Consider $\gamma_{4}$ at a given point in the region (4.14). Its integral formula (4.11) can be split into two parts, corresponding to far away sources $|(y_{1}-y_{1}^{\prime},y_{2}-y_{2}^{\prime},\mu)|_{a}^{\prime}\gtrsim A^{1/4}$ and nearby sources $|(y_{1}-y_{1}^{\prime},y_{2}-y_{2}^{\prime},\mu)|_{a}^{\prime}\lesssim A^{1/4}$.

For far away sources, we use Lemma 4.1 to write the integrand $\gamma$ as a dominant term $-\frac{1}{4\pi\sqrt{\mathbb{A}}|(y_{1}-y_{1}^{\prime},y_{2}-y_{2}^{\prime},\mu)|_{a}^{\prime}}$ plus a remainder term estimated by $\frac{C}{|(y_{1}-y_{1}^{\prime},y_{2}-y_{2}^{\prime},\mu)|_{a}^{\prime 3}}$. The dominant term does not contribute to $\gamma_{4}-\bar{\gamma}_{4}$ because it is constant in the $x_{1},x_{2}$ direction. The remainder term contribution to $\gamma_{4}$ is bounded by

The contribution from nearby sources only arises if our given point of interest is too close to $S$ along one of $\mathfrak{D}_{1}$, $\mathfrak{D}_{2}$ or $\mathfrak{D}_{3}$ directions; we focus on $\mathfrak{D}_{1}$. Lemma 4.2 allows us to write $\gamma$ as $-\frac{1}{8\pi^{2}|(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)|_{a}^{3}}$ plus a well controlled remainder term. By the exponential decay property of the measure $\text{Im}\sqrt{-1}d\eta_{2}^{\prime}\wedge d\bar{\eta}_{1}^{\prime}$,

Combining the above shows $|\gamma_{4}-\bar{\gamma}_{4}|\leq CA^{-1/4}$.

All these arguments carry through to $\gamma_{1},\gamma_{2},\gamma_{3}$ except the exponential decay of the measure. This is compensated by staying sufficiently far from $\mathfrak{D}_{i}$. ∎