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  "target_id": "eq:cSfbnd1",
  "proof_status": "defective",
  "summary": "The displayed odd-modulus evaluation conditionally yields the claimed bound. However, the proof expressly proposes including even moduli, and the bound with constant 1 fails at q0=2 for the normalized sum displayed. The omitted \u201cabove conditions\u201d must be supplied to determine whether any hidden restriction excludes this counterexample. This is a local assessment, not a verdict on the paper\u2019s main theorem.",
  "used_dependencies": [
    "eq:ABCdef",
    "eq:Fv",
    "eq:disc",
    "eq:SfEval"
  ],
  "missing_context": [
    "The preceding definition of S_f, including its normalization, additive-character convention, and all conditions on q0,r,n,m and the form family.",
    "The orbit-primitivity statement connecting primitivity of the gasket to primitive curvature quadruples.",
    "The invertible coordinate reduction ensuring a unit quadratic coefficient for each odd prime power, with corresponding transformation of n,m.",
    "Any omitted restriction that excludes the q0=2 counterexample; otherwise the even-modulus assertion cannot hold as stated.",
    "The actual cited 2004 text and equation (3.38), rather than only its bibliography entry, if the exact Gauss-sum evaluation is to be accepted as an external input."
  ],
  "findings": [
    {
      "kind": "error",
      "source_quote": "Assume for simplicity $(q_{0},2)=1$; similar calculations are needed to handle the $2$-adic case.",
      "explanation": "The advertised even-modulus extension is false without additional restrictions or a changed constant. The proof identifies S_f with q0^(-2) times the complete sum of e_q0(r f(k,l)+nk+ml). Set q0=2, r=1, n\u2261A and m\u2261C modulo 2. Since f(k,l)\u2261Ak+Cl modulo 2, every summand equals 1. Thus S_f=1>2^(-1/2). This works for every integral form of the displayed shape, including primitive ones. It is a direct calculation, not an imported result."
    },
    {
      "kind": "context",
      "source_quote": "With the above conditions,",
      "explanation": "The preceding definition and complete hypotheses are absent. The proof explicitly recalls (r,q0)=1, but the supplied statement does not specify the permitted moduli or restrictions on n,m. If earlier conditions exclude q0=2 or the chosen residues, that must be documented; otherwise the counterexample applies."
    },
    {
      "kind": "unverified",
      "source_quote": "we have that $(A,B,C)=1$;       \n          \n assume henceforth that $(C,q_{0})=1$, say.",
      "explanation": "Primitivity does not imply that the original coefficient C is a unit at every prime. For odd p, the case p dividing A and C but not B requires an invertible change of variables, not merely exchanging coordinates. The argument needs to state that reduction and its effect on the linear coefficients n,m. Binary primitivity itself also uses the prerequisite that each orbit quadruple is primitive, not merely the definition gcd(B_gasket)=1."
    },
    {
      "kind": "context",
      "source_quote": "The claim then follows trivially.",
      "explanation": "Conditional on the displayed odd-modulus evaluation, this final inequality is correct: all phase and symbol factors have modulus at most 1, and sqrt(tilde q0)/q0\u2264q0^(-1/2), since tilde q0=(a\u00b2,q0)\u2264q0. The square completions and the divisibility condition nC\u2261mB modulo tilde q0 are consistent with AC\u2212B\u00b2=a\u00b2. This does not establish the omitted unit-coordinate reduction or the even case."
    },
    {
      "kind": "error",
      "source_quote": "where $({\\cdot\\over q_{0}})$ is the Legendre symbol.",
      "explanation": "The argument permits q0=p^j, so the standard symbol in the stated Gauss-sum formula is the Jacobi symbol, not the Legendre symbol unless j=1. This is a notation correction rather than the substantive obstruction to the bound."
    },
    {
      "kind": "map_correction",
      "source_quote": "This sum also\nappears naturally\nin the minor arcs analysis, see \\eqref{eq:IQbnd1} and \\eqref{eq:IQbnd2}.",
      "explanation": "These references concern the subsequently defined paired average S. They are applications/context, not proof inputs for the individual-sum bound."
    },
    {
      "kind": "map_correction",
      "source_quote": "\\begin{proof}\nWrite $\\cS_{f}$ for $\\cS_{f}(q_{0},r;n,m)$.",
      "explanation": "An explicit proof is present in the supplied slice despite proof_anchor_missing=true. That flag is a mapping limitation, not a proof defect. Moreover, eq:SfEval is an intermediate result inside this same proof, not an independently established prerequisite."
    }
  ],
  "dependency_proposals": [
    {
      "label": "eq:SfEval",
      "role": "conditional",
      "source_quote": "\\label{eq:SfEval}",
      "explanation": "Treat as a same-proof intermediate evaluation valid under the odd-modulus and unit-coefficient assumptions, not as an independent established premise."
    },
    {
      "label": "eq:IQbnd1",
      "role": "excluded",
      "source_quote": "see \\eqref{eq:IQbnd1} and \\eqref{eq:IQbnd2}.",
      "explanation": "Downstream application of the paired average; not needed for this bound."
    },
    {
      "label": "eq:IQbnd2",
      "role": "excluded",
      "source_quote": "see \\eqref{eq:IQbnd1} and \\eqref{eq:IQbnd2}.",
      "explanation": "Downstream application of the paired average; not needed for this bound."
    },
    {
      "label": "external:IwaniecKowalski",
      "role": "conditional",
      "source_quote": "\\newblock American Mathematical Society, Providence, RI, 2004.",
      "explanation": "Needed result: for odd Q and gcd(t,Q)=1, the complete quadratic Gauss sum equals \u03b5_Q sqrt(Q)(t/Q), with the Jacobi symbol; the proof applies it at Q=q0 and Q=q1. available_from: 2004, according to the supplied bibliography; date_evidence: the quoted publication record; date_basis: bibliography publication year; version: AMS Colloquium Publications 53, 2004, cited equation (3.38), text uninspected. The bibliographic date precedes the recorded 2012-05-24 cutoff, but exact equation and hypotheses remain unverified. No later result is used; model recollection is not treated as dated evidence."
    }
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