{
  "target_id": "lemtec2",
  "proof_status": "defective",
  "summary": "The supplied proof is defective as printed: its induction replaces the vector fe_1 by the whole matrix f, and its final paragraph asserts an incorrect invariance identity for C. The intended lemma may hold in the intended companion-matrix setting, but the packet also omits structural definitions needed to verify that setting. This is a local proof assessment, not a counterexample to the intended lemma. No external results were imported.",
  "used_dependencies": [
    "lemtec1"
  ],
  "missing_context": [
    "Definitions of A, B, v and the chosen basis, including the hypotheses establishing the asserted rank-one formula for \\tilde{A}-\\tilde{B}.",
    "The definition \\tilde{B}=(B^t)^{-1} and the applicable statement or hypotheses of the B-analogue of lemtec1.",
    "The structural identities for C=A^{-1}B used in the final calculation, in particular its action on v and on e_1,...,e_{n-1}.",
    "The exact remaining proof obligation is to justify the simultaneous power equalities on fe_1 as equivalent to the stated orthogonality conditions, without upgrading them to matrix annihilation, and then justify the final C-action calculation."
  ],
  "findings": [
    {
      "kind": "error",
      "source_quote": "$(\\tilde{B}-\\tilde{A})\\tilde{B}^{i-2}f=0$",
      "explanation": "The preceding power equalities concern their action on fe_1, whereas this displayed condition concerns every column of f. That is a strictly stronger condition, not justified by the induction. For example, if f is nonsingular and B is invertible, the displayed matrix equation forces \\tilde{A}=\\tilde{B}. The subsequent description of f itself as Euclidean-orthogonal to a vector repeats this vector/matrix mismatch."
    },
    {
      "kind": "error",
      "source_quote": "$C^tfC=C$.",
      "explanation": "Using the supplied definition C=A^{-1}B, the two invariance hypotheses give C^tfC=f, not C. The final scalar calculation requires invariance of f under C; the identity printed here does not supply it."
    },
    {
      "kind": "context",
      "source_quote": "As the first row of\n$\\tilde{A}-\\tilde{B}$ is $-v^t$ while the other rows are zeros",
      "explanation": "The packet does not define v or give A and B in their chosen common basis. Consequently, this rank-one difference identity cannot be checked. The introductory monodromy definitions alone do not imply it. The supplied lemtec1 edge is relevant but does not account for this additional structural input."
    },
    {
      "kind": "unverified",
      "source_quote": "$v^tfe_i=(Cv)^tfCe_i=-v^tfe_i$",
      "explanation": "Besides the correct C-invariance identity, this calculation needs structural information such as Cv=-v and Ce_i=e_i for 1 <= i <= n-1. Those identities are not supplied. Once the displayed scalar equality is justified, its conclusion follows over the integers because 2v^tfe_i=0 implies v^tfe_i=0; symmetry or nonsingularity of f is not needed for that last step."
    }
  ],
  "dependency_proposals": [
    {
      "label": "lemtec1",
      "role": "conditional",
      "source_quote": "Then $A^tfA=f$ if and only if $\\tilde{A}^{i-1}fe_1=fe_i$ for every $2 \\le i \\le n$",
      "explanation": "This supplied but unverified premise supports the first reduction, conditional on its intended hypotheses and basis. Its proof is not included; that omission is not itself a proof defect in lemtec2."
    },
    {
      "label": "B_analogue_of_lemtec1",
      "role": "conditional",
      "source_quote": "Lemma \\ref{lemtec1} and its analog with respect to $B$",
      "explanation": "The proof explicitly invokes a second premise not represented by the single supplied dependency statement. Its applicability requires the corresponding structural hypotheses for B."
    },
    {
      "label": "rank_one_difference_setup",
      "role": "proof_input",
      "source_quote": "As the first row of\n$\\tilde{A}-\\tilde{B}$ is $-v^t$ while the other rows are zeros",
      "explanation": "This structural identity is an additional proof input. The frozen map's lemtec1 edge does not by itself cover it."
    },
    {
      "label": "C_action_on_v_and_basis",
      "role": "proof_input",
      "source_quote": "$v^tfe_i=(Cv)^tfCe_i=-v^tfe_i$",
      "explanation": "The final implication requires the appropriate action identities for C, which are neither stated nor established in the supplied excerpts."
    }
  ],
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      "evidence": "Publisher PDF p.1617 states Received August 11, 2013; publisher.html independently records Submitted 11 August 2013. Reviewed native source is arXiv v2 dated 2013-08-12; that does not advance the external proof-input cutoff. ArXiv v1 is 2013-05-03. Later sources may provide retrospective context, not proof inputs for the original paper.",
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