{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R101",
  "content_hash": null,
  "slug": "casf4-attempt-boundary-profile-program",
  "type": "attempt",
  "title": "Build a certified boundary-profile splice system",
  "summary": "The next useful route is to classify prefix and suffix Parikh-difference profiles of morphic blocks, then certify length intervals closed under a finite set of splices.",
  "relevance": "For Eventual existence of four-letter circular abelian-square-free words, record casf4-attempt-boundary-profile-program (“Build a certified boundary-profile splice system”) documents a concrete method, search boundary, or failed route. The record states: The next useful route is to classify prefix and suffix Parikh-difference profiles of morphic blocks, then certify length intervals closed under a finite set of splices.",
  "relevance_source": "recorded",
  "body": "A linear abelian-square-free word can fail the circular condition only through a factor that crosses the chosen seam. Write P_i for the prefix Parikh vector, extended by P_{i+n}=P_i+P_n. A circular violation at start s and half-length h is exactly\n`P_s - 2 P_{s+h} + P_{s+2h} = 0`.\nThis makes the seam obstruction a boundary-profile question.\n\nA next experiment should enumerate several structurally varied witnesses per length. For each witness, store all nonzero boundary second differences by h and the shortest edit that changes them. Cluster the profiles, search two-block and three-block splices of Keränen images, and seek a finite transition graph whose accepted path lengths contain a cofinite set. Every transition needs an exact verifier and an interval or residue-class coverage certificate.\n\nA successful finite system would separate the proof into a local seam lemma, a semigroup or automaton argument covering all large lengths, and a finite checked list of gaps. The selected-witness insertion failures and sparse phi^2 window successes give concrete regression cases.",
  "status": "next_experiment",
  "evidence_grade": "self_reported",
  "scope": {
    "kind": "family",
    "statement": "proposed certified splice search using seam Parikh-difference profiles",
    "family": "Keranen-image blocks, two-block and three-block splices, and multiple witnesses per length"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "locator": "Proposed from the exact boundary identity and packet computations on 2026-07-28"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": null,
    "locator": "Proposed from the exact boundary identity and packet computations on 2026-07-28"
  },
  "relations": [
    {
      "slug": "R108",
      "title": "The source reports no cyclic counterexample below length 150",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R105",
      "title": "Every length-eight orbit blocks one-letter insertion",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R103",
      "title": "A complete phi-squared window scan gives sparse extra witnesses",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R106",
      "title": "Direct cardinality SMT timed out on four selected lengths",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R104",
      "title": "Restricted-growth backtracking produced the bounded witness table",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "circular-abelian-square-free-four-eventual",
      "title": "circular abelian square free four eventual",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}