{
  "target_id": "thm:cMNtoMNU",
  "proof_status": "needs_context",
  "summary": "The truncation argument is conditionally sound for admissible n, but the packet does not establish several estimates needed by its displayed L1 bound. The principal missing input is a uniform bound for the finite modular factor for every pair occurring in the discarded M\u00f6bius tail; the stated bound for the completed singular series does not itself supply this. The lattice-tail count and parameter-error absorption check directly. Completing the exceptional-set argument also requires the omitted scale relation and a power-growth statement for U. No supplied dependency is treated as established.",
  "used_dependencies": [
    "eq:Q0is",
    "eq:cMNUn",
    "eq:cMNbnd",
    "eq:fMdecomp",
    "eq:fSbnd",
    "eq:nZbnd"
  ],
  "missing_context": [
    "A uniform finite-modular-factor estimate for all x,y in the discarded M\u00f6bius tail, including its congruence-orbit hypotheses and any required tail estimate.",
    "The definition and decay properties of hat(t), and a uniform normalized sampling bound.",
    "The definition and upper cardinality estimate for F, including multiplicities.",
    "Equation TXNis and the T-versus-N scaling needed to make U a fixed positive power of N.",
    "Clarification whether the target is restricted to admissible integers.",
    "A dated, version-specific source record for any external result actually used to close the modular obligation."
  ],
  "findings": [
    {
      "kind": "unverified",
      "source_quote": "using \\eqref{eq:fSbnd} and \\eqref{eq:Q0is}.",
      "explanation": "The second displayed inequality requires a bound such as |S_{Q0,x,y}(n)| \u226a\u03b5 N^\u03b5, uniformly over the supported x,y, including those with gcd(2x,y) \u2265 U. Equation fSbnd bounds the completed singular series S(n), not this finite factor. A completed-series bound alone does not control partial sums without a tail or absolute-convergence estimate. Moreover, the transitivity passage used earlier to remove x,y must apply to these nonprimitive pairs, or a separate uniform estimate is needed. The relevant congruence-orbit hypotheses are omitted, so applicability cannot be checked."
    },
    {
      "kind": "context",
      "source_quote": "\\sum_{\\ff\\in\\fF}\n{K_{0}\\over N}\n\\left[\n\\sum_{n<N}\n\\hat\\ft",
      "explanation": "Reducing this expression to O(T^\u03b4) requires both #F \u226a T^\u03b4, with the intended multiplicities, and the uniform sampling bound (K0/N) \u03a3_{1\u2264n<N} hat(t)((a\u2212n)K0/N) \u226a 1. Positivity is stated in the surrounding excerpt, but the definition and decay of hat(t), the requisite scale K0/N, and the upper cardinality bound for F are not supplied."
    },
    {
      "kind": "context",
      "source_quote": "K_{0}^{2}Q_{0}^{4}T^{\\gT_{1}}",
      "explanation": "The parameter absorption is correct: writing \u0394 = \u03b4\u2212\u03981 > 0 gives K0\u00b2Q0\u2074T^\u03981 = T^(\u03981+2\u0394/5) = T^(\u03b4\u22123\u0394/5) \u2264 T^\u03b4. Also, for nonzero y on support of size O(X), swapping the u and y sums bounds the lattice tail by \u03a3_{U\u2264u\u2264CX} O(X/u)O(X/u) \u226a X\u00b2/U; the condition u | 2x changes only an absolute constant. These checks support the last inequality once the modular and sampling inputs are supplied."
    },
    {
      "kind": "context",
      "source_quote": "The rest of the argument is identical to that leading to \\eqref{eq:nZbnd}.",
      "explanation": "For admissible n, conditional on the supplied lower bounds, let D(n)=|M_N(n)\u2212M_N^U(n)|. Outside D(n) \u2265 (c/2)S(n)T^(\u03b4\u22121), the desired lower bound follows from M_N(n) \u2265 cS(n)T^(\u03b4\u22121). The displayed L1 estimate and S(n) \u226b\u03b5 N^(\u2212\u03b5) give an exceptional count \u226a N^(2\u03b5)TX\u00b2/U. Obtaining a power saving requires TX\u00b2 \u224d N and U \u2265 N^\u03ba for some \u03ba>0. U is defined in the packet, but the required relation between T and N is not."
    },
    {
      "kind": "unverified",
      "source_quote": "the bound \\eqref{eq:cMNbnd} holds with $\\cM_{N}$ replaced by $\\cM_{N}^{U}$",
      "explanation": "The referenced theorem is stated for N/2<n<N without an admissibility restriction, whereas its singular series vanishes at non-admissible n. The L1 transfer using a positive singular-series lower bound proves the assertion only for admissible n. It does not establish M_N^U(n) \u2265 0 for most non-admissible n. The surrounding major-arcs objective explicitly concerns admissible n, so the intended scope should be supplied rather than assumed."
    },
    {
      "kind": "map_correction",
      "source_quote": "The rest of the argument is identical to that leading to \\eqref{eq:nZbnd}.",
      "explanation": "The edge from eq:nZbnd records a proof-method analogy, not a sufficient numerical premise for this target. Its earlier argument concerns different functions and exceptional sets. Direct proof inputs omitted from the proposed dependency list include the definition of U, the scale relation TXNis, and the sampling/cardinality estimates."
    },
    {
      "kind": "unverified",
      "source_quote": "See, e.g. \\cite[\\S4.3]{BourgainKontorovich2010}.",
      "explanation": "This external text has not been inspected. If used to discharge the modular estimate, its exact result, hypotheses, available_from, date_evidence, date_basis, and version remain unresolved against the recorded cutoff 2012-05-24. No external theorem is imported in this assessment. Model training may contain later knowledge; that knowledge is not being used as historical evidence."
    }
  ],
  "dependency_proposals": [
    {
      "label": "eq:fuIs",
      "role": "proof_input",
      "source_quote": "U=Q_{0}{}^{(\\eta_{0})^{2}/100}",
      "explanation": "Needed, together with T-versus-N scaling and \u03b70>0, to convert the L1 saving into a power-saving exceptional count."
    },
    {
      "label": "eq:TXNis",
      "role": "proof_input",
      "source_quote": "using \\eqref{eq:Q0is} and \\eqref{eq:TXNis}.",
      "explanation": "The omitted scale relation must identify TX\u00b2 with N and ensure polynomial growth of U."
    },
    {
      "label": "uniform_finite_modular_bound",
      "role": "conditional",
      "source_quote": "\\fS_{Q_{0}}(n)",
      "explanation": "The proof needs a uniform upper bound for this finite factor, not merely the supplied completed-series statement."
    },
    {
      "label": "archimedean_sampling_and_F_size",
      "role": "proof_input",
      "source_quote": "\\sum_{\\ff\\in\\fF}",
      "explanation": "The normalized n-sum and the upper size of F are necessary to obtain the claimed L1 estimate."
    },
    {
      "label": "eq:nZbnd",
      "role": "background",
      "source_quote": "The rest of the argument is identical to that leading to \\eqref{eq:nZbnd}.",
      "explanation": "Provides the transfer template, not an independently sufficient premise."
    }
  ],
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