Focal–Transport Ray Law in Pseudohyperboloidal Cavities with Variable Negative Gaussian Curvature: Exact Focal Contraction, Event-Order Transport, Skew-Ray Unfolding, and Finite-Wavelength Computational Tests

We formulate a focal–transport ray law for a family of open cavities generated by revolving a radially shifted hyperbola — referred to here as pseudohyperboloidal cavities (this is a constructed surface-of-revolution family, not a standard quadric hyperboloid). The geometry possesses variable negative Gaussian curvature on the smooth horn, while the transition at |x|=a acts as a sharp event-boundary edge where the reflecting wall terminates. The law has an exact core — a focal-contraction recurrence on the invariant meridional focal family — together with distinct computational layers for event-order transport, finite-skew unfolding, and finite-wavelength tests. In the invariant meridional plane (J_x=0), every regular ray belonging to the inherited focal-conjugate family obeys the exact recurrence t_{n+1}=κt_n, where t=tanh(u/2) and κ=(√(1+β²)−1)/(√(1+β²)+1). This contraction is geometric: the entire inter-horn flight segment converges uniformly toward the internal focal-line segment between the two transition rings, with a contraction factor κ that depends only on the shape ratio β=b/a and is independent of the radial shift R. The axial endpoint excess decays asymptotically with the squared factor κ², outpacing the radial approach. The limiting trajectory approaches the transition rings (|x|=a, ρ=R), rather than the rotation axis or the external foci; the parent conic's focal property organizes the reflected directions, while the axial angular momentum J_x is conserved, so that the meridional sector defines the exact property and finite-J_x rays unfold it. Beyond this exact skeleton, finite-size effects introduce distinct layers: - Event-order transport: Adaptive resolution identifies multiple PASS/RETURN boundaries and isolated, section-specific cases of focal–transport inversion. - Finite skew: A targeted three-dimensional audit reveals an even, near-quadratic scaling of the first virtual focal miss (∝ j²) and confirms that small axial angular momentum perturbs the ideal skeleton rather than destroying it. - Finite-wavelength tests: Axisymmetric Maxwell/FDTD simulations show that focally matched excitation produces a reproducible early-time field redistribution near the internal locus at a/λ≈2. A reproducible transient redistribution is indeed observed after source switch-off (t=5): the focally matched excitation yields a larger and narrower response than the matched controls at equal source-current amplitude. This remains a differential, grid-stable redistribution with G_target<1: no absolute energy enrichment or resonant storage is established. These finite-wavelength results are numerical predictions that call for experimental validation. The work establishes a closed-form global recurrence for a defined cavity class and provides a falsifiable hierarchy separating exact focal geometry from computational transport topology and diffraction-scale observations.