  We introduce a new holographic map for encoding black hole interiors by
including both fundamental and effective dynamics. This holographic map is
constructed by evolving a state in the effective, semiclassical gravity
description of the interior backwards in time to pull the degrees of freedom
outside the black hole, before evolving forwards in time in the fundamental
description. We show this ``backwards-forwards'' map is equivalent to a
post-selection map of the type introduced by Akers, Engelhardt, Harlow,
Penington, and Vardhan, and in the case of trivial effective interactions
reduces to their model, while providing a suitable generalization when those
interactions are nontrivial. We show the map is equivariant with respect to
time evolution, and independent of any interactions outside the black hole.
This construction includes interactions with an infaller in a way that
preserves the unitarity of black hole evolution exactly and does not allow for
superpolynomial computational complexity.
