{
  "target_id": "eq:modBndaEqAp",
  "proof_status": "defective",
  "summary": "The prime-partition argument and final exponent comparison are sound conditional on a correctly formulated, well-defined intermediate bound. However, the printed intermediate bound uses a support condition inconsistent with the supplied evaluation of the quadratic sum, so it does not justify the subsequent use of integral L and L'. Additional unresolved prerequisites concern modular inverses, the prime 2, and the asserted extension to composite moduli. This is a local proof assessment, not a disproof of the lemma.",
  "used_dependencies": [
    "eq:15",
    "eq:Ldef",
    "eq:cP1",
    "eq:cP2",
    "eq:cS1",
    "eq:cSdef",
    "eq:fmnfp",
    "eq:modBndaEqAp1",
    "eq:pr2",
    "eq:tildQdef"
  ],
  "missing_context": [
    "The definition of S_f and the inherited hypotheses ensuring that C and C' are units modulo q_0.",
    "A consistent support condition linking eq:SfEval, eq:cS1, eq:modBndaEqAp1, and the integrality of L and L'.",
    "The interpretation of inverse expressions when v_p(a\u00b2)>j, and the scope of the omission of the prime 2.",
    "A CRT derivation of the composite-modulus estimate with uniform constants, together with the exact dated source/version for any external bound actually used."
  ],
  "findings": [
    {
      "kind": "error",
      "source_quote": "nC\\equiv 2mB(\\tilde q_{0})",
      "explanation": "This indicator, appearing in both eq:cS1 and eq:modBndaEqAp1, disagrees with the supplied evaluation eq:SfEval, whose support is nC\u2261mB modulo tilde q_0. The latter condition is also required for L=(Cn\u2212Bm)/tilde q_0 to be integral. The two conditions are not equivalent under the supplied hypotheses. Consequently, the intermediate bound as printed can discard supported terms and does not supply the integrality used later. The analogous primed indicator has the same problem."
    },
    {
      "kind": "context",
      "source_quote": "a^{2}=a_{1}(a^{2},p^{j})",
      "explanation": "The proof uses the inverse of a_1 modulo p^j. If v_p(a^2)>j, then a_1=a^2/p^j remains divisible by p, so this inverse does not exist. Eq:tildQdef guarantees invertibility only modulo q_1, which is 1 in this case. A convention or separate treatment for the entire coefficient involving (a^2,p^j) is missing. Invertibility of C and C' modulo the relevant prime powers is also needed but not supplied."
    },
    {
      "kind": "context",
      "source_quote": "and subtracting $a$ from both sides of \\eqref{eq:15}",
      "explanation": "Conditional on integral L, the determinant identity, and invertible C, the left side of eq:15 already equals f(m,\u2212n) modulo p^{ceil(j/2)}: C^{-1}[(AC\u2212B\u00b2)m\u00b2+(Cn\u2212Bm)\u00b2]=Am\u00b2\u22122Bmn+Cn\u00b2. The primed identity is identical. Thus eq:fmnfp follows directly. Subtracting the common a is unnecessary; it also preserves the congruence because a=a'."
    },
    {
      "kind": "context",
      "source_quote": "Combining \\eqref{eq:cP1} and \\eqref{eq:cP2} in \\eqref{eq:modBndaEqAp1} gives the claim.",
      "explanation": "This final inference checks out conditionally. For odd p in P2, noncongruence gives valuation at most ceil(j/2)\u22121\u2264j/2. For P1, the congruences imply that the product of p^{ceil(j/2)} divides the nonzero integer Z, giving eq:cP1. Writing R=q/q_0\u22651, the resulting bound is R^{9/2}(a\u00b2,q_0)q^{-5/4}q_0^{1/8}Z^{1/2}=R^{35/8}(a\u00b2,q_0)q^{-9/8}Z^{1/2}, which is at most the claimed R^5 bound. Both a=a' and Z\u22600 are used essentially."
    },
    {
      "kind": "unverified",
      "source_quote": "which is valid now without the assumption\n                                          \nthat $q_{0}$ is a prime power.",
      "explanation": "The passage from a prime-power calculation to the displayed uniform composite-modulus bound is asserted without supplying the CRT factorization, transformed local parameters, or control of accumulated constants. This remains an exact proof obligation, rather than an established dependency."
    },
    {
      "kind": "context",
      "source_quote": "continuing to omit the prime $2$.",
      "explanation": "The statement supplied does not explicitly restrict q_0 to odd integers. The proof requires this restriction for the inverse of 4 and the odd-modulus evaluations. Either an inherited odd-modulus hypothesis or a separate 2-adic argument is needed."
    },
    {
      "kind": "unverified",
      "source_quote": "Hence we again apply Kloosterman's $3/4$th bound to \\eqref{eq:cS1}",
      "explanation": "The exact bound and its hypotheses have not been supplied or inspected. No external theorem is imported in this assessment. Its admissibility must be checked against the recorded cutoff, 2012-05-24. Model knowledge may include later developments and is not evidence of historical availability."
    }
  ],
  "dependency_proposals": [
    {
      "label": "eq:SfEval",
      "role": "proof_input",
      "source_quote": "nC\\equiv mB(\\tilde q_{0})",
      "explanation": "This supplied evaluation is needed to audit the support conditions and the definition of L. It is missing from the frozen dependency list despite being central to the displayed derivation."
    },
    {
      "label": "Determinant identities and coefficient conventions",
      "role": "proof_input",
      "source_quote": "AC-B^{2}  \n=   \nA'C'-(B')^{2}",
      "explanation": "These identities, together with f(x,y)=Ax\u00b2+2Bxy+Cy\u00b2, are prerequisites for converting eq:15 into the congruence of form values. Their originating definitions should be anchored."
    },
    {
      "label": "Kloosterman1927",
      "role": "conditional",
      "source_quote": "\\cite{Kloosterman1927}",
      "explanation": "available_from: unresolved; date_evidence: citation key only; date_basis: unresolved; version: unresolved. After a=a' cancels the characters and linear phase, the needed local estimate is |sum over units r mod M of e_M(B r^{-1})|\u226aM^{3/4}(M,B)^{1/4}, uniformly for odd M and integer B. The cited theorem's exact statement and containing version remain unverified."
    },
    {
      "label": "eq:pr2, eq:15, eq:fmnfp, eq:cP1, eq:cP2",
      "role": "conditional",
      "source_quote": "By assumption $Z\\neq0$.",
      "explanation": "These are internal steps of this proof, not independently established premises. Their use is conditional on the corrected support and well-defined modular expressions; the elementary transitions after those prerequisites have been checked above."
    }
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