Easy puzzle, just look at the FEN: 4k2r/8/5B1P/3R1KpP/8/8/8/8 w - - 0 1

Castling is impossible, so the key is 1.Ke6!  :)

Now this is what a puzzle is. it leaves you puzzled!!

There is a similar idea in philosophy where to get somewhere one has to first get halfway there and to get there he has to get halfway there, and so on.  It's an idea that doesn't prevent the final destination, but it does put up a potential circular conversation if one allows the loss of the big picture.  Purely intellectual, but fun nonetheless.

I tend to think it's rather moot to talk about "what happened before" the position in the study, as, of course, nothing needs to have happened before in a study -- in a study, unlike a game, you don't reach a position; you simply set one up arbitarily whether it could be reached or not. Who says a composed problem even has to be possible to achieve with play?

So it seems plausible to assume that no moves occurred before in a problem because they wouldn't be necessary if you're just setting up the position you want. I guess that would mean castling would be fair game if it fits in with all the other rules.

Because of the ambiguous nature of these kinds of puzzles, as you have demonstrated, I think it makes sense to have it as a convention to tell the solver whether castling is meant to be possible or impossible. If this actually became the convention, then I don't think it would ever give away the point of the puzzle -- if you solved 100 puzzles, with each one hearing whether castling was possible or impossible prior, it wouldn't give anything away because there would be no bias -- whether castling is significant in the situation or not, you would still be told about it.

When it comes to en passant... either deal with the fact that knowledge of it gives away the puzzle or just don't make puzzles like that! Puzzles with an ambiguous en passant possibility mostly just rely on the one simple trick (that you might not consider the move if, you unlike a game, hadn't seen a pawn move before), and don't go much deeper than that. The first time I fell victim to that trick, though, it was cute, I admit. Puzzles where you have to just realize you might be able to capture en passant are like riddles: the difficulty doesn't necessarily rely on the depth but rather the ambiguity of the question, and the exploitation of it in an unexpected way -- often in a riddle, the question isn't answered in a way you would normally answer a question.

Retrograde analysis has never been high on my list, but the ideas that you point out here really are thought provoking. This certainly falls well within the definition of curious!

Loved this puzzle!