Finite Hadamard Tests and Nonpolyhedral Immanant Cones

This preprint studies finite testing for total nonnegativity under entrywise multiplication and the geometry of numerical immanant cones. It presents fixed binary totally nonnegative test families for the Hadamard core in orders four and five, including singular matrices and zero entries, and gives rectangular extensions. It also establishes minimum test counts for three-row matrices. The paper proves closedness of the numerical permutation-product cone in every order and characterizes its polyhedrality, including extensions with a rank bound. A moment face and a continuous family of exposing functionals show why finite fixed tests cannot characterize nonnegativity for arbitrary generalized immanants in order four and above. A distinct-shape dual Jacobi-Trudi example provides a counterexample to a coefficient-positivity conjecture, together with an exact equal-alphabet sign threshold. The files include the manuscript PDF, LaTeX source and a computational supplement containing exact polynomial certificates and verification programs.