Complementary Phase Geometry: Lagrangian Dynamics and Deterministic Wave-Like Trajectories

Within the Complementary Phase Geometry (CPG) framework introduced in the preceding Hamiltonian formulation, the present work develops the corresponding Lagrangian and trajectory-level dynamics of coupled observable and complementary degrees of freedom on the real seven-dimensional Clifford geometry Cl(3,4). Starting from the invariant transported velocity norm, the extended particle action, canonical momenta, and coupled Euler-Lagrange equations are derived. A distinguished transport-invariant branch admits bounded periodic motion with$\\omega_{\\rm eff}=\\sqrt{\\omega^2-\\mu^2}$ for $\\omega^2>\\mu^2$, producing a deterministic oscillatory modulation of the observable trajectory through mixed observable--complementary coupling. The resulting dynamics are further applied to aperture configurations through an effective macroscopic boundary approximation. Under the assumed mapping $\\Delta\\theta_{\\rm B}=k\\Delta y\\sin\\alpha$ and an approximately uniformcomplementary phase distribution, the standard Fraunhofer functional forms for single-, double-, and multiple-slit configurations are recovered while each particle remains localized and follows one deterministic spacetime trajectory. The microscopic particle-boundary interaction, the physical normalization of the effective complementary response, and the quantitative mapping from deterministic trajectory ensembles to detector distributions remain open problems.