This work investigates whether projection-origin discrete geometryremains consistent with the conservation structure of generalrelativity when the shift sector of the ADM formulation is activated. A scalar potential generated independently from Einstein’s equationsthrough a projection-origin Poisson kernel is embedded into aweak-field metric. The analysis extends previous shift-freeconfigurations by introducing a minimal shift vector within astatic ADM framework. All geometric quantities, including the full four-dimensionalBianchi residual, are evaluated using the same lattice-consistentspectral operator that defines the projection-origin kernel.This unified operator framework avoids mixing discretizationschemes between the kernel and geometric sectors. For each shift amplitude the Bianchi residual continues to exhibita clear minimum with respect to the lapse parameter.The optimal lapse value deforms smoothly as the shift sector isactivated, while the residual suppression remains stable underlattice refinement. The analysis does not derive the Einstein field equations.Instead it provides numerical evidence that projection-origindiscrete geometry admits a consistent ADM extension in whichlapse and shift form a continuous consistency structure preservingthe discrete Bianchi closure in the weak-field regime. Note: Parts of the manuscript were linguistically and structurally refinedwith the assistance of AI-based tools.All scientific content, analysis, and conclusions are the author's own.
John Jude Hathway (Thu,) studied this question.