M21h The Complex HyperCore: True Complex Ranks of the Elevator ONS: Koenigs-Abel Generators, Core Collapse, Ghost Units, and the HC Rank Manifold

This monograph introduces true complex ranks in HyperCore, extending the rank parameter to z = R + iT. Unlike sideways morphing, which only rotates a fixed Abel coordinate, complex rank evolves the Abel coordinate itself through a Koenigs-Abel generator. This creates a genuinely new direction in the HC hierarchy. The main result is the first-order formula at the canonical anchor R = 2: HC_ (2+iT) (a, b) ~ ab * Apow (a, b) ^ (2iT), where Apow (a, b) = a^ (ln b). Thus imaginary rank does not merely deform multiplication; it reveals the Symmetric Core operation Apow inside the HyperCore complex plane. In NC coordinates this gives the bilinear structure NC = (ln a + ln b) + 2iT (ln a) (ln b), which cannot be produced by ordinary fiber morphing. The central thesis is that Complex Ranks in HyperCore perceive Etages. At R = 2 + iT, imaginary rank penetrates the SC Etage; conjecturally, at R = 3 + iT it reaches the TC Etage. This motivates the HC Rank Manifold conjecture, where the real Elevator and all Etages form one complex-analytic structure: the Elevator as the real axis, and Etages as imaginary fibers over integer ranks. The monograph also clarifies the lower HC taxonomy, shows why AddCore and MultCore collapse under linear tamers, derives complex-rank Hermit units on the unit circle, and explores ghost units, Zeration flatness, and Jacobi elliptic proto-trigonometry at R = 1. 5.