{
  "id": "average-sensitivity-of-threshold-functions",
  "title": "Average sensitivity of threshold functions",
  "description": "OpenAI's proof (2026) of the asymptotic Gotsman–Linial conjecture: a polynomial threshold function of degree d on the n-dimensional Boolean cube has average sensitivity at most 8d√n. The course proves a one-sided coupling inequality, builds a grading of the functions on the cube weighted by the polynomial, counts sensitive edges through an anticommutator, and turns the degree bound into a binomial coupling; it also derives a bound on noise sensitivity.",
  "units": [
    {
      "id": "threshold-functions-and-a-one-sided-coupling",
      "title": "Threshold functions and a one-sided coupling",
      "reader": "threshold-functions-and-a-one-sided-coupling.html",
      "source": "src/threshold-functions-and-a-one-sided-coupling.md",
      "source_sha256": "3476204373991051f4217cd3bbe7640cae04f983f4d19e798994343397fb59d8",
      "status": "author_self_checked"
    },
    {
      "id": "a-weighted-grading-of-the-cube",
      "title": "A weighted grading of the cube",
      "reader": "a-weighted-grading-of-the-cube.html",
      "source": "src/a-weighted-grading-of-the-cube.md",
      "source_sha256": "8c409e1f2966c0c786fdc0efca5292b0c0b2d1ef6f1a27232e114b994516830a",
      "status": "author_self_checked"
    },
    {
      "id": "the-gotsman-linial-bound",
      "title": "The Gotsman–Linial bound",
      "reader": "the-gotsman-linial-bound.html",
      "source": "src/the-gotsman-linial-bound.md",
      "source_sha256": "23759136baa2608edbe49f7b14877e004b8b9ed07f18867a949111954df45f65",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core course Linear Algebra; foundations-of-von-neumann-algebras lesson hilbert-spaces-and-compact-operators (Theorem 2.2, projection theorem; Section 3, adjoints), used in finite dimensions",
    "sensitivity-and-block-sensitivity lesson sensitivity-block-sensitivity-and-composition (definitions, for comparison)"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}