{
  "target_id": "eq:hatRfuBnd",
  "proof_status": "needs_context",
  "summary": "The final algebra is correct conditional on the displayed Poisson identity, exponential-sum bound, integral estimate, and uniform control of the discarded Poisson tail. The packet does not supply all hypotheses needed for those applications or quantify the tail. The citation to the family-cardinality bound does not justify the discriminant lower bound. No counterexample to the proposition is established by this bounded review.",
  "used_dependencies": [
    "eq:Mis",
    "eq:cJfIs",
    "eq:cJfbnd",
    "eq:cRfuIs",
    "eq:cSfbnd1",
    "eq:disc",
    "eq:nmRest",
    "eq:tilQis"
  ],
  "missing_context": [
    "The complete parameter regime relating X,T,N,U,Q_0,q,u and beta, including whether beta is nonzero; the displayed right-hand side is undefined at beta=0 without an explicit convention.",
    "Definitions identifying a_gamma with the first coordinate of gamma v_0 and establishing uniform bounds for A_gamma,B_gamma,C_gamma.",
    "The definition of cS_f and all hypotheses preceding eq:cSfbnd1.",
    "A quantitative, summable nonstationary-phase remainder compatible with the claimed bound, and the uniform hypotheses behind eq:cJfbnd.",
    "The odd-u calculation if the proposition is intended to include odd u.",
    "The exact applicable statement and hypotheses from the cited 2004 edition of Iwaniec\u2013Kowalski; the external text was not supplied or inspected."
  ],
  "findings": [
    {
      "kind": "unverified",
      "source_quote": "has negligible contribution unless",
      "explanation": "Nonstationary phase supplies decay, not an exact frequency cutoff. The proof must bound the sum of all discarded terms uniformly by O(u/(sqrt(q)|beta|T)). After scaling x,y by X, the relevant frequency spacing is X/(u q_0). The supplied packet does not give sufficient parameter inequalities or a quantitative remainder estimate to verify this truncation, particularly when |beta|TXu q_0 is small."
    },
    {
      "kind": "map_correction",
      "source_quote": "\\#\\fF_{T}\\ll T^{\\gd}.",
      "explanation": "This cardinality bound, eq:fFTbnd, cannot imply |a| >> T for an individual form. The relevant candidate input is the coordinate cutoff in eq:fFdef, together with the identification a_gamma = <e_1,gamma v_0>. That identification is not explicitly supplied. The discriminant identity itself gives |discr(f)| = 4a^2, conditional on its stated premise."
    },
    {
      "kind": "context",
      "source_quote": "\\ll\n|\\gb|\\cdot |\\nabla f|\\ll |\\gb|\\cdot TX,",
      "explanation": "The gradient estimate requires uniform coefficient bounds, for example |A|+|B|+|C| << T on the support x,y in [X,2X]. The packet omits the coefficient definitions and norm/scale relations needed to check this. The discriminant lower bound alone does not imply this upper bound."
    },
    {
      "kind": "context",
      "source_quote": "With the above conditions,",
      "explanation": "The hypotheses and normalization of eq:cSfbnd1 are omitted. Conditional on a reduced r/q, the supplied reduction does verify gcd(r u_0,q_0)=1, as required for a coprime numerator. Any additional conditions on f and q_0 remain unchecked."
    },
    {
      "kind": "context",
      "source_quote": "From now on, we focus exclusively on the case $u$ is even, the other case being handled similarly.",
      "explanation": "The displayed Poisson formula and proof cover the even-u lattice. If the proposition includes every positive u, its odd-u application requires checking the lattice with spacings 2u and u, including the resulting integral normalization and complete-sum bound. This adaptation is asserted but not supplied."
    },
    {
      "kind": "context",
      "source_quote": "from which the claim follows, using \\eqref{eq:tilQis}.",
      "explanation": "This transition checks directly: O(u^2) retained pairs cancel the prefactor u^(-2), giving O(1/(sqrt(q_0)|beta|T)). Since q_0=q/gcd(u^2,q) and sqrt(gcd(u^2,q))<=u, the claimed bound follows. This calculation does not establish the preceding truncation or its error."
    }
  ],
  "dependency_proposals": [
    {
      "label": "eq:fFTbnd",
      "role": "excluded",
      "source_quote": "\\#\\fF_{T}\\ll T^{\\gd}.",
      "explanation": "Exclude this cardinality estimate as support for the individual discriminant lower bound; it is not used in the final algebra."
    },
    {
      "label": "eq:fFdef",
      "role": "conditional",
      "source_quote": "\\<e_{1},\\g_{1}\\,\\g_{2}\\,v_{0}\\>>T/100",
      "explanation": "This is the appropriate candidate source of a >> T, conditional on the omitted identification of a with this coordinate."
    },
    {
      "label": "eq:nmRest",
      "role": "conditional",
      "source_quote": "|n|,|m|\\ll  |\\gb|\\cdot TX\\cdot \\d q_{0}",
      "explanation": "This is an intermediate assertion inside the target proof, not independently established support. Its use requires quantitative control of the complementary infinite sum."
    },
    {
      "label": "eq:cJfbnd",
      "role": "conditional",
      "source_quote": "\\ll\\min\\left(X^{2},{1\\over |\\gb| T}\\right)",
      "explanation": "The final inference uses this bound conditionally. Its uniform stationary-phase justification remains part of the local obligation, rather than a verified external premise."
    },
    {
      "label": "external:IwaniecKowalski",
      "role": "conditional",
      "source_quote": "\\newblock American Mathematical Society, Providence, RI, 2004.",
      "explanation": "Availability record: available_from=2004 at year precision for the cited book; date_evidence=the supplied bibliography; date_basis=bibliographic publication year; version=AMS Colloquium Publications, volume 53, 2004. This predates the 2012-05-24 cutoff, but the exact needed theorem and its hypotheses are unverified. No external theorem is imported here. Model knowledge cannot be chronologically erased and is not being used as dated evidence."
    }
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