Discreteness of non‑spherical cross‑section points in 4‑dimensional Ricci limits

Prove that in a noncollapsed Ricci limit space (X,d) of dimension 4, the set of points at which some tangent cone has a cross‑section not homeomorphic to S^3 is discrete.

Background

Theorem 1.1 shows that in four dimensions, the cross-sections of tangent cones are all homeomorphic to a fixed spherical space form S^3/Γ, and the non-spherical set has Hausdorff dimension zero.

Strengthening this dimension‑zero result to discreteness would sharpen the picture of singularities in 4D Ricci limits, paralleling stronger regularity known under two‑sided Ricci bounds and recent Kähler settings.