{ "target_id": "eq:modBndaNeqAp", "proof_status": "defective", "summary": "The supplied proof contains an incorrect congruence indicator when substituting eq:SfEval into eq:cS1. This invalidates that displayed identity, but does not disprove the lemma's bound. Conditional on a correctly obtained Kloosterman/Sali\u00e9 estimate, the final algebra gives exactly the claimed powers of q and q0. The external estimate, the prime-power reduction, and the omitted 2-adic case remain unverified. This assessment uses direct algebra and supplied excerpts, not model knowledge as dated external evidence.", "used_dependencies": [ "eq:Ldef", "eq:SfEval", "eq:tildQdef", "external:Kloosterman1927" ], "missing_context": [ "Complete definition of S and all inherited restrictions on q, q0, u0, the forms, and the family F.", "Earlier reductions establishing multiplicativity, handling p=2 and q0=1, and permitting C and C' to be units.", "Definitions of tilde q and tilde q', inverse-lifting conventions, and the q1=1 convention.", "A dated, inspected source location containing the exact required Kloosterman/Sali\u00e9 estimate and its uniformity hypotheses.", "Justification that local constants yield the global bound without an additional arithmetic factor." ], "findings": [ { "kind": "error", "source_quote": "\\bo_{\nnC\\equiv 2mB(\\tilde q_{0})\n\\atop\nn'C'\\equiv 2m'B'(\\tilde q'_{0})\n}", "explanation": "Both factors of 2 conflict with eq:SfEval, whose support condition is nC \u2261 mB modulo tilde q0, and with L=(Cn\u2212Bm)/tilde q0. The displayed replacement condition does not even ensure L is integral. As an algebraic check of the substitution, take q=q0=9, u0=1, a=a'=3, f=f'=10x\u00b2+2xy+y\u00b2, and n=m=n'=m'=1. Then AC\u2212B\u00b2=a\u00b2 and the displayed coprimality conditions hold. Eq:SfEval gives |Sf|=1/3 for each of the six unit r, so the correlation is 2/3; the new indicator instead makes eq:cS1 zero. Membership of this example in the manuscript's unspecified family is not asserted; the example checks the claimed algebraic substitution." }, { "kind": "context", "source_quote": "Observe that $\\cS$ is multiplicative in $q$, so we again consider the prime power case $q=p^{j}$, $p\\neq2$; then $q_{0}$ is also a prime power, since $q_{0}\\mid q$. As before, we may assume $(C,q_{0})=(C',q_{0})=1$.", "explanation": "The packet does not supply the CRT factorization, a restriction to odd q, treatment of the p=2 factor, or the earlier reduction making both C and C' units. These are prerequisites, not consequences of the supplied notation. The case q0=1 also needs a convention or separate treatment. The full definition of S is truncated in its defining excerpt." }, { "kind": "unverified", "source_quote": "\\overline{a_{1}'C'}\n(L')^{2}\n\\tilde q'\n \n-\n\\overline{a_{1}C}\nL^{2}\n\\tilde q", "explanation": "The packet defines tilde q0 and tilde q0', but does not identify the un-subscripted tilde q and tilde q' used here. Also, the inverses originally occur modulo q1 and q1', whereas this phase is written modulo q0. Their lifted interpretation must be specified. In particular, when q1=1, a1 can share a prime factor with q0, so an inverse of a1C modulo q0 need not exist; the original modulo-1 phase nevertheless vanishes." }, { "kind": "unverified", "source_quote": "The term in brackets $\\big[\\cdot\\big]$ is a Kloosterman- or Sali\\'e-type sum, for which we have an elementary bound \\cite{Kloosterman1927} to the power $3/4$:", "explanation": "The needed result is a bound |\u03a3* \u03c7(r)e_q(Ar+B r^{-1})| \u226a q^(3/4)(A,q)^(1/4), with A=a'\u2212a, arbitrary relevant B, odd prime-power q, and \u03c7 the product of the two induced quadratic characters. The source must cover degenerate coefficients and imprimitive/trivial characters. External text was not supplied or inspected. Availability record: available_from for this exact result is unresolved; date_evidence is the bibliography's 1927 publication year; date_basis is bibliographic publication year only; version is the cited Acta Math. 49 paper, uninspected. No later result is used to close this obligation under the 2012-05-24 cutoff." }, { "kind": "unverified", "source_quote": "Observe that $\\cS$ is multiplicative in $q$", "explanation": "Even if multiplicativity is granted, prime-power bounds with an unspecified absolute constant do not automatically give the stated global \u226a bound: multiplication can introduce a factor C raised to the number of prime divisors. A uniform global estimate or an explanation controlling these local constants is required; inserting an unstated q^epsilon loss would change the displayed claim." }, { "kind": "map_correction", "source_quote": "In the case $a=a'$, \\eqref{eq:modBndaNeqAp} only saves one power of $q$, and in \\S\\ref{sec:QXT} we will need slightly more; see the proof of \\eqref{eq:cI42bnd}.", "explanation": "The references to narrative:115416 and eq:cI42bnd describe later applications and motivation, not proof inputs for this lemma. Their incoming dependency edges should not be used to justify the lemma. Also, the supplied slice contains an explicit proof despite the missing-proof-anchor flag; that flag is a mapping limitation." } ], "dependency_proposals": [ { "label": "eq:tildQdef", "role": "proof_input", "source_quote": "\\tilde q_{0}:=(a^{2},q_{0}), \\qquad q_{1}:=q_{0}/\\tilde q_{0}", "explanation": "Supplies the gcd prefactors and reduced moduli." }, { "label": "eq:Ldef", "role": "proof_input", "source_quote": "L:=(Cn-Bm )/ \\tilde q_{0} .", "explanation": "Supplies the phase parameter and exposes the incorrect support condition." }, { "label": "eq:SfEval", "role": "conditional", "source_quote": "Inputting the analysis from\n \n \n \\eqref{eq:SfEval} into both $\\cS_{f}$ and $\\cS_{f'}$,", "explanation": "Direct proposed input, assessed conditionally rather than established here." }, { "label": "external:Kloosterman1927", "role": "conditional", "source_quote": "\\newblock {\\em Acta Math.}, 49(3-4):407--464, 1927.", "explanation": "Historically plausible citation, but the exact theorem and applicable version remain unverified." }, { "label": "eq:cI42bnd", "role": "excluded", "source_quote": "see the proof of \\eqref{eq:cI42bnd}", "explanation": "Later application, not an input to this proof." }, { "label": "narrative:115416", "role": "background", "source_quote": "\\section{Minor Arcs III: Case $X\\le Q