Leynstrinsic Field Theory: LFT Paper 4 - Connectivity Field Dynamics and the Origin of Gravitational Interaction in Leynstrinsic Field Theory

This work develops the dynamical sector of Leynstrinsic Field Theory (LFT), in which spacetime geometry and gravitational interaction emerge from an underlying connectivity field governing relational structure between physical degrees of freedom. We derive the effective action and equations of motion for the connectivity field and demonstrate how large-scale geometric structure arises from collective field behavior. Within this framework, the effective metric is generated through coarse-grained correlations of the connectivity configuration, linking microscopic dynamics to macroscopic spacetime structure. Gravitational interaction is shown to emerge as a collective phenomenon associated with large-scale connectivity fluctuations, rather than as a fundamental force. The resulting effective field equations reproduce geometric curvature dynamics analogous to Einstein gravity while remaining rooted in the underlying connectivity framework. The theory remains consistent with ultraviolet dimensional crossover, in which the effective spectral dimension flows toward two at high energies, modifying propagation behavior and stabilizing the ultraviolet regime. These results provide the dynamical foundation of Leynstrinsic Field Theory, connecting the emergence of geometry, gravitational interaction, and dimensional flow within a single unified structure.