When a company says a model solved hard math problems, the first hard question is verification. On August 1, OpenAI introduced Astra through ten results in mathematics and theoretical computer science, alongside a 249-page paper, a GitHub repository of Lean certificates and reasoning walkthroughs.
The results span high-dimensional geometry, coding theory, group theory, arithmetic circuit complexity, quantum complexity, lattice cryptography and extremal combinatorics. OpenAI says the main results had seen no progress for at least ten years, often longer, and that the token cost of finding all ten solutions would be roughly $2,000 at Sol API rates.
The evidence comes first
The list includes bounds for sphere packing, stronger binary and spherical code bounds, a construction of non-sofic groups, a counterexample to Connes's rigidity conjecture, lower bounds for the permanent, quantum parallel repetition, hardness for the closest vector problem, Ehrhart's volume conjecture and several Erdős graph problems.
The technical claims are concrete: the paper gives an n^4/log n formula lower bound for the permanent, an n^1/400 hardness factor for Euclidean CVP and a k^Theta(k) lower-bound form for multicolor triangle Ramsey numbers. The GitHub repository uses Lean 4.32.0, mathlib and Lake, with named files such as NonSoficGroup.lean, ConnesRigidity.lean and GapCVP.lean.
“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.”
The $2,000 figure adds pressure
The cost figure does not include problem selection, prompt design, human manuscript work, formalization effort or community review. It should not be read as ten theorems for $2,000. It does, however, price part of the search process and suggests that candidate proofs may become much cheaper to generate.
For Chinese labs and model teams, the lesson is practical: future research claims will need proof artifacts, code and reproducible checks, not just benchmark scores. Lean may become part of the infrastructure for AI-assisted mathematics, theoretical computer science and security proofs.
What to watch next
Astra remains internal, so outsiders cannot replay OpenAI's search process. The next checks are whether third parties can build the Lean repository, whether domain experts accept the stated results and attribution, and whether any of the ten proofs need boundary-condition or literature-position revisions. If those checks hold, Astra becomes more than a model teaser. It becomes a precedent for verifiable AI-produced research.
Sources: OpenAI research release, OpenAI 249-page proof paper, GitHub ten-proofs repository, CocoLoop, IT Home and Developers Digest; sources verify Astra scope, the ten-result list, the roughly $2,000 Sol API token-cost estimate, Lean 4.32.0 build path, and technical figures such as n^4/log n and n^1/400.