{
  "target_id": "lemma j=3 2",
  "proof_status": "defective",
  "summary": "The supplied proof has two local defects: the referenced lemma does not define u or m for the target congruence class, and the final bilinear expansion uses the wrong pairing, contradicting an earlier computed value. The numerical conclusion is not thereby disproved, but it is not established as written. Additional lattice and Gram-matrix context is also required to verify the asserted entries. This assessment uses only supplied text and elementary algebra, with no imported external results.",
  "used_dependencies": [
    "norm 2 vector"
  ],
  "missing_context": [
    "An applicable definition of m and u for n congruent to 5 modulo 6, including the admissible range of n and valid index bounds.",
    "An applicable proof or premise that (u,u)=2 in that congruence class.",
    "Definitions of L, the vectors v_i, their coordinate convention, and f_{ij}, together with the relevant Gram-matrix entries or a formula deriving them.",
    "A corrected final bilinear identity distinguishing (u,v_{n-1}) from (v_{n-2},u)."
  ],
  "findings": [
    {
      "kind": "error",
      "source_quote": "Assume $n\\equiv 5 \\ (\\Mod 6)$ and let $u$ and $m$ be\nas in Lemma  \\ref{norm 2 vector}.",
      "explanation": "The supplied dependency defines m only when n is congruent to 1 or 3 modulo 6. Neither case applies here. Consequently, m and the truncation defining u are unspecified for the target hypothesis, and the dependency's norm conclusion cannot be applied in this congruence class. This is an explicit applicability mismatch, not merely a missing proof anchor."
    },
    {
      "kind": "error",
      "source_quote": "$$(w,v_{n-1})=(v_{n-1},v_{n-2})+(v_{n-2},u)=-4+1=-3.$$",
      "explanation": "Since w=u+v_{n-2}, bilinearity requires the second summand to be (u,v_{n-1}), not (v_{n-2},u). The proof previously computes (u,v_{n-2})=-1, so its written second summand is -1, not +1; together with the asserted first summand -4 it would give -5. The earlier displayed calculation of (u,v_{n-1})=1 indicates the apparent intended term, but the displayed equality is incorrect as written."
    },
    {
      "kind": "unverified",
      "source_quote": "\\begin{proof} Note that $f_{n-2,m}=-11$ and\nwhile $f_{n-2,i}=-12$ for $1 \\le i \\le m-1$.",
      "explanation": "The packet does not define the vectors v_i, identify f_{ij} with their pairings, or supply the relevant Gram-matrix formula. Thus these entries, the analogous row n-1 entries, the norm (v_{n-2},v_{n-2})=-2, and the adjacent pairing (v_{n-1},v_{n-2})=-4 cannot be checked. The two asserted sums also require the coefficient identities sum_i [u]_i=0, [u]_m=-1, and [u]_1=1. These follow from the supplied pattern for positive m congruent to 4 modulo 6, but no applicable m is supplied under the target hypothesis. Apart from these missing inputs, the two middle numerical calculations are elementary and arithmetically consistent."
    }
  ],
  "dependency_proposals": [
    {
      "label": "norm 2 vector",
      "role": "conditional",
      "source_quote": "Denote $m:=n-3$\nif $n\\equiv 1 \\ (\\Mod 6)$ and $m:=n-5$ if $ n\\equiv 3 \\ (\\Mod 6)$.",
      "explanation": "The frozen reference edge records an attempted proof input, but this statement does not cover the target congruence class. Its norm conclusion remains an unverified proposed premise and cannot be used here without an applicable definition and statement."
    },
    {
      "label": "Family-specific lattice basis and Gram matrix",
      "role": "proof_input",
      "source_quote": "Also, $f_{n-1,1}=-11$ while $f_{n-1,i}=-12$ for $2 \\le i \\le m$.",
      "explanation": "The local computation requires the family-specific Gram data and the identification of these entries with basis pairings. No corresponding source statement is supplied; this is an additional necessary proof-input proposal, not a claim that the data have been established."
    }
  ],
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      "cutoff_date": "2013-08-11",
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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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