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We introduce a new holographic map for encoding black hole interiors byincluding both fundamental and effective dynamics. This holographic map isconstructed by evolving a state in the effective, semiclassical gravitydescription of the interior backwards in time to pull the degrees of freedomoutside the black hole, before evolving forwards in time in the fundamentaldescription. We show this ``backwards-forwards'' map is equivalent to apost-selection map of the type introduced by Akers, Engelhardt, Harlow,Penington, and Vardhan, and in the case of trivial effective interactionsreduces to their model, while providing a suitable generalization when thoseinteractions are nontrivial. We show the map is equivariant with respect totime evolution, and independent of any interactions outside the black hole.This construction includes interactions with an infaller in a way thatpreserves the unitarity of black hole evolution exactly and does not allow forsuperpolynomial computational complexity.

Non-isometric codes for the black hole interior from fundamental and effective dynamics