Astra solves 10 open math problems; formal proofs at $2,000 prompt cost
OpenAI's unreleased Astra model solves ten open math problems with Lean certificates at $2,000 cost while pending U.S. regulatory review
On August 1, 2026, OpenAI announced that an unreleased, internal version of its new Astra model family produced ten new results in mathematics and theoretical computer science, resolving or making substantial progress on long-standing open problems across high-dimensional geometry, coding theory, group theory, operator algebras, arithmetic circuit complexity, quantum complexity, lattice cryptography, and extremal combinatorics. The results include a disproof of Connes’s rigidity conjecture in operator algebras and one solution in group theory establishing the existence of non-sofic groups, with OpenAI stating that every proof was formalized in Lean as a machine-checkable certificate alongside published reasoning walkthroughs. The total inference cost across all ten solutions came to approximately $2,000 at API rates, and the span of domains covered demonstrates capabilities that extend well beyond single-shot problem solving into sustained formal verification workflows.
The problems include three originally posed by Paul Erdős, while reporting on the announcement’s context, Quanta Magazine highlights these findings alongside details regarding OpenAI’s compliance with the Leiden Declaration on AI and Mathematics. OpenAI released these solutions without claiming human proof authorship, a practice that aligns with precedents set earlier in the year, including an AI-discovered counterexample to the unit distance conjecture in May. The emphasis on machine-verifiable output alongside the absence of human attribution marks a shift toward results that can be consumed directly by verification pipelines rather than serving merely as hypothesis generators for external mathematicians.
Astra’s ability to navigate these complex structures stems from its underlying design, where independent reporting confirms that Astra employs a multi-agent architecture intended for long-running tasks and stands as the first model to be tested under a planned U.S. government pre-release review framework. Mathematician Thomas Bloom characterized the results as “big news,” reflecting broader interest in how multi-agent systems might operate within the formal constraints that govern mathematical publication. The architecture enables the kind of extended reasoning required for proofs in systems like Lean, which typically rely on automation guided by the principles of the interactive theorem prover to manage the immense complexity of rigorous logical structures.
The inference cost of roughly $2,000 for ten highly specialized proofs places Astra’s performance on a trajectory where machine-assisted research could become economically viable even for difficult theoretical work, yet the model remains strictly internal at this stage. Independent reporting notes that Astra is being withheld pending potential U.S. regulatory review before reaching the public or wider research community, meaning verified proofs from this architecture are currently available only as published data rather than as a tool accessible to external users. This pause introduces a new variable in the adoption curve for AI-generated mathematics: the models may eventually produce sufficient value to justify the cost and complexity, but their deployment will be subject to external oversight just as heavily as any safety-critical system.
What remains to be seen is how quickly this architecture can be adapted once regulatory conditions allow release, particularly given Astra’s status as a testbed for review processes that could set precedents for how high-impact models are evaluated. The combination of formal Lean certificates, substantial cost efficiency per proof, and the multi-agent approach marks a clear pivot toward results that satisfy the structural demands of formal mathematics. With verification infrastructure already in place and costs constrained to API-scale figures, the gap between AI capability and public utility appears defined principally by policy timelines rather than technical limitations.