Sharp Coefficient Bounds and Constant-Term Fibers in the Order-Five Nonnegative Inverse Eigenvalue Problem

The order-five nonnegative inverse eigenvalue problem asks which real monic quintics occur as characteristic polynomials of nonnegative matrices. This two-paper project gives a constructive characterization through directed-cycle optimization. The main paper proves that every realizable quintic has a nonnegative lower-Hessenberg realization with unit first superdiagonal. For each feasible choice of the first four coefficients, the admissible constant terms form a complete downward ray with a finite attained ceiling. This ceiling equals the maximum of a compact independent-cycle relaxation, whose feasibility is itself equivalent to prefix realizability. The resulting criterion includes repeated eigenvalues, nonreal conjugate pairs, and exceptional prefixes. The proof combines cycle expansions, support reductions, and constrained curvature. On the final boundary configuration, second-order conditions yield an explicit two-loop completion, supported by exact rational Bernstein certificates. Changing one corner entry then supplies every smaller constant. The companion develops explicit realization families, moment boundaries, complementary witnesses, spectral flags, collision models, and isospectral perturbations. Sharp regional ceilings and a scalar envelope with at most four possible least-support boundary maxima further illuminate the coefficient geometry. The characterization retains auxiliary variables. Short coefficient-only criteria, efficient general ceiling computation, and universal realization with a prescribed diagonal remain open. The construction may change diagonal and support; no Jordan form is prescribed. The project includes both papers, editable sources, complete certificates, reproducible verification programs, independent internal checks, and separate corrections and scope records.