Determinacy Witnesses in the Completely Monotone--Stieltjes--Bernstein Hierarchy

Abstract

Membership problems in the completely monotone--Stieltjes--Bernstein hierarchy often become explicit after passage to a determinate representation. We organize that passage as a witness calculus. For the Laplace, Stieltjes, and Hausdorff transforms used here, W f denotes the unique admissible signed representing datum; a certificate either proves its positivity or exhibits an obstruction. A concrete dictionary for completely monotone, logarithmically completely monotone, Bernstein, complete Bernstein, and Stieltjes functions and for Hausdorff moment sequences is combined with transport rules for products, shifts, sampling, and Bernstein increments. The applications are consequences of this common calculation. A triple-product identity for a digamma bridge controls ordinary Bernstein and logarithmic complete monotonicity ranges for gamma ratios and gives the strict power cutoff in Szabo's problem. For integer gamma quotients, the inverse Laplace density is a damped Jacobi polynomial, so classical zero geometry decides broad parameter regions. Boundary signs distinguish Bernstein from complete Bernstein behavior and separate completely monotone functions from the Stieltjes class. On the discrete side, signed atoms and finite differences settle interpolation and coefficient questions. The Ramanujan integral and a fractional Volterra symbol provide positive recognition examples. Across the applications, the workflow is the same: normalize the class question, compute or transport its witness, and decide one explicit sign problem. Every exclusion is accompanied by an exact certificate.