{
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  "proof_status": "locally_supported",
  "summary": "The stated combination is locally valid conditional on the five supplied, unverified premises. Substituting the off-diagonal decomposition into the first decomposition gives three contributions. Taking \u03b7 to be the minimum of their three positive saving exponents yields I_Q \u226a N T^{2(\u03b4\u22121)} N^{\u2212\u03b7}. All five mapped dependencies are relevant. A uniform conclusion throughout Q\u2080 \u2264 Q < X requires the component estimates to be uniform there. This assessment does not establish those estimates or any broader theorem. No external result is imported; latent knowledge of later work is not used as evidence.",
  "used_dependencies": [
    "eq:cIQ11bnd",
    "eq:cIQ12",
    "eq:cIQ12bnd",
    "eq:cIQ1break",
    "eq:cIQ2bnd"
  ],
  "missing_context": [
    "The complete statement immediately following the target sentence, to check its exact quantifiers and any additional conclusion.",
    "For a range-uniform conclusion, confirmation that all three component bounds apply throughout Q\u2080 \u2264 Q < X with compatible parameter choices, including the same H in the off-diagonal split, and with Q-independent saving exponents, implied constants, and asymptotic thresholds. Definitions of the underlying sums are not needed for the finite-sum inference, but their substantive bounds remain unverified."
  ],
  "findings": [
    {
      "kind": "context",
      "source_quote": "\\cI_{Q}\\ll \\cI_{Q}^{(=)}+\\cI_{Q}^{(\\neq)},",
      "explanation": "Together with the supplied off-diagonal equality, this reduces the conclusion to the three stated bounds. If their exponents are \u03b7\u2081, \u03b7\u2082, \u03b7\u2083 > 0, set \u03b7 = min(\u03b7\u2081, \u03b7\u2082, \u03b7\u2083). For N \u2265 1, each N^(\u2212\u03b7\u1d62) \u2264 N^(\u2212\u03b7), so summing preserves the claimed power-saving shape. No equality of the three exponents is required."
    },
    {
      "kind": "unverified",
      "source_quote": "There is an $\\eta>0$ such that",
      "explanation": "Each component proposition supplies a positive saving exponent, but these statements are proposed premises, not established results in this review. The elementary combination does not verify their proofs. For a single estimate valid across the section's Q-range, their exponents, implied constants, and sufficiently-large-N thresholds must be independent of Q after fixing the permitted parameters."
    },
    {
      "kind": "context",
      "source_quote": "we have proved the following",
      "explanation": "The full statement announced by this sentence is outside the source_slice. Relation evidence shows its opening concerns Q\u2080 \u2264 Q < X and a bound for I_Q. The supplied premises support that three-term power-saving conclusion, conditional on their validity and required uniformity; they do not by themselves support claims about summing over Q, other Q-ranges, or the paper's headline density theorem."
    }
  ],
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