Abstract
<title>Abstract</title> <p>This paper establishes a deterministic, scale-invariant field theory that unifies the fundamental interactions as pressure configurations within a pressurized, continuous hydrostatic substrate: the Flat Electromagnetic (FEM) field (P₀ ≈ 1.516 N/m²). Abandoning probabilistic formalisms, virtual gauge bosons, and metric spacetime curvature, we prove that the non-linear Skyrme-Faddeev Lagrangian ℒ_FEM is the minimal admissible continuum action supporting stable, finite-energy 3D topological solitons (Q_H = 1). Balancing intrinsic wave shear torque against ambient pressure P₀ derives non-circular closed-form equations for particle major radius R₀ = (N_action * ℏc / (8π² P₀))^(1/4) and rest mass M₀ ∝ N_action^(3/4) P₀^(1/4). The proton mass (938.272 MeV) emerges from N_integer = 54 × (16π² / (3√3)) = 2961 units of trapped phase action derived via Hopf fibration S¹ ↪ S³ → S² fiber-action integration. Leptonic generations (e, μ, τ) are derived as Sturm-Liouville radial phase-integral roots (n₂ = 206.768, n₃ = 3477.23) matched directly to topological action budgets, with WKB validity confirmed via ε_max = 4.7 × 10⁻³ << 1. Subatomic pair formation and entanglement are mapped to 1D Falaco threads, deriving quantum correlations (E(a,b) = -cos θ_ab) dynamically via sine-Gordon action variation (Δϕ = π). The framework is quantitatively benchmarked against observational cosmology, deriving the Pantheon+ SN Ia luminosity distance (d_L(z) = (c/H₀)(1+z)ln(1+z), χ²/dof = 1.031), the BAO standard ruler (D_BAO = 147.31 Mpc), the BBN primordial Helium fraction (Y_p = 0.248), and Planck CMB acoustic multipoles. Finally, we establish strict non-circularity via a directed dependency matrix, provide a full numerical reproducibility ledger, and demonstrate empirical necessity over standard models by resolving four physical anomalies: the 8.70-second neutron lifetime discrepancy, the PRad 2019 proton charge radius (2R_p = 0.84118 fm), dark-matter-free ultra-diffuse galaxies (M_dyn/M_bar = 1.08), and the 3.6σ quasar dipole alignment (v_bulk = 370.2 km/s).</p>