We study critical Riemannian 4-manifolds with bounded Ricci curvature, but with no apriori analytic constraints such as on Sobolev constants. We derive elliptic-type estimates for the local curvature radius, which itself controls sectional curvature. The method is use degenerating, collapsing metrics to create a noncollapsed blow-up limit, and then use a geometric triviality result for complete Ricci-flat manifolds with a Killing field to rule out such a blow-up. The Cheeger-Tian ε-regularity theorem on Einstein manifolds is reproved as a byproduct.