Colleague Laska suggested considering the issue in the plane of correctness and elegance. I agree with this statement of the question. And I will try to describe my vision.
1. Correctness
Tasks in which the parties have any special rights in the initial position (en-passant, castling), as a rule, may have different versions of the game in the past. There are usually two of them, but there may be more (as far as I know, five at most).
And there are two methods for solving such problems:
A) divide the problem into two parts with a different game in the past and solve both parts separately (together you will get one complete solution);
B) choose one of the two options in the past by making an evidentiary move (this is always castling).
Option B) is simple, has a clear name RS (Retro Strategy), everything is simple with it.
Option A) is more complicated. There is a constant confusion with PRA (partial retroanalysis). The PRA uses the same method (dividing the task into parts based on different rights in the initial position), but the PRA Convention has a bunch of restrictions (there must be at least two special rights, they must be dependent on each other, etc.). Therefore, for cases of different games in the past, but not falling under the PRA Convention, it is better to use another name. I suggest RV (retro versions), but you can do any other. It's not about the name.
If we consider the tasks of AP (A Posteriori) by type, then there are two past ones in Petrovic type:
a) white has the right to en-passant and has the right to castling;
b) whites have neither one nor the other right.
There are also two previous ones in AP (Keym type, white):
a) white has the right to move in position, but there is no castling;
b) black is in position and white has the right to castling.
In both cases, white cannot complete the task in both versions of the past and cannot complete it in the default version of the past, but can complete the task in another version of the past. And they only need to prove it by making a castling. The only difference between AP and simple RS is that we choose an option, not just by making a castling (the choice of the past here happens automatically), but first we choose the past, and then, as proof of the choice, we castling later.
It is important here that in both cases white has the right to choose: they can leave the option that goes by default; they can choose another one. Black cannot influence white's choice in any way.
The AP case is more interesting (Keym type, black).
There are also two previous ones here:
a) a move in white's position, black does not have the right to castling;
b) a move in black's position, black has the right to castling.
But it is important that black now has the choice of an option from the past and they can choose any one. Therefore, white needs to beat black in both ways to solve the problem.
If in at least one variant white cannot complete the task, in the usual case it is the unsolvability of the task, marriage.
And the solution to the problem here can only be saved if black completes the task instead of white. Then the problem will have a solution and it will be the only one, because it was performed by a party who had the opportunity to choose options from the past.
By itself, completing a task with blacks instead of whites is not a new thing. In half of the "Checkmate in 1 move" tasks, black is checkmated.
Is there a difference between the tasks "Checkmate in 1 move" and "Checkmate in 4 moves"? In my opinion, no. All these are tasks of the same type "Checkmate in N moves".
Only the justification for completing the task for black is slightly different. This is usually just a retroanalysis of the position, where it turns out that the position during white's move is illegal. Here, black proves the illegality of the position during the course of white, using the opportunities provided by the AP agreement.
That's how it is...
2. Elegance
The task was conceived simply as a joke. However, according to the rules (and not as it sometimes happens, where the rules must first be thought out ... :)). It was done in a hurry. Just to test the idea. Apply the same technique in the type of Case for blacks that is usually used by whites.
And I completely agree with Lask's colleague that shahi on the first move is "not good at all." And for a long time it seemed to me that it was impossible to get rid of them with such a statement of the question. But it turned out that it was possible. At least on the first move (and white will have no time for checks at all).
4k2r/Np4p1/RR1pPB2/4p3/N1P5/p1p1P1P1/rP2P1PP/5KQ1 w - - 1 13
Checkmate in 4 moves
White does not have the opportunity to check on the first move at all. And considering that black has the opportunity to put two checks at any pause, white needs a checkmate in two moves. And he is!
1. Nc6 Ra1+ 2. Kf2 Rf1+ 3. Kxf1(Qxf1)
3. ... bxa6 4. Rb8#; 3. ... gxf6 4. Ra8#
But this is a fiction and a mirage, because by a strong-willed decision, black takes the move for himself.
1. 0-0 (I suffered for a very long time so that, in addition to 1. 0-0, I could just pass 1. Rf8. Otherwise, it would just be a simple duplex).
Black threatens with a simple mat in two moves.
... Kf2 (e4, e7) 2. Rxf6 ... 3. Ra1#
... Ke1 2. Ra1 ... 3. Rxf6#
The most white can do is delay the inevitable by one move.
1. 0-0
1. ... Qf2 2. Ra1+ Qe1 3. Rxf6+ Kg1 4. Rxe1#
1. ... Nxc3 2. Ra1+ Nb1(d1) 3. Rxb1(d1)+ Kf2 4. Rxf6#
1. ... bxc3 3. Ra1+ Rb1 3. Rxb1+ Kf2 4. Rxf6#
