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When comparing descriptions of a dynamical system across multiple prediction tasks, it is natural to ask whether one description is globally best — a stable winner over the full target family. We give an exact answer for finite-dimensional linear Gaussian dynamical systems (LGDS): a stable winner exists within the rank-r observer class if and only if the target family is r-coherent, meaning that all target information matrices share a common top-r eigenspace. When coherence fails, no rank-r observer is simultaneously optimal for all targets and the global-privilege gap is strictly positive. We then extract an empirical diagnostic from the theorem: supported pairwise sign flips across a target family diagnose empirical incoherence and motivate a prediction of winner-absence. We test this diagnostic in an individual-based evolutionary simulation battery spanning six regimes, four descriptions, and two task families (predictive and causal). In a pre-aggregated adapter analysis, all 11 classified incoherent regime-family cells are winner-absent (0 mismatches; pre-registered pass condition met). Current support is strongest for the negative half of the diagnostic: empirically incoherent cells are winner-absent. No coherent cells were observed, so the positive half remains untested.