Geometric problems require the passage from very natural energy bounds on curvature to very unnatural pointwise bounds. The long-standing approach has been to use the technology of analysis, yet this relies on the persistence of a geometric-analytic nexus, expressed concisely as the Sobolev constant, that is dicult or impossible to control in nature. In this talk we discuss a more intrinsically geometric way of approaching regularity questions on critical 4-manifolds.