(Exponential decay for higher Fourier modes in the first order ansatz) If $\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D}_{1})\gtrsim A^{1/2}$, then

(3.8)

$$|\tilde{\alpha}_{1}-\bar{\alpha}_{1}|\leq CA^{-3/4}\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D}_{1})\exp(-2\pi A^{-1/2}\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D}_{1})).$$

Similar bounds hold for $\tilde{\alpha}_{i}-\bar{\alpha}_{i}$ for $i=1,2,3$.