{
  "id": "matrix-multiplication-with-exponent-at-most-nine-fourths",
  "title": "Matrix multiplication with exponent at most 9/4",
  "description": "OpenAI's theorem (2026) that the exponent of matrix multiplication is at most 9/4: two n x n complex matrices can be multiplied with O(n^(9/4+eps)) arithmetic operations. Tensors, rank and Strassen's characters; a Fourier separation of blocks sharing one leg and an entropy inequality; a symmetrized profile of polynomial multiplication that is discretely concave and satisfies shifted tripling, which forces lambda(T_d) <= d^(9/4) for every character; characters that are large on matrix multiplication, from a catalytic comparison and Schauder's fixed-point theorem.",
  "units": [
    {
      "id": "tensors-rank-and-the-exponent",
      "title": "Tensors, rank and the exponent",
      "reader": "tensors-rank-and-the-exponent.html",
      "source": "src/tensors-rank-and-the-exponent.md",
      "source_sha256": "112bbeec3625d7ac52f6e94ac1fc7dbd0d1f05a5a84d418d264ea822f4b90cd8",
      "status": "author_self_checked"
    },
    {
      "id": "characters-and-degenerations",
      "title": "Characters and degenerations",
      "reader": "characters-and-degenerations.html",
      "source": "src/characters-and-degenerations.md",
      "source_sha256": "42e9629323583e8f2c488de65abed7bc659206237f095c0d798d9920d44f14f9",
      "status": "author_self_checked"
    },
    {
      "id": "separating-a-shared-leg",
      "title": "Separating a shared leg",
      "reader": "separating-a-shared-leg.html",
      "source": "src/separating-a-shared-leg.md",
      "source_sha256": "22edcc066fd57660ff482c2b8a877500459fb11e6a8bdaa6c61689f72fe725be",
      "status": "author_self_checked"
    },
    {
      "id": "two-inequalities-for-polynomial-multiplication",
      "title": "Two inequalities for polynomial multiplication",
      "reader": "two-inequalities-for-polynomial-multiplication.html",
      "source": "src/two-inequalities-for-polynomial-multiplication.md",
      "source_sha256": "af620bfaf76ab7780d8b9156da28d13a577cfefc0f9d13612a140c87e5be8745",
      "status": "author_self_checked"
    },
    {
      "id": "diagonal-growth-and-the-bound-on-characters",
      "title": "Diagonal growth and the bound on characters",
      "reader": "diagonal-growth-and-the-bound-on-characters.html",
      "source": "src/diagonal-growth-and-the-bound-on-characters.md",
      "source_sha256": "fb5fdea11ce6b10d07d0fd8cbe92f51d3bf6d065ba728e25b04ad1e30e4bb41c",
      "status": "author_self_checked"
    },
    {
      "id": "states-and-a-catalytic-comparison",
      "title": "States and a catalytic comparison",
      "reader": "states-and-a-catalytic-comparison.html",
      "source": "src/states-and-a-catalytic-comparison.md",
      "source_sha256": "d9de41ebbfbcdbe4081521b9b8c446e417b967cb1beb460783a43ed55ca4131a",
      "status": "author_self_checked"
    },
    {
      "id": "multiplicative-states-and-the-proof-of-theorem-a",
      "title": "Multiplicative states and the proof of Theorem A",
      "reader": "multiplicative-states-and-the-proof-of-theorem-a.html",
      "source": "src/multiplicative-states-and-the-proof-of-theorem-a.md",
      "source_sha256": "a868ccf0e3224c246933a087603b10d17abab69e1b2f422a51925b559ab1d1c2",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "linear algebra over a field: bases, matrices of bilinear maps, rank, row reduction, Vandermonde matrices (core course Linear Algebra)",
    "metric spaces, compactness and sequences; Schauder's fixed-point theorem for compact convex subsets of normed spaces, Theorem 5.1 of the lesson Ultrapowers and Darbo's fixed-point theorem (course Tingley's problem)"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}