Traveling?
 

A1 catches the ball just above the three point line at the top of the key, with both feet on the ground (can pivot with either) and goes to take a shot and becomes airborne. While airborne he realizes that B1 will block it if he continues with his motion and opts to throw the ball to the ground with one hand begins his dribble.

Is this legal? I looked through the rule 4.44, but didn't find much. Maybe the case book or your veterans can shed some light.

Traveling - Case Book Play 4.44.3

Traveling. Pivot foot cannot leave the floor before the dribble is started.

This doesn't apply, technically, since a pivot hasn't been established. They (NFHS) closed this loophole with 4.44.3. "The fact that no pivot foot had been established does not alter this ruling."

Actually, I believe...
 

I don't have my rule book with me, but I seem to recall wording to the effect that one of the the feet is assumed to be the pivot foot. They traveled in the OP because they picked up their pivot foot before starting a dribble.

I checked before posting
 

I didn't quote it, but I checked it before I posted. Here's the relevant portions of the rule:

The rule itself says nothing about a player who can establish a pivot foot but jumps instead. The case play seems based more on the intent and spirit of the traveling rule.

Interesting follow-on to this, same situation (no pivot foot established), but instead of jumping to shoot, the player just jumps to jump stop - think for instance setting feet for a three point shot. I had a camp facilitator tell me that a player who has not established a pivot foot is allowed to jump off of two feet and land on both feet in this situation in order to set the feet. I can't find anything specific to address this, but I've got travelling under 4.44.3.

A camp facilitator told you that? Good grief. I've got traveling. Now, what you do see a lot of is the three point shooter doing a jump stop to set his feet as he catches the ball. I called a shooter for trying this but he did it incorrectly and traveled. Then, just my luck, on the other end a player executed it perfectly and made the shot. I no called it and just ignored the coach's comment.

Agreed which is how the situation came up in the first place. I of course followed up with the question what if the player just bunny hopped off of two feet all the way down the floor, never technically establishing a pivot foot. He just looked at me funny..but yeah it never fails that is going to happen exactly how you describe!

Based on this crazy logic, a player should be able to continue to jump stop from 25' all the way to the basket?!?!?!?

This particular move has been "overlooked" at the varsity boys level in Central Ohio for the past several years -- you can call it without question at the Jr. High, FR/JV level, BUT to call it at the varsity boys level will yield screams from the coaches, players and parents!!!

No referee will argue that a player who catches the ball in the air, lands with both feet roughly simultaneously (jump stop), jumps to shoot a shot, realizes the defender would block the shot, and comes down with the ball still in his/her possession, has committed a violation -- traveling (we would have called it ups-and-downs when we were kids).

From a footwork standpoint, the "hop" into the shot and the jump and return to the floor because the shot would be blocked is the exact same in each case.

4-44 TRAVELING
Traveling (running with the ball) is moving a foot or feet in any direction in
excess of prescribed limits while holding the ball. The limits on foot movements
are as follows:
ART. 1 . . . A player who catches the ball with both feet on the floor, may pivot, using either foot. When one foot is lifted, the other is the pivot foot.
ART. 3 . . . After coming to a stop and establishing a pivot foot:
a. The pivot foot may be lifted, but not returned to the floor, before the ball is released on a pass or try for goal.

How this is not a travel by book rule is a mystery. Not called in many cases, but a travel nonetheless.

Since it was not being called, the NFHS committee added this:
2009-10 POINTS OF EMPHASIS
1. TRAVELING. The traveling rule has not changed; however, the committee is
still concerned that the rule is not being properly enforced. Consequently,
offensive players are gaining a tremendous advantage. Areas of specific concern are: the spin move, the step-through move, the jump stop, perimeter shooters taking an extra "hop" prior to releasing the try and ball handlers lifting the pivot foot prior to releasing the ball on the dribble.