Task-Relative Descriptive Privilege in Linear Gaussian Dynamical Systems

In a finite-dimensional linear Gaussian dynamical system, the rank-r linear observer minimizing squared-error prediction loss is uniquely determined—up to rotation within the eigenspace—by the top-r eigenspace of the target information matrix. Two prediction targets with distinct dominant eigenspaces admit no jointly optimal observer. Global privilege—a single observer simultaneously optimal for all tasks—holds exactly when the target family is r-coherent: every information matrix shares a common top-r eigenspace. This is an exact algebraic condition. Numerical illustrations confirm a no-global-privilege instance (Γ = 0.026 > 0) and a structurally coherent construction, both consistent with the theorem.