Z4

Z4 is a reductionbased method for solving the 4x4x4 using ZZ principals: minimal rotations and F/B moves.
Contents
Steps
A full description is available here.
 Solve Centres
 EOpair DF/DB and place
 EOpair 10 remaining edges
 Flip final bad dedges (including orientation parity)
 ZZF2L
 COLL
 EPLL including PLL parity (1 step)
Necessary Principles
EO Pairing
The basic idea is to orient edges while pairing them. This is achieved by using a varient of Robert Yau's edge pairing method, and solving edges using only the substet <l,L,r,R,U>.
Take the following as an example:
As you can see, the 2 dedges in UF are oriented, and so is the white sticker in UB and the blue sticker in FR. The other 2 dedges are not oriented, so these would need to be flipped.
The moves used to EOPair all three of these edges would be Lw U' R U Lw. Let me explain how this works. the first L move EOPairs the YO dedge, which you can see because the unoriented orange sticker is matched up with the oriented yellow sticker in FUr. This pair is then moved out of the way and replaced with the FR dedge by the U' R U moves. These moves also orient the dedge correctly, so that the two remaining edge pairs are created by the Lw' move, and are also oriented correctly.
Clarification
There are 3 possible scenarios which can arise (Note: all of these cases can be mirrored):
 1) Both dedges are oriented: Place them in the same positions as the BR dedges above.
 2) One dedge is oriented, the other is not: This is the easiest case to deal with. Place them either in the GW or YO places above.
 3) None of the dedges are correctly oriented: For this case another algorithm is used. First, you place the unoriented dedges in UBl and FRd and then perform: (Lw U' R U Lw2 U R' U' Lw)
Algorithms
Note: Some of the algs on this page are in SiGN, some are in WCA notation.
EOPairing
Standard Pairing (use mirror for left)
l U' R U l'
r' U' R2 U r
Pair 2x unoriented edges (place edges in UBl and FRd positions)
l U' R U l2 U R' U' l
Final 6 edges where all orientations match:
F {pairing alg} F'
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'
Flip single edge (in UR)
y' r U2 r' U2 r' D2 r D2 r' B2 r B2 r' y
EPLL+Parity
Opposite Swap
r2 U2 r2 Uw2 r2 u2
Adjacent Swap
(R U R' U') r2 U2 r2 Uw2 r2 u2 (U R U' R')
Name: HPLL+Parity, W4PLL 
Alg  F2 Uw2 R2 F2 Uw2 R2 F2 U F2 R2 F2 U R2 Uw2 
Alg  F2 Uw2 L2 F2 Uw2 L2 F2 U' F2 L2 F2 U' L2 Uw2 
Name: ZPLL+Parity, O4PLL 
Alg  r2 Uw2 r2 b2 U' r2 y r2 U r2 Uw2 y' r2 
Alg  l2 Uw2 l2 b2 U l2 y' l2 U' l2 Uw2 y l2 
Guide
Step 1a: First 2 centres
For this step you need to create 2 opposite centres, which will be your R/L face centres. This can be done using any types of moves, but wide turns are faster than inner layer slices.
Step 1b: Remaining 4 Centres
First of all, rotate your cube so that your R/L face centres are on the correct sides, then using any moves except for F, B, or S to form the other 4 centres in the 2 central layers. You need to ensure that your chosen U colour is on top, D face on bottem etc. This step can be very fast becuase it only uses 3 types of moves
Step 2: EOPair abd Place DF and DB
This is when we start to use the EOPair concept explained above. You want to EOPair DF and DB, the 2 EOLine edges and then move them to the correct places to give a EOLine.
Step 3: EOPair remaining 10 dedges
Using the EOPair concept above, the remaining edges need to be EOPaired. To make lookahead easier, After pairing edges, move them into the RD,RB,LB or LD positions, so that you can see the other edges that need to be paired.
Step 4: Finish EO (Including OLL Parity)
Similar to fixing EO at the End of the 2x2x3 block in petrus, You need to finish Orienting all of the edges using F/B moves. If an odd number of edges is flipped, you can use the Flip Single Edge (in UR) algorithm above.
Step 57: ZZF2LCOLLEPLL
This is the final step and involves solving the 3x3 stage as in the ZZ method, without the EOLine becuase it has already been formed. COLLEPLL is recommended becuase then it is easier to deal with PLL parity becuase it can be dealt with in a single step. For the 4 cases that arise with PLL parity, see the algorithms section above.
Walkthrough Solves
Walkthrough solve 1: (SiGN notation, lower case = double layer turn)
Scramble: D' r2 D u' r2 u2 d' f' D' B F' r2 D' f2 R2 D' U' B2 D2 l2 b2 u2 F' r2 U' B2 f2 U2 F' l2 b U' F u' d r D2 d' F2 L
L+R Centres: u2 r' R' L2 f U l U' l' z' (9) Finish Centres: r' U r U l' U l2 U2 l' x U2 r2 U2 r2 (13/22) First line dedge (+dedge): L U D' L' U r' U L' U' r (10/32) Second line dedge: R U' R r' U' R2 U r (8/40) Place line: R' L' D' (3/43) 1x dedge: U2 R' l U' R U l' (7/50) 2x dedges: U2 r' U' R2 U r (6/56) 3x dedges: L U L' r' U L' U' r (8/64) Final 3 dedges: L' U' l U' R U l' U R' U' l U' R U l' (15/79) Finish EO: B L' B' (3/82) ZZF2L: R2 L' U2 L2 R2 U2 R U' R' U R' U' R U' R' U R U2 L' U L U L' (23/105) OCLL: y2 R U2 R' U' R U' R' (7/112) PLL: y x R D' R U2 R' D R U2 R2 x' U' (10/122)
Walkthrough solve 2: (WCA notation)
Scramble: D2 L' Lw' Rw' R' Fw2 B2 Uw2 D2 Bw' Rw Bw B2 L' Dw2 Rw Dw2 Lw Uw' F' B' Rw' Fw' R' Dw' Fw' Bw B' Uw B2 Uw Fw2 U2 F' Uw Dw2 D' Fw Dw2 F
See also
External links
 Speedsolving.com: Simultaneous EO and edge pairing for ZZ on 4x4