The Vortex Framework Part IV: An Action-Based Derivation of Emergent Gravitation

Previous works in the Vortex Framework identified matter as topological vortex structures in a superfluid substrate (Part I), derived gauge interactions from vortex linking and phase topology (Part II), and demonstrated that gravitation emerges as an effective acoustic geometry governing excitation propagation (Part III). However, the emergent metric in Part III was treated kinematically rather than dynamically. In this work, we derive the dynamics of the emergent metric from a variational principle applied to the superfluid substrate. Starting from a Galilean-invariant superfluid action, we integrate out density fluctuations to obtain a low-energy effective action containing an Einstein–Hilbert term. The resulting field equations reproduce Einstein-like dynamics, with an effective gravitational coupling determined by substrate parameters and vortex network energy densities. Strong-field corrections, including singularity avoidance via evacuated cores, emerge naturally. This establishes gravitation as a universal infrared consequence of the same substrate that generates fermions and gauge interactions.