4Z4

The 4Z4 method is a 4x4 speedsolving method for ZZ method solvers. It starts off with a redux stage very similar to the Yau method's, but it ends with a ZZ finish, giving the preferable method for ZZ solvers while retaining the advantages that Yau has. It is a definite improvement over other ZZ based 4x4 methods such as Z4 and NS4 due to a lower movecount and better ergonomics.
Currently, there are no notable solvers with 4Z4 as it is quite a new method, as well as there being very few ZZ solvers who are fast at 4x4.
Contents
Steps
 Solve 2 opposite centres. These will be your L and R colours that you use for ZZ on 3x3.
 Solve 3 pseudo cross edges on L. (These have to meet a specific requirement and will be explained later.)
 Solve last 4 centres
 Solve final pseudo cross edge
 Pair last 8 edges using 323 edge pairing or your preferred edge pairing technique
 Solve line. You do this by positioning the line edge that isn't in the pseudo cross in DR, and using an L move to place the other one at DL. You then do a D/D' to solve the line, then an L2 to positiont the pseudo cross in DL.
 Solve edge orientation and do OLL parity
 ZZ F2L
 COLL
 EPLL+Parity
The Pseudo Cross
The pseudo cross needs to have these specific edges:
 1 line edge
 The left cross edge (and should match with your left centre)
 2 edges that don't meet the other requirements (any E slice edge, any U layer edge and the opposite cross edge)
It is generally considered better if you solve the line edge and the cross edge in the same relationship to each other every time (for example, opposite). Also, you don't technically have to solve the cross edge, but it limits the amount of L2 moves you have to do later on.
Edge Orientation
The way you do edge orientation is identical to the way in Petrus, just you need to stretch the definition of a bad edge to any edge. You hold the cube so that the line and pseudo cross are in BD (which means tha your L/R colours are on F/B).
There are 2 types of edges:
 E slice
 U/D layer
If an E slice edge is in E, and the colour on F is opposite or the same as the centre colour, it is good. If it is not, it is bad. The inverse is true if the edge is on U/D. If the colour on U/D is opposite or the same to the colour of the centre on F, it is bad. If not, it is good.
For a U/D edge, if the U/D colour is the same/opposite as the U/D centre and is on U/D it's good. If the U/D colour isn't matching to the F centre, it is good, but if it is, it's bad.
You use R/L to flip edges, U/F to replace them, then undo the flip.
The parity algorithm flips one edge and doesn't preserve F2L. You hold the edge in UF and with the line in the normal position, so you essentially orientate as many edges in petrus style as possible, then rotate to put your line in the normal position, do parity, then continue. This is the algorithm: Rw U2 Rw2 U' Rw' U2 Rw U2 Rw' U Rw2 U2 Rw2 U R2 U' Rw'
NB. If you are good at LEOR, the line and EO step can be merged into one.
Last Layer
For the last layer, you can use the same COLL algs as on 3x3, as you don't need to worry about orientation parity due to it being solved already.
After COLL, you have EPLL+parity. There are the 4 normal EPLL cases (Hperm, Uaperm, Ubperm, Zperm), plus the additional opposite swap, adjacent swap, Oa, Ob and W permutations.
r2 U2 r2 Uw2 r2 u2  (12,6)  N  Chris Hardwick  [1] 
(Rw2 F2 U2) r2 (U2 F2 Rw2)  (14,7)  N  Stefan Pochmann  [2] 
(R U R' U') r2 U2 r2 Uw2 r2 Uw2 (U' R U' R')  (20,14)  N  Chris Hardwick  [3] 
(R U R' U') (Rw2 F2 U2) r2 (U2 F2 Rw2) (U R U' R')  (22,15)  N  Stefan Pochmann  [] 
M2 U M2 U M' U2 l2 U2 r2 Uw2 r2 u2 M'  (22,13)  N  [4] 
M2 U' M2 U' M' U2 l2 U2 r2 Uw2 r2 u2 M'  (22,13)  N  [5] 
(U') R' U R' U' R' U' R' U R U' Uw2 r2 Uw2 r2 U2 Rw2  (21,16)  SP04  N  Stefan Pochmann  [6] 
Do the correct algorithm out of the 9, and you're done!
Example solve
Scramble: R Uw B' Uw' Rw' B2 Uw2 B' Uw' L2 D' Fw' B F2 Uw' B2 F' U2 F D' Uw2 R' Rw L2 Fw B' Rw' U' L2 Rw' B' U D Uw R' F' Rw2 F2 B' Uw'
z//inspection
D U Rw' F' U2 Rw2 U Rw Uw2 U' y Rw U2 Rw'//F2C, L&R colours
z F R U Rw U Rw' L' F//Pseudo cross
L2 3Rw' U' 3Rw2 U Rw U' Rw2 3Rw U 3Rw2 U Rw U' Rw' 3Rw2 Rw2 U2 Rw2 U' 3Rw' Rw U Rw' Lw' L U2 Rw// L4C
z' y' Uw2 R' U' R Uw2 F2//Solve pseudo cross
Uw' F' U F L' U2 L y2 Uw2 U F U' F' Uw' y R U R' Uw' R U2 R' Uw//Pair edges
z R2 L2 D L2//Solve line
y L F L' y' R2 U Rw U2 Rw2 U' Rw' U2 Rw U2 Rw' U Rw2 U2 Rw2 U R2 U' Rw'//Edge orientation
R' U R U L U L' R' U' R' U2 R' U R' U2 R U2 R' U R L' U' L//ZZ F2L
U R' U L U' R U L'//COLL
Rw2 F2 U2 Rw2 R2 U2 F2 Rw2 U2 //EPLL+Parity [7]
142 moves with double parity.
Advantages and Disadvantages
Advantages:
 Gives a ZZ finish for a similar movecount to Yau, with all of the advantages Yau has.
 Better ways to deal with parity. The number of algs used in a solve with double parity is one less than with standard Yau with double parity, and PLL parity recognition is easier due to corners being solved.
 The EO recognition is suitable for doing midsolve, unlike with standard ZZ.
 Due to the freedom of having a wider selection of edges for pseudo cross, it is more efficient than yau cross.
Disadvantages:
 Pseudo cross is more abstract than Yau cross, and therefore could be potentially slower as the recognition is harder.
 More steps to do after reduction is done (2: line and EO).
 Fewer pieces are directly solved in pseudo cross and EO compared to Yau and Hoya and Meyer.