{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R121",
  "content_hash": null,
  "slug": "cdg-attempt-periodic-three-coloring",
  "type": "attempt",
  "title": "Search for a periodic finite coloring",
  "summary": "Every periodic coloring fails immediately because a period \\(q\\) is also crossed by the allowed distance \\(q^3\\).",
  "relevance": "This records a line of attack on the target problem, including what it establishes and where it stops.",
  "relevance_source": "derived",
  "body": "Suppose a coloring has period \\(q\\). Then \\(x\\) and \\(x+q^3\\) receive the same color because \\(q\\) divides \\(q^3\\), while their difference is an allowed cube. This rules out every periodic finite coloring before any search begins.\n\nThe original entry suggested a periodic coloring as the route to a three-color upper bound. The polynomial van der Waerden theorem rules out every finite coloring, including aperiodic ones. Finite-window colorings therefore cannot extend to the full graph.",
  "status": "failed_as_complete_proof",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "all periodic colorings of the integers with any finite number of colors"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://arxiv.org/abs/1606.00418",
      "locator": "The general obstruction follows from Corollary 4; the periodic obstruction is elementary"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1606.00418",
    "locator": "The general obstruction follows from Corollary 4; the periodic obstruction is elementary"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R120",
      "title": "Exact cube-distance colorings on small windows",
      "object_type": "artifact",
      "relation": "constrains",
      "direction": "outgoing"
    },
    {
      "slug": "chromatic-number-cube-distance-graph",
      "title": "chromatic number cube distance graph",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}