Deriving the Gravitational Constant from Electromagnetic Coupling and Z₃ Wave Geometry

The gravitational constant G is derived from the electromagnetic fine structure constant α, the proton mass, and the geometric structure of Z₃ strip topology. The formula G = √3 · α¹⁸ · ℏc/m²_p reproduces the measured value to 0.99% with zero adjustable parameters. The factor √3 is the amplitude amplification at a three-phase convergence point. The exponent 18 is derived from group theory: the gravitational interaction must be invariant under Z₃ × Z₃ × Z₂ — the full symmetry group of two three-mode convergences coupled through a chiral channel — whose order is 3 × 3 × 2 = 18. Each group element contributes one factor of α, yielding α¹⁸. Within the complete framework, α is not an independent input. It derives from the observer's angular position on the strip: α = 6sin²δ, where δ = 2° is the offset from the anti-gravity node (0.14% error). The formula for G is therefore fully geometric — G = √3 · (6sin²δ)¹⁸ · ℏc/m²_p — with one mass scale as the only external input. The hierarchy problem dissolves: the 10³⁶ ratio between electromagnetic and gravitational strength is α⁻¹⁷/√3, a geometric identity, not a fine-tuning puzzle. Verified against twelve independent gravitational measurements, all within 1.4%. The predicted G = 6.740 × 10⁻¹¹ matches Cavendish's 1798 value exactly. A complete mass chain follows: proton mass M_Pl · 3^(1/4) · α⁹ (0.5%), pion mass m_p × 4/27 (0.4%), proton radius 4ℏ/(m_p·c) (0.05%). The full 14-constant derivation chain is developed in the companion paper "From One Axiom to Fourteen Constants."