A Universal Minimax Lower Bound for Causal Lagrangian TrackingFrom empirical flow-family classification to an algorithm-independent impossibility certificate with object-shape-aware tolerance

Abstract

We develop a deterministic minimax lower bound for causal tracking of a Lagrangian object across an observation gap. Earlier work in this project identified an empirical dimensionless coordinate, Pi_U, that organized synthetic tracking difficulty across multiple flow families. The present work is deliberately different: it does not claim universality from classifier performance. Instead, it defines an admissible trajectory class containing observationally indistinguishable alternatives with opposite unresolved acceleration radius A and jerk radius J. For every causal point predictor based on the same pre-gap information, the worst-case endpoint error is at least (1/2)A Delta t^2 + (1/6)J Delta t^3. The bound is sharp for the constructed two-point uncertainty set. Defining U=[(1/2)A Delta t^2+(1/6)J Delta t^3]/L yields a sufficient impossibility certificate: U>1 implies that no causal predictor can uniformly guarantee endpoint error <=L over the stated admissible class. Object shape is incorporated only through the tolerance scale L, using a characteristic length or an explicit user-specified tolerance; shape does not alter the theorem. A reproducible 100,000-case synthetic dataset and Python implementation supporting sphere, ellipsoid, box, cylinder, irregular-volume approximations, and optional mesh-derived characteristic length are provided. This result is a theorem about an explicitly defined information pattern and admissible trajectory class, not an unconditional statement that all physical fluid trajectories with large measured RMS acceleration or jerk are untrackable. Keywords: Lagrangian tracking; minimax lower bound; causal prediction; tracking impossibility; uncertainty radius; fluid mechanics; object geometry