System-Independent Order-Projection Principle and the Emergence of Complex Low-Mode Geometry

⚠️ 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 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.