{
  "target_id": "lemtec1",
  "proof_status": "defective",
  "summary": "The displayed proof contains an incorrect Cayley\u2013Hamilton substitution: its final matrix A should be a constant-coefficient term. The column argument also requires a companion-matrix normalization of A that is not supplied in this packet. The cyclotomic reciprocal-spectrum assertion is correct, conditional on the stated cyclotomic premise. This assessment identifies a local defect in the written proof, not a counterexample to the intended lemma.",
  "used_dependencies": [],
  "missing_context": [
    "The definition or preceding normalization of A establishing Ae_i=e_{i+1} for 1\u2264i<n, together with the convention that e_i are the standard basis vectors.",
    "The definition of the coefficients a_i and the characteristic polynomial P(x).",
    "If an external normal-form theorem is needed rather than an internal matrix definition, supply its exact statement, hypotheses, version, available_from, date_evidence, and date_basis establishing availability by 2013-08-11. External texts were not inspected; model background knowledge is not dated evidence."
  ],
  "findings": [
    {
      "kind": "error",
      "source_quote": "fAe_n=fA^{n}e_1=-f(a_1A^{n-1}+\\cdots +a_{n-1}A+A)e_1=",
      "explanation": "For P(x)=x^n+a_1x^{n-1}+...+a_n, Cayley\u2013Hamilton supplies the final term a_n I, not A. The same error occurs in the following expression involving \\tilde{A}. This is an actual algebraic failure: take n=3, f=I, and the companion permutation matrix satisfying Ae_1=e_2, Ae_2=e_3, Ae_3=e_1. Its characteristic polynomial is x^3-1, and all the lemma's column conditions hold, but this displayed equality would assert I=-A. This example tests the printed substitution; it does not refute the intended equivalence."
    },
    {
      "kind": "context",
      "source_quote": "The only if part follows from comparing \nthe first $n-1$ columns of both sides of \n$\\tilde{A}f=fA$.",
      "explanation": "This argument requires Ae_i=e_{i+1} for 1\u2264i<n, hence A^{i-1}e_1=e_i. Those identities also underlie fAe_n=fA^ne_1 and the final equality of the proof. The supplied context identifies A as a monodromy matrix but does not give its companion-matrix realization. The criterion is basis-sensitive and does not hold for arbitrary invertible A. This missing prerequisite is a packet/mapping limitation, not evidence that the manuscript never supplies it."
    },
    {
      "kind": "context",
      "source_quote": "The characteristic polynomial $P(x)$ of $A$ is also the characteristic polynomial of $\\tilde{A}$ since it is a a product of cyclotomic polynomials.",
      "explanation": "Conditional on the cyclotomic premise, this assertion checks directly: inverse transpose reciprocates eigenvalues, and the roots of each cyclotomic polynomial are closed under inversion, with multiplicities preserved. That premise also guarantees invertibility. No symmetry or nondegeneracy of f is required for the matrix argument, and its integrality is not used. No external theorem is imported for these checks."
    }
  ],
  "dependency_proposals": [
    {
      "label": "Companion-matrix normalization of A",
      "role": "proof_input",
      "source_quote": "fAe_n=fA^{n}e_1",
      "explanation": "Locate the internal definition or result giving A^{i-1}e_1=e_i. This is an essential omitted input to both directions of the argument; the introductory identification of A as local monodromy does not itself establish this basis normalization."
    },
    {
      "label": "Cyclotomic characteristic polynomial of A",
      "role": "conditional",
      "source_quote": "The characteristic polynomial $P(x)$ of $A$ is also the characteristic polynomial of $\\tilde{A}$ since it is a a product of cyclotomic polynomials.",
      "explanation": "The proof explicitly uses the cyclotomic premise to obtain invertibility and a common characteristic polynomial. Its consequence was checked directly, but the application of that premise to the particular normalized A remains conditional on the surrounding setup."
    }
  ],
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      "cutoff_date": "2013-08-11",
      "basis": "journal_received",
      "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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