{
  "target_id": "eq:cI41bnd",
  "proof_status": "needs_context",
  "summary": "Conditional on the supplied dependency statements and omitted notation and scale bounds, the displayed summation argument is algebraically consistent and yields I_Q^(=,=) \u226a_\u03b5 N^\u03b5 U^4 X^2 T^\u03b4. The packet does not supply enough information to verify the initial restriction, the gcd and family-counting estimates, or the final power saving. The eq:gls dependency is incorrectly mapped from a reference to eq:dim. No external theorem is imported; later knowledge potentially present in model weights is not used to fill these gaps.",
  "used_dependencies": [
    "eq:cIQ3def",
    "eq:eqeq",
    "eq:modBndaNeqAp"
  ],
  "missing_context": [
    "Definition of I_Q^(=,=), including its collision restriction and its relation to the bound eq:cIQ3def.",
    "Definition eq:tilQis and surrounding hypotheses for eq:modBndaNeqAp, establishing q0|q, (q/q0)^2\u2264u^4, and applicability when a=a'.",
    "Bounds for the family F and its coefficients, including #F\u226aT^\u03b4, nonzero a, and polynomial size control needed for divisor losses.",
    "Parameter definitions relating N, X, T, U and the global M, including U\u22651 and a sufficiently large frequency cutoff for eq:eqeq.",
    "The statement eq:dim and quantitative parameter choices implying U^4 T^(1\u2212\u03b4)\u2264N^(\u2212c) for some c>0."
  ],
  "findings": [
    {
      "kind": "context",
      "source_quote": "\\cI_{Q}^{(=,=)}",
      "explanation": "The packet defines the split by a=a' but not the second equality in this notation. The first displayed estimate additionally restricts to ff(m,-n)=ff'(m',-n'); its validity requires the missing definition of this contribution. The section supplies Q\u2265X, so the explicit hypothesis of eq:cIQ3def is satisfied within the section's range."
    },
    {
      "kind": "context",
      "source_quote": "u^{4}\n{(a^{2},q)\\over q^{5/4}}q^{1/4}",
      "explanation": "At a=a', the stated modular estimate gives (q/q0)^2 (a^2,q0)/q, since (0,q)=q. The displayed bound follows if q0 divides q and (q/q0)^2\u2264u^4. Neither the definition of q0 nor these consequences is supplied. The omitted 'above notation' must also be checked to ensure the modular lemma allows a=a'."
    },
    {
      "kind": "unverified",
      "source_quote": "\\sum_{\\tilde q_{1}\\mid a^{2}\\atop\\tilde q_{1}\\ll Q}",
      "explanation": "The divisor step is valid conditionally: (a^2,q)\u2264\u03a3_{d|a^2,d|q}d, and for d\u226aQ there are O(Q/d) multiples in q\u223cQ. Thus the weighted q-sum is O(Q \u03c4(a^2)). Absorbing this into Q N^\u03b5 requires nonzero a of polynomially bounded size. The next line also requires #F\u226aT^\u03b4, up to an absorbable loss. These prerequisites are absent."
    },
    {
      "kind": "context",
      "source_quote": "\\left( {UQ\\over X}\\right)^{2}\n+\nT\n{UQ\\over X}",
      "explanation": "The application of eq:eqeq correctly counts all four integer variables for each fixed ff; no additional factor for n,m is needed. Its hypotheses still require the frequency range to be covered by a sufficiently large bound K\u224dUQ/X, and (TK)^\u03b5 to be absorbable into N^\u03b5. Expanding the resulting line gives N^\u03b5[U^4 XQ T^(\u03b4\u22121)+U^3 X^2 T^\u03b4]. For Q<XT and U\u22651, this is bounded by N^\u03b5 U^4 X^2 T^\u03b4, as displayed."
    },
    {
      "kind": "unverified",
      "source_quote": "T^{1-\\gd}\nU^{4}\nN^{\\vep}",
      "explanation": "The final factorization is exact, but the claimed saving needs a relation such as X^2T\u224dN and a quantitative bound U^4 T^(1\u2212\u03b4)\u2264N^(\u2212c) for some c>0, followed by \u03b5<c. Merely knowing \u03b4>1 would not establish an N-power saving without growth restrictions on T and U. These parameter statements are not supplied."
    },
    {
      "kind": "map_correction",
      "source_quote": "From \\eqref{eq:dim}, this is a power savings.",
      "explanation": "The source references eq:dim, not eq:gls. The supplied spectral theorem neither asserts \u03b4>1 unconditionally nor supplies the required parameter scales. The eq:gls edge should therefore not be treated as a proof input here; the missing eq:dim statement and relevant scale definitions are the appropriate candidates. This is a mapping/context limitation, not itself a demonstrated proof defect."
    }
  ],
  "dependency_proposals": [
    {
      "label": "eq:dim",
      "role": "proof_input",
      "source_quote": "From \\eqref{eq:dim}, this is a power savings.",
      "explanation": "Explicitly invoked for the final saving, but its statement is missing. It must be assessed together with the parameter choices."
    },
    {
      "label": "eq:gls",
      "role": "excluded",
      "source_quote": "The latter is empty unless $\\gd>1$",
      "explanation": "Not referenced by this proof and insufficient to establish its quantitative saving. The frozen edge appears to misresolve eq:dim."
    },
    {
      "label": "eq:tilQis",
      "role": "proof_input",
      "source_quote": "recall the notation \\eqref{eq:tilQis}",
      "explanation": "Candidate source for the missing q0 definition needed to justify the u^4 modular bound."
    },
    {
      "label": "family-size and coefficient bounds",
      "role": "proof_input",
      "source_quote": "T^{\\gd}\nQ",
      "explanation": "The counting transition needs a family-cardinality bound and coefficient-size control for the divisor estimate; neither is among the supplied dependencies."
    },
    {
      "label": "parameter scales and choice of U",
      "role": "proof_input",
      "source_quote": "X^{2}T\\,\nT^{2(\\gd-1)}",
      "explanation": "Relating this normalization to N and proving a negative N-power from the remaining factor are necessary final inputs."
    }
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