Law of Ideal Geometric Ray Dynamics in Second-Order Pseudoellipsoids: Exact Focal Structure and Topological Transitions, with Numerically Identified Periodic Ray Classes and Path-Density Concentration Zones

Geometric Wave Engineering (GWE) is a research program in which deliberately constructed geometry is treated as a passive means of organising ray and, at later stages, wave propagation. This preprint gives the first dedicated and unified ray-geometric law, within that program, for the second-order pseudoellipsoid (PEB-2), building on the constructive pseudoellipsoid geometry introduced earlier within GWE [9]. A PEB-2 is not a classical ellipsoid. Its meridional boundary is assembled from two selected quarter-elliptic parent branches and is then revolved about a common body axis. Each smooth parent branch locally retains the familiar focal-reflection property of an ellipse, but the composite surface creates a different mathematical problem: the physically reached parent must be selected, the two parent focal pairs must be organised as one geometry, and the planar point foci become coaxial focal rings after revolution. The law formulated here states that, at every smooth physically active parent point, a meridional ray whose incident line belongs to one focal line of that parent is mapped by specular reflection into the conjugate focal line of the same parent. In a composite pseudoellipsoid this local conjugation becomes a physical event only after an earliest-hit rule selects the first admissible boundary intersection. The shape ratio K=b/a organises the focal topology of the surface. For 0 1 the regular vertical type has radially oriented parent focal pairs. Within the horizontal family, the exact condition h=-f produces a horizontal three-focus type in which the two inner parent focal rings coincide while the two outer rings remain distinct. Revolution of the meridional construction converts the parent focal points into circular focal loci and carries the exact focal-conjugation law into every invariant meridional plane of the three-dimensional body. Axisymmetry additionally supplies an exact axial-angular-momentum invariant for general three-dimensional specular rays, while the manuscript deliberately does not extend meridional focal-ring conjugation to arbitrary skew trajectories without proof. Nonsmooth seams are likewise excluded from the smooth-point specular theorem unless an explicit local boundary law is specified. The resulting formulation separates the exact geometric law from numerical verification, long-time billiard dynamics, finite-wavelength behaviour, resonator performance, and experiment. Its purpose is to establish the pseudoellipsoid as a mathematically defined ray-control surface and as a base geometry for subsequent studies in Geometric Wave Engineering.