{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R730",
  "content_hash": null,
  "slug": "skolem-attempt-formalize-one-sided-boundary",
  "type": "attempt",
  "title": "Formalize the reversible one-sided boundary lemma",
  "summary": "Formalize companion-state invertibility, finite modular recurrence, and the Fibonacci shift so later sieve records cannot conflate integer-index and nonnegative-index zeros.",
  "relevance": "For Decidability of zeros in integer linear recurrence sequences, record skolem-attempt-formalize-one-sided-boundary (“Formalize the reversible one-sided boundary lemma”) documents a concrete method, search boundary, or failed route. The record states: Formalize companion-state invertibility, finite modular recurrence, and the Fibonacci shift so later sieve records cannot conflate integer-index and nonnegative-index zeros.",
  "relevance_source": "recorded",
  "body": "Define an order-\\(d\\) companion state over \\(\\mathbb Z\\) and \\(\\mathbb Z/m\\mathbb Z\\). Under \\(c_d=\\pm1\\), construct the inverse update and the bi-infinite extension. Prove that every state modulo \\(m\\) is periodic. Transport a negative-index zero to a nonnegative congruent zero.\n\nState the broader coprime-modulus variant as a separate lemma. For \\(c_d\\ne0\\), the backward extension lies in \\(\\mathbb Z[1/c_d]\\). Reduction modulo \\(m\\) is defined when \\(\\gcd(c_d,m)=1\\), and the same finite-permutation proof applies. Keep the all-modulus conclusion as the \\(c_d=\\pm1\\) corollary.\n\nSpecialize the statement to \\(u_0=u_1=1\\) and \\(u_{n+2}=u_{n+1}+u_n\\). Prove \\(u_n=F_{n+1}\\), positivity for \\(n\\ge0\\), and \\(u_{-1}=0\\). Use the selected replay rows only as tests. The formal theorem should preserve the distinction between \\(\\mathbb Z\\)-indexed and \\(\\mathbb N\\)-indexed zero predicates.",
  "status": "next_experiment",
  "evidence_grade": "self_reported",
  "scope": {
    "kind": "family",
    "statement": "formal statement and proof for integer LRS with nonzero final coefficient and a negative-index zero, including the all-modulus reversible corollary",
    "family": "integer linear recurrence sequences with a negative-index zero in the localized backward extension"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "locator": "Next-work specification derived from skolem-claim-reversible-negative-zero-modular-boundary and the 2026-07-28 exact replay"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": null,
    "locator": "Next-work specification derived from skolem-claim-reversible-negative-zero-modular-boundary and the 2026-07-28 exact replay"
  },
  "relations": [
    {
      "slug": "R733",
      "title": "A negative zero defeats one-sided modular exclusion for reversible recurrences",
      "object_type": "claim",
      "relation": "depends_on",
      "direction": "outgoing"
    },
    {
      "slug": "R728",
      "title": "Exact Fibonacci and order-two modular replay",
      "object_type": "artifact",
      "relation": "uses",
      "direction": "outgoing"
    },
    {
      "slug": "skolem-problem-decidability",
      "title": "skolem problem decidability",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}