Emergent quantization from a dynamic vacuum

We show that adding quadratic temporal dispersion to a dynamic-vacuum acoustic model yields a fully analytic, exactly isospectral mapping to the hydrogenic Coulomb problem. In the regime ω = D q 2 with D = ℏ / ( 2 m eff ) , a proton-imprinted constitutive profile produces an inverse sound speed 1 / c s 2 ( r ) = A ( ω ) + C ( ω ) / r and hence a time-harmonic operator ( ∇ 2 + k eff 2 ) that is Coulombic at each bound eigenfrequency. Separation of variables yields the exact hydrogenic eigenfunctions R n ℓ ( r ) Y ℓ m ( θ , ϕ ) ; the angular labels ( ℓ , m ) emerge naturally from the Laplace-Beltrami spectrum on S 2 via rotational symmetry and boundary conditions (as in standard quantum mechanics), while localization follows from A ( ω n ) < 0 in a reactive stop band consistent with causal, passive dispersion. While angular-momentum quantization follows directly from rotational symmetry and boundary conditions in standard quantum mechanics (consistent with Noether's theorem), here it emerges within a classical-like dispersive acoustic framework without introducing additional wave-mechanical postulates beyond symmetry and self-adjointness. This highlights dispersion's role in bridging a hydrodynamic description to quantumlike spectral structure. Identifying q n ≡ κ n maps spatial scale to frequency, giving ω n = D κ n 2 ∝ 1 / n 2 and reproducing the Rydberg ladder. Calibration to the reduced-mass Rydberg frequency ( ω * = 2 π c R H ) fixes D = ℏ / ( 2 μ ) and m eff = μ , with no free parameters. We determine the frequency dependence of A (