Requirement-Induced Predictive Geometry for Finite-Resource Prediction in Dynamical Systems

Abstract

In nonlinear prediction, two state representations with the same local uncertainty volume can have radically different predictive value. We consider differentiable finite-time dynamics together with a quadratic terminal requirement that specifies which terminal state differences matter. Pulling this requirement back through the tangent map defines a requirement-induced predictive geometry on the present tangent space. This geometry is fixed by the dynamics and the requirement; under the fixed local uncertainty-volume constraint used here, the capacity parameter fixes only the scale of the representation that realizes it. We show that the unique optimal covariance has the inverse predictive shape and, equivalently, that optimality is characterized by isotropization of propagated uncertainty in the terminal requirement geometry. The gain over a baseline-isotropic representation is the arithmetic-to-geometric mean ratio of the predictive spectrum, making it a coordinate-invariant measure of predictive anisotropy. In Euclidean coordinates this yields a general finite-time spectral law, and for two-dimensional area-preserving dynamics an exact identity between predictive gain and the condition number of the optimal geometry. Stronger directional stretching therefore increases potential predictive gain only together with the anisotropy required to realize it; bounding that anisotropy imposes a finite gain ceiling. Standard-map and double-pendulum calculations demonstrate these relations in nonlinear Hamiltonian systems. The framework provides a basis for adaptive sensing, state estimation, and model-predictive computation in which representation geometry is matched to predicted task consequences.