"Be in this universe"(1) vs "be mathematically possible" should cover most cases, though other times it might not quite match either of those and be much harder to explain.
Defining 'existence' by using 'interaction' (or worse yet the possibility of interaction) seems to me to be trying to define something fundamental by using something non-fundamental.
As for "mathematical possibility", that's generally not what most people mean by existence -- unless Tegmark IV is proven or assumed to be true, I don't think we can therefore taboo it in this manner...
Here's the new thread for posting quotes, with the usual rules: