Electric–Magnetic Duality Breaking and the Vacuum Impedance in the Cl(6) Bivector Algebra

The 2019 SI redefinition of the ampere exposes the vacuum impedance Z0 = μ0/ε0 as a derived quantity: Z0 = 2αRK, with RK = h/e2 the von Klitzingconstant. The question of why α ≈ 1/137 therefore reduces to the question of why Z0 takes the value it does, a question with no accepted first-principlesderivation. We identify an algebraic setting in which this question becomes tractable. The fifteen bivectors of the Clifford algebra Cl(6) contain the six components of theelectromagnetic field tensor Fμν as a natural subspace, completely decoupled from the internal Cl(2) sector. We prove that electromagnetic duality, the rotationF → cos θ F + sin θ Fγ5, is implemented by right multiplication by the chiral element γ5 and cannot be realized as a Spin(6) conjugation. The central result is the commutativity [T3, ∗] = 0 of the Cartan generator with the Hodge dual, establishing that the Hopf decomposition and the electromagnetic duality decomposition admit simultaneous diagonalization. In the neutral sector, Landau gauge fixing breaks the continuous U(1)EM-dual by selecting a preferred orientation in the (B3,E3) plane. The order parameter ρ = ⟨|Bgf3 |2⟩/⟨|Egf3 |2⟩ is proved equal to 1 in Euclidean SU(2) by O(4) symmetry. We conjecture ρ = 2α in Minkowski signature, making the vacuumimpedance a geometric output of the fiber orientation rather than an independent input.