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) ⊃ GSM applied to the tangent fibre (56, 2) identifies two anomaly-free 16ₒ₎ (₁₀) multiplets with standard quantum numbers (two Standard-Model generations) and their vector-like 1̄6 partners; the SU (5) -normalised weak mixing angle is sin²θW|₌₆ₔₓ = 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 16ₒ₎ (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.