Non-Singular Emergent Gravity: A Framework for the Transport-Induced Einstein Equations

We present a rigorous macroscopic framework wherein space-time geometry and relativistic field equations emerge from an underlying, non-equilibrium, dissipative transport process regulated by a local entropy maximum principle. By extending the Maxwell-Cattaneo-Vernotte relativistic transport model to a generic curved background metric, we show that a fundamental coherence density field driven by irreversible gradient flows naturally satisfies a massive Klein-Gordon equation possessing an explicit geometric coupling to the background Ricci tensor. Using an effective field theory (EFT) formulation, we map the short-scale statistical fluctuations of this coherence sector into a coarse-grained effective action. Through the principle of least action, the metric tensor acts as a dynamic variational field, avoiding theoretical double-counting and yielding the exact, non-linear macroscopic Einstein field equations without enforcing artificial algebraic constraints. Finally, we show that the non-relativistic static infrared limit of this model unconditionally preserves the uncoupled, massless geometric structure of the classical Poisson operator. We prove that the flat, constant velocity profiles observed in galactic rotation curves emerge exactly from the internal, dual-mode transport convolutions embedded within the effective energy-density tensor ($T_{00}^{\mathrm{eff}}$), successfully resolving the galactic dark matter paradox within the observational error margins of the SPARC database while naturally suppressing quadratic curvature corrections over large spatial scales. This framework bridges the conceptual divide between irreversible entropic transport processes and conservative macro-gravitational field dynamics.