Predictive Holonomy and Minimal Completion of Coarse Dynamics

A coarse description of a dynamical system generally fails to evolve autonomously: evolving and coarse-graining in different orders gives different predictions, and the compressed propagator fails the semigroup law. We show that this path dependence is the observable boundary of missing state. For a finite continuous-time Markov chain with generator L, coarse channel C, and stochastic section R satisfying CR=I, define P=RC, Q=I-P, and K=CLR. The exact coarse propagator T t=C e tLR obeys a Volterra equation with memory kernel M(t)=CLQ, e tQLQ,QLR. Every memory jet M(k)(0) is recursively recoverable from the short-time derivatives of T t alone. The rank of the block Hankel operator H infinity=(M(i+j)(0)) i,j at least 0 is exactly the smallest number of auxiliary linear state variables required to turn the coarse dynamics into an exact autonomous, time-homogeneous system. A minimal completion is constructible by Ho--Kalman factorization and is unique up to similarity. If d= k: M(k)(0) not equal to 0, then every smooth classical f-divergence generically scales as t 2d+4. Exact Markov examples realize exponents 4,6,8 and closure ranks 1,2,3. We also derive finite-sample singular-value certificates for hidden predictive coordinates, including a matrix-Bernstein bound with intrinsic-variance control. Finally, histories are identified when they induce the same future law under every admissible intervention, yielding the minimal counterfactual predictive state. The results turn coarse-graining into a constructive state-completion problem and separate ordinary linear realization from the harder requirement of a positive or Markov realization.