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  "target_id": "th_sg",
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  "summary": "Theorem th_sg is the main combinatorial spectral gap theorem of the Appendix. Because proof_anchor_missing is true, the supplied source_slice contains only the statement of Theorem th_sg and the introductory outline that defines the subgroups \\Ga_1 and \\Ga_2 and defers the proof to the 'following two Lemmata' after line 5483. Consequently, verifying Theorem th_sg requires the omitted remainder of the Appendix containing those two lemmata and the concluding deduction of Theorem th_sg.",
  "used_dependencies": [
    "external:Brooks1986",
    "external:Brooks2007",
    "external:SarnakToLagarias",
    "external:Selberg1965"
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    "The remainder of the Appendix after line 5483 (char 131279), specifically the statements and proofs of the 'following two Lemmata' and the concluding deduction of Theorem th_sg from them.",
    "Exact statements and hypotheses of Selberg (1965) and Brooks (1986, Theorem 1; 2007, Theorem 1) if the subsequent lemmata rely on the prose paragraph in lines 5469-5476 as the sole justification for the combinatorial spectral gap of \\Ga_1 and \\Ga_2."
  ],
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      "source_quote": "We transfer the combinatorial spectral gap property\nof $\\Ga_1$ and $\\Ga_2$ to the Apollonian  group $\\bar\\Ga$\nand conclude Theorem \\ref{th_sg}.\nThis is done in following two Lemmata:",
      "explanation": "The attached prose is a narrative outline rather than a complete proof of Theorem th_sg. It explicitly defers the proof of Theorem th_sg to two subsequent lemmata (and any concluding argument) located after line 5483, outside the supplied source_slice."
    },
    {
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      "source_quote": "As we will see later, $\\bG_1$ and $\\bG_2$ are isomorphic to $\\SL(2,\\R)$.\nMoreover $\\Ga_1$ and $\\Ga_2$ are lattices inside them.\nThis feature of the\nApollonian group was pointed out by Sarnak \\cite{SarnakToLagarias}.",
      "explanation": "The reference edge from external:SarnakToLagarias to th_sg arises from the candidate prose following the theorem statement. In the text, Sarnak (2007) is cited for historical attribution of a structural observation that the author states will be proved later in the section ('As we will see later...'), while the actual proof inputs to th_sg are the two subsequent lemmata not yet in the map."
    },
    {
      "kind": "unverified",
      "source_quote": "Due to a result going back to\nSelberg \\cite{Selberg1965}, $\\Ga_1$ and $\\Ga_2$ have geometric spectral gaps with\nrespect to the congruence subgroups.\nFrom here we can deduce the combinatorial spectral gap using\nBrooks \\cite[Theorem 1]{Brooks1986} (see also \\cite[Theorem 1]{Brooks2007}, where\nthe non-compact case is considered.)",
      "explanation": "The prose sketches why congruence quotients of \\Ga_1 and \\Ga_2 have combinatorial spectral gaps by invoking Selberg (1965) for the geometric spectral gap of congruence subgroups of lattices in SL(2,R) and Brooks (1986, Theorem 1; 2007, Theorem 1) for the equivalence with combinatorial spectral gap in the non-compact case (note \\Ga_1 and \\Ga_2 contain the parabolic element \\g_1, so their quotients are non-compact). All three external sources predate the 2012-05-24 cutoff (published in 1965, 1986, and 2007 respectively), but their external texts are not supplied in the packet and it remains to check the subsequent lemmata to see whether this deduction is formalized there or used directly as an unexpanded premise."
    }
  ],
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    {
      "label": "following_two_lemmata_after_line_5483",
      "role": "proof_input",
      "source_quote": "We transfer the combinatorial spectral gap property\nof $\\Ga_1$ and $\\Ga_2$ to the Apollonian  group $\\bar\\Ga$\nand conclude Theorem \\ref{th_sg}.\nThis is done in following two Lemmata:",
      "explanation": "Theorem th_sg depends directly on the two lemmata immediately following the source_slice (and the proof assembling them)."
    },
    {
      "label": "external:SarnakToLagarias",
      "role": "background",
      "source_quote": "This feature of the\nApollonian group was pointed out by Sarnak \\cite{SarnakToLagarias}.",
      "explanation": "Cited as attribution for the observation that \\Ga_1 and \\Ga_2 are lattices in copies of SL(2,R), which the text indicates is verified directly later in the appendix. Chronology: Letter to J. Lagarias, 2007 (available before the 2012-05-24 cutoff)."
    },
    {
      "label": "external:Selberg1965",
      "role": "conditional",
      "source_quote": "Due to a result going back to\nSelberg \\cite{Selberg1965}, $\\Ga_1$ and $\\Ga_2$ have geometric spectral gaps with\nrespect to the congruence subgroups.",
      "explanation": "Used in the bridging prose to assert the geometric spectral gap for congruence subgroups of \\Ga_1 and \\Ga_2. Chronology: Proc. Sympos. Pure Math. VII, pp. 1-15, 1965 (available before the 2012-05-24 cutoff)."
    },
    {
      "label": "external:Brooks1986",
      "role": "conditional",
      "source_quote": "From here we can deduce the combinatorial spectral gap using\nBrooks \\cite[Theorem 1]{Brooks1986} (see also \\cite[Theorem 1]{Brooks2007}, where\nthe non-compact case is considered.)",
      "explanation": "Cited for Theorem 1 on the equivalence of geometric and combinatorial spectral gaps (originally for compact manifolds, supplemented by Brooks 2007 for the non-compact case). Chronology: J. Differential Geom. 23(1):97-107, 1986 (available before the 2012-05-24 cutoff)."
    },
    {
      "label": "external:Brooks2007",
      "role": "conditional",
      "source_quote": "From here we can deduce the combinatorial spectral gap using\nBrooks \\cite[Theorem 1]{Brooks1986} (see also \\cite[Theorem 1]{Brooks2007}, where\nthe non-compact case is considered.)",
      "explanation": "Cited for Theorem 1 extending the relation between geometric and combinatorial spectral gaps to non-compact Riemann surfaces. Chronology: In Topology in Molecular Biology, Springer, 2007 (available before the 2012-05-24 cutoff)."
    }
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