Observable Dynamics as M = G U

Observable dynamics arise from the composition of an underlying evolution with an observation operator. If U denotes the intrinsic evolution of a system and G denotes an observation or coarse-graining map then the effective observable update is given by M = G U. This short note records the factorization and emphasizes that the spectral properties of M determine which structures persist under repeated observation. Eigenvalues near the unit circle correspond to long-lived observable modes.