Part XXVII: Geometric Vortex Loop Gravity - A Geometric Mass Gap Theorem for Sobolev Mappings in Four Dimensions

This paper establishes a geometric mass gap in a class of Sobolev mappings between compact, oriented manifolds of equal dimension. The model is supported by a discrete backbone Λ4 whose projection hierarchy gives rise to eight independent activation variants, each representing a distinct structural mode of geometric response. The analytic proof relies on a quantitative degree–energy inequality for Sobolev maps, which links the topological invariant deg(F) to the gradient norm. This inequality shows that any configuration with nonzero degree must incur irreducible geometric distortion, enforcing a positive lower bound on the energy in every nontrivial sector. The mass gap follows directly from the energy decomposition and holds uniformly across all activation variants. No physical or field-theoretic claims are made; the argument is purely geometric and topological.