One-loop vacuum energy from the (2,4) sector of an E₈ group field theory on the Wolf space EIX

The companion paper on the foundations of an adjoint E₈ group field theory adopts a first-derivative constraint which restricts the Lagrangian to at most first derivatives of the field. Together with the remaining postulates, this constraint fixes a unique eleven-coefficient action and, in particular, eliminates the entire (2,4)-derivative sector generated by S_d = Tr(L_A L_B Φ · L^A L^B Φ). We examine the consequences of relaxing this selection. At the quantum level, higher-derivative operators are generically present in the effective action; their absence would require a non-trivial cancellation rather than being a natural default. The one-loop vacuum-energy cancellation, proven for the constrained action on the Wolf space EIX = E₈/(E₇ × SU(2)), is broken by the (2,4)-sector perturbation. The residuum is controlled by two independently computable suppression factors: the one-loop volume factor (4π)^{−56} from the 112-dimensional internal space and the Sakharov induced-gravity ratio N_EIX^{−2}. With the second Seeley–DeWitt coefficient a₂(EIX) = 1 175 384/15 as the geometric prefactor, the canonical normalisation κ_d/κ_2 = 1, and the electroweak-scale calibration of the Sakharov mechanism (m_rad = v_EW), we obtain δ(Λℓ_P²) ≈ 2.8 × 10^{−122}, within a few percent of the PDG 2024 value Λℓ_P² ≈ 2.9 × 10^{−122}. The principal assumption is the electroweak calibration of the radion mass; with the Planck-scale alternative, the same formula overshoots by approximately 62 orders of magnitude. The quasi-geometric structure of the heat-kernel coefficients on EIX yields a testable prediction: a₃(EIX) ≈ 280³. Verification scripts are available at https://github.com/lukasbednarik/E8-GFT.