GENERIC Consistency Of A Global Hybrid Quantum–Classical Operator Density

This paper develops the first thermodynamically consistent GENERIC formulation of the Gay-Balmaz–Tronci hybrid quantum–classical closure by constructing a global hybrid operator density that unifies reversible geometric dynamics with irreversible GKSL and Fokker–Planck dissipation. It establishes rigorous disintegration and completeness theorems for the admissible state space, extends the Berry-curvature-corrected hybrid entropy to mixed conditional quantum states, and proves its Abelian gauge invariance on the non-degenerate admissible manifold. Under an explicitly stated well-posedness hypothesis, the fused hybrid dynamics is shown to satisfy all four GENERIC consistency conditions, while recovering both classical GENERIC and the standard GKSL/Spohn theorem as limiting cases. These results provide a rigorous mathematical framework for nonequilibrium hybrid quantum–classical systems and lay the thermodynamic foundation for future studies of correlated and non-Markovian noise in open quantum devices.