The Fully Inhomogeneous Non-linear Regime of Dynamic Scalar Field Theory (DSFT): Complete Field Equations, Mathematical Proofs, and Astrophysical Signatures

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).