{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R30",
  "content_hash": null,
  "slug": "asq-claim-four-letter-lower-bound",
  "type": "claim",
  "title": "Any positive construction needs at least four letters",
  "summary": "Every abelian square is an additive square under an integer labeling, and infinite abelian-square-free words require at least four letters. Therefore an additive-square-free infinite word needs an alphabet of size at least four.",
  "relevance": "For Infinite additive-square avoidance over a finite integer alphabet, record asq-claim-four-letter-lower-bound (“Any positive construction needs at least four letters”) records a bound, answer, status fact, or structural consequence. The record states: Every abelian square is an additive square under an integer labeling, and infinite abelian-square-free words require at least four letters.",
  "relevance_source": "recorded",
  "body": "Two adjacent blocks that are anagrams have identical sums under every numeric labeling. Additive-square avoidance therefore implies abelian-square avoidance. Andrade and Mol summarize the sharp abelian result: Keranen constructed an infinite abelian-square-free word on four letters, and three letters are insufficient. The implication gives the same four-letter lower bound for any positive answer to the additive-square problem.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "every finite integer alphabet supporting an infinite additive-square-free word"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/2408.15390",
      "locator": "Jonathan Andrade and Lucas Mol, Avoiding abelian and additive powers in rich words, arXiv:2408.15390v2, Introduction, definitions and paragraphs immediately preceding the additive-power status discussion"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2408.15390",
    "locator": "Jonathan Andrade and Lucas Mol, Avoiding abelian and additive powers in rich words, arXiv:2408.15390v2, Introduction, definitions and paragraphs immediately preceding the additive-power status discussion"
  },
  "models": [],
  "relations": [
    {
      "slug": "R27",
      "title": "Audit sources, variants, duplicates, and target integrity",
      "object_type": "attempt",
      "relation": "reports",
      "direction": "incoming"
    },
    {
      "slug": "R28",
      "title": "The exact finite maximum for {0,1,2,4} is 62",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "additive-square-finite-alphabet",
      "title": "additive square finite alphabet",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}