{
  "target_id": "narrative:93454",
  "proof_status": "needs_context",
  "summary": "The M\u00f6bius rewriting, unfolding, Fourier factorization, and Q0^4 error count are locally correct conditional on the supplied counting premises. The packet does not fully verify the scale and subgroup hypotheses needed to apply Lemma spec1, or the stated parameter normalization. This passage supports the displayed modular\u2013archimedean decomposition, not yet the announced major-arc lower bound. No external theorem is imported; later model knowledge is not treated as evidence, and external availability by the 2012-05-24 cutoff is not certified here.",
  "used_dependencies": [
    "eq:G0q",
    "eq:G0qBnd",
    "eq:Q0K0intro",
    "eq:Uis",
    "eq:cMNUdef",
    "eq:cMNdef",
    "eq:cRNhatIs",
    "eq:cRNhatUis",
    "eq:fMis",
    "eq:ffToInProd",
    "eq:gT1def",
    "eq:hatFuncN",
    "eq:spec1",
    "eq:spec2",
    "lem:spec1",
    "lem:spec3"
  ],
  "missing_context": [
    "The T,T1,T2,N definitions and bounds establishing T^{delta-Theta1}<=T2 and making the chosen U a positive fixed power of N.",
    "Support for choosing the two counting exponents so that their maximum lies strictly between 1 and delta.",
    "The exact subgroup condition eq:G1G and its verification for the stabilizer.",
    "The definition of w_{x,y}, or an explicit congruence-preservation statement sufficient to make the rational phase constant on stabilizer cosets."
  ],
  "findings": [
    {
      "kind": "context",
      "source_quote": "making a change of variables $\\gt=r/q+\\gb$, and unfolding the integral from $\\sum_{m}\\int_{0}^{1}$ to $\\int_{\\R}$",
      "explanation": "This transition is valid for integer n and integer-valued forms: periodicity removes the integer translates, and compact support of the hat function justifies unfolding. The unrestricted divisor sum equals the indicator of gcd(2x,y)=1; retaining u<U gives the truncated version exactly. The smoothing support keeps the relevant x,y sums finite and within the size range required by spec1."
    },
    {
      "kind": "context",
      "source_quote": "O\\left({T^{\\gT_{1}}\\over N}K_{0}^{2}Q_{0}^{4}\\right)",
      "explanation": "Conditional on applying spec1 with K=K0, this error is correct: each coset contributes O(T^{Theta1}K0), integration contributes O(K0/N), and summing over reduced residues and cosets costs at most sum_{q<Q0} q\u00b7q^2=O(Q0^4). The main term factors because the rational phase is constant on each stabilizer coset. Fourier inversion gives the stated W; the potential sign reversal is harmless because the hat transform is even. No cancellation is needed."
    },
    {
      "kind": "unverified",
      "source_quote": "Applying Lemma \\ref{lem:spec1} to the latter sum",
      "explanation": "The lemma requires 1<K<=T2^{1/10}. Here K=K0=T^{(delta-Theta1)/10}, so the exact outstanding scale check is T^{delta-Theta1}<=T2. The supplied packet omits the defining T,T1,T2 relations and numerical bounds needed to establish this. The beta support satisfies the lemma's strict inequality except at endpoints, where the hat vanishes; that causes no problem."
    },
    {
      "kind": "context",
      "source_quote": "1<\\gT_{1}<\\gd\n\\ee\nto be the larger of the two.",
      "explanation": "The supplied spec1 and spec2 statements assert only Theta<delta, not that their maximum exceeds 1. The normalization therefore needs supporting context. One cannot automatically enlarge the exponent in spec2, since it appears in a positive lower-bound term. Also, compatibility of the chosen U with U=N^u requires the omitted T-versus-N scaling. In contrast, normalizing eta0 to lie below 1 is legitimate by decreasing the positive exponent in spec3."
    },
    {
      "kind": "context",
      "source_quote": "As in \\eqref{eq:G0q}, let $\\tilde\\G(q)$ be the stabilizer of $v_{0}(\\mod q)$.",
      "explanation": "The coset decomposition is valid, including multiplicities in F. For the application of spec1, however, the exact condition G1G is not supplied; an attached passage asserts that the stabilizer satisfies it but does not expose the condition. Constancy of the rational phase also requires that pairing with w_{x,y} preserve congruences modulo q, for example because w_{x,y} is integral. Its definition is omitted. The index estimate remains an unverified proposed premise."
    },
    {
      "kind": "map_correction",
      "source_quote": "We return to the setting and notation of \\S\\ref{sec:outline} with the goal of establishing \\eqref{eq:cMNUbnd}.",
      "explanation": "The map converts a reference to sec:outline into an edge from thm:Main. That is not a proof-input edge supported by this sentence. Likewise, cMNUbnd is the announced goal, not a premise used to derive fMdecomp. The passage supplies neither positivity nor approximation of the truncated singular series, nor the subsequent transfer to M_N^U."
    },
    {
      "kind": "map_correction",
      "source_quote": "First recall the two numbers $\\gT<\\gd$ appearing  in \\eqref{eq:spec1}, \\eqref{eq:spec2}",
      "explanation": "Spec2 is used here only to select a common exponent, not through its counting lower bound. Eq:spec1 and lem:spec1 duplicate the same premise. Eq:fMis and eq:gT1def are definitions within the target itself, so their edges should not be read as independent supporting results. These are mapping distinctions, not proof defects."
    }
  ],
  "dependency_proposals": [
    {
      "label": "eq:TT1T2",
      "role": "proof_input",
      "source_quote": "Let $1<K\\le T_{2}^{1/10}$",
      "explanation": "The omitted scale definitions are needed to verify the actual application with K=K0."
    },
    {
      "label": "eq:G1G",
      "role": "proof_input",
      "source_quote": "any group $\\tilde\\G(q)$ satisfying \\eqref{eq:G1G}",
      "explanation": "This is an explicit hypothesis of the invoked lemma, not merely background."
    },
    {
      "label": "definition of w_{x,y}",
      "role": "proof_input",
      "source_quote": "using \\eqref{eq:ffToInProd}.",
      "explanation": "The pairing identity supports the phase rewriting, but the omitted definition must also justify reduction modulo q."
    },
    {
      "label": "thm:Main",
      "role": "excluded",
      "source_quote": "We return to the setting and notation of \\S\\ref{sec:outline}",
      "explanation": "The cited outline is contextual; the main theorem is not an input to this decomposition."
    },
    {
      "label": "eq:cMNUbnd",
      "role": "background",
      "source_quote": "with the goal of establishing \\eqref{eq:cMNUbnd}",
      "explanation": "This identifies the downstream goal rather than an established premise."
    },
    {
      "label": "lem:spec1",
      "role": "conditional",
      "source_quote": "Applying Lemma \\ref{lem:spec1} to the latter sum",
      "explanation": "This is the operative analytic premise, conditional on its hypotheses and validity; eq:spec1 is its duplicate equation anchor."
    }
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