Complex Zero Decrease in Laguerre and Jacobi Bases: Spectral Root Chains and Exact Classifications

This preprint classifies diagonal operators in generalized Laguerre and Jacobi polynomial bases that do not increase the number of nonreal zeros, counted with multiplicity. It covers all real diagonal sequences for orthogonal generalized Laguerre bases and polynomially interpolated sequences for nonpositive Laguerre parameters. For orthogonal Jacobi bases it classifies polynomially interpolated sequences, including repeated spectral factors and polynomial kernels. The analysis proves rigidity of the quadratic Jacobi spectral parameter, identifies the initial root chains and sharp residual intervals, and determines the least universal Jacobi compensator. It also gives the Jacobi-to-Laguerre contraction and the strict separation between the affine zero-count and hyperbolicity-preserving ranges. The three files comprise the paper, the matching standalone LaTeX source, and a computational supplement. Three Python programs reproduce the exact rational identities, four zero-count certificates, and 432 Taylor-coefficient checks of the exterior identity.