Foundations of UD Theory: Axioms, Field Equations, Projections, and the Emergence of General Relativity

We establish the foundations of UD theory from first principles. The $U$ and $D$ attributes are defined through their observable consequences: the extension of space and the localization of matter. The axioms, formulated as mathematical constraints, determine the action, coupling constants, and structural constant $\pi e^\pi$, derived from field equations in the extreme limits of black hole collapse and vacuum dilution. Four fields, classified by attribute effect ($E = U_u + D_u$, $C = U_d + D_d$) and existence form ($S = U_u + U_d$, $M = D_d + D_u$), are mapped to physical entities via phenomenological parallels. Scalar, spinor, vector, and tensor projections are constructed from the UD action. Spacetime geometry is the macroscopic manifestation of the UD field dynamics: the energy-momentum tensor of the UD fields sources a metric perturbation, whose full nonlinear dynamics reproduce the Einstein-Hilbert action and the Einstein field equations when the $U$-$D$ competition is in a state of relative rest. Geometry is a descriptive tool for the underlying UD dynamics, not the fundamental ontology. The equivalence principle and geodesic motion are derived from first principles. Diffeomorphism invariance is proved from the conservation of the UD energy-momentum tensor. Quantitative predictions that differ from standard $\Lambda$CDM cosmology are given. All derivations are self-contained.