{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R855",
  "content_hash": null,
  "slug": "zs4fz4-claim-extremal-fiber-structure",
  "type": "claim",
  "title": "Every extremal set has two points in each parity fiber",
  "summary": "Equality in the upper-bound count fixes all 16 fiber sizes and all 16 within-fiber pair sums.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "When \\(|A|=32\\), the convexity minimum of \\(\\sum_c{n_c\\choose2}\\) is 16. The proved upper inequality is also 16, so equality holds throughout. Every parity fiber contains exactly two points. Their 16 sums are distinct elements of \\(H\\), hence they use every element of \\(H\\) exactly once. The displayed construction has this forced structure.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "every 32-element four-zero-sum-free subset of (Z/4Z)^4",
    "bounds": {
      "cardinality": {
        "min": 32,
        "max": 32
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/1607.05718",
      "locator": "Equality case of the parity-fiber proof in zs4fz4-claim-exact-value-32"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1607.05718",
    "locator": "Equality case of the parity-fiber proof in zs4fz4-claim-exact-value-32"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R854",
      "title": "The maximum size is 32",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R852",
      "title": "Exhaustive certificate for the 32-point construction",
      "object_type": "artifact",
      "relation": "tests",
      "direction": "incoming"
    },
    {
      "slug": "zero-sum-four-free-z4-four",
      "title": "zero sum four free z4 four",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}