Result 277, Mathematical physics

Threshold repetition for entangled games

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+δv+\delta of k independent repetitions decays exponentially in k, for 0<δ<1−v0\lt \delta\lt 1-v. Arbitrary joint finite-dimensional entangled strategies and correlated question distributions are allowed.

Lean formalization New or sharp bound

The bigger picture

Why it matters

Quantum players can coordinate across repeated games using shared entanglement. This result says that such coordination cannot make scores above their best single-game success rate persist: the chance falls exponentially as repetitions increase.

What changes?

A finite two-player, one-round game asks each player a question and scores their answers. Its entangled value v is the best winning probability using finite-dimensional quantum entanglement. For v below one, the manuscript reports exponential decay in the chance of winning at least a fraction v + delta of k independent repetitions. This holds for every positive integer k and delta strictly between zero and one minus v, allowing arbitrary joint finite-dimensional strategies and correlated question pairs.

What does that help mathematicians do?

The universal exponential rate is proportional to delta to the fifth power divided by one plus the logarithm of the product of the two answer-set sizes. For each fixed question distribution, the manuscript also gives an explicit rate cubic in delta. These bounds quantify how rapidly an above-value score becomes unlikely, even when the players use one entangled strategy across all repetitions rather than independent strategies.

Are there practical applications?

Its immediate value is foundational: it limits what quantum correlations can accomplish across many trials and supplies a tool for analyzing repeated games without assuming independent player behavior. The demonstrated guarantee concerns winning probabilities, not experimental performance or an efficient way to compute the entangled value or implement optimal strategies.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Threshold parallel repetition for finite-dimensional entangled games

September 25, 2026 22 pages Main result formalized in Lean

For every finite two-player game with entangled value v < 1, we prove exponential decay for the probability of winning at least a v+δv+\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<δ<1−v0\lt \delta\lt 1-v and k ≥ 1. Its universal rate is proportional to δ5/(1+log⁡(∣A∣∣B∣))\delta^5/(1+\log(|\mathcal A||\mathcal B|)), where A,B\mathcal A,\mathcal B are the answer alphabets. For each fixed question distribution, we also obtain an explicit cubic rate in δ.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Threshold-parallel-repetition-for-finite-dimensional-entangled-games-September-25-2026,
  author = {{OpenAI}},
  title = {{Threshold parallel repetition for finite-dimensional entangled games}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Threshold-parallel-repetition-for-finite-dimensional-entangled-games-September-25-2026/paper.pdf}{OAI:Threshold-parallel-repetition-for-finite-dimensional-entangled-games-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/277.md.

Threshold repetition for entangled games

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization gives exponential threshold parallel repetition for every finite two-player game with arbitrary correlated questions. If its finite-dimensional entangled value is v<1v<1, its answer sets are A,BA,B, and 0<δ<1−v0<\delta<1-v, then the probability of winning at least the fraction v+δv+\delta of kk repetitions is at most exp⁡(−κδ13k/(1+log⁡(∣A∣∣B∣)))\exp(-\kappa\delta^{13}k/(1+\log(|A||B|))) for one universal κ>0\kappa>0. The bound applies to every joint finite-dimensional strategy and to their supremum, without assuming attainment. The separate distribution-dependent cubic bound is outside this scope.

Comparator links

Result Comparator statement
Threshold parallel repetition for entangled games EntangledGames.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.