{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R119",
  "content_hash": null,
  "slug": "cd1685-claim-lower-104",
  "type": "claim",
  "title": "The recursive lower bound is 104",
  "summary": "Schonheim's inequality, seeded by C(13,5,2)=10, gives C(16,8,5) >= 104.",
  "relevance": "For Covering every five-set with eight-sets on sixteen points, record cd1685-claim-lower-104 (“The recursive lower bound is 104”) records a bound, answer, status fact, or structural consequence. The record states: Schonheim's inequality, seeded by C(13,5,2)=10, gives C(16,8,5) >= 104.",
  "relevance_source": "recorded",
  "body": "For every covering number,\n\\[\nC(v,k,t)\\ge \\left\\lceil\\frac vk C(v-1,k-1,t-1)\\right\\rceil.\n\\]\nTo see this, fix any point. The blocks containing it, with that point deleted, cover all \\((t-1)\\)-subsets of the other \\(v-1\\) points. Every point therefore occurs in at least \\(C(v-1,k-1,t-1)\\) blocks. Summing point-block incidences gives the displayed inequality.\n\nThe LJCR base record gives the exact value \\(C(13,5,2)=10\\), with its lower-bound method listed as `Bounds for fixed number of blocks`. Three recursive applications now give\n\\[\n\\begin{aligned}\nC(14,6,3)&\\ge\\left\\lceil\\frac{14}{6}\\,10\\right\\rceil=24,\\\\\nC(15,7,4)&\\ge\\left\\lceil\\frac{15}{7}\\,24\\right\\rceil=52,\\\\\nC(16,8,5)&\\ge\\left\\lceil\\frac{16}{8}\\,52\\right\\rceil=104.\n\\end{aligned}\n\\]\nThe executable record replays the integer arithmetic. The mathematical input at the base of the chain is the sourced exact value 10.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "all (16,8,5) coverings, using the published exact base value C(13,5,2)=10 and the recursive Schonheim inequality",
    "bounds": {
      "v": {
        "min": 13,
        "max": 16
      },
      "k": {
        "min": 5,
        "max": 8
      },
      "t": {
        "min": 2,
        "max": 5
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://ljcr.dmgordon.org/cover/show_cover.php?k=5&t=2&v=13",
      "locator": "La Jolla Covering Repository, exact C(13,5,2)=10 record and its fixed-block lower-bound attribution; recursive inequality from Johanan Schonheim, On Coverings, Pacific Journal of Mathematics 14 (1964), 1405-1411"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://ljcr.dmgordon.org/cover/show_cover.php?k=5&t=2&v=13",
    "locator": "La Jolla Covering Repository, exact C(13,5,2)=10 record and its fixed-block lower-bound attribution; recursive inequality from Johanan Schonheim, On Coverings, Pacific Journal of Mathematics 14 (1964), 1405-1411"
  },
  "relations": [
    {
      "slug": "R117",
      "title": "The current table interval is 104 through 115",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R116",
      "title": "Exhaustive replay of the 115-block covering",
      "object_type": "artifact",
      "relation": "tests",
      "direction": "incoming"
    },
    {
      "slug": "covering-design-16-8-5",
      "title": "covering design 16 8 5",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}