Conformal Field Dynamics from Complex Relativistic Kinematics

Building directly on the kinematic metric isomorphism $g_{\mu\nu}(x) = \left(1 - \frac{\mathcal{V}^2(x)}{c^2}\right)\eta_{\mu\nu}$ established in recent work (Im, 2026), we construct the field dynamics governing the complex velocity field $\mathcal{V}^\mu(x) = v^\mu(x) - i u^\mu(x)$. We demonstrate that a scale-free spacetime dynamically acquires physical proper length $ds^2$ and metric measure $g_{\mu\nu}$ from local quantum osmotic velocity fluctuations $u^\mu$, which manifest directly as the Bohmian quantum potential energy $Q(x) = -\frac{\hbar^2}{2m}\frac{\nabla^2 R}{R}$. By performing functional variations of the Pure Weyl Action with respect to $\mathcal{V}^\mu(x)$, we derive the scale-covariant gravitational field equation $\square^2 \mathcal{V}^\mu(x) + \mathcal{R}_{\text{conformal}}\mathcal{V}^\mu(x) = \kappa J^\mu_{\text{matter}}(x)$, where the matter source current $J^\mu_{\text{matter}}$ is explicitly driven by $Q(x)$. Finally, we prove that this single field equation provides a unified physical axis connecting microscopic Schrödinger wave mechanics (as local quantum scale ripples), macroscopic gravitational curvature (as smooth metric scale deformations yielding Schwarzschild metrics), and global cosmological expansion (as background scale dissipation yielding Hubble friction).