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+\delta of k independent repetitions decays exponentially in k, for 0\lt \delta\lt 1-v. Arbitrary joint finite-dimensional entangled strategies and correlated question distributions are allowed. (Lean)
Manuscript
Threshold parallel repetition for finite-dimensional entangled games
September 25, 2026 · source · Lean formalization available · cite as \cite{OAI:Threshold-parallel-repetition-for-finite-dimensional-entangled-games-September-25-2026}
For every finite two-player game with entangled value v < 1, we prove exponential decay for the probability of winning at least a v+\delta fraction of k independent repetitions, uniformly over all finite-dimensional joint strategies. The result allows arbitrary correlated question distributions and holds for every 0\lt \delta\lt 1-v and k ≥ 1. Its universal rate is proportional to \delta^5/(1+\log(|\mathcal A||\mathcal B|)), where \mathcal A,\mathcal B are the answer alphabets. For each fixed question distribution, we also obtain an explicit cubic rate in δ.
Result 277 · Mathematical physics. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md