{
  "target_id": "narrative:119956",
  "proof_status": "needs_context",
  "summary": "Conditional on the supplied lemmas and the intended convention \\ff=f-a, the diagonal partition and applicability of eq:modBndaEqAp are locally supported. The packet does not establish the comparative claims that a=a' alone is insufficient or that the stronger estimate produces a useful additional saving after summation. No quantitative estimate for the new off-equality contribution is proved within this slice. No external result is imported; possible later knowledge in model weights is not treated as historical evidence.",
  "used_dependencies": [
    "eq:cIQ3def",
    "eq:modBndaEqAp"
  ],
  "missing_context": [
    "Explicit definitions of \\ff=f-a, q0, and the precise diagonal contributions being partitioned.",
    "The preceding diagonal estimate and the numerical saving required, to assess the claim that a=a' alone is insufficient.",
    "Parameter and family bounds controlling the arithmetic factors and evaluated-form difference, if the claimed additional cancellation is to be verified quantitatively."
  ],
  "findings": [
    {
      "kind": "context",
      "source_quote": "Assume $Q\\ge X$.",
      "explanation": "The surrounding section treats X\u2264Q<M, so the stated range hypothesis of eq:cIQ3def is satisfied. Conditional on that lemma, its diagonal summation can be partitioned by equality or inequality of the two evaluated forms. This elementary partition requires no external theorem."
    },
    {
      "kind": "context",
      "source_quote": "As before, open the square, setting $\\ff'=f'-a'$",
      "explanation": "With the corresponding intended convention \\ff=f-a, the diagonal condition a=a' gives \\ff(m,-n)-\\ff'(m',-n')=f(m,-n)-f'(m',-n'). Thus the unequal-value branch satisfies both explicit hypotheses of eq:modBndaEqAp. The unprimed convention is not explicitly defined in the packet, although the displayed primed convention strongly indicates it."
    },
    {
      "kind": "context",
      "source_quote": "\\cI_{Q}^{(=,=)}\n+\n\\cI_{Q}^{(=,\\neq)}",
      "explanation": "The two restrictions are disjoint and exhaustive. The displayed identity is therefore valid if these quantities denote the restricted contributions to the same diagonal expression. Their precise definitions, and whether they partition the original integral expression or the nonnegative majorant supplied by eq:cIQ3def, are not included. This is a definition/scope limitation, not a demonstrated algebraic error."
    },
    {
      "kind": "unverified",
      "source_quote": "We no longer save enough from $a=a'$ alone. But recall that here more cancellation can be gotten from \\eqref{eq:modBndaEqAp}",
      "explanation": "The supplied estimate can be applied on the unequal-value branch, but its usefulness depends on the factors (q/q0)^5, gcd(a\u00b2,q0), and |f(m,-n)-f'(m',-n')|^{1/2}. No baseline estimate, required saving, or uniform controls for these factors are supplied. Consequently, neither comparative assertion can be checked here. The exact remaining obligation is to show that this bound, after summation in eq:cIQ3def, improves the previous diagonal treatment sufficiently for the intended target."
    },
    {
      "kind": "map_correction",
      "source_quote": "We now estimate $\\cI_{Q}^{(=,\\neq)}$ using the extra cancellation in \\eqref{eq:modBndaEqAp}.",
      "explanation": "This sentence announces an estimate; it does not provide one. The third supplied relation includes subsequent text asserting a power saving under Q<XT, which lies outside the bounded source slice. The dependency edge is relevant to the announced method, but its extended evidence cannot establish that later quantitative conclusion within this review."
    }
  ],
  "dependency_proposals": [
    {
      "label": "eq:cIQ3def",
      "role": "proof_input",
      "source_quote": "\\sum_{\\ff'\\in\\fF\\atop a'\\square a}",
      "explanation": "Conditional input supplying the summation framework and diagonal restriction. Its proof is not established by this packet."
    },
    {
      "label": "eq:modBndaEqAp",
      "role": "proof_input",
      "source_quote": "Assuming\n$a=a'$ and $f(m,-n)\\neq f'(m',-n')$",
      "explanation": "Conditional input applicable to the unequal-value diagonal branch once the affine-shift convention is confirmed. Applicability alone does not establish a useful aggregate saving."
    },
    {
      "label": "Affine-shift convention",
      "role": "conditional",
      "source_quote": "setting $\\ff'=f'-a'$",
      "explanation": "The corresponding unprimed convention is needed to translate the narrative's inequality of shifted forms into the lemma's inequality of quadratic forms."
    }
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