What is a proper noun between fence Thursday and cat?

This is a bit of a trick question! There's no proper noun between "fence Thursday" and "cat" because "fence Thursday" isn't a noun at all. It's a combination of a noun ("fence") and a day of the week ("Thursday"), making it a phrase, not a noun.

If you're looking for a proper noun between "fence" and "cat", then there isn't one in that sequence. Proper nouns are specific names of people, places, or things (like "John", "London", "Amazon").