For some reason, I missed the part "Because inner-layer moves suck" when I first saw your post, but looking at it again since unsolved replied,

Standard algorithms have

**5** inner layer slice turns, because, for example, instead of having

**7** inner slice turns with the "old standard" alg:

3R2 B2 U2 3L U2 3R' U2 3R U2 F2 3R F2 3L' B2 3R2
we can instead make the first and last inner slice turns into wide turns and achieve the same result:

3r2 B2 U2 3L U2 3R' U2 3R U2 F2 3R F2 3L' B2 3r2
[HR][/HR]It seems like you want algorithms which handle this "pure" case which have fewer than 5 inner layer slice turns.

I will list algorithms that I found by hand which conform this this idea.

Here are two very short algorithms (the first I named "cmowlaparity") which only contain

**3** inner layer slice turns. (Note that you may convert the inner slice turns of the first alg into wide turns for traditional speedsolving use, but not in the second one.)

x' 3r2 U2 3l' U2 3R U2 3r U2 x' U 3R U' F2 U 3R' U 3r2 x
x' 3r2 U2 3l' U2 3R U2 3r U2 x' U' 3R U F2 U' 3R' U' 3r2 x
If you want algorithms which only contain 3 inner layer slice turns that only contain face turns of U, here is one such alg (longer):

3r U2 3r2 U L' U 3R U' L U2 L' U 3R' U' L U 3r' U2 3R U2 3r' U2 3r'
[HR][/HR]If you want algorithms which only contain

**2** inner layer slice turns, here are two such algs. (Even longer).

3r U2 3r U 3r U2 3r U2 3r' U2 3r' U 3R' U' 3r U2 3r U2 3r' U2 3r' U 3R2 U2 3r' U2 3r'
3r U2 3r U' 3r U2 3r U2 3r' U2 3r' U' 3R' U 3r U2 3r U2 3r' U2 3r' U' 3R2 U2 3r' U2 3r'
[HR][/HR]I'm pretty sure algorithms which only contain 1 inner slice turn for this case do not exist/are impossible,

despite that **0** inner layer slices is possible for the 4x4x4 case and

the 5x5x5 case.

(Note, however, that 1 inner slice turn is sufficient for the

adjacent double parity case (I also found this by hand), for example.)

Note that I did not find these two linked 0 inner layer slice algs by hand: they were found using k-solve--the 4x4x4 alg by Bruce Norskog and the 5x5x5 alg by Ben Whitmore.

I do note, Lucas Garron as my witness, that I was the first to publicly post such

an alg for the 4x4x4 case (which I did find by hand) which is nearly 200 moves long (my first was about 1100 moves long, but then I decreased it to about 200 moves before the optimal algorithms were found with k-solve). You can find my explanation for how I found this in the "Derivation" spoiler within the "2-Gen 2-Cycles" spoiler in

this post.

Happy solving!

[HR][/HR]EDIT (hopefully my final one):

Just in case you're curious, based on the 0 inner slice turn algs I linked to above, I did find a relatively brief pure 3 dedge flip algorithm by hand which contains

**0** inner layer slices.

r U2 r U2 r U R U r U2 r' U' R' U' r' R' U R U r U2 r' U' R' U' r U2 r' U2 r'
In addition, if you solve the last layer of the 4x4x4 by solving corners first and dedges last, here is a pretty fast 3 flip double parity I found by hand using a similar idea to the above which only contains only

**2** inner layer slices.

r' U2 2R U2 r' x' U2 2R' U' R' U' r' U2 r U R U' r R U2 x