{
  "id": "threshold-repetition-for-entangled-games",
  "title": "Threshold repetition for entangled games",
  "description": "For every finite two-player one-round game with entangled value v < 1, the probability of winning at least a fraction v + delta of k parallel repetitions decays exponentially in k, uniformly over all finite-dimensional entangled strategies and correlated question distributions. The proof: sampling tested coordinates turns a conditioning estimate into a threshold bound; resolvent purifications put conditioned states in common spaces with exact answer probabilities; an entropy telescope controls how they depend on the other player's question; one reference vector per support component yields legal one-game strategies and a delta^3 rate. A universal delta^5 rate rests on Song's conditioning estimate, a named hypothesis. The course is an exposition of OpenAI's proof (2026).",
  "units": [
    {
      "id": "entangled-games-and-threshold-repetition",
      "title": "Entangled games and threshold repetition",
      "reader": "entangled-games-and-threshold-repetition.html",
      "source": "src/entangled-games-and-threshold-repetition.md",
      "source_sha256": "9bfccd449ecff9e79453f04bdef88eb3ea4f0309d549fe050b18399d75020690",
      "status": "author_self_checked"
    },
    {
      "id": "from-conditioning-to-a-threshold-bound",
      "title": "From conditioning to a threshold bound",
      "reader": "from-conditioning-to-a-threshold-bound.html",
      "source": "src/from-conditioning-to-a-threshold-bound.md",
      "source_sha256": "6018361b87b6b06a1ec0b7da930c6a146146b0b25718bc62ca9863d74eb116db",
      "status": "author_self_checked"
    },
    {
      "id": "a-common-purification-of-effects",
      "title": "A common purification of effects",
      "reader": "a-common-purification-of-effects.html",
      "source": "src/a-common-purification-of-effects.md",
      "source_sha256": "e442f988cb9442f9523cd0027a70aab53ba30516ba9b7b3489a394edd9f58475",
      "status": "author_self_checked"
    },
    {
      "id": "conditioning-on-a-tested-set",
      "title": "Conditioning on a tested set",
      "reader": "conditioning-on-a-tested-set.html",
      "source": "src/conditioning-on-a-tested-set.md",
      "source_sha256": "6c8142dd400d595d2bf12eaa5a0e932724e051d109bfb95dc132014e37d99d8d",
      "status": "author_self_checked"
    },
    {
      "id": "a-cubic-bound-for-a-fixed-question-distribution",
      "title": "A cubic bound for a fixed question distribution",
      "reader": "a-cubic-bound-for-a-fixed-question-distribution.html",
      "source": "src/a-cubic-bound-for-a-fixed-question-distribution.md",
      "source_sha256": "35f741f88f243f8346f3978325af4cfae0b511354f0f3488f1a6d2cf6d921ddf",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "course The Gaussian moat problem, lesson Entropy of finite random variables: chain rule and conditioning for entropy (Propositions 2.2 and 2.3), relative entropy and total variation (Definition 5.1), Pinsker's inequality (Theorem 5.3)",
    "course The Gaussian moat problem, lesson Concentration and the discrete cube: Hoeffding's inequality (Theorem 2.2)",
    "finite-dimensional linear algebra: the spectral theorem for self-adjoint matrices, positive operators, tensor products",
    "the named hypothesis (S) of the first lesson, used only for the universal fifth-power theorem and the soundness half of the amplification corollary; the lessons prove every other fact they use"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}