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This is a method that has certainly been considered by a few people. It was also the original version of WaterRoux for a very short time until it was abandoned in favor of a different set of steps. However, I see a lot of potential in the method. I have also spent 4-5 months total and around 200 hours developing this method. For L5C I developed all 614 L5C algorithms, another 614 algorithms for a new type of L5C called SL5C, worked out additional L5C concepts, and put together L5C reduction techniques. For L7E, I worked out how to do L7E iterative EO, developed a complete L7E method, and planned out an expansion for the L7E method. So I have added this method to my site because of all of the work that I have done and because I think this method has potential and needs to be looked into more by the community.
This method in its advanced form can average under 40 moves in speedsolves. Below is a breakdown of the steps and their move count. The move count for the first two steps is based on the averages of top Roux solvers. The move count for the last two steps is based on the actual generated algorithms in the algorithm sheets.
Step 1: 7 moves - There is a reconstruction of an average of 100 by Fahmi where he averaged 6.5 moves for FB.
Step 2: 8 moves.
Step 3: 9.75 moves - This is with the use of SL5C, which is described on this page. SL5C averages 9 moves for the speedsolving optimal algorithms in the algorithm sheet. Then add .75 for the initial AUF before applying the algorithm.
Step 4: 15 - 17 moves. Current L7E methods average around 17 moves. Advanced ones will average lower.
Total: 39.75 - 41.75 moves. Further move reduction can be achieved by building any of the four possible right side 1x2x2 blocks and the use of non-matching corner recognition. This would mean a reduction of around 2 moves on average. Blockbuilding skills are also always improving, with Fahmi's 6.5 move FB being an example. So considering all of this, the method can average 37 - 38 moves.
Step 1: 1x2x3
In the first step, a 1x2x3 block is solved just as in the Roux method and many other methods.
Step 2: 1x2x2
In the second step, a 1x2x2 block is built on the opposite side of the 1x2x3.
Step 3: Last Five Corners (L5C)
Next the last five corners are solved. Solving the corners in a single step provides the lowest move count and includes 614 algorithms.
SL5C
In addition to developing all 614 L5C algorithms, I discovered a new concept and applied it to L5C, resulting in a new type of L5C that I call Shortened L5C (SL5C). In L5C, the corners are always solved. But in SL5C, the algorithms are generated in a way that allows the end state to be one of several possible end states - U* R' away from solved, U* R' U, U* R' U', U* R' U2, or the solved state. Each of those is a natural state for the corners to be in since it allows for easy edge solving in the next step. The major benefit of SL5C is that it averages around 2 moves fewer than L5C. This means that the overall move count of the method is reduced and there is a reduced memorization burden.
Additional possible end states can be added and it would still be SL5C. For now the algorithms were created by truncating any algorithms that ended in U* R'. If a program was developed that could solve to any desired end state, SL5C could be developed to its full potential and have an even lower move count and better algorithms than it already contains.
Below is a link to my original post that describes SL5C and L5C and includes the algorithms that I developed.
Case Reduction
There are options to reduce the number of algorithms at the cost of increasing the number of moves.
The 42 method: 42 is pretty much the sibling of this method. In 42, a U layer corner is placed on the bottom layer along with the right side 1x2x2. Then the four corners on the U layer are solved relative to the U layer corner that was placed on the bottom layer. The benefit of 42 is that it reduces L5C to just 42 algorithms and those algorithms are the exact same as CMLL. The algorithms can be slightly better than CMLL since the algorithms can take advantage of the empty edge slot above the U layer corner that was placed on the bottom layer. The downsides are that recognition is more difficult than CMLL and the move count is 3 - 4 moves longer than SL5C. Check out ATCRM on this site for a recognition method for the corners.
CPFB: If early corner permutation is used, L5C is reduced to 104 algorithms. The algorithms would be slightly shorter than full L5C, but there is an increase in FB move count when CP is added and there is a greatly reduced ability to plan ahead for the right side 1x2x2 during inspection.
Pre-orientation: The DFR corner can be oriented or positioned a certain way during the final moves of the right side 1x2x2. For example, the DFR corner can always be oriented a certain way on the U layer, reducing the corner cases to 162. Or users can solve the DFR corner along with the right side 1x2x2 then just have the 42 last layer corner cases.
Two look: L5C can be solved in two looks, such as the OL5C and PL5C algorithm sets that are provided in the L5C document linked above.
Step 4: Last Seven Edges (L7E)
In the final step, the last seven edges are solved. I have developed an L7E method that is the extension of the standard method for Roux LSE. The seven edges are oriented using iterative EO while trying to place one of the three LR edges. Then the remaining LR edges are solved and finally the M slice is permuted. This L7E method averages around 17 moves in its basic form and could potentially average around 15 with the advanced EOLR extension. There is still a lot to be discovered about L7E.
Below are some additional L7E methods developed by others:
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