Gravitational Reconstruction in $AdS_{4}$

We reconstruct linearized gravitational perturbations in $AdS_{4}$ from a conserved, traceless stress tensor on a fixed conformally flat boundary metric, extending our earlier real Lorentzian construction. For admissible data with no spacelike Fourier support in a flat conformal frame, the physical response reduces to an integral over boundary light-cone cuts. Boundary improvements restore Fefferman–Graham (FG) gauge while preserving normalizability and the linearized Weyl tensor. We then reconstruct perturbations with general spacelike boundary momentum, including nonzero frequency and its zero-frequency limit, using explicit position-space generalized functions on real stress data. Boundary kernels are constructed for normalizable perturbations and sufficient conditions are derived for the reconstruction to converge. The results motivate controlled flat-space limits relevant to Bondi–Metzner–Sachs symmetries and celestial holography.