Geometric Wave Engineering for Passive Multisource Electromagnetic Direction Finding in a Second-Order Pseudohyperboloidal Cavity

Geometric Wave Engineering (GWE) is introduced here as a geometry-first research framework for wave systems: an analytically designed boundary is used not only to confine, guide, or scatter a field, but to perform a reproducible physical transformation of its spatial amplitude and phase before digital processing. GWE does not require a new electromagnetic field equation; it uses ordinary Maxwell propagation under deliberately engineered boundary geometry. As a concrete example, this work develops a passive multisource electromagnetic direction-finding receiver based on a second-order pseudohyperboloid (PHB-2), obtained by revolving a radially translated hyperbola. Its smooth horns possess variable negative Gaussian curvature and two external ring foci. For a specified inherited meridional ray family the horn geometry obeys an exact focal-transport contraction law, which identifies an internal cylindrical transport neighborhood as a non-arbitrary region for coherent field sampling. The receiver is therefore not designed around one brightest focal point. Different arrival directions produce different reflection sequences and phase accumulations, and hence different complex spatial field fingerprints. Twenty-four interior finite-volume stations sample the vector electric field in the near-R transport band. The exact PHB is compared with equal-volume focal-law-broken controls, including a smooth PHB-like deformation and a nonconic surface that retains negative Gaussian curvature without the PHB focal law. Across the tested electromagnetic configurations, exact PHB exhibits a larger incident-referenced internal response and a more favorable local geometry of calibrated complex directional states. On the frozen common readout it also gives the smallest complex-map residual for all three held-out directions. The favorable calibration-state ordering survives analysis of the full stored field atlas and an equal-budget test in which every geometry is allowed to choose its own calibration-only sensor locations. The executed receiver is a one-port open PEC cavity: radiation enters through a circular axial inlet, and a frozen 24-zone near-R measurement operator records the three complex electric-field components at each zone, yielding 72 coherent complex channels. Direct two-source and three-source Maxwell/FDTD scenes then test the physical multisource model without relying only on synthetic post-processing. The simultaneously calculated fields closely match coherent superpositions of independently calibrated single-source fingerprints, including a scene containing a third source substantially weaker than the primary. A separate reproducible inverse diagnostic under one common absolute-noise convention also gives the highest exact support-recovery rate for PHB in the tested scene. Taken together, the results provide computational evidence for a focal-law-guided passive analogue spatial encoder for multisource electromagnetic direction finding. The claim is deliberately limited to the tested models: general blind off-grid recovery, broadband and polarization generalization, higher-resolution replication of the simultaneous-source cases, calibrated hardware probes, and experiment remain open.