A10 has not caught the ball, because he did not return to the ground inbounds while firmly holding/controlling the ball. By interpretation from John Adams a long time ago, and documented by AR 7-3-6-VIII, when he then propelled the ball (even after grasping the ball in his hands), A10 has batted the ball, and a pass (forward or backward) may be batted in any direction in the field of play, by any player eligible to touch the ball. See Legacy’s post below. The batting does not change the status of the ball. Thus, the resultant loose ball is a ball loose from a backward pass (the snap), is alive and in play, and may be recovered by anybody. When B20 falls on, and possesses, the ball while grounded, the ball is dead and belongs to Team B, with a series at that spot.
B, 1/10, 50, snap (25).
I was fixated on the fact that the interpretation from Adams so many years ago is that such ‘propelling’ of the ball by an airborne player that grasps a pass but doesn’t complete the catch by returning to the ground inbounds is to be considered batting of the ball, and not a pass. This interpretation came about as a result of a play in the Peach Bowl the previous season, where a defensive player secured the opponent’s forward pass while airborne and then propelled the ball forward to a teammate. The crew (Jon Bible, Mike Liner, et al) conferred and ruled illegal forward pass, penalized B 5 yards from the spot, and awarded B a series. Both teams were OK with the ruling on the field. But Adams made the interpretations in the following months, to clarify the rule, making it batting, and not a pass.
Yes, batting a backward pass forward by the passing team is illegal, made so to prevent A from batting a snap forward, particularly on 4th and short, for a recovery beyond the NZ for a first down.
Thanks to Legacy for the correction.