{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R729",
  "content_hash": null,
  "slug": "skolem-attempt-fixed-modulus-complete-sieve",
  "type": "attempt",
  "title": "A fixed-modulus exclusion sieve fails as a complete one-sided method",
  "summary": "The Fibonacci shift has no zero at a nonnegative integer index, yet every modulus has a zero somewhere on its nonnegative modular orbit.",
  "relevance": "For Decidability of zeros in integer linear recurrence sequences, record skolem-attempt-fixed-modulus-complete-sieve (“A fixed-modulus exclusion sieve fails as a complete one-sided method”) documents a concrete method, search boundary, or failed route. The record states: The Fibonacci shift has no zero at a nonnegative integer index, yet every modulus has a zero somewhere on its nonnegative modular orbit.",
  "relevance_source": "recorded",
  "body": "A zero-free orbit modulo one integer \\(m\\ge2\\) is a sound certificate that an integer LRS has no zero. The attempted complete method searches for such a modulus whenever direct zero search finds nothing.\n\nThe Fibonacci shift \\(u_n=F_{n+1}\\) ends this route as a general one-sided procedure. Positivity proves \\(u_n\\ne0\\) for every \\(n\\ge0\\). Its recurrence is reversible, and its bi-infinite extension has \\(u_{-1}=0\\). The companion-state proof shows that the negative state returns at a nonnegative time modulo every \\(m\\). Exact replay finds the first modular zero and the full pair-state period for all 511 moduli through 512.\n\nThis failed route does not challenge the Exponential Local-Global Principle in STACS 2026. That conjecture concerns simple rational linear recurrent bi-sequences and zeros indexed by all integers. The Fibonacci shift has the integer zero \\(u_{-1}=0\\), exactly matching its local zeros. Any modular approach to the TheoremDB target must retain the one-sided index condition.",
  "status": "failed",
  "evidence_grade": "self_reported",
  "scope": {
    "kind": "family",
    "statement": "fixed-modulus zero-free-orbit certificates applied as a complete method to one-sided reversible integer LRS",
    "family": "one-sided reversible integer LRS with a negative-index zero"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://doi.org/10.4230/LIPIcs.STACS.2026.8",
      "locator": "Bacik et al., On the p-adic Skolem Problem, Section 3.7 and Conjecture 19, pages 8:16-8:17. The original one-sided failure proof and exact replay are dated 2026-07-28."
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.4230/LIPIcs.STACS.2026.8",
    "locator": "Bacik et al., On the p-adic Skolem Problem, Section 3.7 and Conjecture 19, pages 8:16-8:17. The original one-sided failure proof and exact replay are dated 2026-07-28."
  },
  "relations": [
    {
      "slug": "R728",
      "title": "Exact Fibonacci and order-two modular replay",
      "object_type": "artifact",
      "relation": "uses",
      "direction": "outgoing"
    },
    {
      "slug": "R733",
      "title": "A negative zero defeats one-sided modular exclusion for reversible recurrences",
      "object_type": "claim",
      "relation": "reports",
      "direction": "outgoing"
    },
    {
      "slug": "R733",
      "title": "A negative zero defeats one-sided modular exclusion for reversible recurrences",
      "object_type": "claim",
      "relation": "refutes_as_complete_method",
      "direction": "incoming"
    },
    {
      "slug": "skolem-problem-decidability",
      "title": "skolem problem decidability",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}