Reduced Ray-Wave Verification of an Open Pseudohyperboloid Telescopic Resonator with an Annular Output Slot

This computational preprint presents a reduced ray-wave verification study of an open second-order vertical pseudohyperboloid resonator with one annular output slot. The purpose is to test whether the hyperbolic generating law produces a reproducible focus-guided output mechanism and whether this mechanism can be interpreted as a candidate laser-resonator effect. The study combines a closed scalar C2-C3 eigenmode benchmark, open R2-slot coupling proxies, focus-guided ray tracing, phase-selected ray-family analysis, Huygens/Fresnel far-field reconstruction, a reduced Fox-Li/Fresnel comparison, and matched-aperture diffraction tests. The positive result is geometric and dynamical. In the main ray-telescope configuration, with a = 0.3, b = 0.3, R = 3.0, R1 = 0 and R2 = 0.08, the high-statistics focus-guided ray calculation gives 94.10% slot exit and 65.35% multi-bounce slot exit. Within the multi-bounce subset, the angular metric is θ80 = 86.02 mrad, and 93.80% of the selected rays remain below 100 mrad. The corresponding same-length linear-horn analogue is substantially weaker in the tested multi-bounce output metrics. This supports the interpretation that the external focal structure of the hyperbolic generatrix is not merely decorative, but contributes a real focus-guided selection mechanism. The negative result is equally important. The same verification sequence does not establish that the present annular-slot PHB geometry is a superior CO₂ narrow-beam telescopic laser resonator. The Huygens/Fresnel reconstruction of the phase-selected PHB family remains broad, and the reduced Fox-Li/Fresnel comparison favors classical filled-aperture proxies in the ordinary narrow-beam metric. Even an idealized flat-phase annular slot is broader than a comparable filled aperture in the matched diffraction comparison. Therefore, the scientifically supported conclusion is conditional: the open PHB is a strong focus-guided annular ray selector and a candidate geometry for annular, conical, ring, or OAM-like output studies, but it is not yet validated as a superior replacement for a classical CO₂ telescopic resonator in the nearly diffraction-limited Gaussian-like beam regime. The article is intended as a reproducible reduced-model decision record. It does not claim experimental demonstration, full-vector Maxwell validation, gain-saturated laser operation, or industrial readiness. The next required validation level is a high-fidelity open-boundary vector wave model with wall impedance, output coupling, gain, saturation, and a self-consistent round-trip mode.