For every $\epsilon>0$ there exists $\delta({\rm v},\epsilon)>0$ such that if $M^{4}$ satisfies $|{\rm Ric}_{M^{4}}|\leq 3\delta$, ${\rm Vol}(B_{1}(p))>{\rm v}>0$ and $|\mathcal{V}^{\delta}_{4}(p)-\mathcal{V}^{\delta}_{1/4}(p)|<\delta$, then there exists a discrete subgroup $\Gamma\subseteq{\rm O}(4)$ with $|\Gamma|\leq N({\rm v})$ such that the following hold:

1. (1)

   For each $x\in A_{\epsilon,2}(p)$ we have the harmonic radius lower bound $r_{h}(x)>r_{0}({\rm v})\epsilon$.

2. (2)

   There exists a subset $A_{\epsilon,2}(p)\subseteq U\subseteq A_{\epsilon/2,2+\epsilon}(p)$ and a diffeomorphism $\Phi:A_{\epsilon,2}(0)\to U$, with $0\in\mathds{R}^{4}/\Gamma$, such that if $g_{ij}=\Phi^{*}g$ is the pullback metric then

   $\displaystyle||g_{ij}-\delta_{ij}||_{C^{0}}+||\partial_{k}g_{ij}||_{C^{0}}<\epsilon\,.$

   (8.4)