(Periodic Euclidean region) In the situation of Lemma 3.19, in the region $\{|\vec{\mu}|_{a}\gtrsim A^{-1/4}\}\cap\mathcal{B}^{+}_{\nu}$

$$\left\lVert\nabla^{2}_{g_{a}}\Delta_{a}^{-1}f\right\rVert_{C^{k,\alpha}_{\delta,0}}\leq C\nu.$$

The constant only depends on $k,\alpha,\delta,\kappa$ and the scale-invariant uniform ellipticity bound on $a_{ij}$.