{
  "target_id": "lm_local",
  "proof_status": "defective",
  "summary": "The supplied proof contains false intermediate assertions and an unstated congruence invariant needed for its 2-adic induction. This does not disprove the lemma. Conditional on the subgroup and unipotent premises, the p\u22655 matrix calculation, unit-entry factorization, and more-than-half argument check algebraically. The exceptional-prime argument requires the corrections and missing definitions identified below.",
  "used_dependencies": [
    "eq:unip",
    "eq_Gamma1"
  ],
  "missing_context": [
    "The complete definitions of \u03931, \u03932, their congruence subgroups, and A_k, including any \u00b1 generator convention and the quotient-generation assertion.",
    "A stated and proved congruence invariant validating determinant preservation and product linearization in the exceptional-prime inductions.",
    "If the proof of eq:unip is to be certified rather than treated conditionally, dated source/version evidence for its four-square-theorem input, available by 2012-05-24. No external text was inspected; model recall alone is not dated historical evidence."
  ],
  "findings": [
    {
      "kind": "error",
      "source_quote": "It is well-known and easy to check that the group generated by $\\g_1$ and\n$\\g_2$ is",
      "explanation": "The ensuing identification with all integral determinant-one matrices having b\u22610 mod 4 is false for the two displayed generators alone. Modulo 4, \u03b31 is the identity and \u03b32 is lower unipotent, so every generated matrix has both diagonal entries 1. The asserted right-hand side contains \u2212I, whose diagonal entries are 3 modulo 4. An earlier definition including \u00b1 generators could resolve the discrepancy, but that definition is not supplied. The odd-prime reduction assertion does not by itself resolve this integral-group error."
    },
    {
      "kind": "error",
      "source_quote": "$\\det(h_i+x_i)\\equiv1 \\;\\mod 2^m$",
      "explanation": "This does not follow from the stated conditions h_i\u2261I mod 4, x_i\u22610 mod 2^(m\u22124), and Tr(x_i)\u22610 mod 2^m. For example, take h_i=[[1,0],[4,1]] and x_i=2^(m\u22124)E12. Their sum has determinant 1\u22122^(m\u22122), not 1 modulo 2^m. Likewise, the displayed product linearization discards terms involving (h_i\u2212I)x_j that need not vanish under the stated bounds. The exact remaining obligation is to establish a sufficiently strong invariant for the chosen factors\u2014such as h_i\u2261I mod 16\u2014and verify that it is preserved throughout the induction. The analogous p=3 argument also needs an identity-modulo-3 invariant for its chosen factors."
    },
    {
      "kind": "error",
      "source_quote": "2^{m-1}\\mat{0}{0}{1}{0}+\\g_3\n0\\g_3^{-1}+\\g_3^2\n0\\g_3^{-2}\n2^{m-1}\\equiv \\mat{0}{0}{1}{0} \\mod 2^m,",
      "explanation": "As printed, this says 2^(m\u22121)E21\u2261E21 modulo 2^m, which is false for the relevant m\u22658. The misplaced factor appears typographical: the intended right-hand side presumably has the factor 2^(m\u22121). The other five displayed 2-adic spanning identities check directly."
    },
    {
      "kind": "error",
      "source_quote": "|\\SL(2,\\Z[i]/(2^7))|=46\\cdot64^6.",
      "explanation": "The ambient group order is 48\u00b764^6, not 46\u00b764^6. At modulus 2, reduction to SL(2,F2) has kernel of size 8, giving 6\u00b78=48; each subsequent reduction has a traceless-matrix kernel of size 64. This numerical error alone does not invalidate the final constant: even 3\u00b748\u00b764^6 is less than 10^13. The kernel-containment argument must nevertheless be justified first."
    },
    {
      "kind": "context",
      "source_quote": "h_1,\\ldots h_k\\in \\Ga_2/\\Ga_2(q)",
      "explanation": "The packet supplies the alternating-product description but not the definition of \u03932. Membership claims in A3, A12, and A36 require, in particular, the appropriate powers of \u03b33 to lie in \u03932. The final index-to-word-length step also requires \u03931 and \u03932 to generate the quotient. These are missing prerequisites, not established facts in this assessment."
    },
    {
      "kind": "map_correction",
      "source_quote": "\\begin{proof}[Proof of Lemma \\ref{lm_local} for $p\\ge5$]",
      "explanation": "The eq:unip dependency is an intermediate result inside this branch and must retain the restriction p\u22655. It cannot support the p=2 or p=3 cases. The eq_Gamma1 edge is relevant to both branches, but its statement needs the generator/sign clarification noted above."
    }
  ],
  "dependency_proposals": [
    {
      "label": "eq:unip, restricted to p\u22655",
      "role": "conditional",
      "source_quote": "\\mat{1}{0}{a i}{1}\\in A_{12}(p^m)",
      "explanation": "This supplies imaginary lower unipotents only in the p\u22655 branch. Its use remains conditional on the subgroup premises and its own derivation."
    },
    {
      "label": "eq_Gamma1 with generator convention clarified",
      "role": "conditional",
      "source_quote": "\\Ga_1=\\left\\{\\mat{a}{b}{c}{d}\\in\\SL(2,\\Z): b\\equiv0\\; \\mod 4\\right\\}.",
      "explanation": "Used for real congruence-group membership and odd-prime reductions. As written, its identification with the group generated by \u03b31 and \u03b32 is incorrect."
    },
    {
      "label": "\u03932 membership and generation",
      "role": "proof_input",
      "source_quote": "g_1,\\ldots g_k\\in \\Ga_1/\\Ga_1(q),\nh_1,\\ldots h_k\\in \\Ga_2/\\Ga_2(q)",
      "explanation": "Supply \u03932's definition, \u03b33-membership, and generation of the finite quotient by the two subgroup images."
    },
    {
      "label": "Four-square theorem invocation",
      "role": "conditional",
      "source_quote": "any positive integer\nis a sum of at most 4 squares",
      "explanation": "Exact invoked result: every positive integer is a sum of at most four integer squares. Historical record: available_from=unverified; date_evidence=none supplied; date_basis=unresolved; version=unspecified. It is not admitted here as a verified external input."
    },
    {
      "label": "Simple-root lifting step",
      "role": "proof_input",
      "source_quote": "Then by Hensel's lemma (recall that $p\\neq2$)",
      "explanation": "This elementary step can be checked directly without importing an external theorem: from y\u00b2\u2261T mod p^r with p\u2224y, choose t satisfying 2yt\u2261(T\u2212y\u00b2)/p^r mod p and replace y by y+tp^r. The stated odd-prime and unit hypotheses suffice. For a zero residue, the preceding square representation must be chosen nonempty before lifting."
    }
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