{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R32",
  "content_hash": null,
  "slug": "asq-claim-open-status-2026",
  "type": "claim",
  "title": "The finite-integer additive-square problem remains open",
  "summary": "The 2026-07-28 audit found neither an infinite additive-square-free word over a finite integer alphabet nor an impossibility proof; the exact finite maximum over \\(\\{0,1,2,4\\}\\) is 62, and the infinite existence question remains open.",
  "relevance": "For Infinite additive-square avoidance over a finite integer alphabet, record asq-claim-open-status-2026 (“The finite-integer additive-square problem remains open”) records a bound, answer, status fact, or structural consequence. The record states: The 2026-07-28 audit found neither an infinite additive-square-free word over a finite integer alphabet nor an impossibility proof; the exact finite maximum over \\(\\{0,1,2,4\\}\\) is 62, and the infinite existence question remains open.",
  "relevance_source": "recorded",
  "body": "Vukusic states the discrete question explicitly and calls it open, then proves a Lebesgue-integral analogue. Andrade and Mol, revised in February 2025, also state that the finite integer problem remains unknown and distinguish the known Z^2 construction. Popoli, Shallit, and Stipulanti give the same status in FSTTCS 2024. A dated search on 2026-07-28 checked these papers, the original formulations, exact-title queries, citation indexes, and nearby additive-power work. This record is a dated literature report and makes no claim about sources published after the search.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "dated literature status of the exact finite-integer additive-square existence question as checked 2026-07-28"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/2506.21200v1",
      "locator": "Ingrid Vukusic, A Lebesgue variant of the additive square problem, arXiv:2506.21200v1, abstract and opening paragraph; cross-checked against Andrade and Mol, arXiv:2408.15390v2, Introduction, and Popoli, Shallit, and Stipulanti, LIPIcs.FSTTCS.2024.32, Introduction"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2506.21200v1",
    "locator": "Ingrid Vukusic, A Lebesgue variant of the additive square problem, arXiv:2506.21200v1, abstract and opening paragraph; cross-checked against Andrade and Mol, arXiv:2408.15390v2, Introduction, and Popoli, Shallit, and Stipulanti, LIPIcs.FSTTCS.2024.32, Introduction"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R27",
      "title": "Audit sources, variants, duplicates, and target integrity",
      "object_type": "attempt",
      "relation": "reports",
      "direction": "incoming"
    },
    {
      "slug": "additive-square-finite-alphabet",
      "title": "additive square finite alphabet",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}