Covariant Holographic Entanglement Entropy Inversion to Reconstruct Bulk Geometry

We study when covariant holographic entanglement entropy determines a bulk radial geometry. We focus on stationary homogeneous three-dimensional geometries for which the Hubeny--Rangamani--Takayanagi (HRT) problem reduces to a one-dimensional radial variational problem. In this sector, the renormalized interval entropy $S(\Delta t,\Delta x)$ is an on-shell Hamilton--Jacobi functional. Its endpoint derivatives determine the conserved charges of the corresponding extremal geodesic, and their ratio organizes the data into fixed-$\kappa$ families. For each fixed $\kappa$, the entropy data define an Abel-type reconstruction of a radial metric block. A single classical geometry is obtained only when the reconstructions from different fixed-$\kappa$ families agree as functions of one common radial coordinate. This cross-family compatibility condition is the integrability condition of the covariant inverse problem. When it is satisfied, the reconstructed block determines the projected Lorentzian light cone, including frame dragging, the horizon generator, and the stationary-limit surface. We illustrate the construction with pure anti--de Sitter (AdS) space, rotating Ba\~nados--Teitelboim--Zanelli (BTZ) geometry, a static warped metric, a boosted Einstein--scalar black brane, a higher-dimensional strip example, and a thin-shell obstruction. The analysis is restricted to the classical HRT area term and to smooth single-branch data within the assumed radial reduction.