OpenAI reported three significant discoveries spanning mathematics, computer science and physics, while also introducing a novel partnership framework that assigns external experts, termed “shepherds,” to oversee the dissemination of results. The announcement follows a recent wave of AI-produced solutions to longstanding mathematical problems.
In mathematics, the system generated a proof of the local smoothing conjecture for the Euclidean wave equation in every spatial dimension of two or higher. The proof simultaneously settles the Kakeya maximal and dimension conjectures and confirms Stein’s spherical restriction estimates throughout the interior of the conjectured range. OpenAI released the raw model output directly, replicating its October 6 approach.
The computer-science breakthrough presents a deterministic algorithm that counts satisfying assignments of fixed-width CNF formulas in subexponential time while using only polynomial space, thereby refuting the Exponential Time Hypothesis, especially for the 3-SAT case. An independent group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams will scrutinize the proof, and OpenAI posted the AI-generated write-up and Lean formalization on GitHub.
OpenAI also announced a major physics discovery, though the release did not include technical details. The result is being guided by shepherds, mirroring the oversight model applied to the computer-science finding.
The company’s release strategy diverges from earlier practices: for the mathematics result it followed advice to make the AI output publicly available for researchers to develop their own explanations, while for computer science and physics it employed shepherds to manage verification. The AGMAI panel of math advisers warned against preferential early-access schemes, citing unequal access, borrowed reputation and selection bias, and urged future releases on scholarly repositories with Lean-formalized proofs and dedicated funding for verification work.
To address the verification challenge, OpenAI committed $5 million to support mathematicians in interpreting AI-generated research. The funding aims to cover efforts to check, simplify, explain and extend the new results, and to explore how credit and conflict-of-interest policies should evolve as AI-driven discoveries become more common.
The timing of the math breakthrough aligns with recent progress on the Kakeya conjecture, exemplified by NYU professor Hong Wang’s Fields Medal for resolving the three-dimensional case. OpenAI’s claim of resolving the Kakeya and Stein conjectures in all dimensions underscores the potential impact of AI-assisted proof techniques on longstanding open problems.