{
  "id": "the-entropy-conjecture-for-c1-maps",
  "title": "The entropy conjecture for C¹ maps",
  "description": "OpenAI's theorem (2026) that Shub's entropy conjecture fails for C^1 maps: a noninvertible C^1 self-map of a compact manifold with zero topological entropy and the eigenvalue 2 on second homology; through entropy localization, a clock with prediction registers, an attraction theorem and the degree of squaring.",
  "units": [
    {
      "id": "topological-entropy-and-attraction",
      "title": "Topological entropy and attraction",
      "reader": "topological-entropy-and-attraction.html",
      "source": "src/topological-entropy-and-attraction.md",
      "source_sha256": "731d59075704183e3dac9605093939a1eea12ee675bb99642e5249fef89d4dd1",
      "status": "author_self_checked"
    },
    {
      "id": "a-clock-with-registers",
      "title": "A clock with registers",
      "reader": "a-clock-with-registers.html",
      "source": "src/a-clock-with-registers.md",
      "source_sha256": "782f358a01981ddad41a2f7645e0c48f3bd947ed7978f765a6d56edb391795cf",
      "status": "author_self_checked"
    },
    {
      "id": "every-orbit-is-attracted",
      "title": "Every orbit is attracted",
      "reader": "every-orbit-is-attracted.html",
      "source": "src/every-orbit-is-attracted.md",
      "source_sha256": "e1c0682ebf3010744d8b72d9dbc65f6cea1994f22cda792652b340af16cb30fa",
      "status": "author_self_checked"
    },
    {
      "id": "zero-entropy-and-the-eigenvalue-two",
      "title": "Zero entropy and the eigenvalue two",
      "reader": "zero-entropy-and-the-eigenvalue-two.html",
      "source": "src/zero-entropy-and-the-eigenvalue-two.md",
      "source_sha256": "1afb34e633fbbdb2fa558d95646535dd74025a7244f5687a032f28bb1d0b0fc7",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core courses Real Analysis I and II (C10, C20): Lebl, Basic Analysis I-II, Definition 7.4.7, Theorems 6.2.10, 7.5.6, 7.5.11, 8.3.7, Proposition 8.4.6",
    "core course Algebraic Topology (D60): Fomberg, Algebraic Topology, Proposition 1.9, Theorems 1.13 and 1.23, Definition 1.27, Remark 1.35",
    "Local tools for bundles and transport (DG-FND), Theorem 3.1 (cutoff functions)",
    "Orientations and fundamental classes (poincare-duality-on-manifolds), Section 1, Proposition 3.2, Lemma 3.4, Corollary 3.5, Theorem 4.1, Corollary 5.1",
    "Cap products and cohomology with compact supports (poincare-duality-on-manifolds), equations (1.1)-(1.2)",
    "Poincare duality (poincare-duality-on-manifolds), Corollary 2.1; its proof does not assume finite-dimensional homology",
    "Orientations and fundamental classes (poincare-duality-on-manifolds), Corollary 3.3 (orientation from positive chart transitions)"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}