Einstein Spacetime from the Distinguished Reproducing Kernel

Taking as upstream input the Distinguished Reproducing Kernel fixed-point law, which forces the realized branch together with its exact normalization package, and the already-derived exact-normalized quantum, semiclassical, and gravitational Hamilton–Jacobi sectors, this paper develops spacetime reconstruction from the closure-selected branch. The starting point is the closure-compatible Einstein-value metric sector, whose Hamilton–Jacobi equation, momentum constraint, scalar constraint, and DeWitt-type supermetric have already been selected downstream of exact closure. The paper shows that the associated deformation generators form the correct deformation algebra for a family of embedded hypersurfaces, so that the closure-compatible metric sector admits a genuine spacetime realization. Lapse and shift then emerge as the scalar and tangential deformation parameters of this reconstruction rather than as independent inputs. The paper next proves that the unique opposite-sign channel already selected by the Einstein-value supermetric on configuration space must be realized as the normal direction of the reconstructed spacetime, forcing Lorentzian signature. Under the Einstein-potential condition, the reconstructed Lorentzian sector carries the standard Einstein-form constraint and Hamilton–Jacobi system. The main theorem-level outputs are the hypersurface-deformation reconstruction theorem, the emergence of lapse and shift, the Lorentzian-signature theorem, and the identification of the reconstructed sector with the Einstein spacetime branch.