Threshold repetition for entangled games

What OpenAI claims

Proves exponential threshold repetition for every finite two-player one-round game: if its entangled value is v < 1, the probability of winning at least a fraction v+δ of k independent repetitions decays exponentially in k, for 0 < δ < 1-v. Arbitrary joint finite-dimensional entangled strategies and correlated question distributions are allowed. (Lean)

Paper

1. Threshold parallel repetition for finite-dimensional entangled games

   25 September 2026 · Preprint, not peer reviewed · PDF

   For every finite two-player game with entangled value v < 1, we prove exponential decay for the probability of winning at least a v+δ fraction of k independent repetitions, uniformly over all finite-dimensional joint strategies. The result allows arbitrary correlated question distributions and holds for every 0 < δ < 1-v and k ≥ 1. Its universal rate is proportional to δ⁵/(1+log(|𝒜||ℬ|)), where 𝒜,ℬ are the answer alphabets. For each fixed question distribution, we also obtain an explicit cubic rate in δ.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.