I've enjoyed reading this. The detail, particularly the physical descriptions, are clear and strong. The connection between the narrator and the statue works well and the development of the theme emerges neatly from the narrator's realization of how his sense of duty has turned him into a weathered statue, or at least something similar. I know that feeling well and connected to it.
A question I'd like to ask, for the fun of it, is does the telling also have to be statuesque? Or maybe I should say statue-like? Is there a way you could describe the narrator's state that will the reader to more deeply participate in the mystery of the delivering? I hope that makes sense.
My suggestion(s) may be absolute rubbish. Let's get that stated clearly.
The (archduke's) salute (I like that repeated “u”) was not ostentatious. A quick touch of his fingers to his temple as he passed. (This—going forward—is where I question where the piece is going. What if we are not told that Peter finds the solidity reassuring? What if Peter isn't quite sure what he receives from the statue? To me, Peter knowing what he likes about the statue reveals him to be a sort of middle-class fiction character. Know what I mean? Drudging through ordinary life. Nothing wrong with that if it's what you want. Maybe I'm simply drawn more to characters who are more at drift. Who have no idea why he is on trial, for example, or why he's lost in an endless library. Again, this is me. This piece though made me think it may be worthwhile to at least ask if there is another direction you would rater go.)
I have to stop here. I wanted to make my overall suggestion to unplod it a bit. But of course, that may not be your goal. Even if you can't use my suggestions I've enjoyed making them.