{
  "target_id": "narrative:68030",
  "proof_status": "needs_context",
  "summary": "The announced use in the major-arcs analysis is supported by the explicit application of Lemma spec1 at eq:fMdecomp. Whether spec1 is a slight generalization of spec0 cannot be checked because its statement and proof are absent. The supplied dependencies remain conditional premises. No external theorem is imported, and model training knowledge is not treated as dated evidence under the 2012-05-24 cutoff.",
  "used_dependencies": [
    "lem:spec0",
    "eq:fMdecomp"
  ],
  "missing_context": [
    "The complete statement and relevant proof scope of Lemma spec1, including all restrictions on K, T_2, beta, x, y, q, and coset representatives.",
    "Definitions of the family F, its multiplicities if any, the phase f_gamma, the norm, and the scales N, T, T_2, X, K_0, and Q_0.",
    "The definition eq:gT1def, the index estimate eq:G0qBnd, and the Fourier cutoff properties needed to check uniform applicability and error aggregation.",
    "The group action and stabilizer definitions needed to verify that the congruence subgroups used downstream satisfy the subgroup hypotheses.",
    "If the missing argument imports external counting or spectral-gap results, their exact statements, hypotheses, versions, available_from dates, date_evidence, and date_basis must establish availability by 2012-05-24; no such results are certified here."
  ],
  "findings": [
    {
      "kind": "context",
      "source_quote": "Next we need a slight generalization of Lemma \\ref{lem:spec0}",
      "explanation": "The comparison requires the missing statement of spec1. Spec0 concerns unweighted norm-ball counts for fixed gamma_1 in Gamma, q >= 1, and a subgroup satisfying Gamma(q) < tildeGamma(q) < Gamma, as Y tends to infinity. It asserts an exponent below delta, uniformity in q and gamma_1, and both multiplication orders. To verify the announced generalization, one must identify which hypotheses and uniformities spec1 preserves and how its weighted family specializes to, or extends, this counting statement."
    },
    {
      "kind": "unverified",
      "source_quote": "Applying Lemma \\ref{lem:spec1} to the latter sum, using the definition of $\\gT_{1}$ in \\eqref{eq:gT1def}, and recalling the estimate \\eqref{eq:G0qBnd}  gives",
      "explanation": "This explicitly supports the promised downstream use, but does not independently establish the application. The omitted lemma must control the phase-weighted sum over F intersected with a stabilizer coset, uniformly over the variables being summed and integrated. Its parameter restrictions must hold throughout the beta integration and q < Q_0 range. The missing exponent definition and subgroup-index bound are also needed to check the stated aggregate error O(T^{Theta_1} K_0^2 Q_0^4/N). Spec0 alone does not supply these weighted estimates."
    },
    {
      "kind": "map_correction",
      "source_quote": "which will be used in the major arcs analysis, see \\eqref{eq:fMdecomp}.",
      "explanation": "The eq:fMdecomp reference is a forward application anchor, not a mathematical premise proving the announced generalization. Its edge should be understood as downstream-use evidence. The spec0 reference identifies the precursor statement, not a sufficient proof of spec1. These are role corrections to the proposed dependency interpretation, not changes to the frozen map."
    }
  ],
  "dependency_proposals": [
    {
      "label": "lem:spec0",
      "role": "background",
      "source_quote": "Next we need a slight generalization of Lemma \\ref{lem:spec0}",
      "explanation": "The supplied counting lemma is the comparison baseline. Its truth is not established by this bounded packet, and its statement alone does not justify the weighted extension."
    },
    {
      "label": "eq:fMdecomp",
      "role": "background",
      "source_quote": "which will be used in the major arcs analysis, see \\eqref{eq:fMdecomp}.",
      "explanation": "This is the announced application, whose supplied surrounding text confirms that spec1 is invoked."
    },
    {
      "label": "lem:spec1",
      "role": "proof_input",
      "source_quote": "Applying Lemma \\ref{lem:spec1} to the latter sum",
      "explanation": "Its missing statement is necessary to assess both the claimed generalization and the advertised application."
    },
    {
      "label": "eq:gT1def and eq:G0qBnd",
      "role": "conditional",
      "source_quote": "using the definition of $\\gT_{1}$ in \\eqref{eq:gT1def}, and recalling the estimate \\eqref{eq:G0qBnd}",
      "explanation": "These explicitly cited but omitted inputs are needed if the downstream error estimate is to be checked; they are not evidence establishing spec1 itself."
    }
  ],
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