{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R532",
  "content_hash": null,
  "slug": "minimal105-claim-weight-fourteen-source-uniqueness",
  "type": "claim",
  "title": "Lam and Leung establish the unique weight-fourteen asymmetric type",
  "summary": "Lam and Leung's construction, lower bound, and uniqueness theorem give one rotation class of asymmetric minimal sums at the smallest possible weight 14 for primes 3, 5, and 7.",
  "relevance": "For Minimal vanishing sums of distinct 105th roots, record minimal105-claim-weight-fourteen-source-uniqueness (“Lam and Leung establish the unique weight-fourteen asymmetric type”) records a bound, answer, status fact, or structural consequence. The record states: Lam and Leung's construction, lower bound, and uniqueness theorem give one rotation class of asymmetric minimal sums at the smallest possible weight 14 for primes 3, 5, and 7.",
  "relevance_source": "recorded",
  "body": "Example 2.5 constructs the minimal sum \\((\\alpha+\\cdots+\\alpha^{p-1})(\\beta+\\cdots+\\beta^{q-1})+\\gamma+\\cdots+\\gamma^{\\ell-1}=0\\) for three distinct primes. With \\((p,q,\\ell)=(3,5,7)\\), its support has \\((3-1)(5-1)+(7-1)=14\\) distinct 105th roots. Lower Bound Theorem 4.8 gives 14 as the minimum support of an asymmetric minimal sum in this case. Uniqueness Theorem 6.5 says every asymmetric minimal element at that support or weight is similar to \\(\\sigma(P_1^*)\\sigma(P_2^*)+\\sigma(P_3^*)\\). The paper defines similarity as rotation. This supplies a source-backed uniqueness result at weight 14. In exponent notation, the construction has support \\(\\{7,14,15,28,30,45,49,56,60,75,77,90,91,98\\}\\); translation by 15 gives this packet's canonical representative.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "asymmetric minimal vanishing sums of 105th roots at support or weight 14, up to rotation",
    "bounds": {
      "conductor": {
        "min": 105,
        "max": 105
      },
      "weight": {
        "min": 14,
        "max": 14
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/math/9511209",
      "locator": "T. Y. Lam and K. H. Leung, On vanishing sums for roots of unity, Example 2.5 on pp. 5-6, Lower Bound Theorem 4.8 on pp. 11-12, and Uniqueness Theorem 6.5 on p. 15; J. Algebra 224 (2000), DOI 10.1006/jabr.1999.8089"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/math/9511209",
    "locator": "T. Y. Lam and K. H. Leung, On vanishing sums for roots of unity, Example 2.5 on pp. 5-6, Lower Bound Theorem 4.8 on pp. 11-12, and Uniqueness Theorem 6.5 on p. 15; J. Algebra 224 (2000), DOI 10.1006/jabr.1999.8089"
  },
  "models": [],
  "relations": [
    {
      "slug": "R529",
      "title": "Audit the literature and current TheoremDB state",
      "object_type": "attempt",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R531",
      "title": "There are seven affine-Galois orbits through weight nineteen",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "minimal-vanishing-105th-root-sums",
      "title": "minimal vanishing 105th root sums",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}