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Everything you can do on a real 3x3 you can also do on a pseudo 3x3 (let's say on a 4x4). But a pseudo 3x3 allows more moves than a real 3x3. Can this save moves? Is there a 3x3 case that can be solved on a pseudo 3x3 in fewer moves than on a real 3x3?

If there is one I can't see it, any non-3x3 move done on a pseudo3x3 state will incorporate some 3x3 moves but also require something else to be undone. So it's ultimately going to be less efficient.

So we now have two more examples besides Stefan's.

EDIT:
I used cuBerBruce's single slice turn 4x4x4 optimal solver with my PLL parity algorithm as a setup, and it gave the following unique solutions (which are all related to my alg):