Geometric Sampling Does Not Determine Regularized R-Diagonal Laws

For every \(s_0\gt 0\), \(q\gt 1\) and finite integer \(K\geq 1\), we construct a continuum of distinct affiliated \(R\)-diagonal laws with identical joint star-moments and identical first \(K\) alternating cumulants of \(c(1+s c^*c)^{-1/2}\) at every scale \(s=s_0q^k\), \(k\in\mathbb{Z}\). 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 grid and an exact higher separating cumulant. A Blaschke-Jensen argument gives a sufficient uniqueness condition for other sample sets. This is a complete scoped reconstruction counterexample, not a new radial analytic \(R\)-transform or closure of the entire AIM question. Classical inputs are credited. The preprint is self-audited, unrefereed and AI-assisted; independent review, formal verification and absolute priority are not claimed.