This article presents a restored and expanded scientific version of the C1-C3 verification of a closed second-order vertical pseudohyperboloid cavity. The object is a newly proposed Geometric Wave Engineering resonator geometry obtained by offset rotation of a hyperbolic meridional generator. The verification is intentionally restricted to a reduced axisymmetric scalar Helmholtz eigenproblem. It is not a full-vector Maxwell calculation and it does not establish laser operation, output coupling, radiation loss, or gain selection. The central question is narrower and reproducible: does the geometry support scalar eigenmodes whose weighted energy is concentrated near the annular level rho = R between the external-focal reference planes x = -c and x = +c? The article restores the full scientific narrative that is necessary for a new geometry: geometric construction, focal-ring inheritance, comparison with known resonator classes, mathematical formulation, boundary proxies, numerical mapping, C2 and C3 diagnostics, baseline calculations, high-m sweeps, target spectral scans, type-size validation, benchmark controls, uncertainty diagnostics, and stress tests. Results that are numerically reliable are retained. Results that are suggestive but not yet conclusive are explicitly labelled as secondary diagnostics or future-work hypotheses. The most stable result is C2: the target geometry a = 0.3, b = 0.6, R = 3.0, m = 15 places approximately 92-93% of the weighted scalar energy in the +/-10%R diagnostic annulus across the 70 x 48 to 160 x 112 grid sequence. However, matched non-hyperbolic controls also reach roughly 91-92%, so annular high-m localization alone is not a proof of hyperbolic uniqueness. C3 is supported only as a small set of discrete over-threshold modal peaks after eigenmode deduplication; it is not a continuous spectral window or a physical Q-factor claim. The axial confinement factor CF and horn leakage are scientifically interesting because they probe energy outside the focal-reference planes, but their smallest values lie close to an empirical grid-variability floor. They therefore remain secondary, hypothesis-generating diagnostics rather than final proof of hyperbolic superiority. The defensible conclusion is positive but bounded: the second-order vertical pseudohyperboloid is a new analytically reproducible focal-ring cavity that supports strong scalar high-m annular localization and motivates a broader class of optimized pseudohyperboloid-like resonators. Its specific advantage over all controls is not yet fully established; that requires finer grids, independent solvers, slope/volume-matched controls, open-boundary response, and eventually full-vector Maxwell validation.
Vladimir Khaustov (Sun,) studied this question.