{
  "target_id": "eq:cRfuIs",
  "proof_status": "needs_context",
  "summary": "The displayed derivation is algebraically correct conditional on the arithmetic definitions and hypotheses permitting Poisson summation. In the explicitly adopted even-u case, the change of variables has Jacobian 1/(u\u00b2q\u2080\u00b2), produces the positive residue-class phase e_{q\u2080}(nk+m\u2113), and yields the stated frequencies. The q\u2080\u00b2 normalization in S_f then leaves the claimed prefactor 1/u\u00b2. No local sign or normalization error is apparent. The packet does not establish the weight's regularity or all arithmetic domain conditions, so unconditional applicability remains unchecked.",
  "used_dependencies": [
    "eq:cRffuIs",
    "eq:cSfDef"
  ],
  "missing_context": [
    "The definition of Upsilon and its decay or support and smoothness assumptions, sufficient to justify the stated Poisson formula and summation rearrangements.",
    "The relevant part of eq:ffVdef establishing that f has integer coefficients, or otherwise establishing the modular periodicity used in the proof.",
    "The parameter conventions establishing q\u22651, nonzero even u, X\u22600, and gcd(r,q\u2080)=1, typically through gcd(r,q)=1.",
    "A dated source/version for the invoked Poisson theorem is not supplied. Its historical admissibility is not independently documented here. No external text was inspected or later result used; latent model knowledge is not dated historical evidence."
  ],
  "findings": [
    {
      "kind": "unverified",
      "source_quote": "Apply Poisson summation to the bracketed term above:",
      "explanation": "The definition and regularity of Upsilon are absent. For example, Upsilon in C_c^infinity(R), with X and u nonzero, would make the transformed summand Schwartz and justify Poisson summation, absolute convergence, and the subsequent rearrangements. The exact remaining obligation is to supply the actual weight hypotheses and verify that they justify this application. This is missing context, not a demonstrated failure of Poisson summation."
    },
    {
      "kind": "context",
      "source_quote": "f(x,y)=Ax^{2}+2Bxy+Cy^{2}",
      "explanation": "Separating the sum into residue classes requires e_{q\u2080}(r u\u2080 f(x,y)) to depend only on x,y modulo q\u2080. Integer polynomial coefficients suffice, but their integrality is not stated in the supplied excerpts; the referenced definition eq:ffVdef is missing. Homogeneity itself is supplied and correctly used."
    },
    {
      "kind": "context",
      "source_quote": "fix $r$ with  $(r,q_{0})=1$",
      "explanation": "This is the declared domain of S_f. Its use here with second argument r u\u2080 requires gcd(r u\u2080,q\u2080)=1. The supplied reduction proves gcd(u\u2080,q\u2080)=1, but the setup does not explicitly supply gcd(r,q\u2080)=1. An earlier assumption gcd(r,q)=1 would suffice because q\u2080 divides q. Without that context, the argument has not checked the definition's stated domain, although the finite-sum identity itself does not require coprimality."
    },
    {
      "kind": "map_correction",
      "source_quote": "so that $u^{2}/q=u_{0}/q_{0}$ in lowest terms, $(u_{0},q_{0})=1$.",
      "explanation": "The definitions in eq:tilQis are a direct proof input: they justify replacing the rational quadratic phase by e_{q\u2080}(r u\u2080 f(x,y)). They are present in the setup but absent from the supplied dependency list. Both listed dependency edges are otherwise appropriate."
    }
  ],
  "dependency_proposals": [
    {
      "label": "eq:tilQis",
      "role": "proof_input",
      "source_quote": "\\tilde q:=(u^{2},q)\\qquad\nu_{0}:=u^{2}/\\tilde q\n,\\qquad\nq_{0}:=q/\\tilde q",
      "explanation": "These definitions supply the reduced rational quadratic phase and coprimality of u\u2080 and q\u2080. This is a proposed additional edge, not a change to the frozen map."
    },
    {
      "label": "eq:ffVdef",
      "role": "conditional",
      "source_quote": "according to \\eqref{eq:ffVdef}.",
      "explanation": "The omitted definition should establish the coefficient arithmetic needed for residue-class periodicity. Its contents have not been verified."
    },
    {
      "label": "Poisson summation and weight admissibility",
      "role": "conditional",
      "source_quote": "Apply Poisson summation to the bracketed term above:",
      "explanation": "Required theorem version: for H in Schwartz(R\u00b2), sum over z in Z\u00b2 of H(z) equals the sum over frequencies \u03be in Z\u00b2 of its Fourier transform, defined using exp(-2\u03c0i \u03be\u00b7x). The proof applies this to H(s,t)=Upsilon(u(q\u2080s+k)/X)Upsilon(u(q\u2080t+\u2113)/X)e(beta u\u00b2f(q\u2080s+k,q\u2080t+\u2113)). The packet does not verify H is Schwartz. Historical metadata: available_from=unresolved; date_evidence=none supplied; date_basis=unresolved; version=not identified. This records the manuscript's conditional invocation, not an independently imported theorem."
    }
  ],
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