{
  "target_id": "thm:cMNis",
  "proof_status": "needs_context",
  "summary": "The archimedean estimate and final power saving are conditionally consistent, but the supplied proof does not quantify the passage from the truncated modular sum to the infinite singular series. Transitivity, local densities, parameter-range compatibility, and primitive-pair counting also require omitted context. The all-n formulation additionally requires a sign argument when the singular series vanishes. This is a bounded assessment, not a verdict on the theorem.",
  "used_dependencies": [
    "eq:Fuc",
    "eq:Q0is",
    "eq:TXNis",
    "eq:cMNUn",
    "eq:fMdecomp",
    "eq:hatFunc",
    "lem:spec2"
  ],
  "missing_context": [
    "Definitions of w_{x,y}, Gamma_0(q), the relevant orbit action, and Upsilon, including support and positivity.",
    "Uniform local-factor computations, convergence, positivity on admissible integers, and a quantitative singular-series truncation estimate, including primes 2 and 3.",
    "A compatible bound relating delta-Theta1 and C, weighted primitive-pair counting, and a sign argument for the claimed non-admissible cases.",
    "Dated exact external statements and versions supporting the classical singular-series argument. The recorded cutoff is 2012-05-24. No external theorem was imported here; model knowledge alone cannot certify historical availability."
  ],
  "findings": [
    {
      "kind": "unverified",
      "source_quote": "extend the sum on $q$ to all natural numbers,  and use multiplicativity to write the sum\nas an Euler product.",
      "explanation": "The decomposition contains the finite sum S_{Q0}, not the Euler product S. Convergence and bounds for S alone do not justify replacing S_{Q0}. A uniform estimate implying S_{Q0}(n) >= c S(n) for admissible N/2<n<N is still required. No truncation-error estimate is supplied."
    },
    {
      "kind": "context",
      "source_quote": "use the transitivity of the $\\g_{0}$ sum to replace $\\<w_{x,y},\\g_{0} v_{0}\\>$ by $\\<e_{4},\\g_{0} v_{0}\\>$",
      "explanation": "This requires the relevant dual-orbit statement for w_{x,y} modulo every q, with its coprimality hypotheses. The full M\u00f6bius sum restricts to gcd(2x,y)=1, but that restriction is not stated here. Equation Fuc covers only moduli coprime to 6; transitivity and local factors at 2 and 3 require additional justification. The switch from tilde-Gamma(q) to Gamma_0(q) in the Euler product also needs their definitions or an explanation of equivalent averaging."
    },
    {
      "kind": "context",
      "source_quote": "using Lemma \\ref{lem:spec2}.",
      "explanation": "The window requires K=2K0. The hypothesis N/2<n<N matches, but x,y asymptotic to X depends on the omitted cutoff definition. Also 2K0<=T2^(1/10) must be checked. Writing a=delta-Theta1 and T2=T^{C/(1+C)}, this requires 2T^{a/10}<=T^{C/[10(1+C)]}; a<C/(1+C) suffices asymptotically. The supplied data do not establish this inequality."
    },
    {
      "kind": "error",
      "source_quote": "specifically that $\\hat\\ft>0$ and $\\hat\\ft(y)>2/5$ for $|y|<1/2$",
      "explanation": "The Fourier transform vanishes at every nonzero integer, so global strict positivity is false. Nonnegativity is sufficient for the displayed lower bound. The other inequality is correct: on |y|<1/2 the squared sinc is greater than 4/pi^2>2/5."
    },
    {
      "kind": "unverified",
      "source_quote": "N^{-\\vep}\\ll_{\\vep} \\fS(n)\\ll_{\\vep} N^{\\vep},",
      "explanation": "The lower bound needs an admissibility qualification whenever S can vanish. More importantly, the target includes non-admissible n: if S(n)=0, it asserts M_N(n)>=0. An approximation with an unsigned O-error does not establish that sign. Neither the supplied major-arc definition nor the final error estimate supplies this missing argument."
    },
    {
      "kind": "context",
      "source_quote": "Putting everything into  \\eqref{eq:fMdecomp} and then into \\eqref{eq:cMNUn} gives \\eqref{eq:cMNbnd}",
      "explanation": "Conditional on nonnegative cutoff weights, weighted primitive-pair mass comparable to X^2, and S_{Q0}>=cS, the scaling works. There are O(X^2) contributing pairs, so the decomposition error is O(T^{Theta1-1}Q0^8)=O(T^{delta-1-3a/5}), while the main scale is S(n)T^{delta-1}. Since T=N^{1/100}, the proposed admissible lower bound S(n)>>N^{-epsilon} absorbs this error for epsilon<3a/500. The required support, positivity, and primitive-pair mass are not supplied."
    },
    {
      "kind": "map_correction",
      "source_quote": "gives \\eqref{eq:cMNbnd}",
      "explanation": "eq:cMNbnd is the target conclusion, not an independent proof input. Likewise, thm:RNU specifies the desired singular-series properties; its representation lower bound must not be used to prove this intermediate theorem."
    }
  ],
  "dependency_proposals": [
    {
      "label": "eq:cMNbnd",
      "role": "excluded",
      "source_quote": "\\cM_{N}(n)\\gg\\fS(n)T^{\\gd-1}",
      "explanation": "This is the conclusion being reviewed, not a premise."
    },
    {
      "label": "thm:RNU",
      "role": "background",
      "source_quote": "there exists a function $\\fS(n)$ as in Theorem \\ref{thm:RNU}",
      "explanation": "Use only as a specification of singular-series properties, not as an established representation estimate."
    },
    {
      "label": "lem:GGq",
      "role": "conditional",
      "source_quote": "Similarly, the powers of $3$ stabilize at $3$",
      "explanation": "The full supplied quotient lemma, including multiplicativity and exceptional-prime stabilization, is relevant beyond the narrower eq:Fuc edge. Its assertions remain unverified premises."
    },
    {
      "label": "singular-series-truncation",
      "role": "proof_input",
      "source_quote": "extend the sum on $q$ to all natural numbers",
      "explanation": "A uniform finite-to-infinite comparison is needed as a separate proof obligation."
    },
    {
      "label": "external:BourgainKontorovich2010",
      "role": "unclassified",
      "source_quote": "See, e.g. \\cite[\\S4.3]{BourgainKontorovich2010}.",
      "explanation": "available_from: unresolved for the needed result; date_evidence: supplied bibliography lists GAFA 20(5):1144\u20131174, 2010; date_basis: bibliographic publication year; version: cited journal article, exact section uninspected. The exact theorem and hypotheses establishing applicability here remain unresolved, so this citation cannot close the obligation."
    }
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