The only reason we need a length parity algorithm at all is because in reduction we insist that the centers remain solved and all but the last two edges remain intact. Otherwise we would indeed, as in the cage method, simply do an r turn to fix it, and probably not even realize we had parity in the first place (similar to how on the 3x3 we do AUF turns without realizing the parity of edges and corners gets swapped). The problem is that as far as reduction is concerned an r move messes up too many things to quickly fix with commutators - for speedsolving purposes you don't have time to think (or care) about fixing center and edge swaps intuitively; it makes far more sense to memorize an algorithm, however unintuitive, and just do it as quickly as possible.
Here is a simple(r) parity algorithm which does not have any edge commutators. Here I can again explain what every move does, as well as reconstruct the algorithm from the concept even if I have forgotten the moves, so I consider it fully intuitive.
(r2 U2 r U2 r2) (B2 (l U2 l') E2 (l U2 l') (r U2 r') E2 (r U2 r') B2)
The first sequence does an edge wing 4-cycle on U without affecting the edges on the bottom or sides (since parity algs ought to leave those alone). Now look at the cube: four rows of centers are off, in two block two-cycles. The second algorithm simply solves this with a single commutator-like algorithm by performing one two-cycle with E2, placing the other two-cycle in its place, and doing E2 again, fixing both E2 and the second two-cycle. This is set up by B2 to ensure that only two opposite 'slots' on E2 need to be affected.
It is entirely different than all of the algorithms you guys have given me or any "single edge correction" algorithms that I have ever seen. But, it is 24q. However, because it is derived from pure logic (meaning that it is much more logical than the explanations which I have been getting in this thread), I feel that it is too valueable to give away freely on the internet. In the near future, I will probably make and sell a brief book which explains how it was derived. It seems like the "optimal" cube solvers of today just do not match up to reality of what there is to know about cube theory.
This attitude is not really welcome in the cubing community. We're not competing against each other; nobody is trying to publish papers and win some kind of professorship, and there is no real money to be made in cubing. There is nothing to be gained for yourself by keeping some kind of secret withheld. Our goal is to increase shared knowledge and this is the only way cube theory can be developed to its fullest. I'm sure that if people had known you were trying to steal information in order to make money off of it they would have given you a lot less help.
PS: There is nothing wrong with optimal cube solvers and I think many of us would appreciate it if you wouldn't insult our work simply because you think you have improved on it. As I have mentioned to you (although you did not seem to care to listen to ir) they rely only on the assumptions that the users have input. As far as parity algorithms people have been asking for algorithms in <Rw,Lw,U2,F2,D2,B2>, not because this is the way to construct the most optimal algorithms (and it probably isn't, especially in quarter turn metric, because of all the double turns) but because this leads to algorithms that are fast to execute. There is not enough computing power to be able to come up with an optimal parity alg if you allow every possible 4x4 move, so this computation has not been done, and it is wrong to assume that it has.
