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Predictive Holonomy and Minimal Completion of Coarse Dynamics

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Briefings

A new study reveals that coarse-grained descriptions of dynamical systems fail to evolve autonomously due to path dependence, a phenomenon bounded by missing state information. The research defines a minimal completion problem solvable via Ho-Kalman factorization, where the rank of a memory kernel determines the required auxiliary variables. For finite continuous-time Markov chains, the memory jet rank equals the smallest number of linear state variables needed to make the system time-homogeneous. The study identifies specific exponents for classical f-divergences (4, 6, 8) and provides finite-sample singular-value certificates for predictive coordinates.

Key points

• The coarse propagator of a finite continuous-time Markov chain fails the semigroup law, making predictions dependent on the order of evolution and coarse-graining operations.
• The memory kernel of the coarse propagator is recursively recoverable from short-time derivatives of the propagator itself, allowing for the construction of a unique minimal completion via Ho-Kalman factorization.

Context

This research addresses the fundamental issue of path dependence in coarse-grained dynamical systems, where different orders of evolution yield different predictions. It provides a constructive framework for identifying the minimal state variables required to resolve this non-autonomous behavior, turning a state-completion problem into a solvable linear realization task.

Limitations

The study applies to finite continuous-time Markov chains with specific generator and stochastic section constraints. Results regarding classical f-divergence exponents are generic for smooth functions, while the matrix-Bernstein bound applies to finite-sample scenarios. The paper is a preprint (arXiv:2610.13951v1) and does not include experimental validation or long-term simulation data.

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