It seems to hint that UDT should be coded from the start with a "logical prior" over mathematical statements, which encodes the creator's arbitrary "logical degrees of caring", just like its regular prior encodes the creator's arbitrary degrees of caring over physics.

This seems like a type error. Statements are references to collections of structures, they are not themselves the kind of stuff agent should care about, they are elements of agent's map, not territory. Caring about physics generalizes to caring about abstract structures, not to caring about statements that describe abstract structures. So there might be some notion of a prior over abstract structures, possibly expressed in terms of logical statements about them, but it wouldn't be a prior over the statements.