This work reports a system-spanning numerical investigation of order-dependenteffects arising from noncommutative operational histories under irreversibleprojection. Using several representative systems with distinct dynamics—including theIsing model, neural networks, and simulated annealing—we analyze history-dependentdifferences through a minimal, model-agnostic framework based on projection andlow-mode decomposition. We show that a low-dimensional geometric structureconsistently emerges across systems, dominated by a small number of modes, andthat this structure is preserved only when represented in a vector (complex)form rather than by scalar amplitudes alone. The observed phase variability across realizations is interpreted as a gaugedegree of freedom, while directional overlap provides a robust invariant forcross-system comparison. Differences between systems are found not in theexistence of the geometric core itself, but in the visibility of its rotationalresponse under changing conditions. This record is intended as a technical preprint to establish precedence anddocument the numerical and geometric findings. Broader theoretical connectionsand applications are deliberately left outside the scope of the present work.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.
John Jude Hathway (Tue,) studied this question.