Note that column N is the "trunk" in both 3x+1 (top)
and 3x+3 (bottom). The trunk does not attach to any 
other branch (assuming we stop when the termination 
point is reached, otherwise the sequence loops as
shown by the arrow).

In your terminology, you call the trunk a "state"
and anything not part of the state is a "cycle" because
the sequence contains at least one odd number.

Note that the powers of 2 have reversed position between
3x+1 (column N) and 3x+3 (column B). In your state/cycle 
system, it is column N that is signifigant, not the numbers 
that occupy it.

When you say that 3x+1 seperates into infinite pieces when
you remove the powers of two, you are mis-stating what is
happening. The fragmentation happens when you remove the 
trunk, it has nothing to do with the powers of two. If you
were CONSISTENT, you would also remove the trunk in 3x+3.
Then you would get the same fragmentation you see in 3x+1.

That is why I've been telling you the difference you're 
seeing is an illusion.