{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R186",
  "content_hash": null,
  "slug": "cyclic315-claim-certified-interval-9-13",
  "type": "claim",
  "title": "The certified interval is 9 through 13 base blocks",
  "summary": "Nine explicit compatible translation orbits give the lower endpoint, while pair incidences limit every cyclic packing to thirteen base blocks.",
  "relevance": "For Largest cyclic 3-(31,5,1) packing, record cyclic315-claim-certified-interval-9-13 (“The certified interval is 9 through 13 base blocks”) records a bound, answer, status fact, or structural consequence. The record states: Nine explicit compatible translation orbits give the lower endpoint, while pair incidences limit every cyclic packing to thirteen base blocks.",
  "relevance_source": "recorded",
  "body": "Let \\(M\\) be the maximum number of base blocks. The checked construction in this record proves \\(M\\geq9\\). A pair-incidence argument proves \\(M\\leq13\\), improving the raw triple-counting bound of 14. Thus\n\\[\n\\boxed{9\\leq M\\leq13}.\n\\]\nThe exact value remains unresolved by the evidence retained here. The cyclic packing is equivalently a one-dimensional optical orthogonal code of length 31, weight 5, and auto- and cross-correlation at most 2. The literature audit found the general OOC correspondence and Johnson bound, but no primary-source table or construction settling this exact parameter.",
  "status": "open",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "translation orbits of 5-subsets of Z_31 in which each 3-subset occurs in at most one developed block",
    "bounds": {
      "group_order": {
        "min": 31,
        "max": 31
      },
      "block_size": {
        "min": 5,
        "max": 5
      },
      "packing_strength": {
        "min": 3,
        "max": 3
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1016/j.disc.2011.11.039",
      "locator": "Robert F. Bailey and Andrea C. Burgess, Generalized packing designs, Discrete Mathematics 313 (2013), 1167-1190, Definition 1.1.1 and Proposition 1.1.4; parameter-specific construction and upper-bound replay in this fixture"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1016/j.disc.2011.11.039",
    "locator": "Robert F. Bailey and Andrea C. Burgess, Generalized packing designs, Discrete Mathematics 313 (2013), 1167-1190, Definition 1.1.1 and Proposition 1.1.4; parameter-specific construction and upper-bound replay in this fixture"
  },
  "relations": [
    {
      "slug": "R187",
      "title": "Nine base blocks form a cyclic 3-packing",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R188",
      "title": "Pair incidences give an upper bound of thirteen",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R185",
      "title": "Literature and exact-search audit leaves four cases",
      "object_type": "attempt",
      "relation": "contextualizes",
      "direction": "incoming"
    },
    {
      "slug": "cyclic-315-packing-31",
      "title": "cyclic 315 packing 31",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}