The Fine Structure Constant from the Geometry of K₈ Manifold: Paper XVI in MEF Series

We derive the electromagnetic fine structure constant 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 (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 = 0 of the T²/Z₂ orbifold. This localisation nullifies the exponential warp factor (e^6A (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 w^-1 ₂², where ₂ = Rw/Rc is the stabilised aspect ratio of the fibre radii. Step 2 (Topological Normalisation). The normalisation constant C₍₎ₑ₌ is the ratio of the topological intersection weight of the gauge configuration to the global flux budget of the compactification: C₍₎ₑ₌ = ₂₏ℂ c₁ (O (3) ) ² (S²) (T²/Z₂) (K₈) = 9 2 212 = 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 ²W = 1/ (1 + ₂²). By the electroweak unification identity ^-1 = w^-1 ^-2W, the fine structure eigenvalue at Mc is: ^-1 (Mc) = ₂₏ℂ c₁ (O (3) ) ² (S²) (T²/Z₂) (K₈) ₂² (1 + ₂²) = 46. 37 evaluated at ₂ = 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: . ^-1 (0) |₌₄₅ = 136. 93 vs. . ^-1 (0) |₄ₗ₏ₓ = 137. 036 a deviation of 0. 08%, consistent with the precision limitations of two-loop evolution and the three-decimal-place truncation of ₂. 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.