{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R531",
  "content_hash": null,
  "slug": "minimal105-claim-seven-orbits-through-nineteen",
  "type": "claim",
  "title": "There are seven affine-Galois orbits through weight nineteen",
  "summary": "Exact enumeration finds seven affine-Galois orbits through weight 19; weights 20 and above, and therefore the complete fixed-conductor classification, remain open.",
  "relevance": "For Minimal vanishing sums of distinct 105th roots, record minimal105-claim-seven-orbits-through-nineteen (“There are seven affine-Galois orbits through weight nineteen”) records a bound, answer, status fact, or structural consequence. The record states: Exact enumeration finds seven affine-Galois orbits through weight 19; weights 20 and above, and therefore the complete fixed-conductor classification, remain open.",
  "relevance_source": "recorded",
  "body": "Let \\(\\zeta=\\zeta_{105}\\), and identify a subset with its exponents in \\(\\mathbb Z/105\\mathbb Z\\). The equivalence relation is \\(S\\sim a+uS\\), where \\(a\\in\\mathbb Z/105\\mathbb Z\\) and \\(u\\) is a unit modulo 105. Exact enumeration of all nonempty inclusion-minimal vanishing subsets of weight at most 19 gives the following canonical representatives:\n\n- weight 3: \\(\\{0,35,70\\}\\);\n- weight 5: \\(\\{0,21,42,63,84\\}\\);\n- weight 7: \\(\\{0,15,30,45,60,75,90\\}\\);\n- weight 14: \\(\\{0,1,8,22,29,30,43,45,60,64,71,75,90,92\\}\\);\n- weight 16: \\(\\{0,1,5,20,22,30,35,43,45,60,64,65,75,80,90,95\\}\\);\n- weight 18: \\(\\{0,1,3,11,16,24,26,41,42,45,46,61,63,71,76,84,86,87\\}\\) and \\(\\{0,1,3,18,22,29,30,33,45,48,60,63,64,71,75,90,92,93\\}\\).\n\nThe corresponding orbit sizes are 35, 21, 15, 105, 105, 315, and 210. Their stabilizer sizes in the affine group of order 5,040 are 144, 240, 336, 48, 48, 16, and 24. No orbit occurs at weights 1, 2, 4, 6, 8 through 13, 15, 17, or 19. The search uses exact remainders modulo \\(\\Phi_{105}\\), exact Boolean constraints, and complete affine-orbit blocking. It is a bounded classification and does not settle whether further orbits occur at weight 20 or above.",
  "status": "supported",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "all distinct inclusion-minimal vanishing subsets of the 105th roots, modulo translation and multiplication by a unit, at weights 1 through 19",
    "bounds": {
      "conductor": {
        "min": 105,
        "max": 105
      },
      "weight": {
        "min": 1,
        "max": 19
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "locator": "Exact computation and replay in minimal105-artifact-exhaustive-replay-through-nineteen, executed 2026-07-28 UTC"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": null,
    "locator": "Exact computation and replay in minimal105-artifact-exhaustive-replay-through-nineteen, executed 2026-07-28 UTC"
  },
  "models": [],
  "relations": [
    {
      "slug": "R526",
      "title": "Exact affine-orbit replay through weight nineteen",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R527",
      "title": "Enumerate exact affine orbits through weight nineteen",
      "object_type": "attempt",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R532",
      "title": "Lam and Leung establish the unique weight-fourteen asymmetric type",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R530",
      "title": "The proved type classification stops at weight sixteen",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "minimal-vanishing-105th-root-sums",
      "title": "minimal vanishing 105th root sums",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}