Analytical Formalism and Lagrangian Derivation of the Viscoelastic Vacuum

Abstract While previous developments of the Dynamic Substrate Theory (DST) successfully mapped macroscopic phenomena—such as gravitational wave propagation limits and background screening—a formal analytical grounding via variational principles is required to elevate the theory to a complete field-theoretic framework. In this work, we formalize the DST vacuum not as an empty geometric manifold, but as an active continuum endowed with relaxation times and internal stress memory, constructing the foundational field equations from an action-based variational approach. By applying Euler-Lagrange variations with respect to the displacement field, we derive the generalized momentum conservation equations for the vacuum substratum and prove analytically that inertia (F = ma) emerges as the reactive drag force exerted by the Maxwell-Oldroyd-B continuum on localized topological mass defects. Keywords: Lagrangian Formalism, Viscoelastic Vacuum, Emergent Inertia, Maxwell-Oldroyd-B Rheology, Variational Principles, Field Theory.