theory | 1999 (Gromov) |
| Disproof of Connes's rigidity conjecture | Von Neumann algebras | Decades-old |
| Proof of Ehrhart's volume conjecture | High-dimensional geometry | Decades-old |
| Resolution of Erdős problems (incl. #183 on multicolor Ramsey numbers) | Extremal combinatorics | Decades-old |
| Additional results in quantum complexity and lattice cryptography | Theoretical CS | 10+ years each |
The non-sofic group construction is the one drawing the most attention from mathematicians specifically. Mikhail Gromov introduced the concept of soficity in 1999, and whether a non-sofic group actually exists had been an open question in group theory ever since. The Connes rigidity disproof is similarly notable: Astra constructed infinitely many non-isomorphic groups with property (T) that share the same von Neumann algebra, directly contradicting the conjecture.
How It Was Verified
- Every one of the 10 results was formalized as a Lean 4 proof certificate, not just an informal write-up
- The certificate repository reports a 'sorry' count of zero — meaning no step in any of the 10 formalized proofs was left unproven or assumed
- OpenAI published the full 249-page manuscript collection alongside model-written reasoning walkthroughs, not just final answers
- Thomas Bloom, who curates the erdosproblems.com database, publicly called the results 'big news' and ranked them above the Erdős unit-distance counterexample OpenAI's model produced in May 2026
The Cost Is the Other Headline
OpenAI put the total compute cost for generating all 10 solutions at roughly $2,000, calculated at GPT-5.6 Sol API rates. For results that include settling a 27-year-old open question in group theory, that price point is arguably as newsworthy as the math itself — it reframes 'AI does original research' from a moonshot-lab capability into something closer to a line item.
How This Differs From the May Erdős Result
This isn't OpenAI's first math headline this year — an internal reasoning model disproved the Erdős unit distance conjecture back in May 2026. The difference is scale and formality: that result was a single disproof, independently checked by Fields Medalist Tim Gowers and Noga Alon after the fact. This time, OpenAI shipped 10 results at once, each with a machine-checked Lean proof attached from the start, which removes most of the burden from human reviewers having to verify the logic by hand.
Why It Matters
- It's the largest single batch of AI-generated, independently checkable original math results published to date, not a one-off proof
- Machine-verified Lean certificates sidestep the usual 'trust the AI's reasoning' problem — the proofs check out mechanically, regardless of how they were produced
- The ~$2,000 total cost signals that this category of result may become routine rather than a rare, resource-intensive event
- Astra itself is still unreleased, so this is a preview of the model's capability rather than something available to test today
The math checks out — zero unproven steps across 10 machine-verified results, and a leading Erdős-problems curator calling it significant is a real signal, not hype. What's still unknown is how Astra performs outside curated, decades-old open problems with clean formal statements. Treat this as strong evidence that frontier models can now do genuine, verifiable original research on narrow, well-posed math questions — not yet proof they can do research broadly.