OpenAI says its model has disproved the Exponential Time Hypothesis, proved the Kakeya conjecture in all dimensions and established the existence of quantum chromodynamics; the ETH claim is under review by a group that includes the hypothesis’s authors

OpenAI says its model has disproved the Exponential Time Hypothesis, proved the Kakeya conjecture in all dimensions and established the existence of quantum chromodynamics; the ETH claim is under review by a group that includes the hypothesis’s authors

On October 26, the company released manuscripts on algorithms and mathematics. It commissioned groups of researchers to review the algorithmic and physics results; the physics manuscript remains unpublished.

The Exponential Time Hypothesis (ETH) concerns 3-SAT: given a logical formula made up of clauses such as “x or not y or z,” the task is to assign values to n variables so that every clause is satisfied. Exhaustive search takes about 2^n steps, and ETH states that, in the worst case, any algorithm requires at least 2^(cn) steps for some c > 0. According to OpenAI, hundreds of papers use ETH to derive limits on how fast algorithms can run. In the manuscript, the model describes an algorithm that solves 3-SAT in roughly 2^(n/√log n) time by exactly counting the solutions to a formula with short clauses. For sufficiently large n, the exponent n/√log n is smaller than any fixed positive fraction of n, so the algorithm runs faster than 2^(cn) for any positive c, contradicting ETH. Computer verification in Lean succeeded for a version with a slightly weaker running-time bound that also contradicts ETH.

Complexity theorist Scott Aaronson wrote in the comments on his blog post that the algorithm works for formulas with short clauses, while exhaustive search remains the best known method for arbitrary Boolean circuits. He raised his estimate of the probability that P = NP (every problem whose solution can be checked quickly can also be solved quickly) from 3 to 5 %: “Essentially, a foot-high fence has been cleared, while a fence as tall as the Empire State Building is still standing.”

The claimed disproof is being reviewed by Russell Impagliazzo and Ramamohan Paturi, who proposed ETH in the late 1990s, and Ryan Williams of the Massachusetts Institute of Technology. In an open letter, they write that their participation “should not be taken as an endorsement” of OpenAI’s claim. At their request, OpenAI released all the materials.

The second result concerns local smoothing. In 1991, Christopher Sogge conjectured that when a wave converges on a point, its strongest amplification is brief: at a randomly chosen moment, the wave is much weaker than at its peak. This implies the Kakeya conjecture: a set containing a line segment of length 1 in every direction has the same dimension as the ambient space, even though its volume may be zero. Hong Wang, who received the Fields Medal in 2026, and Joshua Zahl proved the Kakeya conjecture in three dimensions in 2025. OpenAI says its model has proved local smoothing, and therefore the Kakeya conjecture, in every dimension. On the advice of an invited expert, the company immediately released the manuscript and the proof formalized in Lean without convening a review group, following the same approach it took when it released 722 manuscripts on October 6.

The third claim is a proof that quantum chromodynamics, the theory of quarks and gluons, exists in four dimensions as a rigorous mathematical object. This includes pure Yang-Mills theory (the theory of gluons alone), whose existence is part of a Millennium Prize Problem. A group of five researchers led by Nobel laureate David Gross is reviewing the result. The group decided to withhold the model’s proof for now and write its own account.

On October 25, the day before the release, the Advisory Group on Mathematics and AI recommended that companies avoid selecting mathematicians to receive results in advance: their names “vouch for the release and shield the AI company from criticism.” On October 26, it called for the physics materials to be made public: since OpenAI had attributed the delay to formalization in Lean and had completed that work, “there are no scientific reasons to delay the release further.”