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We ask whether a single description can be simultaneously optimal for all prediction tasks in a linear dynamical system. In a finite-dimensional linear Gaussian dynamical system, the optimal rank-r observer for a squared-error prediction target is uniquely determined by the top-r eigenspace of a target information matrix derived from the stationary covariance and the dynamics. Two targets whose information matrices have distinct top-r eigenspaces admit no jointly optimal observer. Global privilege holds if and only if the target family is r-coherent: all information matrices share a common top-r eigenspace. r-coherence requires strong structural alignment between the dynamics and the task family. Outside it, privilege is task-relative within this class. We illustrate the theorem numerically: a concrete no-privilege instance yields a global-privilege gap of Γ = 0.026 > 0, and structural coherence is confirmed to be perturbation-robust.