  We propose a CFT definition of local observables in both the exterior and
interior of bulk black holes, whenever such an interior exists. We achieve this
by introducing a small microcanonical black hole as a "probe" and using its
modular flow to propagate operators from the asymptotic boundary to the
interior of other black holes along its worldline, elaborating on the ideas of
[2009.04476]. The key conceptual advance is a CFT criterion for selecting
states whose modular flow acts as geometric proper time translation in the
bulk, which we dub "local equilibrium" states. Our interior reconstruction
depends on the choice of code subspace but not on the specific black hole
microstate and does not suffer from the "frozen vacuum" problem of other
approaches. By virtue of our construction, the question of firewall typicality
reduces to a technical problem we articulate and we identify a CFT correlator
that is expected to signal the approach to the black hole singularity via a
universal divergence. We end with comments on the utility of our framework to
the quest for a quantum description of de Sitter cosmologies.
