Boundary Ball Readouts and Linearized Einstein Gravity in a Source–Readout Architecture

This paper reconstructs the Einstein-gravity branch of Faulkner and colleagues’ derivation of linearized gravity from the entanglement first law within Shadow Theory’s source–readout architecture. A single bulk metric perturbation supplies two readouts for each boundary ball: the first variation of its extremal surface area and the variation of its asymptotic gravitational charge. Their difference is a weighted integral of the linearized Einstein residual. The paper derives this identity by covariant phase space and checks it independently using the scalar-curvature variation on a time-symmetric slice. The paper determines what different families of boundary balls can recover. All centres and radii on one time slice determine the Hamiltonian constraint on that slice. A finite family of suitably chosen Lorentz frames, using all time slices in each frame and the Bianchi and boundary Ward identities, recovers the full linearized Einstein equation. Explicit non-gauge perturbations invisible to fixed-frame readouts show why recovering the field equation and reconstructing the metric are distinct problems.