$L^2$ curvature bounds on manifolds with bounded Ricci curvature

Consider a Riemannian manifold with bounded Ricci curvature $|\Ric|\leq n-1$ and the noncollapsing lower volume bound $\Vol(B_1(p))>\rv>0$. The first main result of this paper is to prove that we have the $L^2$ curvature bound $\fint_{B_1(p)}|\Rm|^2 < C(n,\rv)$, which proves the $L^2$ conjecture. In order to prove this, we will need to first show the following structural result for limits. Namely, if $(M^n_j,d_j,p_j) \longrightarrow (X,d,p)$ is a $GH$-limit of noncollapsed manifolds with bounded Ricci curvature, then the singular set $\cS(X)$ is $n-4$ rectifiable with the uniform Hausdorff measure estimates $H^{n-4}\big(\cS(X)\cap B_1\big)