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Abstract

We give an explicit admissible Fourier kernel for which every continuous root-coordinate Laguerre iterate is positive and uniformly strictly log-concave, although its Fourier transform has nonreal zeros. Every root iterate also defines a strictly totally positive translation kernel of all orders, with explicit signed confluent determinants. The fixed kernel is φ(s)=exp(-e^(s^2)-999s^2). Its Fourier transform is entire of exact order one. Two whole-line rational Gaussian-moment enclosures certify the negative sixth cumulant, the sharper bound being κ_6<-1.8 × 10^(-18). The real-zero canonical product would require that cumulant to be nonnegative. Exact quadratic-tilt invariance explains why the entire root hierarchy misses this obstruction. We also give a general Gaussian-localization formula for higher cumulants, including an effective finite-jet error bound. Newman's Gaussian-damping classification already implies existence of a bad tilt for the underlying double-exponential family; the present conclusion is an explicit certified all-depth counterexample. It concerns general admissible kernels.