The Fine Structure Constant from the Geometry of K_8 Manifold: Paper XVI in MEF Series

We derive the electromagnetic fine structure constant α at the compactification scale Mc from the pure topology and geometry of the internal manifold K₈ = [(CP² × S²) ×w (T²/Z₂)]Spinᶜ with Euler characteristic χ(K₈) = 12. The derivation proceeds in three steps. Step 1 (Brane Localisation). The Standard Model gauge fields are topological defects localised at the Q-wall fixed point ω = 0 of the T²/Z₂ orbifold. This localisation nullifies the exponential warp factor (e6A(0) = 1) and reduces the Kaluza-Klein gauge kinetic integral to the unwarped 6-dimensional fibre CP² × S². The SU(2)L inverse coupling scales as αw⁻¹ ∝ τ₂², where τ₂ = Rw/Rc is the stabilised aspect ratio of the fibre radii. Step 2 (Topological Normalisation). The normalisation constant Cₙₒᵣₘ is the ratio of the topological intersection weight of the gauge configuration to the global flux budget of the compactification: Cₙₒᵣₘ = ∫CP² c₁(O(3))² · χ(S²) · χ(T²/Z₂)χ(K₈) = 9 × 2 × 2/12 = 3. Every factor is a topological invariant computable by the Chern–Weil theorem, the Gauss–Bonnet theorem, and the Kawasaki orbifold formula respectively. Step 3 (Electroweak Assembly). The Spinᶜ holonomy at the orbifold fixed point induces the physical Weinberg angle sin²θW = 1/(1 + τ₂²). By the electroweak unification identity α⁻¹ = αw⁻¹ sin⁻²θW, the fine structure eigenvalue at Mc is: α⁻¹(Mc) = [ ∫CP² c₁(O(3))² · χ(S²) · χ(T²/Z₂)χ(K₈) ] τ₂² (1 + τ₂²) = 46.37 evaluated at τ₂ = 1.861 (derived from Spinᶜ flux stabilisation in Paper VI [16] §8.2). Two-loop Standard Model renormalisation group evolution, with top-quark threshold matching and measured hadronic vacuum polarisation, carries this boundary condition to the Thomson limit: . α⁻¹(0) |MEF = 136.93 vs. . α⁻¹(0) |ₑₓₚₜ = 137.036 a deviation of 0.08%, consistent with the precision limitations of two-loop evolution and the three-decimal-place truncation of τ₂. The construction uses zero continuous free parameters. Every factor in the master formula is either a topological integer or a discrete geometric modulus stabilised by flux quantisation. The result establishes the fine structure constant as a derived output of the K₈ compactification geometry, not an input.