Persistence of Algebraic Orthogonality in Dissipative Stochastic Linear Oscillators with Impulses

We study a linear oscillator on R2 driven by (i) Hamiltonian rotation, (ii) isotropic linear dissipation, (iii) additive Gaussian noise, and (iv) sparse impulses modeled as compound Poisson jumps. We prove (A) a uniform-in-time second-moment bound via a single Foster–Lyapunov drift inequality with explicit constants; (B) an exact algebraic orthogonality identity showing that the skew (Hamiltonian) drift contributes zero to Euclidean-energy drift while dissipation contracts; and (C) existence of at least one non-Dirac invariant probability measure for the resulting Ornstein–Uhlenbeck–Lévy process, with standard sufficient conditions for uniqueness. An appendix provides an audit-ready, computable verification interface: given declared param- eters, one computes explicit drift constants and a certified bound on supt≥0 E ∥z(t)∥2 .