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I will present our theory of the {\it Regularity Transformation (RT-)equations}, an elliptic system of partial differential equations which determines coordinate and gauge transformations that remove apparent singularities in spacetime by establishing {\it optimal regularity} for general connections. This gain of one derivative for the connections above their $L^p$ curvature then suffices to establish {\it Uhlenbeck compactness}. By developing an existence theory for the RT-equations we prove optimal regularity and Uhlenbeck compactness in Lorentzian geometry, including general affine connections and connections on vector bundles with both compact and non-compact gauge groups. As an application in General Relativity, our optimal regularity result implies that the Lorentzian metrics of shock wave solutions of the Einstein-Euler equations are non-singular---geodesic curves, locally inertial coordinates and the Newtonian limit all exist in a classical sense---, resolving a longstanding open problem in the field.