No violation by the jumper?
 

Looking at the rule book and surprised that I did not find it a violation if the jumper crosses over the center line during the jump ball. It always seemed like an obvious violation, in fact I've called it on more than one occasion. Without fouling it seems like the jumper can gain an obvious advantage by jumping into the other jumper and nudging him away from the ball to win the opening tip. Is it there and I'm not seeing it? Or is this a violation only in my head?

If you think it's a violation, then it's in your head. ;)

That said, I'm not sure I understand your play. How can "live ball contact that creates an advantage" not be a foul?

If he 'nudges the other tapper away' that would/could be a foul. There is no requirement that a tapper stay in his half of the court vertically. If there were, it would be impossible for a tapper to tap the ball back, as that requires reaching across to the other side of the ball...

The jumper just has to start in his half of the restraining circle. After the toss if he bumps the other jumper while gaining an advantage, I think I would call a foul.

Put the foul question aside for a moment. How about if the jumper steps across the division line as he begins his jump? I see nothing in the rule book that says this is a violation, though I certainly thought it must be.

6-3-1

Who Can Recite, From Memory, All The Other Jump Ball Limitations ???
 

For any jump ball, each jumper shall have both feet within that half
of the center restraining circle which is farther from his/her basket.