chess - Can White Castle?

This is the first chess puzzle I composed in the retrograde genre. I originally posted this in a chess dedicated forum. Hope you like it!

In the following position, is it possible that White could still castle?

• To prove it's possible, all you have to do is provide a legal game.


• If you believe it's impossible, you need to provide your reasoning.

Answer

Now that we have three increasingly complex proofs (two deleted, one of them mine) that it's impossible, it's pretty clear that it must be

possible after all!

Here's why:

1. b3 Nf6
2. Bb2 Ng4
3. Bf6 gxf6
4. Na3 Ne3
5. Nc4 Nxf1!
6. Ne5 fxe5
7. h4 Rg8
8. h5 Rg6
9. hxg6 Bh6
10. g7 Be3

11. g8=N Bc5
12. Nh6 Ba3
13. Nf5 Ng3
14. Nd4 exd4
15. Qb1 Bc1!
16. Qb2 Nh5
17. Qc3 dxc3
18. Rb1 Nf6
19. Rb2 cxb2
20. Nf3 b1=R

21. Nd4 Ra1
22. Nb5 Ng8
23. Na3 Bb2+
24. Nb1 Bg7
25. e3!! Bf8
26. O-O

In case you haven't already done so, you should totally check out @greenturtle3141's thorough answer (that unfortunately tripped up mere inches before the finish line) to see why the highlighted moves are absolutely essential.
Note that the notation has been edited to work with most PGN viewers, for instance https://chesstempo.com/pgn-viewer.html

This is, without doubt, the most refreshing chess problem I've ever tried to solve. Thanks, OP!