Uhlmann Holonomy and Leakage Structure in the Hybrid Quantum–Classical Closure

This paper develops a mixed-state geometric framework for hybrid quantum–classical dynamics by constructing a Hybrid Uhlmann connection on the Global Hybrid Operator Density of the GENERIC-consistent Gay-Balmaz–Tronci closure. Building on the Stiefel-bundle parallel transport formulation of Avitabile and Bridges together with an exactly unitary Cayley integrator, it proves the Global Hybrid Winding Theorem, establishing that the Uhlmann holonomy spectrum is invariant under memoryless evolution and changes if and only if the recovery cycle carries non-Markovian memory. The paper further characterizes the rank-deficient leakage boundary, proving that the determinant (abelian) channel is exactly blind to leakage while the full non-abelian holonomy develops an intrinsic directional singularity as occupations vanish. Numerical simulations illustrate these geometric signatures, and the paper proposes a general experimental readout protocol for future implementation across quantum computing architectures.