OpenAI's Astra Reportedly Solved Ten Open Math Problems for $2,000 — Including One Unsolved Since 1999
OpenAI says an unreleased internal model, codenamed Astra, has produced novel machine-checked solutions to ten long-standing open problems in mathematics and theoretical computer science. According to the company, the effort cost roughly $2,000 in token usage — a figure OpenAI and outside commentators have pointed to as a marker of efficiency. The headline claim is striking, but it's worth stating upfront: none of the results have gone through independent peer review, and Astra itself has not been released.
Among the reported results is a resolution tied to Erdős problem #183, a question that had remained open since 1999. That detail has drawn particular attention, though as with the rest of the claims, it rests entirely on OpenAI's own account for now.
What Astra Reportedly Solved
OpenAI's list of claimed results includes a non-sofic group construction, a disproof of Connes's rigidity conjecture, and a proof of Ehrhart's volume conjecture. The company also says Astra resolved three Erdős problems, including #183. All of the proofs were reportedly formalized in the Lean 4 proof assistant with zero "sorry" placeholders — a technical detail suggesting the formal logic in each proof is internally complete, without unresolved gaps left in the code.
What Lean Verification Actually Proves — and What It Doesn't
A zero-"sorry" Lean proof is a meaningful technical signal: it confirms that, given a particular formal statement, the logical steps to the conclusion hold up under machine checking. What it does not confirm is whether that formal statement is a faithful, complete translation of the original open problem as mathematicians have understood it for decades. A recurring concern among researchers and mathematicians with AI-generated formal proofs is exactly this gap — formalization choices can subtly narrow or shift a problem's actual claim. Determining whether Astra's proofs are mathematically significant, and whether they truly answer the historical open questions, still requires review by human domain experts.
A Credibility Problem OpenAI Is Still Living Down
The Astra announcement doesn't arrive in a vacuum. In October 2025, OpenAI researcher Kevin Weil publicly claimed that GPT-5 had solved ten Erdős problems. Mathematician Thomas Bloom, who had been independently tracking progress on these problems, pushed back hard, calling the claim a "dramatic misrepresentation" and "embarrassing," noting that the problems in question had, in fact, already been solved in existing published literature. The claim was later retracted. Many observers argue this episode is essential context for evaluating any new OpenAI math claim, including Astra's.
This Time, a Cautious Endorsement
Notably, Bloom — the same mathematician who criticized the October 2025 claim — has offered more favorable comments this time, reportedly describing the Astra results as "big news" and ranking them above a prior verification milestone from May 2026. That said, a recurring theme among commentators is caution: an endorsement from a respected external verifier is meaningful, but it is not a substitute for formal peer review, and the underlying work has not been published in a peer-reviewed venue.
An Unreleased Model, Limited Scrutiny
Astra has no announced release date, pricing structure, or confirmed model lineage. That absence matters: without access to the model or a fuller technical writeup, independent researchers have limited ability to reproduce results, probe edge cases, or stress-test the formalizations themselves. There are also reports that OpenAI CEO Sam Altman has demonstrated the results to policymakers, which some observers read as evidence of a political and public-relations dimension to the announcement's timing and framing.
Pushback From the Math and Research Community
Beyond questions about this specific set of claims, there's broader unease in academic circles about how AI labs are engaging with mathematical research. Efforts like the Leiden Declaration have raised concerns about AI systems training on or drawing from research corpora without consent, and about the risk of bypassing established peer-review norms. Questions of attribution and research integrity are recurring themes in this debate. Some commentators, including writer Nick Borretti, have raised a more speculative concern: that as AI-generated proofs scale in volume and complexity, mathematics itself could become increasingly difficult for human experts to fully verify or understand — a possibility rather than a settled prediction, but one that's shaping how some in the field are reacting to news like this.
What to Watch Next
For Astra's claims to move from provisional to established, most observers agree independent mathematicians would need meaningful access to review the formalizations against the original problem statements, not just the Lean output. Key open questions include when — or whether — Astra will be released, what it might cost, and whether outside researchers get a real opportunity to scrutinize the work before it's treated as settled. The underlying tension is likely to persist regardless: genuine efficiency gains in AI-assisted mathematics are plausible, but they sit alongside a documented pattern of overstated claims that makes skepticism a reasonable default until independent verification catches up.