{
  "target_id": "thm:specGap",
  "proof_status": "needs_context",
  "summary": "The supplied packet does not contain the announced Appendix proof. Conditional on the spectral premises, the excerpts describe spectra at individual levels but do not establish a gap uniform in q. The statement also needs clarification of its inherited dimension hypothesis and of \u03bb\u2081(q) when no second discrete eigenvalue exists. These are bounded contextual and mapping issues, not a verdict against the theorem in its full setting. No external theorem or later development is imported; my stored knowledge is not chronologically filtered and is not used to fill the missing argument.",
  "used_dependencies": [
    "eq:glQs",
    "eq:gls"
  ],
  "missing_context": [
    "The Appendix statement and proof corresponding to thm:specGap, with all hypotheses and its connection to the main-text formulation.",
    "An explicit inherited \u03b4>1 hypothesis, or clarification of the spectral notation outside that range.",
    "The definition of \u03bb\u2081(q) when there is no second discrete eigenvalue below 1.",
    "An argument covering every positive integer q, including prime powers and any exceptional levels, with a constant depending only on \u0393.",
    "For any external result needed by the Appendix: its exact theorem and hypotheses, verification of applicability, and available_from, date_evidence, date_basis, and version establishing availability by 2012-05-24. The supplied bibliography alone does not provide those theorem-level checks."
  ],
  "findings": [
    {
      "kind": "unverified",
      "source_quote": "This is proved in the Appendix by P\\'eter Varj\\'u.",
      "explanation": "The actual proof is explicitly located outside the supplied excerpts. Its absence is a mapping limitation, not evidence of a defective proof. The remaining obligation is to establish one \u03b5(\u0393)>0 valid for every positive integer q, rather than merely isolation of the base eigenvalue at each fixed level."
    },
    {
      "kind": "context",
      "source_quote": "The latter is empty unless $\\gd>1$",
      "explanation": "The supplied setup assumes geometric finiteness, discreteness, non-elementarity, infinite covolume, and \u0393<SL(2,Z[i]); it gives 0<\u03b4<2 but does not explicitly assume \u03b4>1. Conditional on this spectral premise, when \u03b4\u22641 there is no discrete base eigenvalue to which the displayed theorem can refer. Clarify whether an earlier hypothesis restricts \u0393 to \u03b4>1 or whether another interpretation of \u03bb\u2080 is intended."
    },
    {
      "kind": "context",
      "source_quote": "0<\\gl_{0}(q)<\\gl_{1}(q)",
      "explanation": "This enumeration presupposes a second discrete eigenvalue below 1. Even the supplied \u03b4>1 spectral premise only provides a base eigenvalue; it does not guarantee a second one. The definition of \u03bb\u2081(q), including levels with only the base eigenvalue, is missing. This also affects q=1."
    },
    {
      "kind": "unverified",
      "source_quote": "\\gl_{0}(q)=\\gl_{0}=\\gd(2-\\gd).",
      "explanation": "Accepting this supplied equality conditionally identifies the common base, but supplies no uniform separation above it. Likewise, discreteness at each level would give only level-dependent gaps; it does not establish that their infimum over q is positive."
    },
    {
      "kind": "map_correction",
      "source_quote": "Obviously, a similar decomposition holds for $L^{2}(\\G(q)\\bk G)$, corresponding to \\eqref{eq:glQs}.",
      "explanation": "The eq:gls and eq:glQs reference edges arise in a subsequent representation-theoretic discussion. That transition describes individual level decompositions, not uniform exclusion of complementary-series parameters approaching the base parameter. Treat these edges as spectral setup, not as a proof of the gap."
    },
    {
      "kind": "map_correction",
      "source_quote": "By the Duality Theorem of Gelfand, Graev, and Piatetski-Shapiro \\cite{GelfandGraevPS1966}",
      "explanation": "This citation supports the following representation-theoretic interpretation, not the preceding theorem. The Cowling\u2013Haagerup\u2013Howe citation likewise introduces subsequent mixing estimates; the supplied slice states no such estimate, only a Sobolev norm. These reference locations do not establish either citation as a proof input for thm:specGap."
    }
  ],
  "dependency_proposals": [
    {
      "label": "Appendix proof of thm:specGap",
      "role": "proof_input",
      "source_quote": "This is proved in the Appendix by P\\'eter Varj\\'u.",
      "explanation": "This is the explicit proof location. Its anchor and contents should be supplied; this proposal does not change the frozen map."
    },
    {
      "label": "eq:gls",
      "role": "conditional",
      "source_quote": "The spectrum above $1$ is purely continuous, and the spectrum below $1$ is purely discrete.",
      "explanation": "Use only as an unverified supplied spectral premise. It supplies fixed-level spectral context, not uniformity over congruence levels."
    },
    {
      "label": "eq:glQs",
      "role": "background",
      "source_quote": "for the discrete spectrum of $\\G(q)\\bk\\bH^{3}$.",
      "explanation": "This defines the level-dependent notation, subject to the missing-eigenvalue convention; it does not assert the required gap."
    },
    {
      "label": "external:GelfandGraevPS1966",
      "role": "background",
      "source_quote": "the spectral decomposition above is equivalent to the decomposition into irreducibles",
      "explanation": "The visible use concerns spectral duality after the target theorem. The bibliography dates the cited book to 1966, but its exact theorem, hypotheses, and version have not been inspected. No result is imported."
    },
    {
      "label": "external:CowlingHaagerupHowe1988",
      "role": "background",
      "source_quote": "general fact about mixing rates of matrix coefficients",
      "explanation": "The visible use concerns subsequent mixing estimates. The bibliography dates the article to 1988, but its exact theorem and applicability have not been inspected. No result is imported."
    }
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