Predictive Holonomy and Minimal Completion of Coarse Dynamics

Abstract

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 $T_t=C{\mathrm{e}}^{tL}R$ obeys a Volterra equation with memory kernel $\mathcal{M}(t)=CLQ{\mathrm{e}}^{tQLQ}QLR$. Every memory jet ${\mathcal{M}}^{(k)}(0)$ is recursively recoverable from the short-time derivatives of $T_t$ alone. The rank of the block Hankel operator ${\mathbb{H}}_{\infty }=({\mathcal{M}}^{(i+j)}(0){)}_{i,j\ge 0}$ 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 $d=\min \{k:{\mathcal{M}}^{(k)}(0)\ne 0\}$, 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.

arXiv comment13 pages, 3 figures; ancillary symbolic and numerical verification code included