I lied. One more. And, oooh boy; I looove this one.
Edge sees through Papyrus’ constant cheer. Not that it isn’t genuine, but it’s so much more draining than Papyrus lets on. He needs to be strong for Sans, after all—Sans has enough to worry about without fretting over Papyrus! Besides, Papyrus is fine! He’s absolutely fine! He’s the Great Papyrus, right? He’s not struggling with anything. Certainly not the growing certainty that Undyne will never let him into the guard. Or the fear that the few friends he has don’t…don’t believe in him. Which is fine! He’s the Great Papyrus, after all! He has enough belief in himself for all of them! But…. But. It might be nice if someone else believed in him too….
Which is why meeting Edge was so wonderful. Momentous, in fact. Because Edge believes in him. (Edge knows how far he’d go, if he was pushed past breaking.) Edge knows he’s stronger than any of the others thinks he is. (Except, perhaps, Blue.) So Edge is there for him, when Papyrus needs someone to believe in him.
And Papyrus? Papyrus knows how hard it must be for Edge, doing the things he must do to survive. Maintaining an uncaring, cold, cruel facade. So Papyrus gives him a safe place, a place he can be the kind of person he wants to be, but can’t. Around Papyrus, Edge can be “weak”, knowing that Papyrus would never see him that way.