{
  "target_id": "thm:CircMeth",
  "proof_status": "locally_supported",
  "summary": "The theorem is locally supported as a conditional assembly of the five supplied major- and minor-arc estimates. The Fourier definitions give the decomposition, the two major-arc theorems give the main-term bound, and Parseval plus the three minor-arc theorems give the error bound after absorbing the logarithmic number of dyadic blocks. This does not establish those dependency theorems. The missing proof anchor is a mapping limitation, not a demonstrated proof defect.",
  "used_dependencies": [
    "eq:Q0is",
    "eq:IQ0K0",
    "eq:IQ0",
    "eq:IQdef",
    "thm:cMNis",
    "thm:cMNtoMNU",
    "thm:IQ0",
    "thm:IQX",
    "thm:IQM",
    "thm:RNU"
  ],
  "missing_context": [
    "Proofs of thm:cMNis, thm:cMNtoMNU, thm:IQ0, thm:IQX and thm:IQM, including their prerequisite hypotheses and any historically admissible external inputs.",
    "Explicit confirmation that the minor-arc estimates have saving exponents and implied constants uniform over their stated Q-ranges.",
    "The omitted preamble defining macros such as \\foh, and the conventional restriction of r to reduced residue representatives modulo q in the periodized cutoff."
  ],
  "findings": [
    {
      "kind": "context",
      "source_quote": "so that \\eqref{eq:RNcMcE} obviously holds.",
      "explanation": "This transition is correct: adding the two defining integrals cancels the cutoff, and Fourier orthogonality recovers the truncated representation function. The supplied exponential sum has finite support in its summation variables. Parseval applies to the resulting square-integrable periodic function; bounding the sum over all integer n also bounds the stated sum over n<N."
    },
    {
      "kind": "context",
      "source_quote": "where we have dissected the circle into the following regions (using that $|1-\\ft(x)|=|x|$ on $[-1,1]$):",
      "explanation": "The weighted estimate requires nonoverlapping cutoff spikes. This follows from the supplied parameters: writing a=\u03b4\u2212\u03981, the supplied bounds give 0<a<1 and Q0=N^(a/2000). Thus 2K0Q0\u00b2/N=2Q0\u2074/N tends to zero, while Q0\u00b3M/N tends to zero because M=N^(101/200). Consequently the spikes are disjoint and their widths fit inside the relevant small-denominator approximation windows. Away from their supports the weight is 1; Dirichlet coverage supplies the remaining regions up to a null set. Overlap between approximation regions is harmless for this upper bound."
    },
    {
      "kind": "unverified",
      "source_quote": "see Theorems \\ref{thm:IQX} and \\ref{thm:IQM}, respectively.",
      "explanation": "Conditional on these estimates holding uniformly in their stated Q-ranges, the minor-arc conclusion follows. If \u03b7* is the minimum of their saving exponents and that of thm:IQ0, there are O(log N) blocks, giving an error bound with any fixed exponent smaller than \u03b7*. Taking a further minimum with the exceptional-set exponent in thm:cMNtoMNU supplies one \u03b7 for both conclusions. The dependency proofs are not provided or verified here."
    },
    {
      "kind": "map_correction",
      "source_quote": "The singular series $\\fS(n)$ is the same as in Theorem \\ref{thm:RNU}.",
      "explanation": "Only the specification of the singular series is needed from thm:RNU; its representation lower bound must not be treated as an input proving CircMeth. Likewise eq:RNcMcE, eq:cMNUbnd and eq:cEN2bnd are the target's own conclusions, not independent premises. The references to eq:nZbnd and eq:nmRest supply an analogy and parameter motivation, respectively, rather than proofs of this theorem."
    },
    {
      "kind": "context",
      "source_quote": "Proof of Theorem \\ref{thm:RNU} assuming Theorem \\ref{thm:CircMeth}:",
      "explanation": "The immediately attached proof establishes the downstream implication CircMeth\u21d2RNU, not CircMeth itself. Its power-saving count uses the singular-series lower bound and an arbitrarily small \u03b5; the displayed squared estimate requires the usual relabelling of \u03b5. This attachment therefore does not close the target's dependency proofs."
    },
    {
      "kind": "context",
      "source_quote": "This completes our outline of the rest of the paper.",
      "explanation": "The outline supports the assembly described above, not verification of the later estimates. No external text or theorem is imported in this assessment. The recorded cutoff is 2012-05-24. Model knowledge cannot itself be temporally filtered, so possible later knowledge is not used as evidence."
    }
  ],
  "dependency_proposals": [
    {
      "label": "thm:cMNis",
      "role": "proof_input",
      "source_quote": "\\cM_{N}(n)\\gg\\fS(n)T^{\\gd-1}",
      "explanation": "Supplies the untruncated main-term lower bound on the stated n-range."
    },
    {
      "label": "thm:cMNtoMNU",
      "role": "proof_input",
      "source_quote": "except on a set           \nof cardinality $\\ll N^{1-\\eta}$.",
      "explanation": "Transfers that bound to the exact truncated main term used by the target."
    },
    {
      "label": "thm:IQ0",
      "role": "conditional",
      "source_quote": "\\cI_{Q_{0},K_{0}}, \\ \\cI_{Q_{0}}\\ll N\\, T^{2(\\gd-1)}\\, N^{-\\eta}",
      "explanation": "Provides both small-denominator error estimates, conditional on its unverified proof."
    },
    {
      "label": "thm:IQX",
      "role": "conditional",
      "source_quote": "For $Q_{0}\\le Q<X$, there is some $\\eta>0$",
      "explanation": "Provides the intermediate-denominator estimate, with uniformity needed for dyadic summation."
    },
    {
      "label": "thm:IQM",
      "role": "conditional",
      "source_quote": "If $X\\le Q<M$, then there is some $\\eta>0$",
      "explanation": "Provides the remaining denominator range, with the same uniformity qualification."
    },
    {
      "label": "thm:RNU",
      "role": "background",
      "source_quote": "the singular series $\\fS(n)$ is nonnegative",
      "explanation": "Use only its stated singular-series properties, not its downstream lower-bound conclusion."
    },
    {
      "label": "narrative:13050",
      "role": "background",
      "source_quote": "\\begin{conj}[Local-Global Conjecture]",
      "explanation": "Motivates the distinction between averaged and individual conclusions; it is not a proof input."
    },
    {
      "label": "eq:cEN2bnd",
      "role": "excluded",
      "source_quote": "\\sum_{n<N}|\\cE_{N}^{U}(n)|^{2}",
      "explanation": "This labels the error-bound obligation itself and cannot independently support it."
    }
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