It is an open question whether shock wave solutions of the Einstein Euler equations contain "regularity singularities'', i.e., points where the spacetime metric would be Lipschitz continuous ($C^{0,1}$), but no smoother, in any coordinate system. In 1966, Israel showed that a metric $C^{0,1}$ across a single shock surface can be smoothed to the $C^{1,1}$ regularity sufficient for spacetime to be non-singular and for locally inertial frames to exist. In 2015, B. Temple and I gave the first (and only) extension of Israel's result to shock wave interactions in spherical symmetry by a new constructive proof involving non-local PDE's. In 2016, to address most general shock wave solutions (generated by Glimm's random choice method), we introduced the "Riemann flat condition" on $L^\infty$ connections and proved our condition necessary and sufficient for the essential metric regularity to be smooth (i.e. $C^{1,1}$). In our work in progress, we took the Riemann flat condition to derive an elliptic system which determines the essential metric regularity at shock waves (and beyond). Our preliminary results suggest that our elliptic system is well-posed and we believe this system to provide a systematic way for resolving the problem of regularity singularities completely.