{
  "target_id": "thm:Main",
  "proof_status": "defective",
  "summary": "The conditional exceptional-set argument has the correct M\u00f6bius-tail estimate and exponent arithmetic. However, the proof applies Theorem thm:RNU to all admissible n<N although its conclusion is stated only for a restricted interval. The resulting complement estimate is unsupported as written. Effective computability of the final exponent also requires quantitative inputs not supplied here. These are local assessments of the displayed reduction, not a verdict on the full theorem.",
  "used_dependencies": [
    "thm:RNU",
    "eq:RNUbnd",
    "eq:fFTbnd",
    "eq:Uis",
    "eq:TXNis",
    "eq:binQuad",
    "eq:ffToInProd",
    "eq:fFdef"
  ],
  "missing_context": [
    "The definition of the macro \\foh and the omitted introductory assumptions on the fixed gasket, including the applicable integrality/primitivity conditions and formal meaning of admissibility.",
    "A supplied argument passing from the restricted interval in thm:RNU to all positive admissible integers up to N, with uniform estimates at the required scales.",
    "Effective quantitative versions of the exceptional exponent in thm:RNU and the spectral/counting parameters determining u.",
    "Confirmation that eq:fFTbnd bounds the same weighted or multiplicity-counted ensemble used in the representation sums.",
    "Dated, version-specific evidence for any external result actually used upstream, especially the cited Vinogradov theorem, before the recorded 2012-05-24 cutoff."
  ],
  "findings": [
    {
      "kind": "error",
      "source_quote": "Note also from Theorem \\ref{thm:RNU} that $|Z\\cap W^{c}|\n\\le |W^{c}|\\ll N^{1-\\eta}$.",
      "explanation": "Theorem thm:RNU controls only \\foh N<n<N, whereas Z and W are defined over admissible n<N. Its exceptional-set bound therefore does not establish this displayed estimate outside the stated interval. Complements must also be understood relative to the admissible domain. A separate scale-covering argument is required for the up-to-N conclusion; none is supplied in this proof."
    },
    {
      "kind": "context",
      "source_quote": "T^{\\gd}            \n{X^{2}\\over U}",
      "explanation": "The M\u00f6bius-tail bound is elementary and valid conditional on eq:fFTbnd counting the ensemble with the same multiplicities as the representation sums. The smoothing support gives X\u22642x,y\u22642X, hence u\u22642X. For each u, the two divisibility conditions admit O(X/u) choices each, including the factor gcd(2,u)\u22642. Summing X\u00b2/u\u00b2 gives O(X\u00b2/U). Positivity and boundedness of the fixed smooth weight justify removing it in this upper bound."
    },
    {
      "kind": "context",
      "source_quote": "which\n is a power savings since $\\vep>0$ is arbitrary.",
      "explanation": "On the interval actually covered by thm:RNU, the algebra is correct: the tail estimate and singular-series lower bound give O(N^{1\u2212\u03b7_R}+N^{1+\u03b5\u2212u}) exceptions, where U=N^u. Taking \u03b5=u/2 gives exponent saving min(\u03b7_R,u/2)>0. No positivity of the truncated representation function outside the good set is needed. The implication from a nonzero untruncated representation to an actual curvature is supported conditionally by eq:binQuad."
    },
    {
      "kind": "unverified",
      "source_quote": "$O(N^{1-\\eta})$, where $\\eta>0$ is effectively computable.",
      "explanation": "The supplied statement of thm:RNU asserts existence of \u03b7_R>0, not its effective computability. The parameter u likewise depends on spectral-gap and counting exponents. Effectivity of these positive exponents must be supplied to justify the stronger target qualification. Ineffective constants in the N^{\u2212\u03b5} singular-series lower bound would not, by themselves, prevent computing the exponent."
    },
    {
      "kind": "unverified",
      "source_quote": "The following theorem follows easily              \nfrom \\cite[Thm 2.2]{Vinogradov2013}.",
      "explanation": "No external theorem is imported in this assessment; eq:fFTbnd and thm:RNU remain proposed premises. For the cited Vinogradov input, available_from is unknown, date_evidence is only the supplied citation label, date_basis is unverified citation metadata, and version is unknown. A version containing the needed result available by 2012-05-24 is required before historical admissibility can be confirmed. Model knowledge is not intrinsically date-filtered; no later knowledge is used here to close this obligation."
    },
    {
      "kind": "map_correction",
      "source_quote": "The history of this problem is as follows.",
      "explanation": "The historical density bounds, figures, and methodological inspiration citations are not direct premises of the displayed reduction. Its operative inputs are the representation definitions, curvature implication, ensemble count, parameter identities, and thm:RNU. The edge with source_from rmk:locA should identify that remark rather than silently substituting lem:GGq."
    }
  ],
  "dependency_proposals": [
    {
      "label": "thm:RNU",
      "role": "conditional",
      "source_quote": "except for a set of cardinality $\\ll N^{1-\\eta}$.",
      "explanation": "Principal proposed premise; its restricted interval and singular-series lower bound must be retained. Its proof is not established by this packet."
    },
    {
      "label": "eq:fFTbnd",
      "role": "conditional",
      "source_quote": "\\#\\fF_{T}\\ll T^{\\gd}.",
      "explanation": "Needed for the M\u00f6bius-tail estimate, with matching multiplicity convention; not independently established here."
    },
    {
      "label": "eq:cRNis",
      "role": "proof_input",
      "source_quote": "Clearly $\\cR_{N}(n)\\neq0$ implies that $n\\in\\sB$.",
      "explanation": "Add the untruncated representation definition and its nonnegative weights as explicit inputs connecting exceptional representations to missing curvatures."
    },
    {
      "label": "eq:cRNhatUis",
      "role": "proof_input",
      "source_quote": "with corresponding ``representation \n       \nfunction'' $\\cR_{N}^{U}$ (which could be negative).",
      "explanation": "Add the truncated definition explicitly: the difference identity depends on its exact divisor cutoff."
    },
    {
      "label": "eq:Uis",
      "role": "proof_input",
      "source_quote": "U=N^{\\fu}",
      "explanation": "The stipulated positivity of u is what converts the tail estimate into a power saving."
    },
    {
      "label": "eq:TXNis",
      "role": "proof_input",
      "source_quote": "TX^{2}=N.",
      "explanation": "This identity converts the exceptional-set estimate to N^{1+\u03b5}/U; the particular numerical exponents are unnecessary for this reduction."
    },
    {
      "label": "eq:totMass",
      "role": "background",
      "source_quote": "\\widehat{\\cR_{N}}(0)\\ll T^{\\gd}X^{2}.",
      "explanation": "Provides a scale comparison, but is not needed to derive the exceptional-set bound."
    },
    {
      "label": "external:Bourgain2012",
      "role": "background",
      "source_quote": "inspired by\nthat given recently by the first-named author in \\cite{Bourgain2012}",
      "explanation": "Methodological attribution, not a theorem imported into the displayed conditional reduction."
    }
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