Averaging structure in the 3x+13x+1 problem

The 3x+13x+1 problem has resisted analysis from multiple perspectives for many decades. This paper studies the more general qx+rqx+r problem, where q and r   are odd, and finds new, averaging structures for the iterates. This structure supports the conjecture that all orbits enter a cycle if q=1,3q=1,3 but divergent orbits exist if q≥5q≥5.