OpenAI's unreleased Astra model solves ten open math problems
OpenAI announced that an internal version of its next major model, called Astra, produced new results for ten problems in mathematics and theoretical computer science that have seen no progress on their main result for at least a decade, in most cases much longer. The problems span high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography and extremal combinatorics. OpenAI says all of them are of substantial interest to their respective mathematical communities, and several are of broad interest across mathematics as a whole.
The ten results, as described by OpenAI: high-dimensional sphere packing, with new upper bounds on sphere-packing density down to the Cohn-Elkies threshold; binary and spherical codes, with exponentially improved bounds on the maximum size of binary codes at any prescribed minimum distance and analogous results for spherical codes; non-sofic groups, a construction establishing that such groups exist, addressing a central open question in group theory; Connes's rigidity conjecture, disproved, concerning whether certain groups are uniquely determined by their von Neumann algebras; arithmetic circuit complexity, with new lower bounds for computing the permanent using arithmetic circuits and formulas, including an arithmetic-formula lower bound of order n4/log n; quantum parallel repetition, an exponential parallel repetition theorem for general two-player quantum games; the closest vector problem, with polynomial-factor hardness of approximation shown for this lattice question tied to post-quantum cryptography; Ehrhart's volume conjecture, resolved by determining, in every dimension, the maximum volume of a convex body whose centroid is its only interior lattice point; multicolor Ramsey numbers, with a superexponential lower bound for multicolor triangle Ramsey numbers, resolving Erdos problem 183; and extremal number conjectures, with results on the compactness and degeneracy conjectures in extremal graph theory, resolving Erdos problems 146 and 180.
On process and cost, OpenAI says the mathematical arguments were generated by Astra, and that the total number of tokens needed to find solutions to all ten problems would cost roughly $2,000 at Sol API rates. Humans then worked with the same model to prepare the arguments into manuscripts, and the model itself formalized each argument as a Lean certificate, a machine-checkable proof format. OpenAI is also releasing, for each solution, the model's own narration of its thinking process.
The release follows OpenAI's disclosure in May of an AI-generated disproof of the Erdos unit-distance conjecture, found while evaluating an unreleased model, which OpenAI says has already inspired further work in mathematics and theoretical computer science, citing five follow-up papers in a footnote. It also comes alongside ChatGPT for Academic Researchers, an initiative OpenAI says it recently announced that gives 100,000 scientists and mathematicians free access to its best ChatGPT models.
On attribution, OpenAI says credit should honestly reflect how a result was produced, and that 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 company says it helped prepare the manuscripts and formalize the proofs in Lean and takes responsibility for their correctness, while the mathematical arguments themselves were generated by its system. OpenAI says it has deep respect and understanding for those concerned about AI's role in mathematics, naming the signers of the Leiden declaration on AI and Mathematics.
Key facts
- OpenAI says an internal, unreleased version of its next major model, Astra, generated new results for ten problems in mathematics and theoretical computer science that had seen no progress on their main result for at least a decade.
- The ten problems span high-dimensional geometry, coding theory, group theory, operator algebras, quantum complexity, lattice cryptography and extremal combinatorics; two of the results resolve Erdos problems 183, 146 and 180.
- Humans, working with the same model, turned Astra's arguments into manuscripts, and the model itself formalized each proof as a Lean certificate; OpenAI is also releasing the model's narration of its thinking process for each solution.
- OpenAI estimates the total token cost to find all ten solutions at roughly $2,000 at Sol API rates.
- OpenAI says attribution should reflect how a result was produced, crediting the mathematical arguments to its system while taking responsibility itself for the manuscripts and Lean formalizations.
Why it matters
Astra, still unreleased, produced arguments for ten problems that had sat open for at least a decade, in most cases much longer, across geometry, coding theory, complexity theory, group theory, operator algebras, quantum complexity and lattice cryptography, with two of the results resolving named Erdos problems. That extends OpenAI's May disclosure of a single AI-generated disproof of the Erdos unit-distance conjecture into a coordinated batch of ten results, each backed by a formal Lean proof rather than a single headline finding.
Who it affects
Mathematicians and theoretical computer scientists working in the specific subfields named, particularly sphere packing, coding theory, group theory, operator algebras, quantum complexity theory and lattice-based cryptography, now have new results and machine-checked Lean certificates to examine and build on. It also affects the wider research community weighing how AI-generated proofs should be credited, including the signers of the Leiden declaration on AI and Mathematics, whom OpenAI names directly.
How to use it
Astra itself is not being released, and OpenAI gives no release date or specifications for it. What is published now is the write-up of the ten results, a Lean formalization of each proof, and the model's own narration of its thinking process for each solution. Separately, OpenAI's ChatGPT for Academic Researchers program, described as recently announced, gives 100,000 scientists and mathematicians free access to its best current ChatGPT models, distinct from Astra.
How solid is it
Each proof was formalized as a Lean certificate, a machine-checkable format, which is the main verification signal OpenAI offers; it estimates the token cost of finding all ten solutions at about $2,000 at Sol API rates. Humans working with the same model turned Astra's arguments into manuscripts, and OpenAI says it takes responsibility for the correctness of the manuscripts and the Lean formalizations while attributing the mathematical arguments themselves to the system. The source does not mention independent peer review or third-party verification of the results.
Risks and caveats
The source names no individual human mathematicians involved in preparing the manuscripts, only that humans worked with the same model; it gives no year for the May disproof of the Erdos unit-distance conjecture and no specific date beyond today for this release. Astra remains unreleased with no stated timeline, so outsiders cannot yet reproduce the results using the same model. OpenAI itself flags the unresolved question of how AI-generated proofs should be credited, pointing to the Leiden declaration on AI and Mathematics as a marker of ongoing community concern.
“We believe attribution should honestly reflect how a result was produced: 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.”
— OpenAI