{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R858",
  "content_hash": null,
  "slug": "zsf7s-claim-certified-thirteen-to-eighteen",
  "type": "claim",
  "title": "The certified interval is 13 to 18",
  "summary": "An exhaustive replay certifies a 13-point construction, and Olson's exact Davenport constant for C_7^3 gives the upper endpoint.",
  "relevance": "For Zero-sum-free subsets of the unit sphere over F_7, record zsf7s-claim-certified-thirteen-to-eighteen (“The certified interval is 13 to 18”) records a bound, answer, status fact, or structural consequence. The record states: An exhaustive replay certifies a 13-point construction, and Olson's exact Davenport constant for C_7^3 gives the upper endpoint.",
  "relevance_source": "recorded",
  "body": "Let\n\\[\nS=\\{(x,y,z)\\in\\mathbb F_7^3:x^2+y^2+z^2=1\\}.\n\\]\nDirect enumeration gives \\(|S|=42\\). The displayed set\n\\[\n\\begin{split}\nA=\\{&(3,5,4),(2,4,4),(2,5,0),(4,3,5),(4,5,3),\\\\\n&(2,0,5),(4,2,4),(5,0,5),(5,3,4),(0,0,1),\\\\\n&(2,3,3),(0,2,5),(5,5,0)\\}\n\\end{split}\n\\]\nlies in \\(S\\). The executable verifier evaluates all \\(2^{13}-1=8191\\) nonempty subsets and finds no zero sum. This proves that the unknown maximum \\(M\\) satisfies \\(M\\geq13\\).\n\nOlson proved the exact Davenport constant for finite abelian p-groups. Applied to \\(C_7^3\\), it gives\n\\[\nD(C_7^3)=1+3(7-1)=19.\n\\]\nEvery sequence of 19 elements of \\(C_7^3\\) therefore has a nonempty zero-sum subsequence. A 19-point subset of \\(S\\) is such a sequence, with each term appearing once, so \\(M\\leq18\\). Hence\n\\[\n13\\leq M\\leq18.\n\\]\nThe computation reported in the candidate record supplies the lower endpoint. This fixture independently replays it. The exact value remains open in this audit.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "zero-sum-free subsets of the 42-point unit sphere in F_7^3",
    "bounds": {
      "field_order": {
        "min": 7,
        "max": 7
      },
      "ambient_dimension": {
        "min": 3,
        "max": 3
      },
      "sphere_size": {
        "min": 42,
        "max": 42
      },
      "optimum_lower_bound": {
        "min": 13,
        "max": 13
      },
      "optimum_upper_bound": {
        "min": 18,
        "max": 18
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1016/0022-314X(69)90021-3",
      "locator": "John E. Olson, A combinatorial problem on finite Abelian groups, I, Journal of Number Theory 1 (1969), 8-10; lower endpoint replayed in zsf7s-artifact-thirteen-point-verifier"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1016/0022-314X(69)90021-3",
    "locator": "John E. Olson, A combinatorial problem on finite Abelian groups, I, Journal of Number Theory 1 (1969), 8-10; lower endpoint replayed in zsf7s-artifact-thirteen-point-verifier"
  },
  "relations": [
    {
      "slug": "R856",
      "title": "Exhaustive verifier for the 13-point construction",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R857",
      "title": "Three symmetry cases remain in the 14-point search",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "zero-sum-free-f7-sphere",
      "title": "zero sum free f7 sphere",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}