Order Dynamics Beyond Gradient Flow: Evidence for Holonomic Rotation from Noncommutative Projection History

This short study examines the dynamical structure of order-dependent responsesobserved in systems with noncommutative operational history. Using newly generated data from two distinct systems—a two-dimensional Isingmodel and a neural network trained under noncommutative curricula—we constructlow-dimensional complex trajectories that represent the evolution of orderdifferences. Two minimal dynamical descriptions are tested on identical trajectories:a gradient-flow model and a gradient-plus-rotation (holonomic) model.While pure gradient relaxation fails to reproduce the observed dynamics,the inclusion of a minimal rotational term provides a consistent descriptionacross systems. The results indicate that order trajectories can contain a rotationalcomponent associated with noncommutative operational history,suggesting that order evolution may not always be reducibleto gradient relaxation in a scalar potential landscape. The work is intended as a structural dynamical study of order-dependentphenomena and provides a minimal empirical test of dynamical modelsbeyond gradient flow. Note: Parts of the manuscript were linguistically and structurally refinedwith the assistance of AI-based tools.All scientific content, analysis, and conclusions are the author's own.