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  "target_id": "eq:gls",
  "proof_status": "needs_context",
  "summary": "The spectral statement is consistent with the supplied geometrically finite, discrete, non-elementary, infinite-volume setup, but its substantive assertions are attributed to external results whose exact statements and dated versions are absent. The figure is not a proof dependency. The subsequent congruence-spectrum discussion also needs an explicit continuation of the hypothesis \u03b4>1 and clarification when no second discrete eigenvalue exists. These are local verification and scope limitations, not a full-theorem verdict.",
  "used_dependencies": [],
  "missing_context": [
    "Exact cited theorem statements and dated source versions establishing the spectral assertions under all stated hypotheses, available by the recorded cutoff 2012-05-24.",
    "The precise Apollonian subgroup/lift used in SL(2,C), together with support for geometric finiteness, its limit-set identification, and \u03b4>1.",
    "An explicit \u03b4>1 restriction for the congruence-spectrum discussion and treatment of an absent second discrete eigenvalue.",
    "If the Patterson\u2013Sullivan construction is intended as more than exposition, its compactification, convergence statement, normalization, and Poisson-kernel formula."
  ],
  "findings": [
    {
      "kind": "unverified",
      "source_quote": "The spectrum above $1$ is purely continuous, and the spectrum below $1$ is purely discrete.",
      "explanation": "No proof or inspected external theorem is supplied for the spectral decomposition, finiteness of the subthreshold list, simplicity of its lowest eigenvalue, the \u03b4>1 criterion, or the formula \u03bb\u2080=\u03b4(2\u2212\u03b4). The missing proof anchor is a mapping limitation, not itself a proof defect. Verification requires exact results covering geometrically finite, non-elementary, infinite-covolume discrete subgroups of SL(2,C), including any torsion or cusp qualifications."
    },
    {
      "kind": "context",
      "source_quote": "We assume that $\\G$ is non-elementary (not virtually abelian), so $\\gd>0$, and moreover that $\\G$ is not a lattice,",
      "explanation": "Together with the preceding discreteness and geometric-finiteness assumptions, these are the stated hypotheses for the theorem; the additional arithmetic hypothesis appears only afterward. Conditional on the supplied conclusion \u03b4<2, the proposed eigenvalue passes the elementary check: if 1<\u03b4<2, then 0<\u03b4(2\u2212\u03b4)<1. The implications relating geometric finiteness, covolume, and limit-set dimension are not proved in this packet."
    },
    {
      "kind": "map_correction",
      "source_quote": "In our application to the Apollonian group, the limit set is precisely the underlying gasket, see Fig. \\ref{fig:lim}.",
      "explanation": "The edge fig:lim \u2192 eq:gls comes from the application remark, not the proof of the general spectral theorem. The orbit illustration establishes neither spectral assertions nor the numerical dimension. This dependency should be classified as illustrative background rather than a proof input."
    },
    {
      "kind": "unverified",
      "source_quote": "\\gd\\approx 1.3...>1.",
      "explanation": "For the Apollonian application, the needed conclusion is \u03b4>1; the numerical approximation is stronger than necessary. Neither a dimension estimate nor a proof identifying the relevant lifted group's limit set with the gasket is supplied. The illustration cannot establish these application prerequisites."
    },
    {
      "kind": "context",
      "source_quote": "As in \\eqref{eq:gls}, write",
      "explanation": "The ensuing notation \u03bb\u2080(q), \u03bb\u2081(q), \u2026 presupposes \u03b4>1, whereas the general setup also permits \u03b4\u22641. The added condition \u0393<SL(2,Z[i]) alone does not supply \u03b4>1. Even with \u03b4>1, the theorem does not guarantee a second discrete eigenvalue. The intended restriction to the Apollonian application and the meaning of \u03bb\u2081(q) when absent need clarification. Finite index follows directly from the finite reduction target; equality of base eigenvalues additionally uses finite-index invariance of the limit-set dimension and the cited spectral formula."
    },
    {
      "kind": "context",
      "source_quote": "Roughly speaking, $\\mu$ is the weak$^{*}$ limit as $s\\to\\gd^{+}$ of the measures",
      "explanation": "This is explicitly heuristic, not a proof of the theorem. For a precise construction, the displayed point indicators must represent Dirac masses, the weak-* topology must be on the compactification including the boundary, and convergence, boundary support, and the Poisson-kernel exponent must be specified. None of these later explanatory claims independently verifies eq:gls."
    }
  ],
  "dependency_proposals": [
    {
      "label": "Patterson1976 / Sullivan1984 / LaxPhillips1982",
      "role": "conditional",
      "source_quote": "The spectrum is controlled via the following, see \\cite{Patterson1976, Sullivan1984, LaxPhillips1982}.",
      "explanation": "Proposed external support for the stated spectral theorem. Required content: continuous spectrum above 1, finite discrete spectrum below 1, its nonemptiness exactly when \u03b4>1, and a simple lowest eigenvalue \u03b4(2\u2212\u03b4), for the supplied geometrically finite, discrete, non-elementary, infinite-volume setting. For each citation: available_from=unresolved; date_evidence=bibliographic key only; date_basis=no inspected dated source record; version=unresolved. The exact allocation of assertions among these references is also unresolved. No external result is treated here as verified or historically admissible; model background knowledge is not dated evidence."
    },
    {
      "label": "Geometric and covolume setup",
      "role": "proof_input",
      "source_quote": "Let $\\G<G$ be a geometrically finite discrete group.",
      "explanation": "Together with the explicitly stated non-elementarity and nonlattice assumptions, this is essential theorem context omitted from the frozen dependency list. It is accepted only as the proposed local setup, not independently established for the Apollonian application."
    },
    {
      "label": "fig:lim",
      "role": "excluded",
      "source_quote": "The orbit of a point in hyperbolic space under the Apollonian group.",
      "explanation": "Exclude as a proof input to eq:gls; retain only as illustrative background for the application remark."
    },
    {
      "label": "Apollonian dimension bound",
      "role": "conditional",
      "source_quote": "\\gd\\approx 1.3...>1.",
      "explanation": "A bound \u03b4>1 would justify applying the theorem's nonempty-discrete-spectrum branch to the intended Apollonian group. It does not prove the general theorem. No supporting external result is imported: available_from, date_evidence, date_basis, and version remain unresolved pending a dated source."
    }
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