{
  "target_id": "narrative:73512",
  "proof_status": "needs_context",
  "summary": "The definition and the claim that this sum occurs in the later analysis are locally consistent with the supplied excerpts. Those occurrences do not prove an exponential-sum estimate. The cited standard lemma is not supplied or externally verified. Moreover, the bound visible only in the relation evidence fails at q₀=2 under the displayed hypotheses. This is a bounded assessment of the narrative and its mapped claims, not a verdict on the paper's full argument.",
  "used_dependencies": [
    "eq:ffVdef",
    "eq:cRfuIs",
    "eq:RfuNew"
  ],
  "findings": [
    {
      "kind": "context",
      "source_quote": "Fix $\\ff\\in\\fF$, and write $\\ff=f-a$",
      "explanation": "The supplied definition eq:ffVdef supports the decomposition and quadratic expression. However, the definition of the family F and the arithmetic restrictions on A, B, C are missing. Integral coefficients are needed for the summand to be well-defined on residue classes. Primitivity, parity, or other restrictions needed for subsequent estimates cannot be inferred merely from the displayed quadratic expression or discriminant identity."
    },
    {
      "kind": "map_correction",
      "source_quote": "This sum appears naturally in many places in the minor arcs analysis, see e.g. \\eqref{eq:cRfuIs} and \\eqref{eq:RfuNew}.",
      "explanation": "Both supplied formulas do contain S_f, so they support the occurrence claim. Their edges are application/background links, not proof inputs establishing a bound for S_f. The excerpt does not provide the definitions needed to verify the complete formulas themselves. In particular, the omitted q₀,u₀ setup is needed to check q₀ divides q and that the later argument ru₀ satisfies the stated coprimality requirement."
    },
    {
      "kind": "error",
      "source_quote": "| \\cS_{f}(q_{0},r;n,m) | \\le q_{0}^{-1/2}",
      "explanation": "This inequality appears in relation evidence, not in the target slice. Under the displayed hypotheses it is false for integral A,B,C: take q₀=2, r=1, n=A, m=C. Since k²≡k and ℓ²≡ℓ modulo 2, f(k,ℓ)+Ak+Cℓ is always even. Every summand equals 1, so S_f=1>2^(-1/2). Thus the advertised exact bound requires a correction or an additional restriction excluding these parameters. This direct check does not establish what the omitted full lemma actually states."
    },
    {
      "kind": "unverified",
      "source_quote": "Our first lemma is completely standard, see, e.g. \\cite[\\S12.3]{IwaniecKowalski}.",
      "explanation": "The bibliography identifies a 2004 book, predating the recorded cutoff of 2012-05-24, but §12.3 and its precise hypotheses have not been supplied or inspected. Consequently the citation does not verify the advertised bound, especially its constant and behavior at even moduli. No external theorem is imported here; model recollection is not treated as dated evidence."
    }
  ],
  "missing_context": [
    "Definition of F and the established integrality, primitivity, and parity properties of its quadratic forms.",
    "The complete statement of the 'first lemma', including any modulus restrictions, exceptional cases, or implicit constants.",
    "The relevant theorem in the cited 2004 version of Iwaniec–Kowalski §12.3, with exact hypotheses, if it is intended as proof support.",
    "Definitions of q₀ and u₀ from eq:tilQis, needed to check the stated compatibility with later applications."
  ],
  "dependency_proposals": [
    {
      "label": "eq:ffVdef",
      "role": "proof_input",
      "source_quote": "according to \\eqref{eq:ffVdef}.",
      "explanation": "This supplies the quadratic-form notation and shifted-form identity, conditional on the omitted family setup."
    },
    {
      "label": "eq:cRfuIs",
      "role": "background",
      "source_quote": "see e.g. \\eqref{eq:cRfuIs} and \\eqref{eq:RfuNew}.",
      "explanation": "An example of a later occurrence of the defined sum, not a premise proving its local estimate."
    },
    {
      "label": "eq:RfuNew",
      "role": "background",
      "source_quote": "see e.g. \\eqref{eq:cRfuIs} and \\eqref{eq:RfuNew}.",
      "explanation": "An example of finite Fourier completion involving S_f; it supports relevance rather than an exponential-sum bound."
    },
    {
      "label": "external:IwaniecKowalski",
      "role": "conditional",
      "source_quote": "\\cite[\\S12.3]{IwaniecKowalski}",
      "explanation": "Potential proof input only after the exact result and hypotheses are checked. Bibliographic availability record: available_from=2004, year precision; date_evidence='American Mathematical Society, Providence, RI, 2004.'; date_basis=supplied bibliography; version=2004 AMS Colloquium Publications volume 53. The needed §12.3 statement is uninspected, and no theorem from it is imported."
    },
    {
      "label": "definition of F",
      "role": "proof_input",
      "source_quote": "Fix $\\ff\\in\\fF$",
      "explanation": "The family definition is needed to establish the arithmetic hypotheses of the sum and any subsequent estimate."
    },
    {
      "label": "eq:tilQis",
      "role": "conditional",
      "source_quote": "recall the notation \\eqref{eq:tilQis}",
      "explanation": "Needed if the narrative's compatibility claim is extended to verifying q₀|q and coprimality of the later argument ru₀ with q₀."
    }
  ]
}