{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R223",
  "content_hash": null,
  "slug": "dpi-attempt-finite-versus-single-orbit",
  "type": "attempt",
  "title": "Separate the finite-orbit question from Loomis's conjecture",
  "summary": "Universal joining to the orbit of 1 implies the candidate, while the weaker finite-trajectory statement may admit a separate proof.",
  "relevance": "For Merging of orbits under adding the product of nonzero digits, record dpi-attempt-finite-versus-single-orbit (“Separate the finite-orbit question from Loomis's conjecture”) documents a concrete method, search boundary, or failed route. The record states: Universal joining to the orbit of 1 implies the candidate, while the weaker finite-trajectory statement may admit a separate proof.",
  "relevance_source": "recorded",
  "body": "The literature audit found no independent argument that only finitely many disjoint eventual trajectories exist. One route is to bound the number of disjoint increasing orbits using decimal intervals and the small increments near powers of ten. The opposite route would construct infinitely many starts whose forward sets stay disjoint. A proof of Loomis's stronger conjecture would settle both questions at once.",
  "status": "open_strategy",
  "evidence_grade": "proposed",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://oeis.org/A063108",
      "locator": "Open direction relative to the conjecture recorded by Loomis and OEIS A063108"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://oeis.org/A063108",
    "locator": "Open direction relative to the conjecture recorded by Loomis and OEIS A063108"
  },
  "models": [],
  "relations": [
    {
      "slug": "R224",
      "title": "Loomis conjectured that every orbit joins the orbit of 1",
      "object_type": "claim",
      "relation": "uses",
      "direction": "outgoing"
    },
    {
      "slug": "digit-product-iteration-trajectories",
      "title": "digit product iteration trajectories",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}