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  "target_id": "narrative:109314",
  "proof_status": "needs_context",
  "summary": "The proposed diagonal/off-diagonal organization is supported at the bookkeeping level by the supplied expansion and decomposition. The announced power-saving bounds are not established in this packet. Equation eq:cEN2bnd supplies a comparison scale, not a proof of those bounds. No external results are imported; the recorded cutoff is 2012-05-24, and potentially later model knowledge is not used as evidence.",
  "used_dependencies": [
    "eq:cEN2bnd"
  ],
  "missing_context": [
    "The actual statements and proofs of the diagonal and off-diagonal bounds in the range Q_0 <= Q < X, including the quantifiers and uniformity of the positive saving exponent.",
    "Definitions of I_Q and its two contributions, together with the inherited meanings and relations of Q_0, X, M, U, T, N, delta, the form family, and the q_0/u_0 notation needed to check the estimates.",
    "If the announcement is intended to support the full bound eq:cEN2bnd, the relation between these integrals and the error second moment, including summation over Q and treatment of the other ranges."
  ],
  "findings": [
    {
      "kind": "context",
      "source_quote": "where, once $\\ff$ is fixed,  we collect $\\ff'$ according to whether $a'=a$ (the ``diagonal'' case) and the off-diagonal $a'\\neq a$.",
      "explanation": "These conditions are exhaustive and disjoint, so they support the announced separation of the expanded expression into two contributions. This checks the organizational transition, conditional on the preceding expansion and legitimate regrouping of its sums. It does not establish cancellation or a power saving in either contribution."
    },
    {
      "kind": "unverified",
      "source_quote": "We treat $\\cI_{Q}^{(=)}, \\ \\cI_{Q}^{(\\neq)}$ separately, starting with the former; we give bounds of the quality claimed in \\eqref{eq:cEN2bnd}.",
      "explanation": "This announces estimates but supplies neither their statements nor their proofs within the target scope. The remaining local obligation is to bound both contributions at the advertised scale N T^{2(delta-1)} N^{-eta}, for some positive eta and with the required parameter uniformity. Their absence here is a context limitation, not evidence that the estimates are false."
    },
    {
      "kind": "map_correction",
      "source_quote": "we give bounds of the quality claimed in \\eqref{eq:cEN2bnd}.",
      "explanation": "The source-reference edge from eq:cEN2bnd should be understood as a benchmark/background reference, not a logical proof input establishing the diagonal or off-diagonal estimates. The supplied theorem statement is itself an unverified proposed premise."
    },
    {
      "kind": "context",
      "source_quote": "\\section{Minor Arcs II: Case $Q_{0}\\le Q<X$}",
      "explanation": "The announcement concerns this Q-range, not the entire error term. The packet omits the definition of eq:IQdef, the parameter relations, and the subsequent estimates. Thus it does not establish uniformity over this range or explain how bounds here contribute to the full error second moment."
    }
  ],
  "dependency_proposals": [
    {
      "label": "eq:cEN2bnd",
      "role": "background",
      "source_quote": "\\sum_{n<N}|\\cE_{N}^{U}(n)|^{2}\n\\ll \nN\\, T^{2(\\gd-1)} N^{-\\eta}",
      "explanation": "This gives the advertised power-saving scale. It is not established by the supplied theorem statement and does not imply bounds for the individual contributions."
    },
    {
      "label": "eq:cIQ12",
      "role": "proof_input",
      "source_quote": "\\cI_{Q}\\ll \\cI_{Q}^{(=)}+\\cI_{Q}^{(\\neq)},",
      "explanation": "This is the direct local basis for treating the two contributions separately. Its availability supports the narrative transition, conditional on the preceding expansion and definitions."
    },
    {
      "label": "eq:IQdef and inherited parameter setup",
      "role": "conditional",
      "source_quote": "Keeping all the notation from the last section,  we now turn our attention to the integrals $\\cI_{Q}$ in \\eqref{eq:IQdef}.",
      "explanation": "The source explicitly inherits prerequisites not present in the packet. Their definitions and parameter restrictions are required to verify the promised quantitative estimates."
    }
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