OpenAI Astra Tackles Ten Open Problems in Mathematics Using Multi-Agent AI for Just $2,000

OpenAI says its next model, Astra, has solved ten open problems in mathematics

OpenAI says its unreleased Astra model solved ten open maths problems, shipping Lean proofs on GitHub for roughly $2,000 in compute OpenAI says an internal version of its next major model, called Astra, has produced ten new results in mathematics and theoretical computer science. Each of the problems had been open for at least a decade. The company published a 249-page manuscript alongside machine-checkable Lean 4 certificates for every result on GitHub. The headline result is the first-ever explicit construction of a non-sofic group, resolving a central question in group theory that has stood since Mikhail Gromov introduced the concept of soficity in 1999. No mathematician had managed to prove or disprove whether non-sofic groups exist in the 27 years since. The other results span several fields. Astra disproved Connes's rigidity conjecture on von Neumann algebras, proved Ehrhart's volume conjecture, and resolved three problems from Paul Erdos's famous catalogue, including problem number 183 on multicoloured Ramsey numbers. It also produced the first improvement to the general upper bound on high-dimensional sphere-packing density since 1978, proved a parallel repetition theorem for two-player quantum games, and established new lower bounds on the circuit complexity of computing the permanent. OpenAI's head of mathematics research, Sebastien Bubeck, confirmed the results on X, calling them "beautiful" and noting that each ships with a Lean certificate and a chain-of-thought walkthrough. The total compute cost for all ten solutions was roughly $2,000 at Sol API rates, according to OpenAI. The announcement lands against a backdrop of escalating tension between AI companies and the mathematics community. In June, mathematicians issued the Leiden Declaration, endorsed by the International Mathematical Union, warning that AI companies are using published research without consent, bypassing peer review, and threatening the integrity of proof and attribution. The Declaration specifically cited companies that announce results through press releases rather than peer-reviewed journals. OpenAI has form on this front. In May it announced that the same long-horizon model family disproved the Erdos unit distance conjecture, an 80-year-old problem in discrete geometry. Fields Medalist Tim Gowers said at the time that he would recommend that proof for publication in Annals of Mathematics without hesitation. Thomas Bloom, who runs the erdosproblems website, called the latest ten results "big news" on X, saying they are more significant than the unit distance counterexample. OpenAI has not said when Astra will be released publicly, describing it only as its "next major model." Some observers, including investor Mark Kretschmann, have speculated that Astra is the GPT-6 series. The company is also giving 100,000 academic researchers free access to its frontier models through 2027, a move that deepens its ties to the scientific community while concentrating research infrastructure on its own platform. The Lean certificates address a key objection that the mathematical community has raised about AI-generated proofs: that they are difficult to verify independently. Machine-checkable proofs can be validated by anyone with the Lean compiler, without trusting the model or its operators. Whether the broader mathematical community will accept results announced through a blog post rather than a peer-reviewed journal remains an open question, one the Leiden Declaration was written to answer.

OpenAI Smuggled the Announcement of Astra, Its Next AI Model, Into a Blog Post About Math

OpenAI announced its next major AI model Saturday, and it did so in the third paragraph of a blog post called "Ten advances in mathematics and theoretical computer science." The math results the post is touting, OpenAI writes, "were achieved by an internal version of Astra, our next major model." So there you go. It sounds like after GPT-5.6 Sol comes either GPT-5.6 Astra, or GPT-6 Astra, or -- who knows? -- just "Astra" and the whole GPT part gets scrapped? It's not spelled out. OpenAI's naming conventions suggest that this would be another GPT-5.6 release. There's a GPT-5.6 Terra, which, as corny Latin-knowers are well aware, means "earth," Luna which is Latin for "moon," and Sol, which is Latin for "sun." Astra means "the stars." According to an anonymously sourced story in the Information, Astra boasts the ability to do "long-running" work. CEO Sam Altman was, the report claims, in Washington, D.C. over the past week, demoing the model to federal officials. Gizmodo asked OpenAI on Saturday for the official name of the model. We also asked OpenAI about the relationship between this model and another OpenAI model only vaguely described in one of that company's blog posts. We did not receive a reply. The "unprecedented cyber incident" covered in that July 21 blog post is already infamous. An entity described as a "combination of OpenAI models -- including GPT‑5.6 Sol and an even more capable pre-release model, all with reduced cyber refusals for evaluation purposes" compromised the AI resource depository Hugging Face during a model evaluation exercise that was supposed to remain inside OpenAI. A later update to the blog post clarified that the unreleased and unnamed model involved in the incident was an "internal-only research prototype and was never intended for public release," and that it had been "deactivated, encrypted, and restricted." So to be clear, Astra is not the model that broke into Hugging Face. Gizmodo asked OpenAI on Saturday to more fully clarify the relationship or lack thereof between Astra and the never-to-be-released model. We did not receive responses in time for publication, but will update if we receive a clarifying answer. As for the blog post about math, it comes with a paper. There are ten proofs covered, covering such topics as the "asymptotic strength of the Cohn-Elkies linear program," for sphere-packing, which OpenAI purports to be "determined exactly." That sounds very cool, but I'm just the guy who blogs nights and weekends for Gizmodo. For what it's worth, OpenAI published a mathematical disproof back in May, purportedly solved by an unnamed OpenAI model. For the most part, math folks wrote about the model's work approvingly, but didn't seem completely knocked out. For instance, Harvard mathematician Melanie Matchett Wood wrote that OpenAI's proof was, "a beautiful application of number theory to a natural, concrete question," but she also said the problem, described by OpenAI as "a central conjecture in discrete geometry" was nothing she had ever heard of before. She added: "This result does not show us all the times AI has claimed to have a proof of something and been wrong. Without that context (which many of us have just from personal experience), it is also easy to draw incorrect conclusions about the current state of AI and research mathematics."

OpenAI unveils Astra: its next major model family for harder problems

Inside OpenAI, Astra is now the name attached to what appears to be the company's next big model family. The goal is to take on harder problems and stay with them much longer than today's chat-style tools can. It uses multiple agents that can work together for hours, even days. Think less one-shot chatbot, more research team: something that can make a plan, run tests, revise the work, and keep pushing through deep research, coding, or other multi-step analysis without needing much from you. Sam Altman has already shown Astra to policymakers in Washington, which suggests government conversations are starting early. As the industry moves toward more autonomous systems, questions about oversight, reliability, and responsibility are coming up in the US, the EU, and elsewhere. OpenAI says an internal version of Astra solved 10 long-standing open problems in math and theoretical computer science. That included problems in group theory, coding theory, and quantum complexity. Humans checked the results, the work was formalized in Lean, and OpenAI estimates the cost at about 2,000 tokens at Sol API rates. It didn't solve everything. Astra also came up short on other major open questions, including Millennium Prize Problems. So while it looks unusually strong on some reasoning tasks, it's still far from general-purpose. If you use AI for research or coding, this is something to keep an eye on. You can't download Astra yet, and OpenAI still hasn't decided whether it will ship as GPT-6 or as a new GPT-5 variant.