{
  "target_id": "narrative:122162",
  "proof_status": "needs_context",
  "summary": "The cited counting proposition matches the stated equality condition and the section's diagonal restriction a=a'. Conditional on that proposition, it can bound the number of contributing index tuples. The packet does not establish that this counting bound yields savings for the weighted quantity I_Q^(=,=): the relevant weight estimates, parameter relationships, and comparison bound are missing. This is a limitation of the supplied local context, not a demonstrated proof defect.",
  "used_dependencies": [
    "eq:eqeq"
  ],
  "missing_context": [
    "Definitions of the family F, its multiplicities or cardinality, the forms f and ff=f-a, and the parameter T.",
    "Definitions and relationships of q_0, Q, M, X, and u, including the effective ranges of n,m,n',m' and treatment of small counting cutoffs.",
    "The definition of I_Q^(=,=) and its precise relation to the weighted expansion eq:IQbnd2.",
    "Bounds for the modular and archimedean factors on the equality locus, followed by the sums over ff, q, and u.",
    "The benchmark estimate and parameter inequalities against which the claimed savings are measured."
  ],
  "findings": [
    {
      "kind": "unverified",
      "source_quote": "We exploit this condition to get savings using \\eqref{eq:eqeq}.",
      "explanation": "The reference is relevant: eq:eqeq counts tuples satisfying precisely a'=a and ff(m,-n)=ff'(m',-n'). Its statement is supplied, but its proof is not assessed here. Conditional on its validity, for each fixed ff and a sufficiently large cutoff L it gives at most O_epsilon((TL)^epsilon(L^2+TL)) tuples. This supports an unweighted counting reduction, not yet the asserted analytic savings."
    },
    {
      "kind": "context",
      "source_quote": "For large $M$ and  $\\ff=f-a\\in\\fF$ fixed,",
      "explanation": "The fixed-form hypothesis can be matched by applying the proposition separately inside the outer ff-sum. The diagonal condition and equality condition also match. However, the applicable counting cutoff and its required largeness are not established. The displayed analytic sum uses indices below q_0, whereas M also appears as a section-level parameter. Applying the proposition with its cutoff equal to q_0 requires q_0 to be sufficiently large, or a separate treatment of smaller q_0. Definitions of T, q_0, and the family are absent."
    },
    {
      "kind": "context",
      "source_quote": "Yet again the sum has split into modular and archimedean components with the former being exactly equal to $\\cS$ in \\eqref{eq:cSdef}.",
      "explanation": "The expression in eq:IQbnd2 is weighted by a modular correlation and an archimedean integral. A tuple-counting bound controls it only after suitable absolute-value or other weight estimates. Neither those estimates nor the precise definition and normalization of I_Q^(=,=) are supplied. Thus no quantitative saving for that quantity, or downstream headline estimate, follows from this packet alone."
    }
  ],
  "dependency_proposals": [
    {
      "label": "eq:eqeq",
      "role": "conditional",
      "source_quote": "\\ff(m,-n)=\\ff'(m',-n')",
      "explanation": "Retain the supplied edge: the proposition directly addresses the equality locus under a'=a. Its validity remains a proposed premise, and its application requires matching the cutoff and family hypotheses."
    },
    {
      "label": "eq:IQbnd2",
      "role": "proof_input",
      "source_quote": "\\label{eq:IQbnd2}",
      "explanation": "This expansion identifies the weighted quantity to which the counting proposition must be applied. It is an additional local input beyond the single frozen dependency edge; it does not itself provide the missing weight estimates."
    },
    {
      "label": "eq:cIQ34",
      "role": "background",
      "source_quote": "\\cI_{Q}\\ll \\cI_{Q}^{(=)}+\\cI_{Q}^{(\\neq)}.",
      "explanation": "The diagonal/off-diagonal split supplies the condition a=a' needed by eq:eqeq. The further split defining I_Q^(=,=) is not included."
    },
    {
      "label": "historical_scope",
      "role": "excluded",
      "source_quote": "We exploit this condition to get savings using \\eqref{eq:eqeq}.",
      "explanation": "No external result is imported to justify this transition, so no external theorem availability record is asserted. The recorded cutoff is 2012-05-24. Potential later knowledge in model weights is not evidence and is not used to close the missing obligations."
    }
  ],
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