  The backwards-forwards map, introduced as a generalization of the
non-isometric holographic maps of the black hole interior of Akers, Engelhardt,
Harlow, Penington, and Vardhan to include non-trivial dynamics in the effective
description, has two possible formulations differing in when the post-selection
is performed. While these two forms are equivalent on the set of dynamically
generated states -- states formed from unitary time evolution acting on
well-defined initial configurations of infalling matter -- they differ on the
generic set of states necessary to describe the apparent world of the infalling
observer. We show that while both versions successfully reproduce the Page
curve, the version involving post-selection as the final step, dubbed the
backwards-forwards-post-selection (BFP) map, has the desirable properties of
being non-isometric but isometric on average and providing state-dependent
reconstruction of bulk operators, while the other version does not. Thus the
BFP map is a suitable non-isometric code describing the black hole interior
including interior interactions.
