{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R716",
  "content_hash": null,
  "slug": "sdd50-claim-exact-value",
  "type": "claim",
  "title": "The exact maximum is M(50)=8",
  "summary": "The set {8,18,22,23,30,41,47,50} attains eight, and a complete search finds no valid nine-element subset.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "For a \\(k\\)-element set, the combined collection has \\(2\\binom{k}{2}=k(k-1)\\) entries. Every positive difference and every sum lies in \\(\\{1,\\ldots,99\\}\\), so \\(k(k-1)\\leq99\\) and \\(k\\leq10\\).\n\nThe set\n\\[\nS=\\{8,18,22,23,30,41,47,50\\}\n\\]\nproduces 56 distinct combined values, with minimum 1 and maximum 97. This proves \\(M(50)\\geq8\\).\n\nThe executable certificate enumerates every increasing nine-tuple in \\(\\{1,\\ldots,50\\}\\). It carries a bit mask for all sums and differences created by the current partial set. An extension is retained exactly when its new values avoid the mask and one another. The search reaches no depth-nine leaf. The property passes to subsets, so every valid set of size at least nine would contain a valid nine-element subset. Hence \\(M(50)\\leq8\\), and the two bounds give\n\\[\nM(50)=8.\n\\]",
  "status": "established",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "all subsets of {1,...,50} whose off-diagonal sums and positive differences form one collision-free collection",
    "bounds": {
      "ambient_maximum": {
        "min": 50,
        "max": 50
      },
      "set_size": {
        "min": 8,
        "max": 10
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1109/TCOM.1986.1096587",
      "locator": "Independent exhaustive certificate sdd50-artifact-exhaustive-nine, executed on 2026-07-24"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1109/TCOM.1986.1096587",
    "locator": "Independent exhaustive certificate sdd50-artifact-exhaustive-nine, executed on 2026-07-24"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R715",
      "title": "Exhaustive nine-element branch certificate",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R717",
      "title": "The condition strengthens the Golomb ruler and Sidon property",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "sum-difference-distinct-subset-50",
      "title": "sum difference distinct subset 50",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}