Here's my attempt at **Diagram 1**. Will try the others when time permits.

White wants to retract a R to the d-file (e.g. Rd5-g5 or Rd1-a1) as that effectively immobilises the d7-B (in view of Rd8#), and then forces M3 with 1.a7 and 2.a8=Q+ (or 1...Kf8 2.Rxd7). Trouble is that 1...0-0! would refute as castling is currently legal. So the idea is that White must retract specifically to create a position where it can be proved by retro-analysis that castling is illegal.

Black castling is legal in the diagram because we can assume that the h8-R is the KR, but then how did the other R get to h7? If it reached h7 via the 8th rank, then it must have dislodged the BK previously from e8 and castling is actually illegal; the only alternative is that the R got there from the south side, meaning the g6- and h6-Ps have cross-captured to let it through. This cross-capture and the doubled c-Ps require 3 white units to be captured, and that's not a problem since White is missing 6 units.

However, White can start by retracting Rd5xPg5! The g5-P must have come from e7, implying the e6-P is from d7 and the c4-P is from a7. Such a P structure requires 5 captures, leaving only 1 missing white unit as a spare and that's not enough for the g6- and h6-Ps to have cross-captured. Therefore the h7-R must have gotten there via the 8th rank and had dislodged the BK previously. Black castling is thus illegal and White mates in 3 with 1.a7!

Help-Retractorsounds as a fancy concept but it is not. You are playing an OTB game and someone walks by and whispers "you just missed a checkmate in 3". That is a help-retractor. If you missed it one move ago then you need to take back 1 move and forcefully checkmate your opponent in 3 moves.All 4 diagrams have that precise assignment. You take back any white move you like and checkmate black in 3 moves instead. They are not just ordinary moves because you must take into account the retrograde analysis for each position related to the castling rights on both sides.

The 4 diagrams form a quadruplet. Each diagram only differs in 1 detail from the previous diagram, mostly by changing the position of just one piece or pawn. But as said, the similarity is deceptive when looking at the game history.