{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R733",
  "content_hash": null,
  "slug": "skolem-claim-reversible-negative-zero-modular-boundary",
  "type": "claim",
  "title": "A negative zero defeats one-sided modular exclusion for reversible recurrences",
  "summary": "If an integer LRS has final recurrence coefficient ±1 and a negative-index zero in its bi-infinite extension, every modulus has a congruent zero at some nonnegative index.",
  "relevance": "For Decidability of zeros in integer linear recurrence sequences, record skolem-claim-reversible-negative-zero-modular-boundary (“A negative zero defeats one-sided modular exclusion for reversible recurrences”) records a bound, answer, status fact, or structural consequence. The record states: If an integer LRS has final recurrence coefficient ±1 and a negative-index zero in its bi-infinite extension, every modulus has a congruent zero at some nonnegative index.",
  "relevance_source": "recorded",
  "body": "Let the recurrence have order \\(d\\), final coefficient \\(c_d\\in\\{-1,1\\}\\), and companion state\n\\[\nX_n=(u_n,u_{n+1},\\ldots,u_{n+d-1})^{\\mathsf T}.\n\\]\nIts companion matrix \\(A\\) has determinant \\((-1)^{d-1}c_d\\). Hence \\(A\\) is invertible over \\(\\mathbb Z\\) and modulo every integer \\(m\\ge2\\). The initial data therefore has a unique bi-infinite integer extension.\n\nFix a negative index \\(j<0\\) with \\(u_j=0\\). Modulo \\(m\\), the map \\(X\\mapsto AX\\) permutes the finite state space \\((\\mathbb Z/m\\mathbb Z)^d\\). The state \\(X_j\\) lies on a finite cycle, so some \\(P\\ge1\\) satisfies \\(A^P X_j=X_j\\pmod m\\). Choose \\(q\\) with \\(j+qP\\ge0\\). Then\n\\[\nu_{j+qP}\\equiv u_j\\equiv0\\pmod m.\n\\]\nThis holds for every modulus.\n\nFor a simple order-two example, take \\(u_0=u_1=1\\) and \\(u_{n+2}=u_{n+1}+u_n\\). Its bi-infinite extension is \\(u_n=F_{n+1}\\), so \\(u_{-1}=0\\). Every term with \\(n\\ge0\\) is positive. Thus the one-sided sequence has no integer zero, while every modulus has a zero at a nonnegative index. The characteristic polynomial \\(x^2-x-1\\) has distinct roots.\n\nThe conclusion concerns a sieve that seeks one modulus with no nonnegative modular zero. It says nothing about general Skolem decidability. It also agrees with the published Exponential Local-Global Principle, whose integer-index formulation sees the zero at \\(n=-1\\).",
  "status": "supported",
  "evidence_grade": "self_reported",
  "scope": {
    "kind": "family",
    "statement": "integer LRS with final recurrence coefficient ±1 whose unique bi-infinite integer extension has a zero at a negative index",
    "family": "reversible integer linear recurrence sequences with a negative-index zero"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "locator": "Original companion-matrix proof in this record, prepared 2026-07-28; finite cases replayed by tools/skolem_one_sided_modular_replay.py"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": null,
    "locator": "Original companion-matrix proof in this record, prepared 2026-07-28; finite cases replayed by tools/skolem_one_sided_modular_replay.py"
  },
  "relations": [
    {
      "slug": "R731",
      "title": "Primary-source audit through July 2026",
      "object_type": "attempt",
      "relation": "depends_on",
      "direction": "outgoing"
    },
    {
      "slug": "R728",
      "title": "Exact Fibonacci and order-two modular replay",
      "object_type": "artifact",
      "relation": "evidences",
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    {
      "slug": "R729",
      "title": "A fixed-modulus exclusion sieve fails as a complete one-sided method",
      "object_type": "attempt",
      "relation": "reports",
      "direction": "incoming"
    },
    {
      "slug": "R729",
      "title": "A fixed-modulus exclusion sieve fails as a complete one-sided method",
      "object_type": "attempt",
      "relation": "refutes_as_complete_method",
      "direction": "outgoing"
    },
    {
      "slug": "R730",
      "title": "Formalize the reversible one-sided boundary lemma",
      "object_type": "attempt",
      "relation": "depends_on",
      "direction": "incoming"
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    {
      "slug": "skolem-problem-decidability",
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      "object_type": "problem",
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    }
  ]
}