The de Broglie Wave Equation: Unified Vector Forms and Physical Interpretation as Spacetime Curvature.

We derive the complete vector forms of the de Broglie wave equation directly from the fundamental relation δp = hδψ without assumptions or ad hoc additions. The wave function ψ = 1/λ emerges naturally as spacetime curvature. We show that bound states require both cross product (tangential) and dot product (radial) components, while free states exhibit only the cross product form. This framework provides clear physical meaning to quantum mechanical wave functions and explainsthe emergence of the factor ℏ = h/2π in bound state dynamics.