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 Sd = Tr (LA LB Φ · LA LB 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 NEIX^−2. With the second Seeley–DeWitt coefficient a₂ (EIX) = 1 175 384/15 as the geometric prefactor, the canonical normalisation κd/κ₂ = 1, and the electroweak-scale calibration of the Sakharov mechanism (mᵣad = vEW), we obtain δ (ΛℓP²) ≈ 2. 8 × 10^−122, within a few percent of the PDG 2024 value ΛℓP² ≈ 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.