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: M1,3 → EIX yields the spinor bundle of the emergent Lorentz space-time. The Slansky chain E₇ ⊃ E₆ × U(1) ⊃ SO(10) ⊃ SU(5) ⊃ G_SM applied to the tangent fibre (56, 2) identifies two anomaly-free 16SO(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 B_Hopf = 1 Hopfion background: the Atiyah–Singer index on the Sp(1)-factor is 1, yielding 56 net chiral zero modes (a complete 56E₇ containing one 16SO(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.