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…