Geometric Sampling Does Not Determine Regularized R-Diagonal Laws

Abstract

For every s0>0, q>1 and finite integer K>=1, we construct a continuum of distinct affiliated R-diagonal laws having the same joint star-moments and the same first K alternating cumulants of c(1+s c*c)^(-1/2) at every scale s=s0 q^k, k in Z. All absolute moments are finite. A finite combination of classical lognormal Stieltjes perturbations gives an explicit resolvent difference with prescribed finite-order zeros on the entire grid, and an exact formula identifies a higher cumulant separating every pair. Conversely, a Blaschke-Jensen argument gives a sufficient condition under which samples of the first regularized cumulant determine the law without moment assumptions. This is a complete scoped reconstruction counterexample, not a new construction of the established radial analytic R-transform or closure of the entire AIM source question. Classical inputs are credited. The preprint is self-audited, unrefereed and AI-assisted; independent review, formal verification and absolute priority are not claimed.