{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R781",
  "content_hash": null,
  "slug": "tfs-claim-open-interval",
  "type": "claim",
  "title": "Global optimality for fifteen points remains open",
  "summary": "With \\(\\alpha\\) the isolated root near \\(0.5926059029250737\\) of \\(13x^5-x^4+6x^3+2x^2-3x-1\\), the checked bounds are \\(\\arccos(\\alpha)\\le\\theta_{15}\\le2\\arccos(13/15)\\), or \\(53.65785012993268\\ldots^\\circ\\) through \\(59.85286973322849\\ldots^\\circ\\); global optimality of the lower construction remains open.",
  "relevance": "For Tammes separation for fifteen points on the sphere, record tfs-claim-open-interval (“Global optimality for fifteen points remains open”) records a bound, answer, status fact, or structural consequence. The record states: With \\(\\alpha\\) the isolated root near \\(0.5926059029250737\\) of \\(13x^5-x^4+6x^3+2x^2-3x-1\\), the checked bounds are \\(\\arccos(\\alpha)\\le\\theta_{15}\\le2\\arccos(13/15)\\), or \\(53.65785012993268\\ldots^\\circ\\) through \\(59.85286973322849\\ldots^\\circ\\); global optimality of the lower construction remains open.",
  "relevance_source": "recorded",
  "body": "The current spherical-code table marks proved optima with an asterisk. Its 15-point row has no asterisk. Kottwitz's primary paper describes its solutions for 15 through 90 circles as conjectured solutions. Musin and Tarasov later proved the 14-point case by enumerating irreducible contact graphs, and their proof makes no claim for 15 points.\n\nThe strongest construction checked here and the elementary cap-area bound give\n\\[\n\\arccos(\\alpha)\\leq\\theta_{15}\\leq2\\arccos(13/15),\n\\]\nor numerically\n\\[\n53.65785012993268\\ldots^\\circ\n\\leq\\theta_{15}\\leq\n59.85286973322849\\ldots^\\circ.\n\\]\nFor the upper bound, place disjoint caps of angular radius \\(\\theta_{15}/2\\) around the points. Each cap has area \\(2\\pi(1-\\cos(\\theta_{15}/2))\\). Their total area is at most \\(4\\pi\\), so \\(15(1-\\cos(\\theta_{15}/2))\\leq2\\).",
  "status": "open",
  "evidence_grade": "literature_audit",
  "scope": {
    "kind": "bounded",
    "statement": "the global Tammes optimum for exactly 15 points on S^2",
    "bounds": {
      "points": {
        "min": 15,
        "max": 15
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1107/S0108767390011370",
      "locator": "D. A. Kottwitz, The densest packing of equal circles on a sphere, Acta Crystallographica A 47 (1991), 158-165. The abstract calls the reported solutions conjectured. See also O. R. Musin and A. S. Tarasov, The Tammes problem for N=14, Experimental Mathematics 24 (2015), 460-468, arXiv:1410.2536."
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1107/S0108767390011370",
    "locator": "D. A. Kottwitz, The densest packing of equal circles on a sphere, Acta Crystallographica A 47 (1991), 158-165. The abstract calls the reported solutions conjectured. See also O. R. Musin and A. S. Tarasov, The Tammes problem for N=14, Experimental Mathematics 24 (2015), 460-468, arXiv:1410.2536."
  },
  "models": [],
  "relations": [
    {
      "slug": "R780",
      "title": "The best published construction has an exact algebraic separation",
      "object_type": "claim",
      "relation": "bounds",
      "direction": "incoming"
    },
    {
      "slug": "tammes-fifteen-separation",
      "title": "tammes fifteen separation",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}