From quaternion-Kähler holonomy to Standard-Model fermions: spinor bundle, three generations, and emergent statistics on EIX

Within an adjoint E₈ group field theory on the compact real form, whose bosonic sector selects the quaternion-Kähler Wolf space EIX = E₈/(E₇ × SU(2)) as the unique vacuum manifold, we construct the complete emergent fermion sector. No fundamental fermionic field is postulated. The BEC orbit E₈/(E₇ × U(1)) is canonically isomorphic to the Salamon twistor space Z(EIX); pulling back the holonomy Sp(1)-bundle along the sigma-model field ψ: M^{1,3} → EIX yields the spinor bundle of the emergent Lorentz space-time, with an automatic spin structure (w₂ = 0) on any orientable base manifold. The Slansky chain E₇ ⊃ E₆ × U(1) ⊃ SO(10) ⊃ SU(5) ⊃ G_SM applied to the tangent fibre (56, 2) identifies two anomaly-free 16_{SO(10)} multiplets with standard quantum numbers (two Standard-Model generations) and their vector-like 1̄6 partners; the SU(5)-normalised weak mixing angle is sin²θ_W|_{M_GUT} = 3/8. A third generation arises from the topological sector: the Atiyah–Singer index on the instanton background HP¹ ↪ EIX, transferred to the Hopfion via the Atiyah–Manton bridge, gives 54 net chiral zero modes containing at least one complete 16_{SO(10)}. Fermionic statistics follows without an independent postulate: the Finkelstein–Rubinstein mechanism (π₄(S²) = ℤ₂) in the topological sector and the Wightman spin-statistics theorem in the linearised sector both reduce to the same physical chain: quaternion-Kähler holonomy endows the modes with half-integer spin, and the emergent Lorentz structure forces anti-commutation.