{
  "target_id": "thm:IQX",
  "proof_status": "needs_context",
  "summary": "The packet states the desired power-saving estimate but supplies no proof of it. The section setup gives a conditional algebraic reduction to diagonal and off-diagonal contributions; it does not bound either contribution. Missing definitions, parameter relations, and subsequent estimates prevent verification. The absent proof anchor is a mapping limitation, not evidence that the theorem is defective.",
  "used_dependencies": [],
  "missing_context": [
    "The actual proof of thm:IQX, including the diagonal and off-diagonal estimates and their combination.",
    "Definitions and parameter restrictions for I_Q, Q_0, X, M, U, T, N and delta, including their dependence on N.",
    "Definitions of the form family, a, q_0, u_0, J_f and S_f, together with the identities referenced in the section setup.",
    "Support, decay or convergence statements justifying expansion and interchange of sums and the beta-integral.",
    "Any external results used by the missing proof, with exact statements, hypotheses and versions available by 2012-05-24. No external result is imported in this assessment; model background knowledge alone cannot certify historical availability."
  ],
  "findings": [
    {
      "kind": "map_correction",
      "source_quote": "\\begin{thm}\\label{thm:IQX}",
      "explanation": "The target slice contains only the theorem statement and whitespace. The attached section setup precedes the target and cannot be treated as its complete proof. No dependency or proof-reference edges are supplied; the proof location and its actual inputs remain unidentified."
    },
    {
      "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 definition of the quantity being bounded is omitted. Also missing are the meanings and allowed ranges of N, T, X, M, U, Q_0 and delta; the family of forms; and the definitions of the modular and archimedean factors. The standing assumption of a fixed primitive integral gasket does not supply these analytic prerequisites."
    },
    {
      "kind": "unverified",
      "source_quote": "applying Cauchy-Schwarz in the $u$-variable:",
      "explanation": "The factor U is consistent with Cauchy\u2013Schwarz for a sum containing O(U) integer indices. This requires the omitted summation convention and preceding representation. Conditional on those identities and justified sum/integral interchanges, opening the square produces the displayed conjugations and phases a'-a. The packet does not provide the convergence or truncation information needed to verify these operations fully."
    },
    {
      "kind": "unverified",
      "source_quote": "\\cI_{Q}\\ll \\cI_{Q}^{(=)}+\\cI_{Q}^{(\\neq)},",
      "explanation": "Partitioning pairs according to a'=a or a'\u2260a is exhaustive, but the two contributions are not explicitly defined or estimated here. Because the expanded expression contains complex terms, one must also specify whether these contributions are signed subsums, absolute values of subsums, or nonnegative majorants. The remaining substantive obligation is to bound their combined contribution by N T^{2(delta-1)} N^{-eta}, uniformly over the asserted Q-range."
    },
    {
      "kind": "context",
      "source_quote": "For $Q_{0}\\le Q<X$, there is some $\\eta>0$ such that",
      "explanation": "Verification requires the parameter choices as N tends to infinity, the quantifier governing eta across the Q-range, and the permitted dependence of the implied constant. None is specified in this packet. In particular, no supplied estimate explains how a positive saving survives losses involving U or the family of forms."
    }
  ],
  "dependency_proposals": [
    {
      "label": "eq:IQdef and parameter setup",
      "role": "proof_input",
      "source_quote": "the integrals $\\cI_{Q}$ in \\eqref{eq:IQdef}",
      "explanation": "The omitted definition and parameter conventions are necessary to interpret the target and verify the initial reduction."
    },
    {
      "label": "eq:tilQis, eq:cRfuIs and eq:RNRf",
      "role": "conditional",
      "source_quote": "recall the notation \\eqref{eq:tilQis}, and put \\eqref{eq:cRfuIs} into \\eqref{eq:RNRf}",
      "explanation": "These are explicitly cited inputs to the displayed Cauchy\u2013Schwarz calculation, but their statements and hypotheses are unavailable."
    },
    {
      "label": "eq:cSdef",
      "role": "conditional",
      "source_quote": "the former being exactly equal to $\\cS$ in \\eqref{eq:cSdef}",
      "explanation": "This omitted definition is needed to check the claimed identification of the modular bracket and any subsequent use of modular estimates."
    },
    {
      "label": "Diagonal and off-diagonal bounds",
      "role": "proof_input",
      "source_quote": "whether $a'=a$ (the ``diagonal'' case) and the off-diagonal $a'\\neq a$",
      "explanation": "Locate the actual estimates for these contributions. Bounds at the target scale with a common positive saving would complete the displayed reduction, subject to the definitions and parameter restrictions; no such bounds are supplied."
    }
  ],
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