Codimension-four regularity of noncollapsed Ricci limit spaces under an integral volume-deficit bound

Let $(X,d,p)$ be an $n$-dimensional noncollapsed Ricci limit space, where $n\ge4$. Under an integral volume-deficit bound, we prove that the metric singular set has Hausdorff codimension at least four and sigma-finite $(n-4)$-dimensional Hausdorff measure. This establishes a special case of the codi