{
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  "ref": "R1634",
  "content_hash": null,
  "slug": "linear-recurrence-positivity-decidability-claim-status-20260801",
  "type": "claim",
  "title": "Current status and exact unresolved remainder",
  "summary": "OPEN as checked on 2026-08-01. Strongest checked neighboring result: Decidability is known through order five, for simple sequences through order nine, and for additional reversible or dominant-root subclasses. Exact unresolved remainder: A decision procedure for unrestricted order, already beyond the low-order frontier, remains open.",
  "relevance": "This is the dated publication status for the canonical target Decidability of positivity for linear recurrence sequences.",
  "relevance_source": "recorded",
  "body": "The problem was checked as open on 2026-08-01.\n\nThe strongest neighboring result found in the cited sources is: Decidability is known through order five, for simple sequences through order nine, and for additional reversible or dominant-root subclasses.\n\nThe exact unresolved remainder is: A decision procedure for unrestricted order, already beyond the low-order frontier, remains open.\n\nA complete resolution must meet the following acceptance conditions:\n- Give a terminating correct decision algorithm for arbitrary order.\n- Or prove undecidability through a computable reduction.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1137/1.9781611973402.27",
      "locator": "J. Ouaknine and J. Worrell, Positivity Problems for Low-Order Linear Recurrence Sequences, SODA 2014. main decidability theorems"
    },
    "missing": [
      "source",
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    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1137/1.9781611973402.27",
    "locator": "J. Ouaknine and J. Worrell, Positivity Problems for Low-Order Linear Recurrence Sequences, SODA 2014. main decidability theorems"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R1633",
      "title": "Strongest checked neighboring result",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R1631",
      "title": "Dated source and duplicate audit",
      "object_type": "attempt",
      "relation": "evidences",
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    },
    {
      "slug": "R1632",
      "title": "Work at the unresolved boundary",
      "object_type": "attempt",
      "relation": "addresses",
      "direction": "incoming"
    },
    {
      "slug": "linear-recurrence-positivity-decidability",
      "title": "linear recurrence positivity decidability",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}