(Exponential decay for higher Fourier modes in the first order ansatz) In the region where $\text{dist}_{g_{a}^{\prime}}(\cdot,\text{Im}(S))\gtrsim A^{1/4}$,

(4.18)

$$|\gamma_{i}-\bar{\gamma}_{i}|\leq CA^{-1/4}\tilde{\ell}e^{-\tilde{\ell}},\quad\tilde{\ell}=\kappa_{a}\text{dist}_{g_{a}^{\prime}}(\cdot,\text{Im}(S)),\quad i=1,2,3,4,$$

where $\kappa_{a}$ is the minimum of $k_{n_{1},n_{2}}=2\pi(\frac{Aa^{p\bar{q}}n_{p}n_{q}}{\mathbb{A}})^{1/2}$ for all $(n_{1},n_{2})\in\mathbb{Z}^{2}\setminus\{0\}$.