{
  "target_id": "norm 2 vector",
  "proof_status": "defective",
  "summary": "The final displayed computation contains a directly checkable arithmetic error: its double sum equals -1, so the displayed expression equals 3, not 2. This diagnoses the written proof, not a counterexample to the norm assertion. The base cases, compatibility of the forms across dimensions, and the asserted action of B also require definitions or results absent from the packet. No external theorem is imported; the assessment uses only the supplied algebra, not undated model knowledge.",
  "used_dependencies": [],
  "missing_context": [
    "The preceding definition of family j=3 and its admissible dimensions, including the lower bound on n.",
    "The lattice L, the coordinate basis used by [u]_i, and the definitions and normalization of F and f_{i,j}.",
    "The Gram blocks needed for the n=7 and n=9 base computations.",
    "A formula or earlier result establishing the principal-block compatibility and the asserted cross-block entries.",
    "The definition of B, its form-preserving property, and its action on the relevant coordinate vectors."
  ],
  "findings": [
    {
      "kind": "error",
      "source_quote": "$$(u,u)=(a,a)+(b,b)+2(b,a)=4+\\sum_{i=m-3}^{m}\\sum_{j=1}^{m-6}[u]_i [u]_jf_{i,j}=2$$",
      "explanation": "In an induction step, write n=6l+1 or n=6l+3, with l>=2. Then m=6l-2. Both coefficient blocks in the double sum have total coefficient zero. Consequently, replacing every listed matrix entry by -12 gives zero. The sole exceptional entry, f_{m-3,m-6}=-11, changes that sum by (1)(-1)(-11+12)=-1. Thus the displayed double sum is -1 and 4 plus that sum is 3. Under the usual Gram-matrix convention, the transition from 2(b,a) to the double sum omits a factor of 2. The final equality is false as written, independently of any external result."
    },
    {
      "kind": "unverified",
      "source_quote": "The induction base is $n=7$ and  $n=9$ and it is a\ndirect computation.",
      "explanation": "For both base dimensions the specified vector begins (1,-2,2,-1) and is otherwise zero. Checking its norm requires the relevant upper-left 4-by-4 Gram block. No such block or defining formula for the family\u2019s form is supplied."
    },
    {
      "kind": "unverified",
      "source_quote": "Note that if $\\tilde{f}$ is the form corresponding to the $n-6$\ndimensional group in the family $j=3$ then the upper left $m-6\n\\times m-6$ of $\\tilde{f}$ is the same as the upper left $m-6 \\times\nm-6$ of $F$.",
      "explanation": "This principal-block identity would permit applying the lower-dimensional induction hypothesis to a, provided the coordinates and lattice identifications agree. The packet does not define family j=3, F, or its dimension-dependent Gram matrices, so this premise cannot be checked locally."
    },
    {
      "kind": "unverified",
      "source_quote": "and that $(b,b)=(B^{-6(l-1)}b,B^{-6(l-1)}b)=2$.",
      "explanation": "The first equality requires B to preserve the form. The second requires its stated power to carry b to a vector whose norm is already known, presumably the initial four-coordinate block. Neither B nor its coordinate action is supplied. Form preservation alone does not establish the second equality."
    },
    {
      "kind": "context",
      "source_quote": "Denote $m:=n-3$\nif $n\\equiv 1 \\ (\\Mod 6)$ and $m:=n-5$ if $ n\\equiv 3 \\ (\\Mod 6)$.",
      "explanation": "The surrounding parameter restrictions are needed. The proof covers n congruent to 1 or 3 modulo 6 starting at 7 and 9. The statement does not locally specify a lower bound or exclude the other residue class. If n=3 were admitted, m=-2 and the prescribed vector would be zero. This is an omitted-scope issue in the packet, not an established manuscript error."
    }
  ],
  "dependency_proposals": [
    {
      "label": "Family j=3 lattice and dimension restrictions",
      "role": "proof_input",
      "source_quote": "dimensional group in the family $j=3$",
      "explanation": "The earlier family definition is needed to identify the form, coordinate lattice, and admissible induction domain. No supporting anchor is supplied, so this is a mapping limitation rather than proof of a manuscript omission."
    },
    {
      "label": "Dimension-compatible Gram matrices and cross-block entries",
      "role": "proof_input",
      "source_quote": "since $f_{m-6,m-3}=f_{m-3,m-6}=-11$ while all the other $f_{i,j}$ in\nthe above range equal $-12$.",
      "explanation": "An explicit Gram formula or established internal result should support both these entries and the principal-block identity. The quoted entry values suffice to detect the displayed arithmetic error, but their validity for the intended lattice remains unverified."
    },
    {
      "label": "B-isometry and coordinate-shift identities",
      "role": "proof_input",
      "source_quote": "$(b,b)=(B^{-6(l-1)}b,B^{-6(l-1)}b)=2$",
      "explanation": "The definition and established properties of B are needed to justify this norm reduction; the equality itself is not a substitute for those premises."
    }
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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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