Geometric Wave Engineering: First Reproducible Cylindrical Maxwell/FDTD Wave Verification of a Second-Order Pseudohyperboloid — Inter-Focal Annular Confinement, Benchmark-Surface Comparison, and Annular-Slot Response

Engineering. The physical object is a three-dimensional surface of revolution, but the computation is an azimuthal-mode cylindrical model rather than a brute-force Cartesian 3D calculation. The central question is whether the PHB is only a visually unusual resonator outline, or whether its hyperbolic, variable-negative-curvature horn geometry produces a measurable wave-control signature when compared with non-hyperbolic test surfaces. The strict result is positive but bounded. The calculation does not prove that a high-m ring is unique to PHB: in any axisymmetric wave problem, the angular-momentum term m^2/r^2 can drive fields toward annular, equatorial, or near-wall regions. Therefore the PHB-specific claim is not simply ring formation. The stronger claim is that, in selected high-m regimes, the PHB organizes annular energy inside the hyperbolically prescribed inter-focal corridor |z| <= c and suppresses extra-focal energy more strongly than the tested non-hyperbolic controls. The completed table audit contains 143 CSV files. The complete PHB advantage table contains 23 meaningful runs: the PHB advantage in log10(CF) is positive in 21 of them, exceeds 0.5 in 19, and has mean and median values of 1.20 and 1.38. The passive low-mode region is not confirmed: m=0 and m=5 do not show a PHB advantage. The useful passive corridor is m=10-15. Harminv returns no robust Q modes, and no gain, plasma, thermonuclear, or Low-J laser operation is established. These items are formulated as future verification branches, not as results of the present matrix. The geometric characteristics used in this first wave verification - a = 0.3, b = 0.6, and R = 3.0 - are treated as a first reference PHB geometry, not as a universal optimum. They were selected on the basis of prior geometric and reduced ray-wave billiard verification records, because the parameters a, b, and R determine the horn length, curvature distribution, focal-ring separation, diagnostic annular level, and ray-family organization. Since geometric parameters influence ray propagation, they are expected to influence the Maxwell/FDTD wave response as well. Full wave verification over alternative a, b, and R families is therefore identified as a separate next-stage calculation rather than as a completed claim in the present report.