Unless first touched by R beyond the neutral zone, if the kickers recover a scrimmage kick in or behind the neutral zone, the ball remains live and belongs to K and the down counts.
Category: Rule 5 – Series of Downs, Number of Down and Team possessions After Penalty
Rule 5, Section 3, Article 4
A ball touching the goal-line plane, when it becomes dead is in the end zone, even though it is moving away from the nearer end line and his its foremost point in the field of play.
Rule 5, Section 5, Article 7
A receiver shall not give an invalid fair-catch signal.
Rule 5, Section 1, Article 1
The team which next puts the ball in play by scrimmage following a free kick, touchback or fair catch is awarded a series of four consecutively numbered downs in which to advance the ball to the line to gain.
a. Each awarded first down starts a new series of four consecutively numbered downs.
b. The referee shall correct the number of the next down prior to the ball becoming live after a new series of downs is awarded and prior to the declaration of the end of any period.
The Zebra Files 