Deriving the Four Fundamental Interactions from Angular Regimes on S3 ×R

Deriving the Four Fundamental Interactions from Angular Regimes on S³ × ℝ — Angular World Architecture (Unified Interactions) The origin of the Standard Model coupling constants — α_EM, α_s, sin²θ_W, and the gravitational coupling — remains one of the deepest open problems in theoretical physics. Standard approaches (GUT unification, string landscapes, anthropic reasoning) require either large-scale fine-tuning or multiple free parameters, with no first-principles derivation in sight. This article shows that these four couplings — and, with them, the four fundamental interactions themselves — emerge from a single angular law θ = (360/π) · arctan √Ξ defined on the bilayer geometry S³ × ℝ, with zero free dimensionless parameters. Electromagnetism, gravitation, the strong force, and the weak force are four angular regimes of a single geometric field on S³ × ℝ with a Hopf fibration. Each interaction corresponds to a privileged angle: EM at 0°, strong at 45°, gravity at 90°, weak at 135°. The central result is a complete derivation of all Standard Model coupling constants from two axioms (the bilayer field Φ/Φ̄ on S³, and the temporal medium Φ with speed c and granularity ℓ_P), a Wilson-RG-selected potential V = ω₀²(cosh x − 1), and zero free dimensionless parameters. The sole input is the electron-to-Planck mass ratio m_e/m_P — unavoidable in any quantum theory of gravity. The bilayer dihedral angle φ, identified as the Cabibbo angle, is derived from the BPS soliton geometry via tan φ = (α_s^AWA · N_c)², where N_c = 3 follows from the icosahedral representation of Isom(S³)/ℤ₂ = SO(3), and α_s^AWA from a pure mathematical integral over the kink profile. Quantitative predictions (all within 1.1σ of experiment, no adjusted parameter): Higgs mass m_H = 125.216 GeV (0.013 % from PDG) Strong coupling α_s(M_Z) = 0.11790 (0.085 %) Weak mixing angle sin²θ_W = 0.23125 (0.014 %) Hubble constant H₀ = 67.404 km/s/Mpc derived from {G, ℏ, c, m_e} alone (0.01σ from Planck 2018) Eight of nine CKM matrix elements reproduced at better than 1.2 % Observational tests include the PDG median of broad resonances at 45.46° (zero adjusted parameter), atomic parity violation in Cs and Yb mapping to [135°, 172°], and three precision gravity tests (MICROSCOPE, Cassini, GW170817). Limitations are stated explicitly: the bilayer geometry predicts a right-handed W_R boson whose mass relation MW_R = f(φ, m_W) is conjectured from bilayer symmetry but requires a rigorous Lagrangian derivation (open programme); the empirical factor (1 − π/40) in the H₀ prediction is deferred to AWA V; the ninth CKM element lies at 1.2 % from experiment versus better than 0.5 % for the other eight.