Fully lifted blirp interpolation -- a large deviation view

[104] introduced a powerful fully lifted (fl) statistical interpolating mechanism. It established a nested connection between blirps (bilinearly indexed random processes) and their decoupled (linearly indexed) comparative counterparts. We here revisit the comparison from [104] and introduce its a large deviation upgrade. The new machinery allows to substantially widen the [104]'s range of applicability. In addition to typical, studying analytically much harder atypical random structures features is now possible as well. To give a bit of a practical flavor, we show how the obtained results connect to the so-called local entropies (LE) and their predicated role in understanding solutions clustering and associated computational gaps in hard random optimization problems. As was the case in [104], even though the technical considerations often appear as fairly involved, the final interpolating forms admit elegant expressions thereby providing a relatively easy to use tool readily available for further studies. Moreover, as the considered models encompass all well known random structures discussed in [104], the obtained results automatically apply to them as well.