Breakthrough results and verification
OpenAI has disclosed that an internal model designated Astra has solved ten significant open problems in mathematics and theoretical computer science. The problems span high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, quantum complexity, lattice cryptography and extremal combinatorics. According to the company, the token costs required to find the solutions amounted to roughly two thousand dollars at Sol API rates, approximately forty-two thousand Czech koruna.
Among the specific results, Astra constructed a non-sofic group whose existence was posed by Mikhail Gromov in 1999 and had remained unanswered for twenty-seven years, though the community had suspected such groups exist for the past fifteen. The model also refuted the Connes rigidity conjecture, proved the Ehrhart volume conjecture, improved the sphere packing density estimate for the first time since 1978, and resolved three problems posed by Paul Erdős, one of the most prolific mathematicians of the twentieth century. OpenAI released a 249-page manuscript and, for each solution, a certificate in the Lean language that enables machine verification of the proofs by anyone with the appropriate compiler. Human researchers used the same model to prepare arguments for the manuscripts and Astra subsequently formalised them into Lean certificates.
Accelerating pace of AI-driven mathematics
OpenAI describes Astra as its next major model designed for complex, long-running tasks. The model enables AI agents to collaborate on different parts of a larger problem. The Information independently confirmed that OpenAI is developing Astra as a new family of models for sustained workloads. According to BleepingComputer, the company has not yet decided whether the model will be released as GPT-5.7, GPT-6 or under another name, though Astra already qualifies as a breakthrough model and may be subject to policies similar to those of Anthropic, where one version is designated for consumers and a stronger variant requires special approval.
The tempo of progress is accelerating. Only days earlier, Harvard mathematician Levent Alpöge used Anthropic’s Claude Fable 5 model to disprove the 87-year-old Jacobi conjecture over a weekend, finalising the work during the World Championship final. In May, another Erdős conjecture on unit distances fell, and shortly thereafter a Google DeepMind team closed nine further Erdős problems. Thomas Bloom of the University of Manchester, who maintains the catalogue of Erdős problems, called the development major news. Sébastien Bubeck, head of mathematics research at OpenAI, spoke of beautiful proofs.
Community backlash and governance questions
Sharp criticism has emerged from within the mathematics community. Caltech doctoral candidate Tamsin Chu, in an essay titled “Mathematicians Must Act”, described watching leading colleagues she knows collapse and called on the community to stop working for AI companies, verifying their proofs and accepting free gifts and subscriptions. In June, sixteen mathematicians issued the Leiden Declaration, endorsed by the International Mathematical Union and signed by more than three thousand people including several prominent names. The declaration accuses AI firms of exploiting published research without consent, bypassing peer review and announcing results via press releases, precisely as OpenAI has now done. OpenAI acknowledges the declaration but argues that attributing authorship of new proofs to a human when the machine generated the entire proof would distort both the system’s contribution and the nature of genuine human work. As Thomas Bloom notes the paradox, the AI model now taking work from mathematicians was built by mathematicians and trained on everything they have ever written.
Implications for enterprise AI adoption
For companies implementing AI, Astra represents a shift toward models capable of autonomous scientific reasoning with verifiable outputs. The ability to generate Lean certificates reduces the risk of hallucinations in critical applications. At the same time, pressure is growing to establish rules for the use of openly published scientific data in training commercial models and for transparency when presenting breakthrough results outside standard peer review processes. These dynamics intersect with the EU AI Act’s requirements for high-risk AI systems and the broader governance of foundation models trained on public knowledge corpora.