Positive Quadrature Paths in Skew-Normal Mixtures and Failure of a Fourth-Order Inverse Bound

Abstract

This paper constructs an explicit three-component skew-normal mixture that approaches a single non-Gaussian skew-normal distribution while its parameter distance is of first order and its total variation distance is of sixth order. The construction disproves the fourth-order Wasserstein inverse bound conjectured by Ho and Nguyen for two excess components. All mixture weights remain fixed and strictly positive, and the counterexample lies in a compact interior parameter region satisfying the stated condition on the true mixing distribution. The method combines the classical Gaussian convolution representation of a skew-normal distribution with positive Gauss–Hermite quadrature. It preserves the half-normal loading, reduces the independent Gaussian variance, and replaces the missing Gaussian factor by a finite positive distribution. For every component count of at least two, this produces a path whose exact contact order is twice that count. A two-term density expansion is proved with an explicit bound on the integrated absolute remainder, independent of the convolved probability measure and without moment assumptions on that measure. Within the specified fixed-node variance-compensation family, the maximal contact order is attained precisely by Gaussian quadrature. The estimates are uniform on compact interior parameter sets and extend to splitting one component of a separated finite mixture. This characterization is restricted to the stated family; it is not a sharp inverse bound for arbitrary skew-normal mixture paths. A separate appendix gives explicit real, positive-weight solutions to nine polynomial equations for which a later technical report asserts nonexistence at specified parameter values. This algebraic result is distinct from the density construction and does not provide a complete classification of skew-normal singularity indices. The accompanying source archive contains the LaTeX manuscript and two exact verification implementations with expected outputs and a reproducibility entry point. Research methodology and AI assistance:This work was developed using the CARMA-Math research workflow, a cumulative AI-assisted mathematical research methodology using persistent research archives, literature and prior-art investigation, iterative proof exploration, and verification procedures. Generative AI (ChatGPT) was used extensively for mathematical exploration, proof development, computational reasoning, literature research, and manuscript preparation.