Law of Ideal Geometrical Ray Dynamics in Second-Order Pseudoparaboloids. Exact Focal-Return Structure, Periodic Ray Families, and Topological Splitting of the Horizontal and Vertical Types

Geometric Wave Engineering (GWE) considers boundary geometry not merely as a container for propagation, but as a passive geometric operator capable of organizing ray trajectories before any active signal processing is applied. Within this programme, the present work introduces the ray dynamics of the second-order pseudoparaboloid (PPB-2), a new multi-branch surface constructed from opposed, radially shifted parabolic meridians and their surfaces of revolution. The familiar focus-to-parallel property of an individual parabola remains local and classical; the new problem begins when several displaced parent branches act together and repeated propagation is determined by the competition between physically accessible reflections. The central result is the focal-return ray law for the regular meridional dynamics of PPB-2. It gives an exact branch-resolved description of successive specular collisions, including the physical requirement that the next event be the earliest admissible intersection. When a ray is incident through a parent focus, the parabolic reflection property places the outgoing ray on a distinguished axial post-focus state. From that state, repeated branch-to-branch propagation is governed by an exact algebraic map, while return to a parent focus is defined by an additional geometric condition. This structure generates recurrent and periodic ray families and explains how simple focal reflection is transformed into an organized multi-reflection dynamics. The law applies to a distinguished focal-return structure and does not reduce the full meridional billiard to a one-dimensional system. A further part of the law is the distinction between two three-dimensional realizations of the same signed meridional generatrix. The horizontal and vertical pseudoparaboloids possess the same regular meridional branch dynamics after coordinate relabelling, yet they are different spatial surfaces: their azimuthal metrics, curvature fields, aperture topology and nonmeridional ray propagation are not equivalent. Thus the law separates what is universal in the meridional focal-return mechanism from what is created by the three-dimensional embedding of the surface. This preprint is intended as the first systematic presentation within the Geometric Wave Engineering programme of the ray-propagation law associated with the second-order pseudoparaboloid. It establishes the geometry, admissible reflecting domain, exact collision and focal-return conditions, periodic-ray structure, stability framework, ray-density consequences and the horizontal/vertical three-dimensional split in a single mathematical description. The results concern ideal geometrical-ray dynamics. Finite-wavelength diffraction, modal structure, losses and experimental performance require separate wave and experimental verification and are not part of the present law.