{
  "id": "the-dyadic-triangular-hilbert-form",
  "title": "The dyadic triangular Hilbert form",
  "description": "OpenAI's theorem (2026) that the dyadic triangular Hilbert form is bounded on L^3 x L^3 x L^3 uniformly in the scales and with no condition on the inputs: the absolute values of the local forms over admissible triples of dyadic intervals, summed over any finite set of scales, are at most 40 times the product of the L^3 norms. Step functions as matrices; the energy tr(P^(3/2)), its Hessian and its comparison under the matrix order; a dimension-free trace estimate for a triangle of matrices; energy increase under splitting into four children, and telescoping.",
  "units": [
    {
      "id": "dyadic-triangles-and-the-main-estimate",
      "title": "Dyadic triangles and the main estimate",
      "reader": "dyadic-triangles-and-the-main-estimate.html",
      "source": "src/dyadic-triangles-and-the-main-estimate.md",
      "source_sha256": "0bb0ded2cec4ff1ad328e524b47aa2c1497036726439595beb503823f4980b8e",
      "status": "author_self_checked"
    },
    {
      "id": "a-matrix-energy-and-a-trace-inequality",
      "title": "A matrix energy and a trace inequality",
      "reader": "a-matrix-energy-and-a-trace-inequality.html",
      "source": "src/a-matrix-energy-and-a-trace-inequality.md",
      "source_sha256": "620cfb1cbcf4d0d020d71da581b1dedbe28c69c81a16dff85df1d83f25d3b7ee",
      "status": "author_self_checked"
    },
    {
      "id": "energy-increase-and-telescoping",
      "title": "Energy increase and telescoping",
      "reader": "energy-increase-and-telescoping.html",
      "source": "src/energy-increase-and-telescoping.md",
      "source_sha256": "f467a44fbefbc764c7e071c2b20d03d0ce8e7190d68ba522999cd3035a5c066d",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "linear algebra over R: symmetric matrices, the spectral theorem, positive semidefinite order, the trace; the inverse function theorem in finite dimensions",
    "the Lebesgue integral, L^p spaces and Holder's inequality, as in Fremlin's Measure Theory (core course Measure and Integration)"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}