Fair enough. But as I said before, I provided solid proof for my definition of antagonist, and you have provided no proof of yours. It's your turn to prove yourself if you want me to entertain your definition of antagonist any further. And since you just admitted you weren't thinking when you adressed this argument, I'm going to give you one last chance. This is the fourth and last time I will say it, so pay atrention this time.
Careful, I did not say that I did not think, just that my words were not the one I should have used.
Now.. this is not about proof, this is about how we can create the best interaction between a protagonist and an antagonist. I can't "proove" you a storytelling rule here, I would need to make an entire book about that with multiple story analysis, that's nonsens. That's why I suggest for you some ressources like "the anatomy of story" from Truby. you will find enough ressource here to understand my point.
Even if your definition of antagonist was right, it's still undeniable that every strawhat has had a significant connection with the arc villain in the arc where they joined. Why is Carrot the exception to that?
Ok, so here there is one thing wrong and one right, and one thing I need to explain:
(Wrong) - To mix up Joining and being recruted, you have to correct your sentence here by saying : "[...] had a significant connection with the arc villain in the arc where
they were recruted".
(Right) - Each strawhat did have a connection with the main arc villain in the arc they were recruted in.
Now, the question is : what was the nature of that connection. Antagonistic (Usopp/Nami/Chopper/Robin) or because of connection through the situation (Sanji/Franky/Brook/Jinbe).. the real thing is .. in our subject.. well.. it doesn't matter at all lol.
Because if we are talking about Carrot here and a potential main villain,.. the answer canonly come from potentiality.. in the future.
So to the question "Why is Carrot an exception to that", I reply : "She is not an exception .. yet.." in fact, a strong main arc villain COULD have an interaction (and therefore a connection) with her in the future. Nothing in the story is opposing that possibility.
In fact,I talked also about Kired for a reason : this is the one villain with NO PERSONAL connection with the strawhat in the arc where he is recruited. Yeah.. Kried is just a passing bad guy. When all the strawhats have a personnal connection to the main villain, Sanji had nothing with Krieg.
And if that happenned with Sanji, it can happen again with Carrot, meaning that we could cross the road of a bad guy during the journey, and this bad guy would become the main villain at a moment when Carrot start having her final character moment and is finally accepted by the strawhats as a true member of the family.
So.. until Carrot depart from the crew, we can't say that she won't have a connection with a big villain later on in the story.