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 $\mathbb{H}_{\infty}=(\mathcal{M}^{(i+j)}(0))_{i,j\geq 0}$ 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 $d=\min\{k:\mathcal{M}^{(k)}(0)\neq 0\}$, 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:2610.13951 [quant-ph]

(or arXiv:2610.13951v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.13951 Focus to learn more arXiv-issued DOI via DataCite

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