Positivity and log-concavity of Riesz kernels of hyperbolic polynomials, and cone-geometric regularization

This paper studies positivity and log-concavity of Riesz kernels of hyperbolic polynomials and inverse Laplace transforms on proper convex cones. For a complete homogeneous hyperbolic polynomial in n ≥ 2 variables and α ≥ 4096n², it proves positivity and strict log-concavity of the Riesz kernel in the interior of the dual cone, with relative Gaussian comparison error at most 512n²/α and explicit Hessian bounds. The resulting complete monotonicity of sufficiently large negative powers answers a question of Scott and Sokal. Further results concern cone-geometric regularization of holomorphic functions, quantitative strict log-concavity under perturbations with bounded imaginary part, a gap at zero in the set of completely monotone negative powers, and an operator-valued extension. Examples include the specialized Vámos polynomial, determinantal polynomials, and the exponential cone. This deposit contains the paper in PDF format and riesz-kernels-checks.zip. The ZIP includes the single-file LaTeX source, Python verification scripts, required formula manifests, and instructions. The scripts check explicit scalar and algebraic calculations and their correspondence with the manuscript. General analytic assertions are linked to their written proofs; they are not formally machine-verified.