The reason I ask is this: 4-44-2 states A player who catches the ball while moving or dribbling may stop and establish a pivot foot as follows:
b. If one foot is on the floor:
1. It is the pivot foot when the other foot touches in a step.(Not your scenario.)
2. The player may jump off that foot and simultaneously land on both. Neither foot can be a pivot in this case.

So the point is if he is running and he catches the ball on his right foot then has neither on the ground and starts his dribble this should be legal as he has not established a pivot foot yet. He would be able to land on both feet (but not have a pivot) or start a dribble and continue moving. So was it really a travel?

Are you saying a player who catches the ball, while running, with his right foot on the floor, can lift his right foot before starting his dribble?

Yep. he hasn't technically established it as a pivot yet.

I will say this, if the player's second foot isn't down yet when he starts his dribble, then I probably can't tell that the first foot has been lifted. But that's not relevant to the rule you're quoting.

What about a player who catches the ball with both feet on the floor, then jumps off of both feet before starting his dribble?

If he starts a dribble that is a travel. So what if the player in the OP, catches the ball on one foot starts a dribble then lands on both feet simultaneously? Then what do you have?

Why is it a travel? He hasn't established a pivot foot yet.

Interesting question. You're talking about what is essentially a jump stop in which the receiver starts dribbling before he lands?

I see your point too. Good one. In my mind catching a ball on a sprint doesn't give the player much time to get the ball out of their hand nor stop on one foot. I understand the jump stop part. Just saying there is no way to know what the player is going to do and if he starts the dribble in the air in this situation I am passing. It is hard to see how you can call a travel if the player has only ever had one foot on the floor.

Wouldn't make sense since after the player lands he can legally start a dribble....

My point exactly.

Just fleshing this out a bit:
1. A1, on the run, catches with the right foot on the floor.
2. A1 lifts that foot in his run.
3. A1 starts his dribble.
4. A1 puts his left foot down as part of his run.

On step four, his right foot became the pivot, retroactively.
He had lifted that pivot before the dribble started.
Travel.

Other play where we retroactively learn of a travel:
A1 catches the ball, pivots, lifts his pivot foot, and then releases a pass towards A2, who has run away from the area.
A1 runs to retrieve the ball.

This is a travel because A1 lifted his pivot before starting his dribble. (The throw in should be at the spot nearest where his pivot foot was lifted.)

I still don't agree with you in the first case and even if I did, I still wouldn't call it because it would be bang bang.

As for the second one, can't you take as many steps during a dribble as you want. Ball is out of his hands, he hasn't used his dribble yet, I think this should be legit as well, as long as he passed the ball before lifting the pivot foot and hasn't used his dribble already.

How bout this one?
Adam Coble Basketball Breakaway Dunk - YouTube Skip to 0:30

Or this one.
C.J. Fair's Dunk Vs. Cornell - YouTube

Sorry, I meant to add that he lifts his pivot foot before releasing that pass.

And I agree about the likelihood of calling it. As I noted before, if the dribble somehow starts in the moment between the pivot foot going up and the other foot coming down, I'm not likely to call it because I don't think I'm that good. It's just too close, and I don't like calling travels where I need slow motion replay to justify it.

Fair enough. Travel.

I still think it is not a travel because that one foot is not "technically" a pivot foot. What did you have for the two videos I posted? Where in the rules does it say you can retroactively call that a travel? Your first scenario is retroactive. The second is not retroactive. They just started a dribble after lifting the pivot foot and the ball bounced multiple times before they touched it. I think these are very different scenarios.