{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R769",
  "content_hash": null,
  "slug": "tc114-claim-mod-nine-filter",
  "type": "claim",
  "title": "Every coordinate is 2 modulo 3",
  "summary": "Cubes modulo nine force each coordinate into one of the residue classes 2, 5, or 8.",
  "relevance": "For A bounded three-cubes search for 114, record tc114-claim-mod-nine-filter (“Every coordinate is 2 modulo 3”) records a bound, answer, status fact, or structural consequence. The record states: Cubes modulo nine force each coordinate into one of the residue classes 2, 5, or 8.",
  "relevance_source": "recorded",
  "body": "Cubes modulo 9 have residues \\(0,1,8\\). Since\n\\[\n114\\equiv6\\equiv-3\\pmod9,\n\\]\nthe three cube residues must all equal \\(-1\\). Hence every solution satisfies\n\\[\nx,y,z\\in\\{2,5,8\\}\\pmod9,\n\\]\nand in particular\n\\[\nx\\equiv y\\equiv z\\equiv2\\pmod3.\n\\]\nThis is a necessary filter. It supplies no global obstruction because the permitted residue classes exist.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "universal",
    "statement": "every integer solution of x^3+y^3+z^3=114"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1073/pnas.2022377118",
      "locator": "Booker and Sutherland, Section 3, Lemma 3.2; exact residue replay in tc114-artifact-exact-verifier"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1073/pnas.2022377118",
    "locator": "Booker and Sutherland, Section 3, Lemma 3.2; exact residue replay in tc114-artifact-exact-verifier"
  },
  "relations": [
    {
      "slug": "R767",
      "title": "The height-10^20 question remains open",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R766",
      "title": "Exact verifier for a proposed representation",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "three-cubes-114-height-1e20",
      "title": "three cubes 114 height 1e20",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}