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

Previous works in the Vortex Framework identified matter as topo-logical vortex structures in a superfluid substrate (Part I), derivedgauge interactions from vortex linking and phase topology (Part II),and demonstrated that gravitation emerges as an effective acousticgeometry governing excitation propagation (Part III). However, theemergent metric in Part III was treated kinematically rather than dy-namically.In this work, we derive the dynamics of the emergent metric froma variational principle applied to the superfluid substrate. Startingfrom a Galilean-invariant superfluid action, we integrate out den-sity fluctuations to obtain a low-energy effective action containingan Einstein–Hilbert term. The resulting field equations reproduceEinstein-like dynamics, with an effective gravitational coupling deter-mined by substrate parameters and vortex network energy densities.Strong-field corrections, including singularity avoidance via evacuatedcores, emerge naturally. This establishes gravitation as a universal in-frared consequence of the same substrate that generates fermions andgauge interactions.