{
  "target_id": "narrative:123431",
  "proof_status": "locally_supported",
  "summary": "The cited decomposition and three component bounds conditionally yield the intended bound for the total contribution, with saving exponent equal to the minimum of the three positive exponents. The supplied archimedean calculation encodes M=XT, which makes the hypotheses Q<XT applicable throughout X\u2264Q<M. This assesses only the aggregation: the component estimates are unverified premises, and the complete following theorem is not included. No external result is imported; model background knowledge is not evidence of availability before the recorded cutoff.",
  "used_dependencies": [
    "eq:cI3bnd",
    "eq:cI41bnd",
    "eq:cI42bnd",
    "eq:cIQ312",
    "eq:cIQ34"
  ],
  "missing_context": [
    "The complete following theorem, including its quantifiers and any uniformity assertion in Q, is needed to compare its exact wording with the aggregate estimate.",
    "The explicit parameter definition eq:Mis would confirm the scale convention M=XT already encoded in the supplied calculation.",
    "The proofs and standing hypotheses of the three component propositions are not supplied for verification; this assessment treats their statements as conditional premises."
  ],
  "findings": [
    {
      "kind": "context",
      "source_quote": "\\cI_{Q}\\ll \\cI_{Q}^{(=)}+\\cI_{Q}^{(\\neq)}.",
      "explanation": "Together with eq:cIQ312, this reduces the total to the three contributions bounded by eq:cI3bnd, eq:cI41bnd, and eq:cI42bnd. Writing their exponents as eta_3, eta_41, eta_42 and taking eta=min(eta_3,eta_41,eta_42)>0 gives I_Q \u226a N T^{2(delta\u22121)} N^{\u2212eta}. There are only three terms, so this aggregation introduces no logarithmic loss. The component estimates are used conditionally, not established here."
    },
    {
      "kind": "context",
      "source_quote": "|\\gb u^{2} f(x,y)|\\ll \\frac1{QM} u^{2} T\\left(\\frac Xu\\right)^{2}= \\frac XQ\\le 1.",
      "explanation": "The off-diagonal estimate assumes X\u2264Q<M, while both diagonal estimates assume Q<XT. For positive scales, the displayed equality simplifies to TX\u00b2/(QM)=X/Q and hence M=XT. Under that supplied scale convention, Q<M gives the required strict inequality Q<XT, and Q=X is permitted. The defining equation eq:Mis is omitted, but its needed consequence is encoded in this calculation."
    },
    {
      "kind": "map_correction",
      "source_quote": "we have the following",
      "explanation": "The target slice ends before the announced conclusion. All five dependency edges are relevant to the aggregation; none is shown to be incorrect. However, the map should distinguish this transition from the following theorem and record the parameter relation used for range compatibility. The missing consequent is a scope/mapping limitation, not a demonstrated proof defect. The transition supports the total minor-arc estimate, not by itself the paper's density-one headline claim."
    }
  ],
  "dependency_proposals": [
    {
      "label": "eq:Mis",
      "role": "proof_input",
      "source_quote": "and using \\eqref{eq:Mis}, we have that",
      "explanation": "Record this parameter input for the implication Q<M \u21d2 Q<XT. The supplied calculation already encodes M=XT, but the defining equation itself is absent."
    },
    {
      "label": "Following total-bound theorem",
      "role": "unclassified",
      "source_quote": "Combining\n\\eqref{eq:cIQ34}, \\eqref{eq:cI3bnd}, \\eqref{eq:cIQ312}, \\eqref{eq:cI42bnd}, and \\eqref{eq:cI41bnd}, we have the following",
      "explanation": "Attach the complete announced theorem as output context, rather than treating the transitional sentence as its full statement. Its intended bound follows conditionally by the finite-sum calculation above."
    }
  ],
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