{
  "id": "borsuks-conjecture-in-dimension-nine",
  "title": "Borsuk's conjecture fails in dimension nine",
  "description": "OpenAI's theorem (2026) that the rank-one projections of R^4, a copy of projective three-space in R^9, cannot be covered by ten sets of smaller diameter; through admissible maps, an odd map on symmetric matrices, degree modulo two and the combinatorics of six-label triangle systems.",
  "units": [
    {
      "id": "diameter-covers-and-admissible-maps",
      "title": "Diameter covers and admissible maps",
      "reader": "diameter-covers-and-admissible-maps.html",
      "source": "src/diameter-covers-and-admissible-maps.md",
      "source_sha256": "feba55b6ff991e60ebe6dd6bc0acf6615594243ecde1c62ab27ad6ed02bf1db7",
      "status": "author_self_checked"
    },
    {
      "id": "regular-values-and-degree-modulo-two",
      "title": "Regular values and degree modulo two",
      "reader": "regular-values-and-degree-modulo-two.html",
      "source": "src/regular-values-and-degree-modulo-two.md",
      "source_sha256": "bc344ee4fbfdaa09a1446696bdd9e7e2cf25b8405cb7434ed8a2f02e1739b788",
      "status": "author_self_checked"
    },
    {
      "id": "double-covers-and-odd-maps",
      "title": "Double covers and odd maps",
      "reader": "double-covers-and-odd-maps.html",
      "source": "src/double-covers-and-odd-maps.md",
      "source_sha256": "c8fc608b29d50311aa548ce15b747738038d9a7438bf25e5446fb8e644ba4e15",
      "status": "author_self_checked"
    },
    {
      "id": "an-odd-map-on-symmetric-matrices",
      "title": "An odd map on symmetric matrices",
      "reader": "an-odd-map-on-symmetric-matrices.html",
      "source": "src/an-odd-map-on-symmetric-matrices.md",
      "source_sha256": "78d831f63f65131c39e943d2f4541bc15d8d981facf34fef9f88214ffb69440b",
      "status": "author_self_checked"
    },
    {
      "id": "supports-at-equality",
      "title": "Supports at equality",
      "reader": "supports-at-equality.html",
      "source": "src/supports-at-equality.md",
      "source_sha256": "172b7ee6796274329d772d8b79b1a64f77167f852ac792d80aba13d35258092c",
      "status": "author_self_checked"
    },
    {
      "id": "six-and-ten-labels",
      "title": "Six and ten labels",
      "reader": "six-and-ten-labels.html",
      "source": "src/six-and-ten-labels.md",
      "source_sha256": "0d0d9daa664ea678b118e1c4d9bd6857dd1ea090feb9282659cd6347bf866487",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core course Algebraic Topology (D60): Fomberg's notes, Theorems 1.13, 1.21, 1.23, Corollaries 1.14, 1.17, Proposition 1.32 (simplicial and singular homology); Roberts's notes, Proposition 8 and Corollary 7 (covering spaces)",
    "core course Real Analysis I (C10): Lebl, Basic Analysis I, Theorem 4.3.2 (Taylor)",
    "core course Measure and Integration (D10): Fremlin, Measure Theory, Vol. 1, 115C-115G, 123C, 123D; Vol. 2, 251N, 252B, 252D, 263A",
    "core course Point-Set Topology (C90): a continuous bijection from a compact space to a Hausdorff space is a homeomorphism",
    "Orientations and fundamental classes (course poincare-duality-on-manifolds): Lemmas 1.1 and 1.2, Proposition 2.3(2), Proposition 3.2, Theorem 4.1, Corollary 5.1",
    "Local tools for bundles and transport (course DG-FND): Lemma 0.4, Theorem 1.2, Corollary 1.3, Theorem 3.1"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}