Shadow E8 Bundle and Gravity as a Gauge Theory in E8 ⊃ E6 × SU(3)F Unification

We extend the E8 ⊃ E6 × SU(3)F unification framework by introducing a secondary "shadow bundle" with structure group E8^(s), defined over a four-dimensional Lorentzian manifold. While the original E8 bundle (Papers I–XIV) unifies all Standard Model fields and their interactions, gravity is encoded in the shadow bundle through an embedding of the Lorentz group SO(1,3) into E8^(s). The connection on the shadow bundle contains the spin connection ω_μ^ab and a set of auxiliary fields; its curvature yields the Riemann tensor upon projection. By coupling the two bundles via a minimal interaction term, we construct a unified action that simultaneously reproduces the Einstein–Hilbert action (from the shadow bundle) and the Yang–Mills action for the Standard Model (from the original bundle). The coupling constant between the bundles is determined by the unification scale M_GUT = 1.2×10^16 GeV from Paper III and the group-theoretic factor c_R from Paper V. This formulation expresses gravity as an internal gauge symmetry while preserving the complete algebraic unification of matter and forces achieved in our earlier work. The shadow bundle construction provides a geometric interpretation of the gravitational coupling constant and opens a pathway toward a fully unified description of all interactions within a single principal bundle structure.