Phase Closure, Spinorial Return, and Invariant-Subspace Protection Under Periodic Evolution

We formulate an exact operator-theoretic criterion for coherence under periodic evolution. A fundamental pathology in the analysis of far-from-equilibrium dynamical systems is the conflation of persistence (the continuous existence of a unitary evolution operator) with coherence (exact topological return under that evolution). In this manuscript, we mathematically sever the two. Let UF denote the monodromy operator associated with a bounded, self-adjoint, time-periodic Hamiltonian on a complex Hilbert space. We prove that phase closure, ray invariance, and rank-one projector invariance are mathematically equivalent. Spinorial identity is strictly characterized by the eigenvalue -1, yielding an exact double-cycle return topology governed by a half-integer fractional winding ratio. Furthermore, we establish perturbative persistence via invariant-subspace protection. We derive a Birman-Schwinger criterion for the breakdown of phase closure under environmental coupling, and apply Kato-Rellich theory to guarantee the stability of isolated quasienergy branches. Finally, a penalized operator construction (the Omega-layer) is introduced. We prove that the application of strict, topologically orthogonal constraint fields successfully suppresses inadmissible macroscopic leakage and preserves structural stability without extinguishing the system's active transport layer. We conclude that dynamic coherence is not the infinite persistence of evolution, but exact topological return under cyclic dynamics. The system is not coherent because it persists; it is coherent because it returns.