Paper IX: Maxwell's Equations, the Gyromagnetic Ratio g = 2, and the Kerr Horizon from a Common Geometric Origin

The complete electromagnetic sector is derived from the axioms of Quantum Geometrodynamics (QGD), with zero additional inputs. Algebraic origin of γEM = 0. The SO (4) ≅ SU (2) L × SU (2) R decomposition of the H-vacuum assigns the Tikbon field to su (2) L (gravitational sector) and the photon field to su (2) R (electromagnetic sector). Left–right orthogonality, ⟨Lₐ, Rb⟩ = 0, implies that the torsion trace T_μ = gT AG_μ has vanishing inner product with AEM_μ. This forces the curvature coupling γEM = 0 — not as a limiting approximation but as an algebraic necessity of the quaternionic structure. Derivation of the Maxwell Lagrangian. From γEM = 0, the Maxwell Lagrangian LEM = − (1/4) F_μν F^μν − A_μ J^μ follows in three steps: (i) γEM = 0 eliminates the curvature term; (ii) U (1) gauge invariance, Lorentz invariance, and the requirement of second-order field equations select the kinetic term uniquely; (iii) local U (1) EM invariance of the matter action forces the minimal coupling −A_μ J^μ. The four Maxwell equations follow: the homogeneous pair from F = dA (implying dF = d²A = 0), and the inhomogeneous pair from the Euler–Lagrange equations. Structural parallel between gravitational and electromagnetic sectors. Both sectors are U (1) gauge theories with identical Lagrangian structure − (1/4) F²; differences arise solely from their distinct Lie algebra sectors (su (2) L vs. su (2) R), coupling constants (gT vs. e), and the composite-graviton mechanism that promotes the gravitational potential to rank-2. The parallel extends to rotating sources: the Kerr–Newman potential A_μ = (kₑ q r / Σ) ℓ_μ shares the same null vector ℓ_μ and the same oblate spheroidal function Σ as the Kerr metric g_μν = η_μν + (2Mr/Σ) ℓ_μ ℓ_ν. Two results from a single geometry. The azimuthal component ℓ_φ = a sin²θ produces the Kerr horizon function Δ = 0 in the gravitational sector and the Dirac gyromagnetic ratio g = 2 in the electromagnetic sector. Both are derived from the same Hopf-fibre geometry; neither is postulated.