Expanded theory of relativity: Demonstrating wave-particle duality as an integral part of relativity theory

We present a geometric framework that promotes time from a scalar parameter to a genuine three-dimensional temporal vector whose local orientation is encoded by a quaternionic rotor field\[R(x)=exp(Q(x)), Q(x)(2).\]Starting from block-boost consistency in a six-dimensional Clifford space, the effective projection onto physical coordinates reproduces the standard Lorentz interval while introducing a new dynamical sector (the flag) that coexists with the usual particle worldline (the pole). Requiring the rotor to be Lie-dragged along particle worldlines yields a universal first-order wave equation for a flag field \(Ψ\),\[i ε^μ Σ^μ∂_μΨ - AQ ρQ(Ψ^Ψ) Ψ = 0,\]valid across spin representations via \(Σ^μ∈\{1,S_1,S_2,S_3\}\). In appropriate representations and limits the formalism reproduces Dirac-, Klein–Gordon- and Maxwell-type dynamics. Separation of variables in the spin-\(12\) flag equation reduces to Kummer’s confluent hypergeometric form and reproduces the Dirac–Coulomb hydrogenic spectrum. The rotor sector generically produces falsifiable signatures — notably small spatially varying clock shifts and anomalous spin-precession terms — which can be constrained by precision atomic clocks and spin-interferometry. We close by outlining paths to determine the temporal potential \(VQ\), couple the rotor to gravity, and quantize the flag sector.