Exact Quadratic Log-Defect Asymptotics, the Taylor–Liouville Barrier, and the Pivot to Multivariate Auxiliaries

For the Type-P quadratic relation, the truncated-log defect has an exact non-zero Taylor asymptotic Δ^log_{m,Q} = ((−1)^m/(m+1)) λ^{m+2} s²(s^m − 1)² Q^{−m} + O(Q^{−m−1}) (no hidden cancellation); a scalar Liouville comparison is structurally short by a factor Q³/log Q; the next step must retain the geometry of multivariate auxiliary functions.