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

This paper extends Uhlmann's parallel-transport geometry for mixed quantum states beyond the conventional full-rank setting to the Global Hybrid Operator Density of a GENERIC-consistent Gay–Balmaz–Tronci hybrid quantum–classical closure, building on the well-posedness and thermodynamic-consistency results established in two companion papers. The central object is a Hybrid Uhlmann connection whose holonomy around a closed recovery cycle yields three main results: the Global Hybrid Winding Theorem, showing that the holonomy spectrum changes if and only if a classical-sector memory kernel is genuinely non-Markovian (sharpened to an exact, non-perturbative statement for a fixed physical direction); an Abelian Blindness theorem, proving the holonomy's determinant channel is exactly and unconditionally insensitive to occupation loss (leakage), even as the underlying state approaches or crosses the rank-deficient boundary; and a Directional Singularity theorem, showing that the full non-Abelian spectrum instead develops a genuine, non-removable singularity there, with the limiting value depending on the direction of approach to the boundary. Together these results show that a single geometric quantity — the holonomy spectrum — can separate non-Markovian memory from leakage, two error mechanisms that existing diagnostics require distinct, purpose-built tools to detect. The theorems are verified on a structure-preserving numerical integration scheme and accompanied by a concrete, falsifiable experimental protocol for superconducting qutrit hardware, extending a recently demonstrated scalar Uhlmann-phase measurement to full non-Abelian holonomy reconstruction.