Passive Geometric Direction-to-State Encoding in Second-Order Pseudohyperboloidal Cavities: Cross-Physics Computational Evidence and Scientific Priority Disclosure within Geometric Wave Engineering

Abstract

A passive geometric direction-to-state transduction architecture is formulated and evaluated computationally for second-order pseudohyperboloidal cavities (PHB-2) using independent ray, acoustic-FDTD, and Maxwell-FDTD models. The central proposition is not universal passive focusing. A radially displaced hyperbolic meridian, revolved about the system axis and completed by a finite operational boundary, acts as a passive analog wave operator that converts incidence direction into a structured internal spatial, spectral, amplitude, phase, and symmetry-resolved state before digital estimation. The two point foci of the parent hyperbola generate two external circular focal sets after revolution. These rings are inherited geometric focal loci rather than material sensors or sinks; their influence is mediated by the complete boundary through its tangent and normal fields, curvature, path families, modal coupling, leakage, and interference. Independent computational branches establish different parts of the proposition. Exact rays verify the hyperbolic reflection law and a focal separatrix, while generic plane-wave occupation is sub-uniform: G=0.792 in 2D and G=0.333 in full 3D. A specifically phase- and focus-matched periodic family reaches G=13.763 in the exploratory post-burn-in diagnostic window spanning reflections 5–10, but its PHB-specific advantage is already lost at the smallest tested angular mismatch of 0.0005° (PHB/C2=0.977) and under minute amplitude/phase perturbations. The exact endpoints 5 and 10 were not established by the archived record as preregistered device-performance bounds. A distributed 3D ray map contains direction information, but burn-in analysis shows that early transport contributes substantially to that code. Acoustic FDTD establishes a reproducible broadband geometry-dependent state. A signed broadband coordinate has R²=0.988 and 0.993 on two grids. After complex empty-reference subtraction, PHB and an equal-area smooth control are separated by 54.659° and 56.083° in normalized complex feature space while their broadband levels remain nearly equal. A fine-grid full-field comparison gives global PHB/control Σ|H|² ratios of 0.9996 at 0° and about 0.9997 at ±15°, yet median local spectral angles remain 26.54° and 30.81°, demonstrating distributed state redistribution rather than energy gain. Maxwell/FDTD supplies direct local inversion. In a small-angle TE branch, PHB maintains an enriched common state G_E=1.144294–1.147149 while the signed odd field is nearly rank one: 99.997% of odd energy lies in the first principal component. Across the complete 11-state small-angle dataset (7 calibration angles and 4 held-out angles), the full odd coordinate follows sinθ with R²=0.999; the calibration-only linear fit uses n=7 angles and has R²=0.999. Four held-out angles are recovered with MAE=0.059° and a maximum absolute error of 0.083° under ideal full-field access. Separately, an inward operational boundary derived from the PHB parent sustains a mode-tracked enriched branch with G=1.224, 1.154, and 1.197 at Resolutions 60, 80, and 120. In a fixed-frequency exact-PHB series, PHB G decreases from 0.900 to 0.340 with refinement; C2 is non-monotonic, so the PHB/C2 values 2.613 and 2.777 on the two finest tested grids are descriptive contrasts, not a converged continuum-limit ratio. The integrated result is a scientific-priority disclosure for a PHB-2-specific pre-digital wave-processing architecture: geometry creates an informative state manifold and a finite calibrated receiver reads it. The evidence supports passive analog preprocessing and geometry-dependent direction encoding; it does not establish a finished 3D direction finder, universal focusing, range, experimental performance, or practical superiority over matched controls.