OpenAI unveils three AI-produced results: a claimed refutation of the Exponential Time Hypothesis, local smoothing in every dimension, and an existence proof for QCD
In an Oct 26 blog post, OpenAI said its internal models produced three new results: a proof of Sogge's local smoothing conjecture in every dimension (which implies the Kakeya and Stein restriction conjectures), a Lean-verified proof that the Exponential Time Hypothesis (ETH) is false, and a rigorous existence proof for four-dimensional QCD and related quantum field theories including pure Yang-Mills. The math and CS papers and Lean proofs are in the openai/math GitHub repo; the physics artifacts are being held back while an external group led by David Gross writes them up. All of this is OpenAI's own claim and still awaits independent verification.

Key Takeaways
- Computer science (per OpenAI): the paper gives a deterministic algorithm that exactly counts satisfying assignments of fixed-width CNF formulas in subexponential time and polynomial space, and the 3-SAT case refutes ETH; the AI found the proof on Oct 16 and it was formalized in Lean on Oct 17
- ETH was proposed by Impagliazzo, Paturi and Zane and has been used in hundreds of papers for conditional lower bounds; the verification group is led by Impagliazzo, Paturi (UC San Diego) and Ryan Williams (MIT), and OpenAI has committed to fund a conference
- Math: local smoothing is proved in every spatial dimension n>=2, which yields the Kakeya and Nikodym maximal conjectures and Stein restriction estimates; earlier work only covered two dimensions (2019) and three-dimensional Kakeya (2025)
- Physics: the natural-language Yang-Mills existence proof was produced on Sep 9 and the QCD proof on Sep 18, with the Yang-Mills Lean formalization finished Oct 16; a five-person group led by David Gross will write it up and the artifacts are not public yet
- OpenAI says it seeks no academic credit, and outside participants are unpaid volunteers with no NDAs; the openai/math repo withdrew 3 manuscripts on Oct 7 over a sign error, so these claims still need independent checking
- Physics follow-up (Oct 28): KITP published the Yang-Mills Working Group statement. The committee is Michael Douglas, David Gross (chair), Arthur Jaffe, Igor Klebanov and Nikita Nekrasov; OpenAI asked Gross on Oct 16 to take custody of the proof, and the committee says it has no responsibility to OpenAI. It will first check that the Lean formalization matches standard definitions before releasing the Lean proof, then publish the full 1,147-page proof with its assessment once satisfied; confinement and the mass gap remain open
