Holonomy Without Curvature in E8 Backgrounds: Bordism Replaces π7-Clutching

We replace a false “π7(E8) ∼= Z120 clutching tower” narrative by the correct low-degreePostnikov truncationBE8 ≃15 K(Z, 4),and locate the genuine torsion relevant to flat-but-holonomic phases in reduced String bordismof BE8. Using Hill’s computation of ˜ΩString∗ (BE8) in low degrees, we show that order-2 torsionexists in degrees 9 and 10 but vanishes in degree 11:˜ΩString9 (BE8) ∼= Z/2, ˜ΩString10 (BE8) ∼= Z/2, ˜ΩString11 (BE8) = 0.We package Hill’s Landweber–Stong detector into a canonical Z/2 anomaly character αLS whosenontrivial value is −1, and we explain how loop-holonomy phases arise from evaluating anomalycharacters on the associated 10-manifolds M 9 × S1 (equivalently, mapping-torus constructions)even when all local Chern–Weil terms vanish on torsion. Finally, we contrast the pure BE8String no-go in degree 11 with Debray’s heterotic 2-group tangential structure, where degree-11torsion reappears and must be analyzed separately.