Emergence of a Noncommutative Phase in Nonlinear Constraint-Driven Projection Dynamics

⚠️ 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 investigates the emergence of ordering dependence in nonlinear projection-driven dynamics governed by constraint-based update rules. We demonstrate numerically that sequential constraint updates generate a noncommutative phase in which the final configuration depends on the ordering of operations. The transition into this regime is controlled by a deformation parameter β and can be characterized by an effective scaling parameter Λ_eff that combines the projection strength and the deformation scale. Across a wide range of numerical experiments, the results show a consistent collapse of the data when expressed in terms of Λ_eff, indicating that the phase boundary is governed by a one-dimensional effective parameter. These results suggest that noncommutativity may arise generically in nonlinear projection-driven systems, providing a structural mechanism for ordering-dependent dynamics without modifying the background geometry. This record is intended primarily to document the numerical and structural observations associated with this phase behavior. Note: Parts of the manuscript were linguistically and structurally refined with the assistance of AI-based tools.All scientific content, analysis, and conclusions are the author's own. Note: This work represents Version 1.0 of an ongoing research program on the Order-Projection Principle (OPP). Minor typographical corrections and clarifications may appear in later versions. The core conceptual claims remain unchanged.