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

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.