{
  "target_id": "thm:IQM",
  "proof_status": "needs_context",
  "summary": "The supplied packet states the large-denominator estimate but does not supply its proof or the preceding diagonal/off-diagonal bounds needed to establish it. The missing proof anchor is a mapping limitation, not evidence that the theorem is false. The closing paragraph identifies IQM, IQ0 and IQX as inputs to the error-energy bound, not as proofs of one another. There is also a sign inconsistency in the supplied setup. No external result is imported under the recorded 2012-05-24 cutoff; although later knowledge may remain in model weights, it is not used to discharge this obligation.",
  "used_dependencies": [
    "thm:IQ0",
    "thm:IQX"
  ],
  "missing_context": [
    "The proof preceding thm:IQM, especially the final bounds for I_Q^{(=)} and I_Q^{(neq)} and their parameter restrictions.",
    "Definitions eq:IQdef, eq:IQ0K0 and eq:IQ0, and the Fourier/Parseval inequality connecting those integrals to eq:cEN2bnd.",
    "Definitions and asymptotic relations for X, M, T, U, Q_0, K_0 and delta, including the allowed Q-blocks and uniformity of eta and implied constants.",
    "Definitions eq:tilQis and eq:cSdef, the family of forms and its coefficient bounds, and the smooth weight hypotheses needed for completion and subsequent estimates.",
    "For the closing headline inference only: established major-arc bounds, the singular-series lower bound, and the reductions connecting the truncated representation function to curvatures."
  ],
  "findings": [
    {
      "kind": "unverified",
      "source_quote": "If $X\\le Q<M$, then there is some $\\eta>0$ so that",
      "explanation": "This states the target but supplies no derivation. The section setup only reaches a diagonal/off-diagonal decomposition. The exact remaining obligation is to bound both resulting contributions by N T^{2(delta-1)} N^{-eta}, accounting for the u, q, frequency and form sums, uniformly over the asserted Q-range. Neither IQ0 nor IQX covers this range."
    },
    {
      "kind": "context",
      "source_quote": "In this section, we continue our analysis of $\\cI_{Q}$ from \\eqref{eq:IQdef}",
      "explanation": "The definition of I_Q is absent, as are the parameter relations and hypotheses on the forms, weights and truncations. Thus the hypotheses X <= Q < M and N tending to infinity cannot be checked against the actual integral. For the subsequent summation, eta and the implied constant must be sufficiently uniform in Q; the statement alone does not document that uniformity."
    },
    {
      "kind": "error",
      "source_quote": "e(\\gb (a-a'))",
      "explanation": "In the displayed expansion IQbnd2, this archimedean phase has the opposite sign from direct multiplication of the factors in cRsq2: e(-a beta) times the conjugate of e(-a' beta) equals e(beta(a'-a)). The modular phase in IQbnd2 already uses a'-a. No compensating change of variables is supplied. This is an inconsistency in the attached setup, not by itself a disproof of IQM."
    },
    {
      "kind": "map_correction",
      "source_quote": "Finally, Theorems \\ref{thm:IQ0}, \\ref{thm:IQX},  and \\ref{thm:IQM} together complete the proof of \\eqref{eq:cEN2bnd}, and hence Theorem \\ref{thm:Main} is proved.",
      "explanation": "These references describe a downstream assembly. They do not support edges IQ0 -> IQM, IQX -> IQM, cEN2bnd -> IQM or Main -> IQM. The proposed proof direction is instead IQ0, IQX and IQM jointly toward cEN2bnd, followed by additional reductions toward Main."
    },
    {
      "kind": "unverified",
      "source_quote": "together complete the proof of \\eqref{eq:cEN2bnd}",
      "explanation": "Conditionally on the three estimates, this transition still requires an inequality controlling the Fourier error energy by these integrals, coverage of all minor arcs, and control of the number of Q-blocks. For polynomially bounded M and logarithmically many blocks, uniform positive savings can absorb the logarithm after decreasing eta; those prerequisites are not supplied. The further conclusion Main also requires the major-arc lower bound and the representation/truncation reductions, not just the error-energy estimate."
    }
  ],
  "dependency_proposals": [
    {
      "label": "thm:IQ0",
      "role": "conditional",
      "source_quote": "\\cI_{Q_{0},K_{0}}, \\ \\cI_{Q_{0}}\\ll N\\, T^{2(\\gd-1)}\\, N^{-\\eta}",
      "explanation": "An unverified sibling estimate for the downstream error-energy assembly, not an input proving IQM. Its coverage cannot be checked without the integral definitions."
    },
    {
      "label": "thm:IQX",
      "role": "conditional",
      "source_quote": "For $Q_{0}\\le Q<X$, there is some $\\eta>0$ such that",
      "explanation": "An unverified estimate for the adjacent smaller-Q range. It supports the closing assembly conditionally, not the target range."
    },
    {
      "label": "eq:cEN2bnd",
      "role": "excluded",
      "source_quote": "together complete the proof of \\eqref{eq:cEN2bnd}",
      "explanation": "Exclude as a premise for IQM: the source explicitly presents it as a consequence of IQM together with the other minor-arc estimates."
    },
    {
      "label": "thm:Main",
      "role": "excluded",
      "source_quote": "and hence Theorem \\ref{thm:Main} is proved.",
      "explanation": "A downstream headline conclusion, not evidence for the target estimate."
    },
    {
      "label": "eq:IQdef",
      "role": "proof_input",
      "source_quote": "insert the result into \\eqref{eq:IQdef}",
      "explanation": "The missing definition is necessary to verify the starting integral, its normalization, and the transition to IQbnd2."
    },
    {
      "label": "eq:cIQ34 and subsequent large-Q estimates",
      "role": "proof_input",
      "source_quote": "\\cI_{Q}\\ll \\cI_{Q}^{(=)}+\\cI_{Q}^{(\\neq)}.",
      "explanation": "This supplies a proposed route to IQM. The definitions and estimates of both contributions, absent here, are the immediate remaining proof inputs."
    }
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