{
  "id": "depth-three-circuits-beyond-the-square-root",
  "title": "Depth-three circuits beyond the square root",
  "description": "OpenAI's theorem (2026) that one explicit polynomial-time language needs depth-three OR-AND-OR circuits with more than 2^(A sqrt n) gates for every constant A: a restriction bound for CNFs that does not depend on the number of clauses, a disjoint sparse partition by sunflowers, slices of a hash-and-polynomial language with limited independence, and a count of all gates.",
  "units": [
    {
      "id": "depth-three-circuits-and-the-square-root-barrier",
      "title": "Depth-three circuits and the square-root barrier",
      "reader": "depth-three-circuits-and-the-square-root-barrier.html",
      "source": "src/depth-three-circuits-and-the-square-root-barrier.md",
      "source_sha256": "fc408cb033707af0ef2b2a9ee4eca479ddc45c21a90b713863bed58a83524e69",
      "status": "author_self_checked"
    },
    {
      "id": "a-finite-expansion-of-a-restricted-cnf",
      "title": "A finite expansion of a restricted CNF",
      "reader": "a-finite-expansion-of-a-restricted-cnf.html",
      "source": "src/a-finite-expansion-of-a-restricted-cnf.md",
      "source_sha256": "1d273dc15eaafbbf68b523e44d7ef41fa33f2f1e1eeceb4fcede6501a21f9d65",
      "status": "author_self_checked"
    },
    {
      "id": "path-weights-and-backward-counting",
      "title": "Path weights and backward counting",
      "reader": "path-weights-and-backward-counting.html",
      "source": "src/path-weights-and-backward-counting.md",
      "source_sha256": "d49b1d69054742054cc7d6f81dc7d099e2c713b423f211f78a9b2d9af6058f5f",
      "status": "author_self_checked"
    },
    {
      "id": "sunflowers-and-a-disjoint-sparse-partition",
      "title": "Sunflowers and a disjoint sparse partition",
      "reader": "sunflowers-and-a-disjoint-sparse-partition.html",
      "source": "src/sunflowers-and-a-disjoint-sparse-partition.md",
      "source_sha256": "eb27fe00c57cfc9c1d0083dc06b9913198c44a30f362a1c86fca2ec79c1b12a5",
      "status": "author_self_checked"
    },
    {
      "id": "an-explicit-language",
      "title": "An explicit language",
      "reader": "an-explicit-language.html",
      "source": "src/an-explicit-language.md",
      "source_sha256": "1ffb3c85778846ade6190dfc6ffa531fa7c9637f2f1e183ca0c9ee894dba6cc8",
      "status": "author_self_checked"
    },
    {
      "id": "hard-slices",
      "title": "Hard slices",
      "reader": "hard-slices.html",
      "source": "src/hard-slices.md",
      "source_sha256": "7ea10d4bf12cb8790aa8170f52fc62dbc41604507778531762b52a6d2dd1e4b7",
      "status": "author_self_checked"
    },
    {
      "id": "the-lower-bound",
      "title": "The lower bound",
      "reader": "the-lower-bound.html",
      "source": "src/the-lower-bound.md",
      "source_sha256": "15219badc652712341c58f351e85d474ffc70297d830d39aeef5114f43309120",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "finite probability: expectation, independence, Markov's inequality, the union bound",
    "the existence of a field with 2^r elements (core course Abstract Algebra II, Judson, Theorem 22.1.6) and division with remainder for polynomials over a field",
    "the course proves every other fact it uses"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}