{
  "id": "the-gaussian-propeller-bound",
  "title": "The Gaussian propeller bound",
  "description": "OpenAI's proof (2026) of the Gaussian propeller conjecture of Khot and Naor: the Gaussian first moments of the cells of any finite measurable partition of a Euclidean space have squared lengths summing to at most 9/(8 pi), with equality for three planar sectors of angle 2 pi/3. Duality with Gaussian maxima, half-space and spherical-cap rearrangement, Ehrhard concavity of gap probabilities, residual covariances and a probability budget with exact margins; a sharp inequality for Gaussian maxima and the threshold for kernel clustering.",
  "units": [
    {
      "id": "gaussian-first-moments-of-partitions",
      "title": "Gaussian first moments of partitions",
      "reader": "gaussian-first-moments-of-partitions.html",
      "source": "src/gaussian-first-moments-of-partitions.md",
      "source_sha256": "6bee026d519fc21e785a5841129cd446018c4c631b1bb5dbdbba7b9ccd0b9390",
      "status": "author_self_checked"
    },
    {
      "id": "rearrangement-bounds-for-one-cell",
      "title": "Rearrangement bounds for one cell",
      "reader": "rearrangement-bounds-for-one-cell.html",
      "source": "src/rearrangement-bounds-for-one-cell.md",
      "source_sha256": "d2ed505043959b350731e11100a4faf1dbe60f8cd08ac964681846afab042238",
      "status": "author_self_checked"
    },
    {
      "id": "ehrhard-concavity-and-the-loss-of-one-score",
      "title": "Ehrhard concavity and the loss of one score",
      "reader": "ehrhard-concavity-and-the-loss-of-one-score.html",
      "source": "src/ehrhard-concavity-and-the-loss-of-one-score.md",
      "source_sha256": "874578ea31725f7566b45c76e2d1122e5419a6a02fa31f3f83c4ff0c8a620f50",
      "status": "author_self_checked"
    },
    {
      "id": "two-competing-scores",
      "title": "Two competing scores",
      "reader": "two-competing-scores.html",
      "source": "src/two-competing-scores.md",
      "source_sha256": "b0342e120e95066e6189e7ddbd2da5bfc4d42d4ada92dc8d66b6b0a4ee903331",
      "status": "author_self_checked"
    },
    {
      "id": "five-or-more-cells-and-the-propeller-bound",
      "title": "Five or more cells and the propeller bound",
      "reader": "five-or-more-cells-and-the-propeller-bound.html",
      "source": "src/five-or-more-cells-and-the-propeller-bound.md",
      "source_sha256": "f32fe07ac45c34e54ffb5b920488ea40c88f13ab51a39924925a16ea3aba71b5",
      "status": "author_self_checked"
    },
    {
      "id": "kernel-clustering-and-the-gaussian-constant",
      "title": "Kernel clustering and the Gaussian constant",
      "reader": "kernel-clustering-and-the-gaussian-constant.html",
      "source": "src/kernel-clustering-and-the-gaussian-constant.md",
      "source_sha256": "33155f463b9c6b9544559fe9af883f50e4d62c82b27c1a482b15fa716b5a612e",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core course Measure and Integration (Fremlin, Measure Theory, Volumes 1 and 2): dominated convergence and differentiation under the integral sign (123C, 123D); Fubini's theorem (252B); linear changes of variable (263A); polar coordinates in the plane (263G); Hausdorff measure under Lipschitz maps (264G); integration over spheres and their surface area (265G, 265H)",
    "Gaussian vectors: covariance, densities, independence of uncorrelated jointly Gaussian variables",
    "for the last lesson: Unique Games instances and polynomial-time reductions",
    "Ehrhard's inequality for closed convex sets: Theorem A and Corollary 3.4 of the programme course ehrhards-inequality-for-convex-sets (complete proof, 17 Oct 2026), used in Lemma 3.1 of the third lesson"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}