{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R187",
  "content_hash": null,
  "slug": "cyclic315-claim-nine-orbit-packing",
  "type": "claim",
  "title": "Nine base blocks form a cyclic 3-packing",
  "summary": "Exhaustive development confirms that the nine orbits contain 279 distinct blocks and 2,790 distinct triples.",
  "relevance": "For Largest cyclic 3-(31,5,1) packing, record cyclic315-claim-nine-orbit-packing (“Nine base blocks form a cyclic 3-packing”) records a bound, answer, status fact, or structural consequence. The record states: Exhaustive development confirms that the nine orbits contain 279 distinct blocks and 2,790 distinct triples.",
  "relevance_source": "recorded",
  "body": "The base blocks are\n\\[\n\\begin{aligned}\n&\\{0,3,9,12,24\\},\\ \\{0,2,6,8,15\\},\\ \\{0,1,15,27,29\\},\\\\\n&\\{0,1,9,18,20\\},\\ \\{0,3,6,14,19\\},\\ \\{0,3,7,13,27\\},\\\\\n&\\{0,1,2,7,24\\},\\ \\{0,1,5,13,26\\},\\ \\{0,1,4,21,22\\}.\n\\end{aligned}\n\\]\nDevelop each block by every translation in \\(\\mathbb Z_{31}\\). Each orbit has length 31, and the resulting 279 blocks are distinct. Exhaustive enumeration of the ten triples in every developed block produces 2,790 distinct triples, each with multiplicity one. This proves the packing condition and the lower bound \\(M\\geq9\\). The canonical newline-delimited base-block list has SHA-256 digest `2323141aa6883f95a8b8ee5b300cc4b8a57f7437d08a43054de576c99e30228f`.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "all 279 translates and all 2,790 triples developed from the nine displayed base blocks in Z_31",
    "bounds": {
      "base_blocks": {
        "min": 9,
        "max": 9
      },
      "developed_blocks": {
        "min": 279,
        "max": 279
      },
      "triple_incidences": {
        "min": 2790,
        "max": 2790
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1109/18.30982",
      "locator": "Construction recorded in the TheoremDB candidate and independently replayed in cyclic315-artifact-exact-verifier; OOC formulation follows Chung, Salehi, and Wei, IEEE Transactions on Information Theory 35 (1989), 595-604"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1109/18.30982",
    "locator": "Construction recorded in the TheoremDB candidate and independently replayed in cyclic315-artifact-exact-verifier; OOC formulation follows Chung, Salehi, and Wei, IEEE Transactions on Information Theory 35 (1989), 595-604"
  },
  "relations": [
    {
      "slug": "R186",
      "title": "The certified interval is 9 through 13 base blocks",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R184",
      "title": "Exact difference and orbit verifier",
      "object_type": "artifact",
      "relation": "verifies",
      "direction": "incoming"
    },
    {
      "slug": "cyclic-315-packing-31",
      "title": "cyclic 315 packing 31",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}