Open macroscopic cavities typically exhibit transient chaos and chaotic escape [1,2], limiting local intensity and precipitating thermal breakdown in high-power optics and plasma confinement architectures. Here we investigate a geometry-driven route to suppress this escape in a class of non-axially generated pseudo-hyperbolic resonators [3]. By rotating a canonical hyperbola around an offset axis, we obtain an open three-dimensional cavity with a spatially structured radius function and a pair of ring-shaped focal zones above the equatorial gap. Throughout the manuscript, all lengths are expressed in dimensionless units normalized to a reference scale ξ; physical dimensionalization is recovered by fixing the product k₀ξ at the operating wavelength. For the optimal topology identified in our parameter scan (R = 20.0, a = 0.05, b = 0.50), non-sequential stochastic ray dynamics yield a global energy retention of 88.9% and a local energy concentration of 15.22 ± 0.25% in the gap region, where the reported uncertainty is dominated by systematic effects rather than Monte Carlo statistics. To interpret this localization beyond the geometric-optics limit, we derive an effective one-dimensional Helmholtz formalism [4,5] in the adiabatic domains of the cavity, under Dirichlet boundary conditions corresponding to TM-polarized modes in a perfectly conducting cavity. The leading-order geometry-induced potential scales as V_eff ∝ 1/r(x)² [4], providing a steeply rising barrier in the horn regions and a low-potential equatorial trapping zone. The reduced wave model predicts a one-dimensional confinement fraction of ~14.5%, of the same order as the stochastic ray result; the two measures probe different observables and their numerical proximity is treated here as qualitative consistency rather than as a quantitative match. Within the limits of the reduced wave model and the macroscopic-ray approximation, these findings identify a geometry-controlled localization mechanism in an open empty cavity and motivate further investigation by full-wave electromagnetic simulation and experiment.