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I've done an analysis of the optimal number of moves (face turns) to solve each case of the supercube cross. By supercube cross, I mean solving the 4 cross edges and orienting the 5 centers adjacent to those edge positions. This is for solving a particular color cross. (It's not for color neutral solving.) The distribution is:

Ooh, cool. I'm kind of surprised there are positions that take 12 moves, because the maximum for normal cross is only 8 moves. Is it possible to do the same calculations for color-neutrality? How about for building a (fixed) 2x2x2 block?

There are a little over 2 quadrillion positions to consider for an exact color neutral calculation. A distributed effort would seem to be required. Of course, you only need to build a table for the fixed color case, and use symmetry to do 6 table lookups to get the best case cross for each position.

The 4 antipodes are in 2 equivalence classes. The essentially distinct antipodes for the D layer cross can be generated by:

R F' R R F' L B R B R B F D D
R F L D F R B D L B F D' F' D

I've also done the 2x2x2 block for the supercube in both FTM and QTM. This means the three centers that are part of the 2x2x2 block must be correctly oriented.