{
  "target_id": "prop form",
  "proof_status": "defective",
  "summary": "The displayed proof has an incorrect invariance equation, an unsupported existence inference, an invalid general normal-subgroup assertion, and an indexing error. Missing preceding definitions also prevent checking the normalization and lattice arguments. These are local findings, not a disproof of the proposition. Lemma lemtec2 supports orthogonality conditionally, but does not itself establish existence. No external theorem is imported; undated model knowledge is not used as historical evidence.",
  "used_dependencies": [
    "lemtec2"
  ],
  "missing_context": [
    "The preceding definitions and assumptions on A, B, C, P(X), Q(X), v, a_i, and B_i, including matrix conventions, integrality, invertibility, constant terms, and the coordinates of v.",
    "The definition of primitive and the justification that H is irreducible; a separate justification that the particular normal subgroup H_r is irreducible.",
    "An existence argument for a nonzero simultaneous invariant bilinear form, beyond the equivalence stated in lemtec2.",
    "If an external result is intended to justify the normal-subgroup step, its exact statement, hypotheses, version, available_from date, date_evidence, and date_basis must establish availability by 2013-08-11. Model knowledge alone cannot certify that chronology."
  ],
  "findings": [
    {
      "kind": "error",
      "source_quote": "such that $A^tfA=A$ and $B^tfB=f$.",
      "explanation": "The required first identity is A^tfA=f, not A^tfA=A. The displayed identity does not express invariance and does not meet the hypothesis of lemtec2. This appears to be a typographical error, but subsequent invariance arguments require its correction."
    },
    {
      "kind": "unverified",
      "source_quote": "By Lemma \\ref{lemtec2} there exists a non-zero $f \\in \\M_{n \\times n}(\\Z)$",
      "explanation": "The supplied lemma assumes A-invariance and characterizes B-invariance by orthogonality conditions; it does not assert that a nonzero matrix satisfying both exists. The remaining obligation is to show that the simultaneous invariant-form linear system has a nonzero rational solution, which can then be scaled to an integral one. No such existence argument is supplied here."
    },
    {
      "kind": "context",
      "source_quote": "Since $v,e_1,\\ldots,e_{n-1}$ span $\\Q^n$",
      "explanation": "This requires v to have nonzero last coordinate, which cannot be checked without its omitted definition. Conditional on invariance, irreducibility, and this spanning assertion, the antisymmetric-form radical argument is sound. The subsequent symmetric form must likewise be shown nondegenerate by its invariant radical. For the claimed normalization, orthogonality gives (v,u)=c u_n; normalizing (v,v)=-2 yields c=-1 precisely when v_n=2. That coordinate identity is not supplied."
    },
    {
      "kind": "error",
      "source_quote": "Since $H$ is primitive, every non-trivial normal subgroup of it is irreducible.",
      "explanation": "With the usual linear meaning of primitive, this is false as a general assertion: a primitive group can contain a nontrivial scalar normal subgroup, which preserves every subspace. The proof needs irreducibility specifically for H_r, using its additional reflection structure or another justified argument. Normality and nontriviality alone do not supply the stated implication. The manuscript's precise definition of primitive is also missing."
    },
    {
      "kind": "context",
      "source_quote": "Thus, $H_r$ preserves the $\\Z$-lattice spanned by $v,Bv, \n\\ldots, B^{n-1}v$ and the proof of $(1)$ is complete.",
      "explanation": "Several elementary prerequisites are unstated in this excerpt. One needs B and B^{-1} integral, and every B^jv for integral j to belong to L; this follows from an integral monic annihilating polynomial with unit constant term, if supplied. Then the reflection coefficients are integral by (3). Irreducibility of H_r would give full rank, while B-stability together with C-stability and A=BC^{-1} would give H-stability. Preservation by H_r alone does not establish all of (1)."
    },
    {
      "kind": "error",
      "source_quote": "From $(3)$ we get $(v,B^iv)=f_{1,i}$ for every $1 \\le i \\le n$.",
      "explanation": "For the ordered basis v,Bv,...,B^{n-1}v, the correct entry is f_{1,i+1} for 1\u2264i\u2264n\u22121. In particular, i=1 in the displayed equation incorrectly identifies (v,Bv) with the diagonal entry f_{1,1}=-2. This is an indexing error in the derivation, not by itself a contradiction to the stated formula in (4). The bottom-row computation must also be read under (4)'s additional coefficient hypothesis."
    }
  ],
  "dependency_proposals": [
    {
      "label": "lemtec2",
      "role": "conditional",
      "source_quote": "Assume that $f \\in \\M_{n \\times n}(\\Z)$ satisfies $A^tfA=f$.",
      "explanation": "Retain the dependency for the orthogonality and uniqueness steps, conditional on the lemma and its surrounding setup. Narrow the mapped existence use: this lemma's supplied statement does not establish a nonzero invariant form."
    },
    {
      "label": "preceding matrix and vector setup",
      "role": "proof_input",
      "source_quote": "Let $A$, $B$, $C$, $P(X)$, $Q(X)$ and $v$ be as above.",
      "explanation": "These omitted definitions are essential inputs for checking v_n=2, the reflection formula, lattice stability, and the coefficient computation. Their absence is packet context missing, not itself a manuscript error."
    },
    {
      "label": "irreducibility of the reflection normal subgroup",
      "role": "conditional",
      "source_quote": "$H_r:=\\langle B^{-i}CB^i\\mid i \\in \\mathbb Z\\rangle$",
      "explanation": "The argument needs an input proving irreducibility of this specific subgroup under the actual hypotheses. The broader assertion about every nontrivial normal subgroup cannot serve as stated. No external theorem supplying this input has been verified."
    }
  ],
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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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