Path Dependence and Confluence in Quantum Coarse-Graining

Does a sequence of quantum coarse-grainings depend only on its initial and final resolutions, or can the intermediate route remain operationally observable? We study finite-dimensional closed systems whose coarse descriptions are partitions of a fixed orthogonal fine decomposition. Each partition defines a non-selective L"uders channel, and a monotone coarse-graining path alternates unitary evolution with nested L"uders maps. Although all projectors commute at equal time, we exhibit a four-state example in which two paths with identical endpoints give final coarse probabilities $3/8$ and $5/8$. We show that arbitrary saturated merge paths are connected by two rank-two local moves, so local channel equality is sufficient for global monotone confluence. For each local comparison we derive an all-time necessary-and-sufficient microscopic criterion: the path defect vanishes exactly when Hamiltonian evolution never generates a coherence sector selected by the difference of the intermediate L"uders maps. Equivalently, no local source block can dynamically feed both ends of a selected coherence pair. The first nonzero order is determined by a two-arm Hamiltonian--Krylov distance. Finally, we distinguish hereditary confluence from fixed-source confluence, where a nonzero local defect can remain inaccessible from the chosen global source. These results characterize when the history of changing coarse resolutions may be removed from an endpoint-based operational description.