OpenAI Claims AI Disproves ETH and Proves Yang-Mills Existence
OpenAI announced that its internal models achieved three major breakthroughs in computer science, physics, and mathematics, including disproving the Exponential Time Hypothesis (ETH), proving the existence of Yang-Mills theory, and solving the Kakeya conjecture. While some results have been formally verified via Lean and code is public, materials for the physics proof remain unpublished and are under independent review by an external expert panel. This marks a new phase for AI in tackling long-standing fundamental scientific problems, though it has sparked controversy regarding peer review and academic attribution.
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OpenAI Claims AI Overturned the "Exponential Time Hypothesis" in Computer Science and Proved the Existence of Yang-Mills Theory; External Expert Panels Begin Verification
On October 26, OpenAI announced that its internal models produced one breakthrough each in mathematics, computer science, and physics. Two of these achievements address decades-old problems. These claims remain unverified by peer review, and as of this writing, no experts involved in the verification process have publicly commented on the validity of the conclusions.
Computer Science: A Foundational Assumption Challenged
The "Satisfiability Problem" (SAT) involves a set of conditions such as "at least one of A or B is true," asking whether they can all be satisfied simultaneously. For every additional variable, the number of combinations to test doubles.
In 1999, Russell Impagliazzo and Ramamohan Paturi proposed the "Exponential Time Hypothesis" (ETH), asserting that there are no fundamentally faster algorithms for such problems and that runtime must grow exponentially. Hundreds of papers have used ETH as a premise to argue that many problems cannot be solved quickly.
A paper published by OpenAI presents a new algorithm that remains slow but grows more slowly than predicted by ETH, which is sufficient to refute the hypothesis. It does not prove that "all hard problems have fast solutions" (i.e., it does not prove P=NP). OpenAI also notes that translating this into practical speedups will require significant further research.
OpenAI states that the proof has been verified using Lean. Lean is a proof assistant that breaks proofs down into minimal steps for line-by-line verification; if any step fails, the entire proof is rejected. The paper and Lean code are available on GitHub for anyone to review. The verification panel is led by Impagliazzo, Paturi, and Ryan Williams from MIT.
Physics: Half of a Millennium Prize Problem
Quantum Chromodynamics (QCD) describes how quarks and gluons bind together via the "strong force" to form atomic nuclei. While calculations align with experimental results, no one has rigorously proven its mathematical consistency. Its simplified version, "Yang-Mills theory," is one of the seven Millennium Prize Problems, carrying a $1 million award from the Clay Mathematics Institute.
OpenAI claims its model wrote proofs for the existence of both theories in September and completed Lean verification in mid-October. However, the Yang-Mills Millennium Problem requires proving two things: the existence of the theory and the presence of a "mass gap" (meaning the lightest particle still has mass). OpenAI only claims to have addressed the first part.
The physics materials are not yet public. An expert panel led by Nobel laureate David Gross is currently drafting their own paper based on these findings, meaning external parties cannot yet inspect the work.
Mathematics: Kakeya Conjecture Extended to All Dimensions
The Kakeya problem asks: What is the minimum size of a shape that can contain a unit line segment pointing in every direction? While the area can approach zero, mathematicians conjecture that its "dimension" is always full. In 2025, Hong Wang and Joshua Zahl proved the three-dimensional case. OpenAI claims its model proved a stronger conjecture, thereby resolving the problem for all dimensions. The paper and Lean proof have been directly published.
Why the Change in Publication Strategy?
In September, OpenAI announced a solution to the Navier-Stokes Millennium Problem, leading to disputes over authorship when some mathematicians claimed it conflicted with their unpublished research. On October 6, OpenAI released 722 AI-generated papers at once, retracting three due to errors the following day. This time, OpenAI has shifted to having external experts lead the effort: they do not sign non-disclosure agreements, OpenAI does not seek academic credit, and the experts participate independently without compensation.
OpenAI Original Post: https://t.co/ANHLKq5ocW Papers and Lean Proofs: https://t.co/aPgNSqXevr
↳ Quoting @gdb (Greg Brockman): Announcing three major scientific discoveries, one each in mathematics (harmonic analysis), computer science (complexity theory), and physics (quantum field theory).
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