{
  "target_id": "narrative:99181",
  "proof_status": "needs_context",
  "summary": "The displayed algebra is locally correct, conditional on finite or absolutely convergent sums. Reversing the truncated divisor sum and using \\ff=f-a gives the stated Fourier decomposition. For even u, the substitutions x_old=(u/2)x and y_old=uy are bijective; for odd u, they are x_old=ux and y_old=uy. Homogeneity gives f(ux,uy)=u\u00b2f(x,y), and dividing u\u00b2 and q by their gcd gives the asserted reduced fraction without requiring (u,q)=1. The minor-arc objectives agree with the supplied reduction. However, the opening claim that the major-arc bound has been established cannot be checked from the supplied proof scope. This is a context limitation, not an identified proof defect.",
  "used_dependencies": [
    "eq:IQ0",
    "eq:IQ0K0",
    "eq:cEN2bnd",
    "eq:cMNUbnd",
    "eq:cRNhatUis",
    "eq:ffVdef"
  ],
  "missing_context": [
    "The proof immediately preceding the opening sentence, including the major-arc lower bound, truncation transfer, exceptional-set estimates, and their parameter hypotheses.",
    "Definitions ensuring that \\fF and the weighted lattice sums are finite, or a justification of absolute convergence.",
    "The later estimates needed to substantiate the assertion that the odd-u case is handled similarly. The target itself supplies only its correct initial reindexing."
  ],
  "findings": [
    {
      "kind": "unverified",
      "source_quote": "This establishes \\eqref{eq:cMNUbnd}, and hence completes our Major Arcs analysis",
      "explanation": "The preceding argument establishing this bound is not supplied. Its exact remaining obligation is the lower bound \\cM_N^U(n)\u226b\\fS(n)T^{\\gd-1} for admissible N/2<n<N, outside a set of size O(N^{1-\\eta}), with the stated parameter choices. The section setup supplies a modular/archimedean decomposition but not the estimates and transfer needed for this conclusion. The claim can therefore be accepted only conditionally here."
    },
    {
      "kind": "context",
      "source_quote": "\\eqref{eq:cRNhatUis} and reverse orders of summation",
      "explanation": "The rearrangement is valid for finite sums, or under absolute convergence. The packet does not define the family \\fF or the support/decay of \\gU sufficiently to verify this prerequisite. No analytic estimate is needed for the rearrangement once that prerequisite holds. The shift a remains attached to its corresponding form, as required."
    },
    {
      "kind": "unverified",
      "source_quote": "From now on, we focus exclusively on the case $u$ is even, the other case being handled similarly.",
      "explanation": "Both displayed parity-dependent reindexings are correct. This does not by itself verify that all subsequent estimates also hold for odd u. That remaining obligation is to establish the needed estimates for the displayed sum involving f(2ux,uy), with the same required uniformity and savings. No failure of that assertion is demonstrated by this packet."
    },
    {
      "kind": "map_correction",
      "source_quote": "We keep all the notation of \\S\\ref{sec:outline}",
      "explanation": "The supplied edge from thm:Main originates from a reference to sec:outline, not to the main theorem. The main theorem is a downstream objective, not an input to this local calculation. Likewise, the references to eq:cMNUbnd, eq:cEN2bnd, eq:IQ0K0 and eq:IQ0 principally identify a completed goal or future goals; they should not be interpreted as proofs of those estimates."
    }
  ],
  "dependency_proposals": [
    {
      "label": "thm:Main",
      "role": "excluded",
      "source_quote": "We keep all the notation of \\S\\ref{sec:outline}",
      "explanation": "Exclude as a local proof input: this section reference does not invoke the main theorem."
    },
    {
      "label": "sec:outline",
      "role": "background",
      "source_quote": "We keep all the notation of \\S\\ref{sec:outline}",
      "explanation": "This is the actual referenced source for notation and the circle-method reduction."
    },
    {
      "label": "eq:cRNhatUis",
      "role": "proof_input",
      "source_quote": "First we return to\n\\eqref{eq:cRNhatUis} and reverse orders of summation",
      "explanation": "The defining truncated M\u00f6bius sum directly supports the displayed rearrangement, subject to finiteness or absolute convergence."
    },
    {
      "label": "eq:ffVdef",
      "role": "proof_input",
      "source_quote": "where $\\ff=f-a$ according to \\eqref{eq:ffVdef}",
      "explanation": "The supplied definition gives both the extracted phase e(-a\\gt) and the homogeneous quadratic form used in scaling."
    },
    {
      "label": "preceding-major-arc-proof",
      "role": "conditional",
      "source_quote": "This establishes \\eqref{eq:cMNUbnd}",
      "explanation": "The opening completion claim depends on the preceding proof, which is outside this packet. Its absence is a mapping/context limitation."
    }
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