Geometric Emergence of Quantum Behavior: MPCT Analysis from Classical Trajectories

This paper presents a novel approach to understanding quantum phenomenathrough geometric analysis of classical-like trajectories. We introduce the MultiPhase Coupling Transform (MPCT) method, which extracts quantum informationfrom complex dynamical trajectories without presupposing quantum mechanicalformalism. Through numerical simulations, we demonstrate that discrete quantumnumbers emerge naturally from topological phase conditions (2π⟨r⟩ = nλ), withexperimental accuracy better than 2%. The action variable shows clearer quantization than energy, suggesting action may be a more fundamental quantum quantity.Our results indicate that quantization may have purely geometric-topological origins, independent of specific dynamical equations. The MPCT method providesa new tool for extracting quantum information from trajectory data and offers acomplementary perspective on the emergence of quantum behavior.Keywords: Quantum emergence, geometric quantization, trajectory analysis,MPCT transform, topological quantum numbers