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  "target_id": "eq:decayMtrx",
  "proof_status": "needs_context",
  "summary": "The estimate is attributed directly to Kontorovich\u2013Oh, Proposition 5.3, but that proposition has not been supplied or verified. The following bisector-counting discussion is not its proof. For \u0398\u22652 the bound follows from Cauchy\u2013Schwarz, provided the Sobolev norm dominates the Hilbert norm. The substantive range 1<\u0398<2 requires verification of the cited result, particularly its treatment of invariant vectors and its exact decay exponent. The missing proof anchor is a mapping limitation, not an established proof defect.",
  "used_dependencies": [
    "external:KontorovichOh2011"
  ],
  "missing_context": [
    "The statement of Kontorovich\u2013Oh Proposition 5.3 in the cited 2011 version, including all hypotheses and norm conventions.",
    "The manuscript's definition and parameter range of complementary series, including whether the trivial representation at the endpoint is covered or invariant vectors are otherwise excluded.",
    "The complete definition of S, its Lie algebra basis, its domain on smooth vectors, and the allowed dependence of the implied constant on \u0398 and the representation.",
    "Only if Vinogradov is proposed as an actual proof input: dated metadata and the relevant statements from arXiv:1204.5498v1. Its present edges appear incidental."
  ],
  "findings": [
    {
      "kind": "unverified",
      "source_quote": "\\begin{thm}[{\\cite[Prop. 5.3]{KontorovichOh2011}}]",
      "explanation": "This is a citation-based assertion, with no local derivation in the supplied scope. It remains necessary to inspect Proposition 5.3 and verify its group, representation hypotheses, parameter normalization, Sobolev order, implied-constant dependence, and exact exponent without an additional epsilon or polynomial factor. No defect follows merely from the absent proof anchor."
    },
    {
      "kind": "context",
      "source_quote": "Let $\\gT>1$ and $(\\pi,V)$ be a unitary representation of $G$ which does not weakly contain any complementary series representation with parameter $s>\\gT$.",
      "explanation": "The definition of the complementary-series parameter and its endpoint convention is missing. For 1<\u0398<2, the trivial representation must be excluded by this hypothesis or separately. If 'complementary series' means only the nontrivial representations with 1<s<2, the trivial representation passes the stated exclusion but violates the conclusion: for unit vectors its coefficient is 1, whereas along a_t the asserted right-hand side tends to zero. Whether this is an actual omission depends on the manuscript's unsupplied conventions."
    },
    {
      "kind": "context",
      "source_quote": "\\cS v\\cdot\n\\cS w",
      "explanation": "The target uses an undefined Sobolev norm. Relation evidence supplies a candidate involving first and second derivatives, but does not identify the Lie algebra basis, vector domains, or constant conventions. These are needed to check uniformity over all smooth vectors. The Frobenius exponent itself is consistent with Cartan coordinates: ||a_t||_F\u00b2=e^t+e^(-t), so the asserted decay is comparable to e^(-(2-\u0398)t). For \u0398\u22652, Cauchy\u2013Schwarz suffices if S(v)\u2265||v||\u2082, since every g in SL(2,C) has Frobenius norm at least \u221a2."
    },
    {
      "kind": "map_correction",
      "source_quote": "\\subsection{Effective Bisector Counting}",
      "explanation": "The Vinogradov references occur after the target theorem ends, in a new subsection introducing a different ingredient. The supplied edges concern bisector counting and a Cartan-coordinate description, not a proof of the matrix-coefficient estimate. They should not be treated as dependencies of eq:decayMtrx."
    },
    {
      "kind": "context",
      "source_quote": "\\newblock Princeton University Thesis 2012, {\\tt arxiv:1204.5498v1}.",
      "explanation": "The recorded cutoff is 2012-05-24. The cited 2013 journal version cannot establish pre-cutoff availability. The preprint identifier suggests an earlier candidate, but the packet supplies neither its dated metadata nor its contents; availability of the needed statements in that version remains unverified. Model weights may retain later knowledge, but no later theorem or proof is used here to discharge an original obligation."
    }
  ],
  "dependency_proposals": [
    {
      "label": "external:KontorovichOh2011",
      "role": "conditional",
      "source_quote": "\\newblock {\\em Journal of the American Mathematical Society}, 24(3):603--648,\n  2011.",
      "explanation": "The sole directly cited proof input is Proposition 5.3, purportedly giving this matrix-coefficient estimate under the displayed spectral exclusion. Its exact original theorem and hypotheses remain unverified. Historical record: available_from=2011 at year-level resolution; date_evidence=the supplied journal citation; date_basis=bibliographic publication year; version=cited JAMS 24(3), 603\u2013648 publication, not inspected. This supports a pre-cutoff candidate, not verification of its contents."
    },
    {
      "label": "external:Vinogradov2013",
      "role": "excluded",
      "source_quote": "The next ingredient which we require is the recent work by Vinogradov \\cite{Vinogradov2013} on effective bisector counting for such infinite volume quotients.",
      "explanation": "This introduces a subsequent counting ingredient rather than supporting the target estimate. Historical record: available_from=2013 for the cited journal version; date_evidence=the supplied IMRN 2013 citation; date_basis=bibliographic publication year; version=2013 journal article. For the separately cited arXiv:1204.5498v1, available_from is unverified, date_evidence is only the identifier and thesis year, date_basis is unresolved pending dated metadata, and the relevant theorem content has not been inspected. Neither version is used to establish the target."
    }
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