{
  "target_id": "lemma j=3 1",
  "proof_status": "defective",
  "summary": "The supplied proof contains two literal indexing errors: an asserted Gram-row entry contradicts an earlier entry, and the final bilinear expansion uses the wrong pairing. These defects do not establish that the lemma itself is false. Conditional on the norm lemma, the intended Gram data, and n>=7, the first pairing sum equals -1 and the calculations of (w,w) and (w,v_{n-1}) are correct. Verification of the final pairing requires the missing entries of the nth Gram row. Assessment uses direct algebra and supplied TeX only; model knowledge is not date-filtered, but no external results are imported.",
  "used_dependencies": [
    "norm 2 vector"
  ],
  "missing_context": [
    "Definitions of v_i, the coordinate convention for u, and the identification f_{a,i}=(v_a,v_i).",
    "The applicable Gram-matrix formula establishing the displayed entries, (v_{n-1},v_{n-1})=-2, and (v_n,v_{n-1})=-4.",
    "Verification that f_{n,i}=-12 for 3<=i<=m-1, which is needed for the claimed sum (u,v_n)=1 but is not what the supplied sentence states.",
    "The local family assumptions and dimension range. The endpoint computations require n>=7, equivalently m>=4 under n congruent to 1 modulo 6; the congruence alone does not state this lower bound.",
    "Proof of the supplied norm lemma if its conclusion is to be established rather than treated as a conditional premise."
  ],
  "findings": [
    {
      "kind": "error",
      "source_quote": "Also, $f_{n,1}=-8$, $f_{n,2}=-11$ and $f_{n,m}=-11$ while\n$f_{n-1,i}=-12$ for $3 \\le i \\le m-1$.",
      "explanation": "For n>=7, m>=4, so this range includes i=m-1. The proof previously asserts f_{n-1,m-1}=-11, whereas this sentence asserts it is -12. Moreover, this sentence supplies entries of row n-1 where the following calculation needs row n. The intended row-n entries remain to be verified from the defining Gram matrix."
    },
    {
      "kind": "error",
      "source_quote": "$$(w,v_{n})=(v_{n},v_{n-1})+(v_{n-1},u)=-4+1=-3.$$",
      "explanation": "Since w=u+v_{n-1}, bilinearity requires the second summand to be (u,v_n), not (v_{n-1},u). The proof already calculated (v_{n-1},u)=-1, so the displayed expression as written gives -4-1=-5, not -3. The claimed numerical result instead uses the preceding value (u,v_n)=1."
    },
    {
      "kind": "context",
      "source_quote": "$$(u,v_{n-1})=\\sum_{i=1}^m[u_i]f_{n-1,i}=-1,$$",
      "explanation": "For n>=7, the supplied coordinate pattern has total coefficient sum zero and endpoint coefficients [u]_1=1, [u]_{m-1}=2, [u]_m=-1. Thus the stated row-n-1 entries give 1+2-4=-1 after subtracting the constant row value -12. This arithmetic is valid conditional on those entries and on f_{a,i}=(v_a,v_i) in the coordinate basis used for u."
    },
    {
      "kind": "unverified",
      "source_quote": "$$(w,w)=(u,u)+(v_{n-1},v_{n-1})+2(v_{n-1},u)=2-2-2=-2$$",
      "explanation": "This calculation is correct conditional on the supplied norm lemma and (v_{n-1},v_{n-1})=-2. The packet does not include the local basis and Gram-matrix definitions establishing that diagonal entry, the asserted value (v_n,v_{n-1})=-4, or the displayed row entries. This is a context/mapping limitation, not by itself a manuscript proof defect."
    }
  ],
  "dependency_proposals": [
    {
      "label": "norm 2 vector",
      "role": "proof_input",
      "source_quote": "Then, $(u,u)=2$.",
      "explanation": "The existing edge is substantive: this lemma provides the coordinate vector and the norm used in the calculation of (w,w). Its statement is supplied, but its proof is not verified here."
    },
    {
      "label": "Local lattice basis and Gram-matrix formula",
      "role": "proof_input",
      "source_quote": "Note that $f_{n-1,1}=-11$, $f_{n-1,m-1}=-11$ and\n$f_{n-1,m}=-8$ while $f_{n-1,i}=-12$ for $2 \\le i \\le m-2$.",
      "explanation": "The mapped norm dependency does not supply these matrix entries. Add an anchor to the applicable internal definition or formula, including the diagonal and adjacent-basis pairings and the nth row. No such anchor is available in this packet."
    },
    {
      "label": "Ambient dimension and family restrictions",
      "role": "conditional",
      "source_quote": "Assume that $n\\equiv 1 \\ (\\Mod 6)$",
      "explanation": "Locate the surrounding hypotheses that ensure n>=7 and select the Gram matrix applicable to this lemma. Their absence from the packet should not be treated as proof that the manuscript omitted them."
    }
  ],
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