PAIRs Selection Theorem

First Theorem

If there is a corner, there are two pairs that share this corner and the positions of the two pairs can be interchanged by move sequence that only include the turn of the R face and the U face, then only the two corner as shown on the left can do this.

Because there are only the R, U layers and these two corners, which are symmetrical to each other in F and B planes, only one of the corners is concerned here, and the other corner can be symmetric with the F and B planes to deduce similar results.

And the two pairs that share one of the corners and meet the above conditions are the two pairs as shown on the right.

Second Theorem

If you find the other three pairs in the R and U layers, and add any one of the two pairs that share the same corner above, you can form 4 pairs, all of which can be used move sequence that only including the R, U face turn under the restriction of no breaking of these 4 pairs, and it can be moved into the position of the pairs as shown on the left, then the three pairs can only be the three as shown on the right.

Third Theorem

If the two pairs described in Second Theorem are two pairs that do not share the same corner, then that three pairs to be found in Second Theorem will not exist.