wave geometry(ver.4)

Wave Geometry is a mathematical framework in which geometric structures are not taken as primitive, but emerge from the relational dynamics of wave phases, transport processes, and resonance stability. Rather than assuming space, distance, or manifolds as fundamental, Wave Geometry begins with a pregeometric foundation consisting of discrete nodes, their connectivity, complex wave states (amplitude and phase), and phase connections. From these primitive elements, notions of distance, locality, curvature, geodesics, and manifolds arise naturally through the stabilization of phase relations and coherent wave transport. The central principle is: Relation → Resonance → Geometry. Geometry is thus understood as stabilized harmonic resonance — a dynamically generated organization of phase coherence rather than a static background. In the appropriate continuous limit, classical Riemannian geometry and nonlinear wave equations are recovered as special cases. This approach offers a unified pregeometric perspective that naturally incorporates synchronization phenomena, discrete differential geometry, topological defects, and emergent spacetime structures. It opens new pathways for research in synchronization theory, analogue gravity, quantum geometry, and foundations of physics.