We present the complete, mathematically rigorous formulation of Dynamic Scalar Field Theory (DSFT) operating within a fully inhomogeneous and non-linear regime. Moving be- yond the restrictive approximations of linear perturbations and spatial homogeneity, we map the exact non-linear cross-couplings of the dynamic scalar field with spacetime curvature and matter fields via the functional couplings α(ϕ) = 1 − ϕ 2/Λ 2 α and β(ϕ) = 1 + ϕ 2/Λ 2 β . We provide rigorous, step-by-step mathematical proofs deriving the comprehensive grav- itational and scalar field equations. Through these exact formulations, we establish the mathematical avoidance of initial and black hole singularities via an emergent repulsive Eu- clidean core (a → 0), and prove a first-principles geometric origin for dark matter profiles (ρDM ∝ r −4 ) via scalar charge polarization in inhomogeneous spacetimes. Furthermore, we derive the exact topological coupling of the scalar field to the Maxwell tensor, predicting a primordial intergalactic magnetic field of ∼ 1 nG and a fine-structure constant dipole of 1.02 × 10−8 . Finally, we provide comprehensive observational concordance matrices vali- dating DSFT against the full 2026 cosmological dataset (Planck, DESI Y1, JWST, Euclid DR2, SH0ES, Pantheon+), demonstrating an exceptional statistical preference over ΛCDM (∆χ 2 ≈ −69.3).