Why must a shared flyweight object be immutable once it has been constructed?

A flyweight instance is deliberately referenced by many different objects at once -- that sharing is the entire point of the pattern. If any code were able to mutate a flyweight's fields after construction, every single object holding a reference to that shared instance would change at the same moment, silently, with no indication at the mutation's call site that it was affecting anything beyond the one object being touched directly. That makes immutability a hard requirement rather than a nice-to-have: without it, sharing state to save memory introduces a correctness bug that is extremely difficult to trace back to its source, since the bug manifests far away from where the actual mutation happened.