Mass Hierarchy of Three Fermion Generations from SU(3)𝐹 Breaking in 𝐸8⊃𝐸6×SU(3)𝐹 Unification: A Quantitative Analysis with Improved Flavor Structure

Why do fermion masses span five orders of magnitude, from the top quark to the electron? This paper provides a quantitative explanation of the fermion mass hierarchy within the 𝐸8⊃𝐸6×SU(3)𝐹 unification framework. The family symmetry SU(3)𝐹 is broken by the vacuum expectation value of its adjoint field Φ. By numerically minimizing the most general renormalizable SU(3)-invariant potential, we determine the VEV ratios ∣𝜙2/𝜙1∣=𝜆=0.22±0.05 and ∣𝜙3/𝜙1∣=0.78±0.05. This hierarchy directly translates into fermion mass ratios. The robust predictions 𝑚𝑐/𝑚𝑡=𝜆=0.22±0.05 and 𝑚𝜇/𝑚𝜏=𝜆=0.22±0.05 agree with experimental values at the 1𝜎 level. For first-generation masses, a natural seesaw-like mixing with exotic states (masses ∼𝜆𝑀GUT) provides the required 10−3 suppression. The exotic states acquire GUT-scale masses and decouple, while the 𝑍2-protected singlet (1,1)0 remains light as a dark matter candidate. This work establishes a deep connection between flavor physics and gauge coupling unification, as the same exotic states contribute to threshold corrections in Paper III.