{
  "id": "the-duality-conjecture-for-metric-entropy",
  "title": "The duality conjecture for metric entropy",
  "description": "OpenAI's counterexample (2026) to the duality conjecture for metric entropy, posed by Pietsch in 1972: covering numbers of convex bodies are not preserved by polarity up to universal constants, already when one body is a cube. The course turns a real matrix into a pair of bodies, proves the duality theorem for convexified packing, approximates all column combinations of a matrix built from partitions by a short list, and constructs the partitions from random directions for symmetric multilinear forms over a finite field.",
  "units": [
    {
      "id": "covering-numbers-and-polar-bodies",
      "title": "Covering numbers and polar bodies",
      "reader": "covering-numbers-and-polar-bodies.html",
      "source": "src/covering-numbers-and-polar-bodies.md",
      "source_sha256": "d94a4e9759338ba9908b7565e1ec52a976d6f6c44a64fd1a9f11c9d22a3870e7",
      "status": "author_self_checked"
    },
    {
      "id": "compression-by-partitions",
      "title": "Compression by partitions",
      "reader": "compression-by-partitions.html",
      "source": "src/compression-by-partitions.md",
      "source_sha256": "a2aedddc609f3661e7485137088d73577015a9038378d4a8383be7b9cfeb0b76",
      "status": "author_self_checked"
    },
    {
      "id": "symmetric-forms-over-finite-fields",
      "title": "Symmetric forms over finite fields",
      "reader": "symmetric-forms-over-finite-fields.html",
      "source": "src/symmetric-forms-over-finite-fields.md",
      "source_sha256": "cf68b8d0a5114097a8ae5a4a4e52aaf44a694f02a263ca7a5f8febf452af9abb",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "foundations-of-von-neumann-algebras lesson hahn-banach-baire-and-the-basic-theorems-on-banach-spaces (Theorems 6.2 and 6.3, separation of convex sets, used in R^n)"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}