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 $n\ge 4$. 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 codimension-four regularity conjecture. Moreover, the nonmanifold locus is closed and has locally finite $(n-4)$-dimensional Hausdorff measure, and its intersection with each bounded ball satisfies a tubular-volume estimate of order $r^4$. In dimension four, the nonmanifold points form a locally finite set. A flat quotient example shows that the codimension bound is sharp.