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 — αEM, αₛ, sin²θW, 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 ℓP), 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 mₑ/mP — 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 = (αₛAWA · Nc) ², where Nc = 3 follows from the icosahedral representation of Isom (S³) /ℤ₂ = SO (3), and αₛAWA from a pure mathematical integral over the kink profile. Quantitative predictions (all within 1. 1σ of experiment, no adjusted parameter): Higgs mass mH = 125. 216 GeV (0. 013 % from PDG) Strong coupling αₛ (MZ) = 0. 11790 (0. 085 %) Weak mixing angle sin²θW = 0. 23125 (0. 014 %) Hubble constant H₀ = 67. 404 km/s/Mpc derived from G, ℏ, c, mₑ 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 WR boson whose mass relation Mₖₑ = f (φ, mW) 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. Keywords: fundamental interactions, Standard Model, coupling constants, electromagnetism, strong force, weak force, gravitation, S³ × ℝ, Hopf fibration, BPS solitons, Cabibbo angle, CKM matrix, Higgs mass, Nc = 3, SO (3), icosahedral symmetry, right-handed W boson, Angular World Architecture, AWA.