  A central question in General Relativity (GR) is how to determine whether
singularities are geometrical properties of spacetime, or simply anomalies of a
coordinate system used to parameterize the spacetime. In particular, it is an
open problem whether there always exist coordinate transformations which smooth
a gravitational metric to optimal regularity, two full derivatives above the
curvature tensor, or whether regularity singularities exist. We resolve this
open problem above a threshold level of smoothness by proving in this paper
that the existence of such coordinate transformations is equivalent to solving
a system of nonlinear elliptic equations in the unknown Jacobian and
transformed connection, both viewed as matrix valued differential forms. We
name these the Regularity Transformation equations, or RT-equations. In a
companion paper we prove existence of solutions to the RT-equations for
connections $\Gamma\in W^{m,p},$ curvature ${\rm Riem}(\Gamma) \in W^{m,p}$,
assuming $m\geq1$, $p>n$. Taken together, these results imply that there always
exist coordinate transformations which smooth arbitrary connections to optimal
regularity, (one derivative more regular than the curvature), and there are no
regularity singularities, above the threshold $m\geq1$, $p>n$. Authors are
currently working on extending these methods to the case of GR shock waves,
when gravitational metrics are only Lipschitz continuous, ($m=0$, $p=\infty$),
and optimal regularity is required to recover basic properties of spacetime.
