{
  "target_id": "narrative:71322",
  "proof_status": "needs_context",
  "summary": "The passage announces a counting estimate at scale N/K and its subsequent use in the major arcs. Neither the estimate nor that application is supplied, so neither transition can be verified. The cited theorem is an intended downstream consumer, not an established premise for the narrative. This is a bounded evidence limitation, not a demonstrated proof defect. No external result is imported; potentially later model knowledge is not used as evidence.",
  "used_dependencies": [],
  "missing_context": [
    "The complete immediately following counting lemma, including its formula, proof, implied-constant dependencies and uniformity ranges.",
    "Definitions of f_gamma, the counting family and its weights or multiplicities, and all scale parameters. The relation evidence mentions N/2<n<N, 1<K<=T_2^(1/10), and x,y asymptotic to X; their applicability cannot be checked without the full statement and parameter setup.",
    "The portion of the proof of Theorem thm:cMNis that applies the counting estimate, including the major-arc expression and the resulting lower-bound calculation.",
    "Theorem thm:RNU, referenced by the supplied theorem statement, if the properties required of the singular-series function are needed to audit that application."
  ],
  "findings": [
    {
      "kind": "unverified",
      "source_quote": "The following is our last counting lemma, showing a certain equidistribution among the values of $\\ff_{\\g}(2x,y)$ at the scale $N/K$.",
      "explanation": "The following lemma and its proof are absent. Consequently, the packet does not specify the counting measure, interval or smoothing window, expected count, error term, or uniformity needed to substantiate 'equidistribution' at the asserted scale. The exact remaining obligation is to compare the complete counting estimate with this description under all its stated hypotheses."
    },
    {
      "kind": "unverified",
      "source_quote": "This bound is used in the major arcs, see\nthe proof of Theorem  \\ref{thm:cMNis}.",
      "explanation": "Only the cited theorem's statement is supplied. Its lower bound for the main term does not establish where or how the announced counting estimate is used. Checking this transition requires the relevant major-arc calculation and verification that its choices of parameters satisfy the counting lemma's hypotheses."
    },
    {
      "kind": "map_correction",
      "source_quote": "This bound is used in the major arcs",
      "explanation": "The supplied edge from thm:cMNis to this narrative records a source reference, but should not be read as a proof-input dependency. The prose describes the counting bound as an input to the theorem's proof, not the theorem as an input to the counting bound. Retain the citation relationship separately from mathematical dependency."
    },
    {
      "kind": "context",
      "source_quote": "Moreover,                                                                                                          \nwe import all the notation from the previous section.",
      "explanation": "The setup identifies the orientation-preserving Apollonian subgroup, but the imported notation is not supplied. Definitions of the polynomial f_gamma, the actual family of gamma being counted, and the relationships among N, K, X, T and T_2 are needed. The fixed primitive integral gasket is specified in the introductory context, but that alone does not determine these objects."
    }
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  "dependency_proposals": [
    {
      "label": "Immediately following counting lemma",
      "role": "proof_input",
      "source_quote": "The following is our last counting lemma",
      "explanation": "Its precise statement and proof are the primary evidence needed for the narrative's equidistribution claim. They are omitted from this packet."
    },
    {
      "label": "thm:cMNis",
      "role": "excluded",
      "source_quote": "\\cM_{N}(n)\\gg\\fS(n)T^{\\gd-1}",
      "explanation": "Exclude this statement as a proof input to the announced counting estimate. It remains relevant as the claimed downstream application, whose proof has not been supplied or established here."
    }
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