Three-Phase Geometry and Phase Unfolding: A Common Geometric Framework for Relativistic and Quantum Relations

This study extends phase unfolding from a single closed phase to a three-directional phase structure composed of three mutually orthogonal phase directions. The three directions are treated as directional components of a single phase structure rather than as intrinsically different physical phases. When a common progression is established, the structure separates geometrically into open progression and a retained closed phase. The retained phase preserves its closed form while remaining connected to the unfolded structure, allowing the geometry to be followed through unfolding and subsequent re-closure. The construction also distinguishes ordinary 2π phase periodicity from a 4π structural return of the connected circular–unfolded trajectory. The unfolded progression is then related to established physical coordinates and wave relations involving t, x, ω, and k, while the retained closed phase provides a structural basis for comparison with relative phase and rotational phase relations in quantum mechanics. The study proposes two connected structural levels within the phase-unfolding framework: phase unfolding as a background geometry associated with relativistic relations, and phase organization within that background associated with wave and quantum-mechanical relations. The resulting three-phase geometry provides a common setting for examining whether these relations can be consistently organized from the same underlying phase structure.