{
  "id": "the-lang-plaut-problem",
  "title": "The Lang–Plaut problem",
  "description": "OpenAI's negative answer (2026) to the Lang–Plaut problem: a doubling subset of Hilbert space that admits no bi-Lipschitz embedding into any finite-dimensional space. The course builds the set from coloured strips at rapidly decreasing scales and proves its doubling bound, differentiates the Lipschitz sheets of a supposed embedding near a lexicographically maximal derivative, proves two energy estimates, and turns crossings of rows and columns into a packing contradiction in Euclidean space.",
  "units": [
    {
      "id": "doubling-spaces-and-coloured-strips",
      "title": "Doubling spaces and coloured strips",
      "reader": "doubling-spaces-and-coloured-strips.html",
      "source": "src/doubling-spaces-and-coloured-strips.md",
      "source_sha256": "14bf68f4f64b1b57f40abad397751e02cfbd515373f23fa2f6299b07d68d07c5",
      "status": "author_self_checked"
    },
    {
      "id": "lipschitz-sheets-and-a-lexicographic-maximum",
      "title": "Lipschitz sheets and a lexicographic maximum",
      "reader": "lipschitz-sheets-and-a-lexicographic-maximum.html",
      "source": "src/lipschitz-sheets-and-a-lexicographic-maximum.md",
      "source_sha256": "02ef4bf7e9d04a4414e4a2dcf75ff7a5a1cad295cd82b7368ca26715db4b097d",
      "status": "author_self_checked"
    },
    {
      "id": "two-energy-estimates",
      "title": "Two energy estimates",
      "reader": "two-energy-estimates.html",
      "source": "src/two-energy-estimates.md",
      "source_sha256": "9c0be5189855a49fbf7ff2670aa633b20fed775004e9ce551b48a6693dc37d3f",
      "status": "author_self_checked"
    },
    {
      "id": "crossings-and-the-lang-plaut-problem",
      "title": "Crossings and the Lang–Plaut problem",
      "reader": "crossings-and-the-lang-plaut-problem.html",
      "source": "src/crossings-and-the-lang-plaut-problem.md",
      "source_sha256": "9b20026152e67750803f870aff455c3eff9dc56f95d587837451c34a46a0c227",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core course Measure and Integration (Fremlin, Measure Theory: Volume 1, 134A; Volume 2, 225E, 251N, 252B, 252D, 261E, 262B, 262Q, 263A)",
    "core course Real Analysis II (compactness of the unit sphere of R^k, minima of continuous functions)"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}