{
  "target_id": "eq:cIQ3def",
  "proof_status": "defective",
  "summary": "The stated prefactor follows from the proposed input eq:IQbnd2, M=XT, and the asserted trivial bound for lambda_f. However, the proof replaces Fourier decay by an unsupported exact frequency cutoff. Consequently, the displayed argument does not establish the lemma as written; this is not a counterexample to the lemma itself. Several definitions needed for uniformity are also absent. No external theorem is imported: the assessment uses displayed algebra and elementary properties of finite Fourier sums. Model knowledge alone is not treated as dated evidence of availability before the recorded cutoff, 2012-05-24.",
  "used_dependencies": [
    "eq:IQbnd2",
    "eq:Mis",
    "eq:glfDef"
  ],
  "missing_context": [
    "Definition and uniform support, smoothness, and derivative bounds of Upsilon; parameter conditions sufficient for the asserted lattice-count bound.",
    "Definitions of f, frak f=f-a, the family frak F, and the uniform O(T) coefficient bounds used in estimating the quadratic phase.",
    "Equation eq:tilQis defining q0 and u0, including the relation between q0 and q.",
    "Definition eq:cSdef, its residue periodicity, and precise definitions of I_Q^(=) and I_Q^(!=).",
    "A quantitative Fourier-tail estimate and a summation argument sufficient to justify the stated four-frequency truncation, including any convention for negligible errors and signed residue representatives."
  ],
  "findings": [
    {
      "kind": "error",
      "source_quote": "there is contribution  only if $nx/q_{0}, my/q_{0} \\ll 1$",
      "explanation": "Smallness of the quadratic phase does not make this finite Fourier sum vanish outside the indicated range. Even at beta=0, a smoothly weighted lattice sum generally has nonzero Fourier coefficients beyond such a cutoff. Appropriate smoothness can give rapid decay, but the proof supplies neither a quantitative decay estimate nor control of the discarded frequencies after summation against |S|. The remaining obligation is to show that the full four-frequency expression is bounded by the asserted truncated expression, or to state and bound a retained remainder. Pointwise rapid decay alone does not automatically establish that sharp truncation."
    },
    {
      "kind": "unverified",
      "source_quote": "\\sum_{0\\le n,m<q_{0}}",
      "explanation": "The supplied expansion uses nonnegative residue representatives, whereas Fourier localization is measured by distance to an integer. Frequencies near q0 represent small negative frequencies and need not be negligible. The restriction n,m << u q0/X therefore requires centered representatives, or explicit inclusion of both endpoints, together with periodicity of the modular factors. The supplied proof does not explain this change of representation."
    },
    {
      "kind": "context",
      "source_quote": "Since $x,y\\asymp X/u$",
      "explanation": "This support assertion requires a definition of Upsilon, which is not supplied. Its smoothness and uniform derivative bounds are also needed for any Fourier-decay argument. The phase estimate additionally uses a uniform coefficient/size bound f(x,y)=O(T(X/u)^2) for every form under consideration; that bound and the definition of the family are absent. These are prerequisites, not consequences of Q>=X."
    },
    {
      "kind": "map_correction",
      "source_quote": "recall the notation \\eqref{eq:tilQis}",
      "explanation": "The three mapped dependencies are relevant but incomplete. To replace u q0/X by UQ/X requires, in addition to u<U and q asymp Q, a relation such as q0<=q. The referenced definition eq:tilQis is absent. Definitions of S, the diagonal/off-diagonal quantities, and the family of forms are also needed to specify exactly what is being bounded. These are missing dependency/context edges, not evidence that the three supplied premises have been established."
    },
    {
      "kind": "context",
      "source_quote": "In this range, we give $\\gl_{f}$ the trivial bound of $X^{2}/u^{2}$.",
      "explanation": "Conditional on this bound and a justified frequency restriction, the numerical factor is correct: the beta interval has length O(1/(QM)), the product of the two lambda bounds is O(X^4/u^4), and M=XT gives U X^3/(QT u^4). Taking absolute values makes this step applicable to either relation a'=a or a'!=a. Thus the obstruction is localization and its prerequisites, not the displayed power counting."
    },
    {
      "kind": "error",
      "source_quote": "\\overline{\\gl_{f'}\\left(X,\\gb;\\frac {n'}{q_{0}},\\frac {m'}{q_{0}},u\\right)}\ne(\\gb (a-a'))",
      "explanation": "There is an upstream sign mismatch in the supplied setup: opening the square of amplitudes containing e(-a beta) produces e(beta(a'-a)), not e(beta(a-a')). This does not affect the absolute-value bound sought here, but eq:IQbnd2 is not literally the displayed square expansion with the supplied conventions. Its use above remains conditional, as required for supplied dependency statements."
    }
  ],
  "dependency_proposals": [
    {
      "label": "eq:tilQis",
      "role": "proof_input",
      "source_quote": "recall the notation \\eqref{eq:tilQis}",
      "explanation": "Needed to verify q0<=q and hence the claimed enlargement of the frequency range to O(UQ/X)."
    },
    {
      "label": "Upsilon and form-family bounds",
      "role": "proof_input",
      "source_quote": "|\\gb u^{2} f(x,y)|\\ll \\frac1{QM} u^{2} T\\left(\\frac Xu\\right)^{2}",
      "explanation": "The displayed estimate needs uniform support and form-size hypotheses. Smoothness assumptions are additionally necessary for the proposed localization mechanism."
    },
    {
      "label": "Fourier localization and tail summation",
      "role": "proof_input",
      "source_quote": "we may restrict to the range\n$$\nn,m\\ll u q_{0}/X",
      "explanation": "This is the missing substantive proof input: establish the restriction in the summed inequality, specify centered representatives, and account for all discarded terms."
    },
    {
      "label": "eq:cSdef and diagonal decomposition",
      "role": "proof_input",
      "source_quote": "the former being exactly equal to $\\cS$ in \\eqref{eq:cSdef}",
      "explanation": "Needed to identify the modular weight, verify periodic reindexing, and specify the two quantities appearing in the target."
    }
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