Well-Posedness of the Gay-Balmaz–Tronci Hybrid Quantum–Classical Closure

This paper establishes the first rigorous well-posedness theory for the exact-factorization hybrid quantum–classical closure introduced by Gay-Balmaz and Tronci. By constructing an admissible operator-valued state space together with a vacuum control condition that resolves the structural singularity at vanishing classical density, the work proves local existence, uniqueness, and a Beale–Kato–Majda-type continuation criterion, together with sufficient conditions for global-in-time existence under physically relevant hybrid Hamiltonians. Numerical verification across fifteen representative quantum-computing platforms confirms preservation of the hybrid closure, operator positivity, and bounded continuation behavior. These analytical results provide the mathematical foundation for subsequent thermodynamic, geometric, and topological extensions of hybrid quantum–classical dynamics, including the irreversible GENERIC framework developed in the companion paper