  We introduce a new algebraic framework for understanding nonperturbative
gravitational aspects of bulk reconstruction with a finite or
infinite-dimensional boundary Hilbert space. We use relative entropy
equivalence between bulk and boundary with an inclusion of nonperturbative
gravitational errors, which give rise to approximate recovery. We utilize the
privacy/correctability correspondence to prove that the reconstruction wedge,
the intersection of all entanglement wedges in pure and mixed states,
manifestly satisfies bulk reconstruction. We explicitly demonstrate that local
operators in the reconstruction wedge of a given boundary region can be
recovered in a state-independent way for arbitrarily large code subspaces, up
to nonperturbative errors in $G_N$. We further discuss state-dependent recovery
beyond the reconstruction wedge and the use of the twirled Petz map as a
universal recovery channel. We discuss our setup in the context of quantum
islands and the information paradox.
