Structural Cosmology v2.1: A Covariant Field-Equation Formulation of the Structural Field

Structural Cosmology v2.1 extends the effective field theory (EFT) framework introduced in v2.0 by developing a covariant field-equation formulation of the structural field Δ. Starting from the effective action Seff[gμν,Δ,Ψi]S_{\mathrm{eff}}[g_{\mu\nu}, \Delta, \Psi_i]Seff[gμν,Δ,Ψi], covariant field equations are derived through metric variation and structural field variation, yielding a modified Einstein-type equation and a generalized structural field equation. The framework introduces an explicit decomposition of the total energy-momentum tensor into matter, structural field, interaction, projection, geometric coupling, and higher-derivative sectors. In a homogeneous and isotropic FLRW background, the coupled equations reduce to a modified Friedmann system governing the evolution of the scale factor, Hubble expansion, and structural field dynamics. Regime-dependent limits are analyzed across multiple cosmological epochs, and a projection framework is developed to connect theoretical quantities to observable signals including expansion history, growth of structure, matter clustering, cosmic microwave background anisotropies, and gravitational-wave observables. This release establishes the field-equation foundation of Structural Cosmology and provides the theoretical bridge from the EFT formulation of v2.0 toward perturbation theory (v2.2), nonlinear structure formation (v2.3), and the long-term Structural Cosmology core theory (v3.0).