{
  "target_id": "narrative:110408",
  "proof_status": "locally_supported",
  "summary": "The passage is locally supported as parameter algebra and a partition of the stated off-diagonal contribution, conditional on the supplied definitions. The two expressions for H agree, and the gcd conditions are disjoint and exhaustive. No estimate for either contribution is established here. No external theorem is imported; possible later knowledge in model weights is not used as evidence.",
  "used_dependencies": [
    "eq:fuIs",
    "lem:spec3"
  ],
  "missing_context": [
    "The exact definition and normalization of the parent contribution I_Q^(\u2260) are not supplied beyond its description as the a\u2032\u2260a portion. The partition is supported for that described sum; positivity, convergence, and subsequent estimates are not certified.",
    "The proof of Lemma spec3 and its underlying parameter definitions are absent. Its counting conclusion remains an unverified premise and is unnecessary for the elementary transitions reviewed here."
  ],
  "findings": [
    {
      "kind": "context",
      "source_quote": "H=U^{10/\\eta_{0}}=Q_{0}{}^{\\eta_{0}/10}",
      "explanation": "Substituting U=Q\u2080^(\u03b7\u2080\u00b2/100) gives U^(10/\u03b7\u2080)=Q\u2080^((\u03b7\u2080\u00b2/100)(10/\u03b7\u2080))=Q\u2080^(\u03b7\u2080/10). The supplied positivity of \u03b7\u2080 permits cancellation. This checks the identity, not the eventual usefulness of this parameter choice."
    },
    {
      "kind": "context",
      "source_quote": "corresponding to whether $(a,a')>H$ or $(a,a')\\le H$, respectively.",
      "explanation": "These conditions partition the index pairs without overlap, including the boundary gcd(a,a\u2032)=H in the second contribution. Keeping the existing condition a\u2032\u2260a in both restricted sums yields the displayed decomposition by definition. The partition itself requires no counting estimate."
    },
    {
      "kind": "context",
      "source_quote": "the constant $\\eta_{0}>0$ comes from Lemma \\ref{lem:spec3}.",
      "explanation": "The lemma supplies the provenance and positivity of \u03b7\u2080, conditionally on its unverified statement. Its congruence-counting bound is not applied in this passage, so its modulus range and parameter hypotheses do not justify any large-gcd estimate here. The supplied reference edge is appropriate as provenance, not evidence that the counting bound has been used."
    },
    {
      "kind": "context",
      "source_quote": "We deal first with the large $\\gcd$.",
      "explanation": "This announces the next step rather than proving a bound. The supplied scope supports only introducing H and splitting the contribution; it does not establish a saving for the large-gcd part or validate the eventual choice of H against downstream requirements."
    }
  ],
  "dependency_proposals": [
    {
      "label": "eq:fuIs",
      "role": "proof_input",
      "source_quote": "U=Q_{0}{}^{(\\eta_{0})^{2}/100}",
      "explanation": "This definition directly supplies the substitution needed to verify the second expression for H."
    },
    {
      "label": "lem:spec3",
      "role": "background",
      "source_quote": "There exists some $\\eta_{0}>0$ depending only on the spectral gap of $\\G$",
      "explanation": "This identifies the positive constant used in H. The lemma's counting estimate is not a proof input to the gcd partition."
    }
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