A Characterization of Minimal Persistent Deviation-Feedback Dynamics

This paper establishes a characterization theorem for finite-dimensional linear translation-invariant deviation-feedback systems admitting persistent bounded solutions with a conserved positive definite quadratic first integral. We prove that scalar systems cannot simultaneously satisfy persistence and conservation, implying that minimal persistent dynamics require dimension at least two. Under the conservation assumption, the system generator must possess purely imaginary spectrum, and the minimal invariant subspace is shown to be two-dimensional with rotational dynamics. After normalizing the conserved quadratic form, the generator becomes skew-adjoint, yielding a natural Hamiltonian formulation without assuming symplectic geometry or variational principles a priori. These results identify rotational structure as the unique minimal mechanism supporting persistent bounded behavior within the stated axiom class.