Unified Einstein–Yang–Mills Theory from 𝐸8⊃𝐸6×SU(3)𝐹 with Gravity

We present a complete proof that the noncompact real form 𝐸8(−24) of the exceptional Lie group 𝐸8, when decomposed as 𝐸8(−24)⊃𝐸6×SU(3)𝐹, naturally contains the Lorentz algebra 𝑠𝑜(3,1) and gives rise to Einstein's general relativity alongside the Standard Model gauge interactions. Using the decomposition 𝐸8(−24)⊃𝑆𝑝𝑖𝑛(7,3)⊃𝑆𝑝𝑖𝑛(1,3)⊗𝑆𝑝𝑖𝑛(6) (Wilson 2025), we identify the subalgebra 𝑆𝑝𝑖𝑛(1,3)≅𝑆𝐿(2,𝐶) as the Lorentz generators, embedded in 𝑆𝐿(4,𝑅) which also provides the translation generators 𝑃𝑎. We show that the tetrad 𝑒𝜇𝑎 and spin connection 𝜔𝜇𝑎𝑏 emerge as components of an 𝐸8(−24) connection. Varying the 𝐸8 Yang–Mills action with respect to these fields yields the Einstein field equations (with possible higher‑curvature corrections suppressed by the unification scale) and identifies the stress–energy tensor with the Standard Model contribution. The derivation is purely algebraic, requires no extra dimensions, and is fully compatible with the previous results on three generations, fermion mass hierarchy, and gauge coupling unification. We also provide a detailed group-theoretic calculation of the coefficient 𝑐𝑅 relating Newton's constant to the unification parameters, using the precise values 𝑀GUT=1.2×1016 GeV and 𝛼GUT−1=40 from Paper III (revised), and show that the observed value emerges naturally from threshold factors without fine-tuning. The paper also discusses the compatibility with the dark matter candidate (1,1)0, pointing to the revised Higgs portal mechanism in Paper IV and the first-principles derivation in Paper VII, and briefly comments on the cosmological constant (addressed fully in Paper VII).