{
  "id": "heilbronns-triangle-problem",
  "title": "Heilbronn's triangle problem",
  "description": "OpenAI's theorem (2026) that n points in the unit square can have every triangle of area at least a constant times n^(-2+eta) for a fixed eta > 0; through field norms written in a large base, residue orbits, a cap at an auxiliary prime, lattice counts and a deletion argument.",
  "units": [
    {
      "id": "small-triangles-and-the-plan",
      "title": "Small triangles and the plan",
      "reader": "small-triangles-and-the-plan.html",
      "source": "src/small-triangles-and-the-plan.md",
      "source_sha256": "1fe13c9949ada33acaa81c92b03770eed80ee134088b56f86507c52f9941327f",
      "status": "author_self_checked"
    },
    {
      "id": "counting-lattice-matrices",
      "title": "Counting lattice matrices",
      "reader": "counting-lattice-matrices.html",
      "source": "src/counting-lattice-matrices.md",
      "source_sha256": "85b8e511ab890f7e29140fa3736fd7c05fd762ef6f28de9efd030ff1186806cb",
      "status": "author_self_checked"
    },
    {
      "id": "a-determinant-obstruction-from-field-norms",
      "title": "A determinant obstruction from field norms",
      "reader": "a-determinant-obstruction-from-field-norms.html",
      "source": "src/a-determinant-obstruction-from-field-norms.md",
      "source_sha256": "99edd6d53d9f5c6ef8842ec488e05a6c3256d27a849415d504b9585af0ec5e9a",
      "status": "author_self_checked"
    },
    {
      "id": "residue-matrices-and-their-orbits",
      "title": "Residue matrices and their orbits",
      "reader": "residue-matrices-and-their-orbits.html",
      "source": "src/residue-matrices-and-their-orbits.md",
      "source_sha256": "9e2ed6685ada00af181a494f0a1c0da145fd6cec67f595d6886e08f713797c55",
      "status": "author_self_checked"
    },
    {
      "id": "a-cap-at-an-auxiliary-prime",
      "title": "A cap at an auxiliary prime",
      "reader": "a-cap-at-an-auxiliary-prime.html",
      "source": "src/a-cap-at-an-auxiliary-prime.md",
      "source_sha256": "98af06672c72a52694fda45f6cc1b4fa2d1ed8ca1003f6205078f29bc1e2df0a",
      "status": "author_self_checked"
    },
    {
      "id": "sampling-integer-columns",
      "title": "Sampling integer columns",
      "reader": "sampling-integer-columns.html",
      "source": "src/sampling-integer-columns.md",
      "source_sha256": "9c071b9ce56f945c730a4f4746397120f4ab9f00ba8cdd1ce33b25d0cc9bcf42",
      "status": "author_self_checked"
    },
    {
      "id": "determinant-zero",
      "title": "Determinant zero",
      "reader": "determinant-zero.html",
      "source": "src/determinant-zero.md",
      "source_sha256": "2239719f15eb2323afd84b24283d468819e08b047c835e181cce9278922846ab",
      "status": "author_self_checked"
    },
    {
      "id": "deletion-and-the-theorem",
      "title": "Deletion and the theorem",
      "reader": "deletion-and-the-theorem.html",
      "source": "src/deletion-and-the-theorem.md",
      "source_sha256": "6661cf7651492adce8d9cd4e645bc148785dff86548c48eacc8167dabb524534",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "Real Analysis II (core course): the extreme value theorem on compact metric spaces",
    "Abstract Algebra I and II (core courses): the orbit-stabilizer theorem, splitting fields, the Frobenius map and finite fields",
    "Measure and Integration (core course): Lebesgue measure of boxes, translation invariance and the linear change of variables",
    "Thom classes and Euler classes (DG-CHAR-06), Lemma 5.1: subgroups of free abelian groups of finite rank are free"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}