{
  "target_id": "eq:cIQ12bnd",
  "proof_status": "defective",
  "summary": "Conditional on the counting lemma and omitted definitions, the intended gcd reduction is sound. However, the displayed proof drops coprimality restrictions needed for its modulus count and asserts a weighted-sum estimate not valid throughout the supplied range 0<eta_0<1. Additional context is also required to convert the resulting parameter saving into N^{-eta}. This is a local proof assessment, not a verdict on the proposition itself. No external result or later development is used; possible later knowledge in model training is not evidence here.",
  "used_dependencies": [
    "eq:His",
    "eq:fuIs",
    "lem:spec3"
  ],
  "missing_context": [
    "The definition or preceding bound eq:cIQ1def, fixing the meaning and normalization of I_Q^{(neq,<=)}.",
    "Definitions identifying the two uses of the family F and preserving multiplicities, together with nonvanishing and polynomial size bounds for a and a'.",
    "The scale relations eq:TXNis, sufficient both for every counting modulus to be below N and for converting the final T-saving into N^{-eta}.",
    "An explicit choice eta_0<1/4 before fixing U and H, or another justified estimate replacing the displayed weighted-sum bound.",
    "Confirmation that pairwise coprimality remains imposed until after the q-count is bounded."
  ],
  "findings": [
    {
      "kind": "error",
      "source_quote": "since $Q\\ge Q_{0}$.",
      "explanation": "The preceding estimate requires controlling Q^{-1/4} sum_{d<=CQ} d^{-3/4-eta_0}. Direct summation gives O(Q^{-eta_0}) only for eta_0<1/4; at eta_0=1/4 there is a logarithm, and for eta_0>1/4 the bound is O(Q^{-1/4}). In the last case, even the d=1 term prevents the asserted O(Q_0^{-eta_0}) estimate when Q=Q_0 tends to infinity. The supplied parameter definition assumes only 0<eta_0<1. An explicit sufficiently small choice of the lemma's exponent is missing."
    },
    {
      "kind": "unverified",
      "source_quote": "\\sum_{g_{1}\\mid a\\atop g_{1}\\ll Q}\n\\sum_{g_{1}'\\mid a'\\atop g_{1}'\\ll Q}",
      "explanation": "For the actual gcd variables, the pairwise-coprimality assertion is correct: writing d=(a-a',q), one has gcd((a,q),d)=h, and similarly for a'. Thus the actual lcm is h*g_1*g_1'*tilde_g. But the displayed enlarged divisor sums omit these coprimality restrictions before replacing the number of q by O(Q/(g_1*g_1'*tilde_g)). That replacement is not valid for unrestricted tuples: g_1=g_1'=tilde_g=p and h=1 gives lcm p rather than p^3. The restrictions must remain in force through the counting step. This is an omission in the displayed argument, not a failure of the actual-gcd identity."
    },
    {
      "kind": "context",
      "source_quote": "\\cI_{Q}^{(\\neq,\\le)}\n&=&",
      "explanation": "The packet does not contain eq:cIQ1def or the precise definition of this contribution. Consequently, the initial equality with the positive arithmetic sum cannot be checked. The neighboring large-gcd argument starts with an upper bound. Independently, the subsequent elementary inequality sqrt(gcd(a^2,q))<=gcd(a,q) is valid."
    },
    {
      "kind": "context",
      "source_quote": "To the last sum, we again apply Lemma \\ref{lem:spec3}, giving",
      "explanation": "This application is conditional on identifying the form-indexed family, including multiplicities, with the gamma-indexed family in the lemma, with a=<e_1,gamma v_0>. Every modulus tilde_g must satisfy 1<=tilde_g<N; Q<X alone does not establish this without the missing scale relations. The divisor bounds absorbed into T^{epsilon} also require nonzero a,a' of polynomially bounded size. Once the indexing is justified, the outer cardinality bound T^delta follows from the supplied lemma at modulus 1."
    },
    {
      "kind": "context",
      "source_quote": "By \\eqref{eq:His} and \\eqref{eq:fuIs}, this is again a power savings, as claimed.",
      "explanation": "Conditional on the asserted Q_0^{-eta_0} estimate, the parameter algebra checks: U^6 H^{9/4} Q_0^{-eta_0}=Q_0^{-c}, where c=31*eta_0/40-3*eta_0^2/50>0 for 0<eta_0<1. Since Q_0=T^kappa with kappa=(delta-Theta_1)/20>0, sufficiently small epsilon leaves a T-power saving. The claimed normalization and N-power saving additionally require X^2*T=N and T to grow at a fixed positive power of N; these relations are not supplied."
    }
  ],
  "dependency_proposals": [
    {
      "label": "eq:cIQ1def",
      "role": "proof_input",
      "source_quote": "using\n$(a-a',q)\\le q$ in \\eqref{eq:cIQ1def},",
      "explanation": "The neighboring proof identifies this missing formula as the source of the arithmetic majorant. It is also needed to check the target proof's starting equality or inequality."
    },
    {
      "label": "eq:TXNis",
      "role": "proof_input",
      "source_quote": "Then \\eqref{eq:fuIs} and \\eqref{eq:TXNis}\nimply the claimed power savings.",
      "explanation": "This supplied neighboring passage identifies a scale dependency omitted from the target's proposed map. Its exact statement is needed for the final N-normalization."
    },
    {
      "label": "small choice of eta_0",
      "role": "conditional",
      "source_quote": "where $0<\\eta_{0}<1$ is the number which appears in Lemma \\ref{lem:spec3}.",
      "explanation": "A smaller positive exponent remains admissible in the lemma because q>=1, but the supplied proof does not explicitly make the choice needed for its weighted-sum estimate. That choice must precede the dependent parameter definitions."
    },
    {
      "label": "family and divisor-size prerequisites",
      "role": "conditional",
      "source_quote": "a=a_{\\g}=\\<e_{1},\\g v_{0}\\>$",
      "explanation": "This supplies the coordinate identification in nearby prose, but not the full family correspondence, multiplicity convention, or size and nonvanishing hypotheses needed in the target argument."
    }
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