The periodicity condition is responsible for the exponential decay. Its effect becomes significant when $\eta\sim 1$, which is compatible with the length scale $|\vec{\mu}|_{a}\sim A^{1/2}$, or $\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D})\sim A^{1/2}$. The geometric significance of exponential decay is that the ansatz models the transition from fully quantum into semiflat behaviour (cf. review Section 1.3).