Emergent fermion sector of an E₈ group field theory: three generations and Hopfion statistics on the Wolf space 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 BEC orbit, we construct the emergent fermion sector. No fundamental fermionic field is postulated. The construction relies on a twistor-geometric identification: the BEC orbit E₈/ (E₇ × U (1) ) is canonically isomorphic to the Salamon twistor space Z (EIX) of the quaternion-Kähler Wolf space EIX = E₈/ (E₇ × SU (2) ). The pullback of 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. 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, corresponding to two Standard-Model generations. A third generation arises from the topologically protected zero modes of the induced Dirac operator on a BHopf = 1 Hopfion background: the Atiyah–Singer index on the Sp (1) -factor is 1, yielding 56 net chiral zero modes (a complete 56₄䃗 containing one 16ₒ₎ (₁₀) ). Fermionic statistics follows from the Finkelstein–Rubinstein mechanism (π₄ (S²) = Z₂) in the topological sector and from the Wightman spin-statistics theorem in the linearised sector.