Paper FBW-E: Cosmological Epoch Structure and Gauge Group Selection from FSC Z4 Symmetry: Derivation of N_epoch = 7, alpha = 28/3, and G_GUT = Spin(10) from First Principles

Abstract We derive three structural theorems of cosmological epoch architecture from the Four-Sector Cosmological (FSC) Z4 symmetry, with zero free parameters. All results flow from the single master relation 9 = |Z4| + |Sub(Z4)| + |Gen(Z4)| = 4+3+2, established as the denominator of the FSC scaling exponent beta in Paper FBW-D. Theorem EQ-1 (Epoch Number Theorem): N_epoch = 9 - |Gen(Z4)| = 7. The |Gen(Z4)| = 2 generators form a CPT-conjugate pair producing no irreversible entropy increase; their exclusion leaves exactly seven macroscopic cosmological epochs, in agreement with the phenomenological count. Theorem EQ-2 (Frequency Hierarchy Theorem): alpha = N_steps/N_boundary = 56/6 = 28/3. A single algebraic exponent, derived as a corollary of Theorem EQ-1, reproduces the 32-decade frequency hierarchy. Six of seven predicted frequencies match observed epoch scales within one decade. The n=6 discrepancy (1.7 decades, BBN) is reinterpreted as a falsifiable prediction of BBN two-stage sub-structure at T~26 keV. Theorem EQ-3 (Gauge Group Selection, Remarkable Consequence): G_GUT = Spin(10), Z(Spin(10)) = Z4. Three independent mathematical bridges (centre isomorphism, McKay correspondence Z4 -> A-hat_3 <-> su(4), representation theory) converge on Spin(10). The Spin(10) breaking chain Spin(10) -> SU(4)xSU(2)xSU(2) -> SU(3)xSU(2)xU(1) maps isomorphically onto the FSC Phase I and Phase II collapses. Observational predictions: BBN two-stage structure (PIXIE/PRISM), proton decay p->e+pi0 (Hyper-K), gauge unification at 2e16 GeV (FCC-ee), N_gen=3 (confirmed at >5sigma). All results from R=5.4, |Z4|=4, and 9=4+3+2, with zero free parameters.