Sharp bounds for nonreal poles of Weniger's delta transform on Stieltjes series

For Weniger's delta transform of a Stieltjes series with n = 0 and β = 1, we establish the sharp upper bound 2⌊(k − 1)/2⌋ on the number of nonreal poles at order k. For every k ≥ 3, equality is attained by measures with smooth, compactly supported densities and by absolutely continuous measures with strictly positive densities and moments asymptotic to ε(2m)!. The upper bound follows from an orthogonality identity for the denominator polynomial. Sharpness is proved by a two-atom limiting construction in which the denominator converges to a constant shift of a combination of consecutive Legendre polynomials; an exact error identity excludes cancellation by the numerator. At order three, a rational-moment example yields a cubic with negative integer discriminant and a coprimality certificate modulo 3. We give exact pole counts at orders three through eight for a single measure attaining all six bounds, together with a study of the effect of a positive tail. A comparison with a Padé approximant formed from the same moments quantifies the local error near a nonreal pole. Thus neither positivity of the Stieltjes measure nor superfactorial moment growth ensures confinement of the poles to the negative real axis.