{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R128",
  "content_hash": null,
  "slug": "cds-attempt-deletion-induction",
  "type": "attempt",
  "title": "One-letter deletion has unbounded square loss",
  "summary": "For w=u^2 with pairwise-distinct letters in u, every one-letter deletion reduces the square count from |u| to one.",
  "relevance": "For The three-halves bound for distinct squares in circular words, record cds-attempt-deletion-induction (“One-letter deletion has unbounded square loss”) documents a concrete method, search boundary, or failed route. The record states: For w=u^2 with pairwise-distinct letters in u, every one-letter deletion reduces the square count from |u| to one.",
  "relevance_source": "recorded",
  "body": "A direct induction on word length would seek a deletion that loses at most the one- or two-unit increment in ceil(3n/2). This route fails on an infinite family.\n\nLet u=a_0...a_{k-1}, where the k letters are pairwise distinct, and let w=u^2. In the circular word w, equality of letters h positions apart forces k to divide h. A square factor has half-length h<=k, so h=k. The k rotations of u^2 are therefore the complete square list and s(w)=k.\n\nDelete either occurrence of a_i and rotate the remaining circular word into x a_i x, where x is u with a_i removed. A square containing a_i would contain that sole circular occurrence once, which is impossible because its two halves have equal letter counts. A square avoiding a_i lies in the block xx across the circular seam. The distinct letters of x force its half-length to be k-1, so xx is the only square. Every deletion leaves square count one. The best deletion loses k-1 squares, an unbounded quantity.\n\nThe exhaustive structural sweep first violates the parity-sensitive ceiling induction at w=012012, where s(w)=3 and every deletion has count one. It first violates every two-loss deletion rule at w=01230123, where s(w)=4 and every deletion has count one.",
  "status": "failed",
  "evidence_grade": "self_reported",
  "scope": {
    "kind": "family",
    "statement": "all doubled words u^2 whose half u consists of k pairwise-distinct letters, for every k at least 2",
    "family": "doubled permutation words u^2 with pairwise-distinct letters in u"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "locator": "Self-contained family argument, with exact finite replay through length 12 on 2026-07-28"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": null,
    "locator": "Self-contained family argument, with exact finite replay through length 12 on 2026-07-28"
  },
  "models": [],
  "relations": [
    {
      "slug": "R129",
      "title": "Strengthen the primitive Rauzy split case",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "circular-distinct-squares-three-halves",
      "title": "circular distinct squares three halves",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}