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

⚠️ CORRECTION NOTICE (added 2026): The author has since identified serious errors in this work through a systematic adversarial audit. The central claims of this record should not be relied upon. Specific issues found across this research program include implementation errors (e.g., conflation of signed cosine with cos²), circular definitions, proof gaps, results that reduce to known theorems, and effects that vanish under proper null models and direct-influence correction. This record is left online for transparency and as a documented case, not as a valid result. For the author's current, audited methodology, see the later work on classical-baseline auditing of quantum-like contextuality claims: https://doi.org/10.5281/zenodo.20680607 — A. Ogasawara 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.