OpenAI's Astra model solves or advances ten open math problems

OpenAI published a blog post presenting ten results in mathematics and theoretical computer science, each of which resolves or makes substantial progress on a long-standing open problem. The results were produced by an internal version of Astra, described as OpenAI's next major model, not yet released. According to OpenAI, the total number of tokens needed to find solutions to all ten problems would cost roughly $2,000 at what the post calls 'Sol API rates.' Humans then worked with the same model to prepare the arguments into manuscripts, after which the model formalized each argument as a Lean proof certificate. OpenAI says it is releasing, alongside each solution, a narration of the model's thinking process.

The ten problems span high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography and extremal combinatorics. The results are: new upper bounds on sphere-packing density down to the Cohn-Elkies threshold; exponentially improved bounds on the maximum size of binary codes at any prescribed minimum distance, with analogous results for spherical codes; a construction establishing the existence of non-sofic groups; a disproof of Connes's rigidity conjecture, that certain groups are uniquely determined by their von Neumann algebras; new lower bounds for computing the permanent with arithmetic circuits and formulas, including an arithmetic-formula lower bound of order n4/log n; an exponential parallel repetition theorem for general two-player quantum games; polynomial-factor hardness of approximation for the closest vector problem, a lattice question tied to post-quantum cryptography; a determination, in every dimension, of the maximum volume of a convex body whose centroid is its only interior lattice point (Ehrhart's volume conjecture); a superexponential lower bound for multicolor triangle Ramsey numbers, resolving Erdos problem 183; and results on the compactness and degeneracy conjectures in extremal graph theory, resolving Erdos problems 146 and 180.

OpenAI frames the release against its 'ChatGPT for Academic Researchers' initiative, which gives 100,000 scientists and mathematicians free access to its best ChatGPT models, and against an earlier disclosure: in May, OpenAI shared an AI-generated disproof of the Erdos unit-distance conjecture, found while evaluating an unreleased model, which the post says has already inspired further published work by other researchers. On attribution, OpenAI states 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; it says it takes responsibility for the correctness of the manuscripts and Lean formalizations, while the mathematical arguments themselves were generated by the system. The post acknowledges the 'Leiden declaration on AI and Mathematics' and its signers as representing views on AI's role in mathematics that OpenAI says it respects.

Key facts

An internal version of Astra, OpenAI's next major model, produced results on ten open problems in mathematics and theoretical computer science, each resolving or making substantial progress on the problem.
OpenAI estimates the token cost to find solutions to all ten problems at roughly $2,000 at 'Sol API rates'; humans then wrote up the manuscripts with the model, and the model formalized each argument as a Lean proof certificate.
Results include a disproof of Connes's rigidity conjecture, new lower bounds for computing the permanent (order n4/log n), a superexponential bound resolving Erdos problem 183, and resolutions of Erdos problems 146 and 180.
The release follows OpenAI's May disclosure of an AI-generated disproof of the Erdos unit-distance conjecture and sits alongside its 'ChatGPT for Academic Researchers' program, which gives 100,000 scientists free access to its best ChatGPT models.
OpenAI says it will not claim human authorship for AI-generated proofs, framing that as misrepresenting both the system's contribution and genuine human intellectual work, while taking responsibility for the correctness of the manuscripts and formalizations it prepared.

Why it matters

This is a claim of AI systems doing original work on hard, previously open problems across eight distinct mathematical fields at once, not a single narrow benchmark result. It follows OpenAI's May disclosure of a disproof of the Erdos unit-distance conjecture, which the post says already spawned follow-on published research from outside mathematicians. If the results hold up, it marks a step from AI solving competition-style or previously-solved problems toward contributing to problems the mathematical community had not been able to crack, using a model, Astra, that has not been released.

Who it affects

Working mathematicians and theoretical computer scientists in the affected subfields (high-dimensional geometry, coding theory, group theory, operator algebras, quantum complexity, lattice cryptography and extremal combinatorics) are the direct audience, since these are the people positioned to verify, extend or build on the results. More broadly, the 100,000 scientists and mathematicians who OpenAI says now have free access to its best ChatGPT models under the 'ChatGPT for Academic Researchers' initiative are the intended beneficiaries of this direction of work.

How to use it

Astra itself is described only as OpenAI's 'next major model' and is not stated to be released or available; there is no public access path to it in this post. The concrete access point OpenAI names is the separate 'ChatGPT for Academic Researchers' initiative, offering free access to its best current ChatGPT models to 100,000 scientists and mathematicians. OpenAI says it is publishing, alongside each of the ten results, a narration of the model's thinking process, in addition to the Lean-formalized proof certificates.

How solid is it

Each argument was, per OpenAI, formalized by the model into a Lean certificate, a machine-checkable proof format, after being written up into manuscripts by humans working with the same model. OpenAI states it takes responsibility for the correctness of the manuscripts and the Lean formalizations, while attributing the mathematical arguments themselves to the system. The post does not name the individual researchers who prepared the manuscripts, nor break down the $2,000 token-cost estimate by problem.

Risks and caveats

OpenAI's own wording is that each result 'resolves or makes substantial progress' on its problem, which is not the same as ten full solutions; some, like the Connes's rigidity conjecture disproof, are stated as resolved, while others, such as the new bounds and hardness results, read as partial advances rather than closed problems. The post names no individual authors for the manuscripts, gives no comparison to how long human mathematicians would take on the same problems, and does not describe what the released 'narration of thinking process' actually contains. OpenAI itself flags that the emergence of AI systems contributing to mathematical research raises questions it says cannot be answered by a technology company alone, and references the Leiden declaration on AI and Mathematics as representing concerns in the field about this shift.

“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 blog post