{
  "target_id": "thm:RNU",
  "proof_status": "needs_context",
  "summary": "The exceptional-set reduction from the proposed circle-method estimates is conditionally valid, with a harmless epsilon-bookkeeping correction. It proves the representation lower bound outside a power-saving exceptional set, provided the singular series already has the stated positivity and lower-bound properties. Those properties are not independently established in this packet. The proof of Theorem Main appearing between the statements is downstream, not a proof of this target. No external results are imported; model knowledge alone is not used to certify historical availability under the recorded 2012-05-24 cutoff.",
  "used_dependencies": [
    "thm:CircMeth",
    "eq:cMNUbnd",
    "eq:cEN2bnd"
  ],
  "missing_context": [
    "An independent definition and proof anchor for the singular series, or an explicit statement that the proposed CircMeth premise includes all singular-series properties asserted in RNU.",
    "In particular, the uniform bound S(n) >= c_epsilon N^{-epsilon} on admissible N/2<n<N, together with nonnegativity and the stated restriction on vanishing, remains unverified by this packet. No inspection of the proofs of the supplied dependency statements has been performed."
  ],
  "findings": [
    {
      "kind": "error",
      "source_quote": "{|\\cE_{N}^{U}(n)|^{2}\n\\over \nN^{-\\vep}T^{2(\\gd-1)}}",
      "explanation": "For the same fixed epsilon used in the preceding pointwise inequality, squaring requires N^{-2 epsilon}, not N^{-epsilon}, in the denominator. This does not invalidate the power-saving conclusion because epsilon is arbitrary. New local repair, separate from the displayed calculation: write eta_0 for the exponent in CircMeth and assume the stated singular-series bound with epsilon=eta_0/4. On Z, the decomposition and R<M/2 imply |E|>M/2, hence |E| is bounded below by a positive constant times N^{-eta_0/4}T^{delta-1}. Summing squares gives |Z|=O(N^{1-eta_0/2}). Adding the O(N^{1-eta_0}) major-term exceptions preserves a power saving. Outside their union, R>=M/2 implies the target representation lower bound."
    },
    {
      "kind": "unverified",
      "source_quote": "The singular series $\\fS(n)$ is the same as in Theorem \\ref{thm:RNU}.",
      "explanation": "The reduction uses the singular-series lower bound on admissible integers, but this sentence does not independently demonstrate its existence, nonnegativity, vanishing restriction, or uniform lower bounds. If CircMeth is intended to include all these properties as part of its proposed premise, the reduction is supported conditional on that interpretation. Otherwise, the singular-series portion of the target remains an additional obligation; the L2 argument cannot establish it."
    },
    {
      "kind": "context",
      "source_quote": "We restrict our attention to the set of admissible $n<N$ so that \\eqref{eq:cMNUbnd} holds",
      "explanation": "The supplied major-term estimate and singular-series properties are only stated for N/2<n<N. The restriction must therefore remain within that interval. This suffices for the target as stated; the paragraph does not justify a claim for all admissible n<N."
    },
    {
      "kind": "map_correction",
      "source_quote": "\\be\\label{eq:RNUbnd}\n\\cR_{N}^{U}(n)\\gg \\fS(n) T^{\\gd-1},\n\\ee",
      "explanation": "eq:RNUbnd is the target's own representation conclusion, not an independent proof input. Its supplied edge to thm:RNU should be understood as a statement/reference edge, not evidentiary support. The decomposition eq:RNcMcE, already contained in CircMeth, is an essential actual input."
    }
  ],
  "dependency_proposals": [
    {
      "label": "eq:RNUbnd",
      "role": "excluded",
      "source_quote": "\\cR_{N}^{U}(n)\\gg \\fS(n) T^{\\gd-1},",
      "explanation": "This is a component of the target, not a premise from which to prove it."
    },
    {
      "label": "eq:RNcMcE",
      "role": "proof_input",
      "source_quote": "\\cR_{N}^{U}(n)=\\cM_{N}^{U}(n)+\\cE_{N}^{U}(n)",
      "explanation": "This identity converts failure of R>=M/2 into |E|>M/2. It is supplied inside thm:CircMeth, so no additional independent theorem is needed."
    },
    {
      "label": "singular-series properties",
      "role": "conditional",
      "source_quote": "the singular series $\\fS(n)$ is nonnegative, vanishes only when $n\\notin\\sA$, and is otherwise",
      "explanation": "These properties, including the following N^{-epsilon} lower bound, must be independently supplied or explicitly included in the CircMeth premise. The exceptional-set argument uses the admissible lower bound and inherits the other properties rather than proving them."
    },
    {
      "label": "adjacent proof of thm:Main",
      "role": "excluded",
      "source_quote": "\\begin{proof}[Proof of Theorem \\ref{thm:Main} assuming Theorem \\ref{thm:RNU}:]",
      "explanation": "This is a downstream application assuming the target. Its L1 truncation argument is not evidence for RNU and is outside the bounded obligation."
    }
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