{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R854",
  "content_hash": null,
  "slug": "zs4fz4-claim-exact-value-32",
  "type": "claim",
  "title": "The maximum size is 32",
  "summary": "A parity-fiber count proves the upper bound 32, and the displayed construction attains it.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "Let \\(G=(\\mathbb Z/4\\mathbb Z)^4\\), let \\(H=2G=\\{0,2\\}^4\\), and partition \\(G\\) into the 16 fibers of reduction modulo 2. Write \\(n_c=|A\\cap c|\\) for a fiber \\(c\\). A pair of distinct elements in one fiber has a sum \\(h\\in H\\). For any fixed \\(h\\in H\\), the pairs with sum \\(h\\) form a matching: an element \\(x\\) has the unique possible partner \\(h-x\\). If two such pairs lay in \\(A\\), their four elements would be distinct and would sum to \\(2h=0\\). Therefore each of the 16 values of \\(h\\) occurs for at most one within-fiber pair, and\n\\[\n\\sum_c {n_c\\choose2}\\leq16.\n\\]\nFor a fixed total \\(\\sum_c n_c\\), this sum is minimized when the 16 fiber sizes differ by at most one. If \\(|A|=33\\), the minimum is \\(15{2\\choose2}+{3\\choose2}=18\\), which contradicts the inequality. Hence \\(|A|\\leq32\\). The certified 32-element construction in zs4fz4-artifact-construction-check proves equality.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "subsets of (Z/4Z)^4 with four distinct zero-sums forbidden",
    "bounds": {
      "modulus": {
        "min": 4,
        "max": 4
      },
      "dimension": {
        "min": 4,
        "max": 4
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/1607.05718",
      "locator": "Parity-fiber proof in this record, paired with the executable construction certificate"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1607.05718",
    "locator": "Parity-fiber proof in this record, paired with the executable construction certificate"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R852",
      "title": "Exhaustive certificate for the 32-point construction",
      "object_type": "artifact",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R855",
      "title": "Every extremal set has two points in each parity fiber",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R853",
      "title": "The problem is the invariant Z_4(C_4^4)",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "zero-sum-four-free-z4-four",
      "title": "zero sum four free z4 four",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}