Fundamental Constants from Grassmannian Geometry: Deriving α = 1/137 via Kaluza-Klein Reduction on Gr(3,16)

Abstract We derive fundamental constants of the Standard Model from the complex Grassmannian Gr(3, 16) = U(16)/[U(3) °ø U(13)]. With only two integers (k = 3, n = 13), we obtain: (1) the electroweak mixing angle sin2 θW = k/n = 3/13 = 0.2308, matching the measured value 0.2312 to 0.2%; (2) the hierarchy ln(MPlanck/vEW) = kn = 39, matching 38.4 to 1.5%; (3) generations = k = 3, exactly; (4) fermions per generation = N = 16, matching the SO(10) spinor exactly. We further derive the fine structure constant via Kaluza-Klein reduction: α(MZ)−1 = dim(u(N))/2 = N2/2 = 128 (0.04% error), and α(0)−1 = N2/2 + Tr(P)2 = 128 + 9 = 137 (0.03% error). The framework is falsifiable: FCC-ee will measure sin2 θW to 10−5 precision by ∼2040. All seven fundamental quantities emerge from one geometric structure with zero free parameters. PACS 12.10.-g · 11.25.Mj · 12.15.-y · 02.40.-k