This monograph develops the spectral theory of the cascade, focusing on how rank geometry interacts with zeta functions, functional equations, and zero structure. The main constructions are spectral rank, Vieta S‑duality, and dual Kneser branches, which organise zeta‑type objects into continuous families parametrised by complex rank. Classical zeta, Dirichlet eta, and polylogarithms appear as special points in the Kneser Euler product family. M17b introduces the LP curve in cascade‑native coordinates and formulates the zero migration ODE, describing how zeros move under rank deformation. An equivariant degeneracy framework shows that dual‑branch births are forced, but does not by itself exclude four‑orbit configurations. The central outcome is a precise reduction: the remaining open condition is the Drift Direction Conjecture, governing the direction of pitchfork splitting at zero births. With this condition, the critical‑line conclusion follows; without it, the result is a disciplined reduction rather than a proof.
Paweł Łukasz Garycki (Fri,) studied this question.