{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R158",
  "content_hash": null,
  "slug": "cpcrt-claim-even-pair-obstruction",
  "type": "claim",
  "title": "A pair at separation two blocks every even order",
  "summary": "The states 0 and 2 remain two steps apart under every word, so they can never merge.",
  "relevance": "For Reset threshold of the cyclic pair-compression automaton, record cpcrt-claim-even-pair-obstruction (“A pair at separation two blocks every even order”) records a bound, answer, status fact, or structural consequence. The record states: The states 0 and 2 remain two steps apart under every word, so they can never merge.",
  "relevance_source": "recorded",
  "body": "Assume \\(n\\) is even and follow the pair \\(\\{0,2\\}\\). The letter \\(a\\) rotates both states and preserves their cyclic difference. Any pair at cyclic difference two has endpoints of the same parity. The letter \\(b\\) fixes both endpoints when they are even and subtracts one from both when they are odd. It therefore preserves the difference two as well. Induction on the word length shows that the two images are always distinct. Hence \\(A_n\\) has no reset word for even \\(n\\).",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "universal",
    "statement": "every even integer n at least 4"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.4213/rm10005e",
      "locator": "Elementary two-state invariant proved in this record"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.4213/rm10005e",
    "locator": "Elementary two-state invariant proved in this record"
  },
  "models": [],
  "relations": [
    {
      "slug": "R156",
      "title": "Exact subset-automaton sweep and weighted lower checks",
      "object_type": "artifact",
      "relation": "tests",
      "direction": "incoming"
    },
    {
      "slug": "cyclic-pair-compression-reset-threshold",
      "title": "cyclic pair compression reset threshold",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}