{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R107",
  "content_hash": null,
  "slug": "casf4-claim-exact-counts-one-through-sixteen",
  "type": "claim",
  "title": "Complete small-length counts are replayable through 16",
  "summary": "Complete enumeration modulo alphabet permutations gives 1,1,1,3,5,14,21,16,12,20,66,177,208,210,405,160 circular abelian-square-free representatives at lengths 1 through 16.",
  "relevance": "For Eventual existence of four-letter circular abelian-square-free words, record casf4-claim-exact-counts-one-through-sixteen (“Complete small-length counts are replayable through 16”) records a bound, answer, status fact, or structural consequence. The record states: Complete enumeration modulo alphabet permutations gives 1,1,1,3,5,14,21,16,12,20,66,177,208,210,405,160 circular abelian-square-free representatives at lengths 1 through 16.",
  "relevance_source": "recorded",
  "body": "A restricted-growth word fixes the first letter as 0 and permits each later letter to be at most one more than the largest earlier letter, capped at 3. Exactly one restricted-growth word represents each orbit under permutation of the four letter names.\n\nComplete depth-first enumeration pruned a branch whenever its newest suffix was a linear abelian square. At full length it applied the circular verifier. The resulting orbit counts for n=1,...,16 are\n`[1,1,1,3,5,14,21,16,12,20,66,177,208,210,405,160]`.\n\nWeighting every orbit by the number of injections of its used letters into a labeled four-letter alphabet gives\n`[4,12,24,72,120,336,504,384,288,480,1584,4248,4992,5040,9720,3840]`.\nA direct enumeration of all 4^n labeled words independently agrees for n=1,...,8. Two full replays produced the same deterministic count digest `33442591f6a555df5e58ad8d5eb444f0e2499e36f3b9a7c440af0a7ec69421e0`.",
  "status": "observed",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "complete circular-word enumeration modulo alphabet permutation for every n from 1 through 16",
    "bounds": {
      "n": {
        "min": 1,
        "max": 16
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "locator": "Original complete enumeration and replay executed 2026-07-28; see casf4-artifact-exact-replay"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": null,
    "locator": "Original complete enumeration and replay executed 2026-07-28; see casf4-artifact-exact-replay"
  },
  "models": [],
  "relations": [
    {
      "slug": "R100",
      "title": "Exact witness, count, and insertion-graph replay",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "circular-abelian-square-free-four-eventual",
      "title": "circular abelian square free four eventual",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}