The Einstein–Completion Equations: A Natural Extension of General Relativity

Einstein's field equations describe how mass-energy determines spacetime curvature, but they leave unresolved the origin of curvature, the nature of inertia, and the localization of gravitational energy. We introduce the Einstein–Completion Equations (ECE), which extend general relativity by adding a covariant gravitational stress tensor Θ_{μν} derived from a substrate free-energy density. This framework unifies curvature, inertia, and conservation into a single momentum ledger. The ECE reproduce all tested predictions of general relativity, while offering a unique falsifiable prediction: a universal weak-lensing curl floor absent in standard GR. Thus, the ECE close Einstein’s original program and provide a concrete path for experimental verification with upcoming surveys such as Euclid and Rubin/LSST.