Family Boson Mass Spectrum in the $E_8$–$\mathbb{X}$ Unified Field Theory

We derive the mass spectrum of the family gauge bosons arising from the SU(3)F family symmetry in the E8–X Unified Field Theory. The symmetry breaking chain E8 → E6 × SU(3)F uniquely determines the vacuum expectation value of the X-field component Φ ∈ (1,8) as ⟨Φ⟩ = diag(φ₁, φ₂, φ₃) with φ₁ : φ₂ : φ₃ = 1 : λ : λ², where λ = 2π/h∨ = 0.2094 is fixed by the E8 dual Coxeter number h∨ = 30. The eight SU(3)F gauge bosons obtain masses Ma = gF |φᵢ - φⱼ|, yielding three distinct mass scales with universal ratios |λ-1| : |λ²-λ| : |λ²-1| ≈ 0.7906 : 0.1656 : 0.9562. All mass scales are controlled by the GUT breaking scale MGUT ≈ 10¹⁶ GeV, consistently with the axion decay constant fa = MGUT/(2π). This provides a parameter-free, falsifiable prediction for future collider searches.