OpenAI Astra has produced machine-checkable proofs for ten long-standing problems in mathematics and theoretical computer science, a claim the company announced on Saturday along with a 249-page manuscript of model-generated reasoning and Lean 4 certificates.
New results span groups, conjectures and coding theory
The most notable achievement is an explicit construction of a non-sofic group, a question that has lingered since Mikhail Gromov introduced the concept in 1999.
A sofic group can be approximated by shuffling a finite deck of cards; every examined group to date fit that pattern, and no proof existed that all groups were sofic.
Astra’s construction provides the first known exception, and it also disproved Connes’s rigidity conjecture, which dates back to 1980.
The conjecture suggested that for a certain rigid class of groups, a related algebraic object—its von Neumann algebra—served as a unique fingerprint.
Astra generated infinitely many non-isomorphic groups that share the same fingerprint, challenging the conjecture’s core premise.
The collection also includes a proof of Ehrhart’s volume conjecture and solutions to three problems from Paul Erdős’s catalog.
They include problem 183 on multicolor Ramsey numbers, and the remaining problems touch on high-dimensional sphere packing, binary and spherical codes, arithmetic circuit complexity, quantum parallel repetition and the hardness of the closest vector problem.
This problem has implications for lattice-based cryptography.
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Formal verification and community response
All ten results are accompanied by Lean certificates hosted on GitHub under an Apache 2.0 license.
The repository reports a “sorry” count of zero, indicating that every step of each formalized proof compiled without gaps.
This technical validation removes reliance on the model’s internal reasoning, though mathematicians still need to confirm that each formal statement aligns with the original problem statements.
They must assess the significance of the findings, and none of the proofs have undergone peer review.
Thomas Bloom, maintainer of the erdosproblems.com database, described the announcements as “big news”.
OpenAI’s former vice president of science, Kevin Weil, had previously asserted that GPT-5 solved ten unsolved Erdős problems, a statement that sparked criticism.
The current Astra results have therefore been met with both enthusiasm for the technical advance and caution about the lack of external validation.
Human researchers turned Astra’s output into publishable papers, but OpenAI emphasized that the underlying mathematical arguments originated from the model.
The compute cost for all ten solutions was roughly $2,000 at GPT-5.6 API rates.
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It is a modest expense given the scale of the achievements.
These developments could reshape how mathematicians approach deep problems.
If AI can reliably generate formal proofs, researchers may delegate routine proof steps to machines.
The timing coincides with growing concerns from the mathematical community.
In June, the International Mathematical Union endorsed the Leiden Declaration, warning that AI firms are using published research without consent.
They are bypassing peer review, which could threaten the integrity of proof attribution.
Software engineer Fernando Borretti argued in a blog post that traditional defenses of human mathematicians may no longer hold.
He suggested the frontier of the field could recede beyond the reach of most practitioners.
Borretti warned of a “demon‑haunted world” filled with marvelous devices whose operation remains opaque.