{
  "target_id": "lem:spec1",
  "proof_status": "defective",
  "summary": "The initial coset reparametrization is correct, but the subsequent kernel evaluation and smoothing ranges use incompatible group products. Consequently, the supplied proof does not establish the stated estimate, even conditional on Lemma spec0. The claimed boundary-strip estimate and uniform smoothing bounds also require additional justification. This is a bounded assessment of the supplied argument, not a verdict that the lemma is false. No external theorem or later development is used to close these obligations.",
  "used_dependencies": [
    "eq:G1G",
    "lem:spec0",
    "eq:fFdef",
    "eq:ffToInProd",
    "eq:FqIs",
    "eq:fgIs",
    "eq:cNq",
    "eq:reg1Lem",
    "eq:reg2Lem",
    "eq:reg3Lem",
    "eq:NqErr",
    "eq:TT1T2",
    "eq:cCis"
  ],
  "missing_context": [
    "The complete multiplicity convention for the family F and its sums.",
    "Definitions and quantitative bounds for the norm, w_{x,y}, v\u2080 and \u03b4; the uniform spectral-gap, matrix-coefficient, Sobolev and integration estimates invoked by the template.",
    "A uniform linear-functional boundary-strip counting estimate, including its hypotheses and dependence on \u03b3\u2081.",
    "Control of smoothing and Sobolev constants independent of q and \u03b3\u2080, with any T\u2081-loss explicitly accounted for.",
    "The cited external theorem-bearing text and dated version needed to check applicability by the recorded cutoff 2012-05-24. Model weights can contain later knowledge; none is used here as historical evidence."
  ],
  "findings": [
    {
      "kind": "error",
      "source_quote": "such as replacing \\eqref{eq:cNq} by\n$F_{q}(\\g_{1}\\g_{0}^{-1},e)$.",
      "explanation": "With the newly defined f, the desired inner sum is \u03a3_{\u03b3\u2082\u2032\u2208tilde\u0393(q)} f(\u03b3\u2080\u03b3\u2082\u2032). The supplied kernel definition instead makes F_q(\u03b3\u2081\u03b3\u2080\u207b\u00b9,e)=\u03a3 f(\u03b3\u2080\u03b3\u2081\u207b\u00b9\u03b3\u2082\u2032). These are not the same expression. The subsequent ranges likewise use \u2016\u03b3\u2081\u03b3\u2080\u207b\u00b9\u03b3\u2082\u2032\u2016 instead of the previously derived \u2016\u03b3\u2081\u207b\u00b9\u03b3\u2080\u03b3\u2082\u2032\u2016, and use \u27e8e\u2081,\u03b3\u2081\u03b3\u2080\u207b\u00b9\u03b3\u2082\u2032v\u2080\u27e9 instead of \u27e8e\u2081,\u03b3\u2080\u03b3\u2082\u2032v\u2080\u27e9. No stated hypothesis identifies these products."
    },
    {
      "kind": "unverified",
      "source_quote": "In the remaining range, \\eqref{eq:reg3Lem} remains unchanged, using $|f|\\le1$.",
      "explanation": "The new remaining range includes a strip around the additional boundary \u27e8e\u2081,\u03b3\u2080\u03b3\u2082\u2032v\u2080\u27e9=T/100. The supplied reg3Lem concerns radial norm shells; it does not supply a uniform count for this linear-functional strip. Boundedness of f controls each summand, not the number of points in that strip. The exact outstanding estimate is a power-saving boundary-strip count, after fixing \u03b3\u2081, of the strength used in the displayed T\u2082-error."
    },
    {
      "kind": "unverified",
      "source_quote": "for a net contribution to the error of $\\ll \\eta K T^{\\gd}$.",
      "explanation": "The phase estimate requires a perturbation bound of size O(\u03b7TX\u00b2); then |\u03b2|<K/N and TX\u00b2=N give O(\u03b7K). Definitions and bounds for w_{x,y} and the perturbation geometry are absent. Moreover, the inherited smoothing inserts a small element between a potentially large coset representative and the lattice variable. Uniformity in arbitrary \u03b3\u2080 requires control of conjugation and translated Sobolev norms, not merely the unweighted counting statement of spec0. The restriction 1<K\u2264T\u2082^{1/10} does not by itself provide those controls."
    },
    {
      "kind": "context",
      "source_quote": "the decomposition $\\g=\\g_{1}\\g_{2}$ in \\eqref{eq:fFdef} need not be unique",
      "explanation": "The asserted equality between the original sum and the double sum requires factorization multiplicity to be retained. It is not an equality for a set of distinct products. The excerpt begins acknowledging multiplicities but truncates the convention; the full family/summation definition is needed."
    },
    {
      "kind": "context",
      "source_quote": "Optimizing $\\eta$ and renaming $\\gT$ gives the bound $O(T_{2}^{\\gT}K^{10/11})$",
      "explanation": "Conditional on the displayed error A=\u03b7KT\u2082^\u03b4+T\u2082^{\u03b4\u2212\u03b5}\u03b7\u207b\u00b9\u2070, its optimization is correct: \u03b7=T\u2082^{\u2212\u03b5/11}K^{\u22121/11} gives A\u226aT\u2082^{\u03b4\u2212\u03b5/11}K^{10/11}. If this is a uniform per-\u03b3\u2081 bound and the outer count is O(T\u2081^\u03b4), summation gives T^{\u03b4\u2212\u03b5C/(11(C+1))}K^{10/11}\u2264T^\u0398K with \u0398<\u03b4. Thus the final exponent conversion is consistent, but does not establish the preceding error estimate."
    },
    {
      "kind": "map_correction",
      "source_quote": "giving \\eqref{eq:spec1}.",
      "explanation": "The eq:spec1 \u2192 lem:spec1 edge is a reference to the target conclusion, not an independent proof input. The kernel and smoothing equations are template references; their applicability to the new weight remains an obligation."
    }
  ],
  "dependency_proposals": [
    {
      "label": "lem:spec0",
      "role": "conditional",
      "source_quote": "The proof follows with minor changes that of Lemma \\ref{lem:spec0}",
      "explanation": "Retain as an unverified template premise. Its unweighted conclusion alone does not imply the weighted, additionally truncated estimate."
    },
    {
      "label": "eq:spec1",
      "role": "excluded",
      "source_quote": "giving \\eqref{eq:spec1}.",
      "explanation": "Self-reference to the conclusion, not a premise."
    },
    {
      "label": "external:BourgainKontorovichSarnak2010",
      "role": "conditional",
      "source_quote": "\\newblock Sector estimates for hyperbolic isometries.\n\\newblock {\\em GAFA}, 20(5):1175--1200, 2010.",
      "explanation": "No result is imported. Availability record: available_from=unverified, with 2010 reported by the citation; date_evidence=the supplied bibliography; date_basis=bibliographic publication year; version=cited GAFA article, theorem-bearing text uninspected. The exact \u00a74 result, its group/dimension, weight, boundary-regularity and uniformity hypotheses remain unresolved. The citation suggests pre-cutoff availability but does not establish applicability."
    },
    {
      "label": "uniform weighted boundary and mixing estimates",
      "role": "proof_input",
      "source_quote": "The error in \\eqref{eq:NqErr} is then replaced by",
      "explanation": "Proposed missing inputs: justify the asserted per-\u03b3\u2081 weighted error, including the added boundary strip and uniform translated-bump estimates. These are not supplied by the mapped radial-shell statement."
    }
  ],
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