{
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  "summary": "Theorem thm:IQX is stated at the end of Section sec:QX as the culmination of the preceding analysis, with only a page break following it. While the section setup (lines 4410-4517) provides the initial Cauchy-Schwarz step and the decomposition I_Q << I_Q^{(=)} + I_Q^{(\\neq)} in (eq:cIQ12), the entire body of the argument bounding I_Q^{(=)} and I_Q^{(\\neq)} (lines 4518-4837) is omitted from the packet and no dependencies are supplied. Verification requires the missing text from lines 4518-4837 as well as the parameter definitions from earlier sections.",
  "used_dependencies": [],
  "missing_context": [
    "Lines 4518-4837 of Section sec:QX, which contain the bounds for the diagonal contribution I_Q^{(=)} and off-diagonal contribution I_Q^{(\\neq)} and the concluding argument for Theorem thm:IQX.",
    "The definition of I_Q in equation (eq:IQdef) and the definitions/growth relations of the parameters N, T, \\delta, Q_0, Q, and X as N \\to \\infty."
  ],
  "findings": [
    {
      "kind": "context",
      "source_quote": "\\begin{thm}\\label{thm:IQX}\nFor $Q_{0}\\le Q<X$, there is some $\\eta>0$ such that \n$$\n\\cI_{Q}\\ll N\\, T^{2(\\gd-1)}\\, N^{-\\eta},\n$$\nas $N\\to\\infty$.\n\\end{thm}",
      "explanation": "Theorem thm:IQX is stated at the very end of Section sec:QX (immediately before \\newpage) without a following proof environment. Its derivation relies on the omitted text of Section sec:QX between lines 4517 and 4838, where the diagonal and off-diagonal terms in (eq:cIQ12) are analyzed."
    },
    {
      "kind": "map_correction",
      "source_quote": "Decompose   \\eqref{eq:IQbnd1} as\n\\be\\label{eq:cIQ12}\n\\cI_{Q}\\ll \\cI_{Q}^{(=)}+\\cI_{Q}^{(\\neq)},\n\\ee\nwhere, once $\\ff$ is fixed,  we collect $\\ff'$ according to whether $a'=a$ (the ``diagonal'' case) and the off-diagonal $a'\\neq a$.",
      "explanation": "The section setup decomposes I_Q into I_Q^{(=)} and I_Q^{(\\neq)}, but the subsequent estimates for these two pieces (lines 4518-4837) are omitted from the packet and no dependency nodes for them are supplied."
    },
    {
      "kind": "context",
      "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 definitions and parameter relations for I_Q, Q_0, Q, X, N, T, and \\delta from earlier sections are referenced but not included in the bounded context."
    }
  ],
  "dependency_proposals": [
    {
      "label": "eq:cIQ12",
      "role": "proof_input",
      "source_quote": "Decompose   \\eqref{eq:IQbnd1} as\n\\be\\label{eq:cIQ12}\n\\cI_{Q}\\ll \\cI_{Q}^{(=)}+\\cI_{Q}^{(\\neq)},\n\\ee",
      "explanation": "Equation (eq:cIQ12) decomposes I_Q into the diagonal term I_Q^{(=)} and off-diagonal term I_Q^{(\\neq)}. This decomposition is a direct setup step toward bounding I_Q in Theorem thm:IQX."
    },
    {
      "label": "eq:IQdef",
      "role": "background",
      "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 definition of the integral I_Q in (eq:IQdef) from the preceding section is required to understand the quantity bounded in Theorem thm:IQX."
    }
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