Phase Closure, Spinorial Return, and Invariant-Subspace Structure in Periodically Driven Systems

We formulate an exact operator-theoretic criterion for coherence under periodic evolution. A persistent ambiguity in the analysis of driven systems is the conflation of continued evolution with coherence. We prove that coherence is not the infinite persistence of evolution, but exact topological return under constrained periodic dynamics. 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. We then show that any coherent constrained dynamical system admits a minimal decomposition into an admissible sector (S), an evolution law (U), and a closure predicate (C). Spinorial identity is characterized exactly by the Floquet eigenvalue -1, yielding strict double-cycle return. Perturbative persistence is established through invariant-subspace protection, and a Birman-Schwinger criterion is derived for closure breakdown. A penalized operator construction (the Omega-layer) is introduced to suppress inadmissible leakage by enlarging the effective spectral gap. Finally, we construct a finite block-compositional symbolic framework whose interpretation map is operator-faithful, utilizing a geometric hexagram state space mapped to orthogonal operator constraints. Within this framework, admissible regime transitions admit a total Lyapunov functional, exhibit strict monotone descent, and are mathematically proven to enter a closure-compatible coherent attractor class in finite time, concluding with a canonical terminal refinement. The symbolic formalism renders this operator structure explicit, proving that governed admissible dynamics refine strictly toward invariant, phase-consistent return. Version 2 Update Notes: This manuscript has been significantly expanded to introduce a block-compositional geometric symbolic language that faithfully encodes the minimal decomposition triple. It defines a total Lyapunov functional on the hexagram state space, mathematically proving that Omega-Sigma admissible transitions exhibit strict monotone descent and force the dynamical system out of critical collapse basins and into the coherent exact-return attractor class in finite time.