{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R732",
  "content_hash": null,
  "slug": "skolem-claim-current-frontier-2026-07-28",
  "type": "claim",
  "title": "General Skolem decidability remains open beyond order four",
  "summary": "General integer-LRS Skolem decidability remains open: order at most four is decidable, and no unconditional algorithm or undecidability proof is known for arbitrary order, beginning with order five.",
  "relevance": "For Decidability of zeros in integer linear recurrence sequences, record skolem-claim-current-frontier-2026-07-28 (“General Skolem decidability remains open beyond order four”) records a bound, answer, status fact, or structural consequence. The record states: General integer-LRS Skolem decidability remains open: order at most four is decidable, and no unconditional algorithm or undecidability proof is known for arbitrary order, beginning with order five.",
  "relevance_source": "recorded",
  "body": "The source audit through 2026-07-28 found no unconditional decision procedure for arbitrary-order integer linear recurrence sequences and no undecidability reduction. Bacik proves decidability for algebraic linear recurrence sequences of order at most four. Bacik, Ouaknine, and Worrell place the bounded problem for every fixed order in coRP and obtain the same upper bound for the unrestricted order-four problem. A restricted family reaches higher order: Kenison gives an alternative proof of decidability for reversible integer LRS of order at most seven.\n\nThe 2026 p-adic algorithm has unconditionally correct output whenever it terminates. Its termination proof assumes the p-adic Schanuel Conjecture. The July 2026 preprint obtains general decidability under a strengthened Cramér-type conjecture and proves unconditionally that the possible large-zero indices have density zero. These results narrow the search without deciding the general target. The first order outside the unconditional general frontier is five.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "the one-sided Skolem decidability status for arbitrary-order integer LRS, with the low-order and reversible subclass frontiers listed in this record"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/2607.15510v1",
      "locator": "Florian Luca, Joël Ouaknine, and James Worrell, Conjectural Decidability of the Skolem Problem, arXiv:2607.15510v1, abstract and Sections 1, 4, and 5, especially Theorems 4.3 and 5.1. Checked 2026-07-28."
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2607.15510v1",
    "locator": "Florian Luca, Joël Ouaknine, and James Worrell, Conjectural Decidability of the Skolem Problem, arXiv:2607.15510v1, abstract and Sections 1, 4, and 5, especially Theorems 4.3 and 5.1. Checked 2026-07-28."
  },
  "relations": [
    {
      "slug": "R731",
      "title": "Primary-source audit through July 2026",
      "object_type": "attempt",
      "relation": "reports",
      "direction": "incoming"
    },
    {
      "slug": "skolem-problem-decidability",
      "title": "skolem problem decidability",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}