Stabilization Hierarchy, Boundary Complexity, and Gauge Group Dimensions

Why read this paper. Three facts, each striking on its own:(1) The electron-to-W-boson mass ratio is governed by the number of SM gauge bosons: me = e−12.0±0.03, dim􏰉U(1) × SU(2) × SU(3)􏰊 = 1 + 3 + 8 = 12. mW (2) The GUT hierarchy splits exactly along the symmetry-breaking tower: ln􏰍mW 􏰎 + ln􏰍MGUT 􏰎 = ln􏰍MGUT 􏰎 . me mW me 􏰝 􏰜􏰛 􏰞 􏰝 􏰜􏰛 􏰞 􏰝 􏰜􏰛 􏰞 ≈ 12 = dim GSM ≈ 33 = dim(SO(10)/GSM) ≈ 45 = dim SO(10) (3) The 􏰅ne-structure constant and the CMB spectral index satisfy: 1 + n = 138.001, accuracy 0.0007%; predictions: n(A) = 0.964001 or n(B) = 0.964632. αsss Behind these coincidences sit three independently proved mathematical theorems 􏰃 on spec- tral gaps (P1), proof complexity (P2), and boundary information (P3) 􏰃 each instantiating the same principle: topological indices determine logarithmic hierarchies. The connection between the mathematics and the physics is a conjecture, not a theorem; we present both with full rigor.