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=Ce^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_∞=(ℳ^(i+j)(0))_{i,j≥ 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:ℳ^(k)(0)≠ 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.

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BibTeX RIS

BibTeX

@misc{churilov2026predictive,
  title = {{Predictive Holonomy and Minimal Completion of Coarse Dynamics}},
  author = {Maxim V. Churilov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.13951},
  eprint = {2610.13951},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.13951},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Predictive Holonomy and Minimal Completion of Coarse Dynamics
AU  - Maxim V. Churilov
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.13951
UR  - https://arxiv.org/abs/2610.13951
AB  - 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=Ce^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_∞=(ℳ^(i+j)(0))_i,j≥ 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:ℳ^(k)(0)≠ 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.
ER  -

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