{
  "target_id": "lem:GGq",
  "proof_status": "defective",
  "summary": "With \u0393(q) interpreted as the ordinary congruence kernel and SO_F as the full determinant-one isometry group, part (2) is false: modulo 7, \u2212I belongs to SO_F but not to the image of \u0393. This is checked directly below, without importing an external theorem. Parts (1), (3), and the following admissibility remark remain unverified from this packet. The absent proof anchor is a mapping limitation, not itself a proof defect.",
  "used_dependencies": [],
  "missing_context": [
    "The exact definition of \u0393(q), the finite-ring group SO_F, and any distinction between the full orthogonal group and the spin image.",
    "The precise Fuchs thesis theorem, page, hypotheses, deposited version, and dated evidence that this version was available by 2012-05-24.",
    "Proof or an accurately stated cited result for part (1) and both assertions in part (3).",
    "Orbit-transitivity, coordinate-lifting, and full-orbit versus even-subgroup coordinate-residue arguments needed for the admissibility remark."
  ],
  "findings": [
    {
      "kind": "error",
      "source_quote": "\\G/\\G(q)\\cong\\SO_{F}(\\Z/q\\Z).",
      "explanation": "Direct countercheck at q=7, which satisfies (q,6)=1: write F(x)=x\u1d40Ax with A=2I\u2212J and B(x,y)=x\u1d40Ay. In the Clifford algebra defined by xy+yx=2B(x,y), each standard basis vector e_i has square 1, and the displayed reflection satisfies S_i(x)=\u2212e_i x e_i. Consequently every even word representing an element of \u0393 acts by conjugation with an even product g of these vectors. If ~ reverses products, then g~g=1. Now take the orthogonal basis u\u2081=e\u2081, u\u2082=e\u2082\u2212e\u2083, u\u2083=e\u2082+e\u2083\u22122e\u2084, u\u2084=3e\u2081+e\u2082+e\u2083+e\u2084. Its squared norms modulo 7 are 1,4,5,2. If conjugation by g were \u2212I, g would anticommute with every u_i. Expansion in the Clifford monomial basis then forces g=c u\u2081u\u2082u\u2083u\u2084: an even monomial anticommutes with u_i exactly when it contains u_i. Hence g~g=5c\u00b2, which cannot equal 1 since 3=5\u207b\u00b9 is nonsquare modulo 7. Thus \u2212I is absent from the congruence image, although it preserves F and has determinant 1. The image is therefore a proper subgroup of the finite group asserted in (2), so even the abstract isomorphism cannot hold. This is an independent elementary diagnostic, not an imported result or a repair."
    },
    {
      "kind": "context",
      "source_quote": "(1) The quotient groups $\\G/\\G(q)$ are multiplicative",
      "explanation": "The supplied excerpts do not define \u0393(q). The countercheck uses \u0393(q)={\u03b3\u2208\u0393:\u03b3\u2261I mod q}, as suggested by the projection and preimage language. Supply the actual definition and the intended meaning of SO_F over finite rings. A nonstandard use of SO_F for a smaller spin-image group would materially change the statement, but is not documented here."
    },
    {
      "kind": "unverified",
      "source_quote": "\\begin{lem}[\\cite{FuchsThesis}]\\label{lem:GGq}",
      "explanation": "The thesis text and exact theorem location are absent. Thus the packet does not verify multiplicativity, either stabilization assertion, or whether the cited theorem actually names the full orthogonal group rather than a smaller congruence image. No external theorem is imported here. The recorded cutoff is 2012-05-24; the bibliography reports 2010, but the version containing the required result has not been inspected. Model knowledge cannot establish historical availability or substitute for dated evidence."
    },
    {
      "kind": "map_correction",
      "source_quote": "This of course explains all local obstructions, cf. \\eqref{eq:locA}.",
      "explanation": "The edge eq:locA \u2192 lem:GGq is not a proof-input edge. The quoted reference occurs in the following consequence remark, and eq:locA concerns one example gasket. It does not establish the general quotient-group assertions."
    },
    {
      "kind": "unverified",
      "source_quote": "The admissible numbers are precisely those\n residue classes $(\\mod 24)$ which appear  as some entry in the orbit of $v_{0}$ under $\\G/\\G(24)$.",
      "explanation": "This conclusion needs more than quotient stabilization stated without orbit analysis. One must show that the relevant coordinate classes lift to every higher 2- and 3-power, that primes \u22655 introduce no coordinate obstruction for the primitive isotropic root vector, and that prime-power choices combine. Also, the supplied setup only gives B \u2287 \u22c3\u2c7c\u27e8e\u2c7c,\u0393v\u2080\u27e9, while admissibility is defined using all curvatures. Equality of the relevant coordinate residue sets for \u0393 and the full Apollonian orbit is not supplied. These are remaining obligations, not established consequences in this packet."
    }
  ],
  "dependency_proposals": [
    {
      "label": "external:FuchsThesis",
      "role": "conditional",
      "source_quote": "Arithmetic properties of {A}pollonian circle packings, 2010.",
      "explanation": "Intended external proof input, presently unverified. Availability record: available_from=unresolved; date_evidence=the supplied bibliography's 2010 attribution; date_basis=bibliographic year only; version=uninspected Princeton thesis. The exact theorem and hypotheses must be recovered, especially its target group and treatment of primes 2 and 3. The citation cannot validate part (2) as currently written."
    },
    {
      "label": "eq:locA",
      "role": "excluded",
      "source_quote": "n\\equiv0, 4, 12, 13, 16,\\text{ or  }21(\\mod 24).",
      "explanation": "Exclude as a proof input to the general quotient lemma. It is an example-level admissibility statement referenced by the subsequent remark."
    },
    {
      "label": "Coordinate-orbit lifting and orbit comparison",
      "role": "proof_input",
      "source_quote": "\\sB\n\\supset\n\\bigcup_{j=1}^{4}\\<e_{j},\\sO\\>",
      "explanation": "Additional input is needed to pass from congruence-group descriptions to all admissible curvature classes: coordinate lifting, absence of good-prime obstructions, and comparison with the full curvature orbit."
    }
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