One of the things with the EOpairing is that when you have the 2 edges swap left, you can fix OLL parity with these 2 algs:
Final 4 edges - oriented (dedge in UL flipped)
r' F R2 U' R' F' U r
r' F U' R' F' R2 U r
Final 4 edges - flipped (dedge in UL flipped)
r' U' F R' U F' r F U2 R' F'
r' F U' R F' U r R' B L' U2 B'
I use these a lot and its great.
None of these cases are actually OLL parity cases. But I'll get to that. This is a very long post, but hopefully you'll be able to make sense of it.
In summary
- Before pairing is finished, recognising OLL parity is not as simple as counting flipped edges.
- Edge orientation on 4x4 is weirder than you think.
First, it is absolutely essential to realise that there is no such thing as edge orientation on a 4x4. To prove it, take a single edge piece (physically pop it out of the cube) and try put it back in the wrong orientation. You'll see it only goes in one way. Only dedges have "orientation", which can be changed by swapping the two edges that make up that dedge. Once you understand that, it's obvious that OLL parity has to be connected to the swapping of individual edge pieces.
And that's exactly how OLL parity is defined. OLL parity is where an odd number of edges need to be swapped to solve the edges. If an even number of edge swaps occur, then it's by definition not OLL parity. It therefore follows that OLL parity cannot be solved by an even number of slice moves (
count them here - every single OLL parity alg has an odd number of slice moves), since slice moves cycle 4 edges (An odd number of swaps: 1&2, 2&3, 3&4).
Before edge pairing is finished, it can be really hard to recognise OLL parity without actually tracing the cycles (have a look at how parity avoidance works - it's a really cool technique if you can handle it - but even if you can't use it in speed solves (I definitely can't), it will help you understand parity a lot better).
Let's look a the first case you give as an example - the one solved by r' F R2 U' R' F' U r
I this case, you can pair and orient the edges with two swaps: swap both orange-blue edges, and swap the back yellow-blue edge with the front yellow-green edge. Because it's an even number of swaps, it's not an OLL parity case. That means any edge pairing alg involving slice, do something, slice back (ie - the natural and most move efficient way) won't change parity.
Here's another case that demonstrates what I'm talking about - it's just a three cycle of edges. Do it three times, and you should get back to the solved cube.
r' R U' R U F R2' F' R' r F R2' F' U' R' U
Do the alg once on a solved cube and you should get this case:
Again, this is not an OLL parity case, even though it looks like it is. If you look at the yellow-blue edge on the front face, it looks flipped. Perform the alg again, and you should get this case
You can see that the two blue-yellow edges look flipped. Do the alg a third time, and you get back to the solved cube.
If the alg actually did change the OLL parity state, then doing it three times would be - solved (no parity) - parity - no parity - solved (parity), and since there's obviously no parity on a solved cube, there's an obvious contradiction here. The alg does not affect OLL parity.
So, none of the cases you gave are parity cases, and the algs do not avoid OLL parity. OLL parity will still occur in 50% of your solves.
Anyway, on to the most important implication - if individual edge pieces don't have orientation, then how does EOpairing work for Z4? Good question. You're recognising the EO of the dedges that will be produced by the pairing, not the orientation of the individual edge pieces. The key to EOpairing is knowing how your EO algs affect EO of dedges. For that to be useful, you also have to look not only at the edges you're pairing, but also at the edges in the places that will be flipped. I find that's a pain and really interferes with lookahead. I can do it early on in the edge pairing, when I don't really care about edges I haven't paired yet, but later on, when I've got to worry about making sure I don't accidentally flip an edge that's already paired and oriented, it's just more hassle than it's worth.
I think a far far better strategy is to prioritise chain pairing (preferably 3-2-3 like in Yau) over EO. It is so simple to fix edge orientation after pairing - almost always 5 moves or less - whereas if you break the chain pairing just once, it's a guaranteed 5 extra moves at least. OLL parity doesn't count here, because as you've hopefully realised from earlier, you can't recognise OLL parity using standard ZZ EO recognition on individual edges until edge pairing is almost complete.
Anyway, I think you're on the right track. I love ZZ - it's why I've also spent so much time trying to get it working. Of the 4x4 ZZ variants I've seen I'm convinced Hoya+ZZ has the most potential. Keep thinking out of the box
