Viewing the problem as an escape velocity problem

Firstly, Interesting problem. Secondly, I am not a mathematician, but when I look at the problem it seems to make sense that it goes to 1. And please, forgive me for my ignorance if this explanation does not make sense but I see it as a spaceship trying to escape the gravity of earth: N/2 is gravity and 3N+1 is the rocket or propelling force.

N/2 has 5 chances (numbers ending in 0, 2, 4, 6, 8) of forming an even number (pulling the rocket down) and 3N + 1 has only 1 opportunity of tossing or propelling the rocket up. Each time 3N+1 manages to propel the number up to infinity, all the 3N+1 efforts are subdued by the 5 times force of N/2 (gravity).

To iterate what I said before differently again, you see both N/2 and 3N+1 are linear functions in a piecewise function where there is also an unseen "rate of application" applied to this piecewise function.

The reason for the cycle is that 1 being the bottom is odd and keeps having the odd function applied to it which has a maximum of 4.

3 x 1 + 1 = 4

At this point, there is not enough power for an escape velocity above 4. Gravity N/2 is strongest closest to the earth or 1 in our case.

I hope that made sense.