  I show here that there are three different kinds of iterations for the
reduced Collatz algorithm; depending on whether the root of the number is odd
or even. There is only one kind of iteration if the root is odd and two kinds
if the root is even. I also show that iterations on numbers with odd roots will
cause an increase in value and eventually lead to an even rooted number. The
iterations on even rooted numbers will subsequently cause a decrease in values.
Because increase in values during the odd root iterations are bounded, I
conclude that the Collatz iteration cannot veer to infinity. Since the sequence
generated by the Collatz iteration is infinite and the values of the numbers do
not veer to infinity it must either cycle and/or converge. I postulate that any
cycling must occur with alternating types of iterations: e.g. odd rooted
iterations which cause the values of the numbers to increase followed by even
rooted iterations which causes the values to decrease. I show here that for
simpler types of cycles, valid values of odd rooted or even rooted numbers are
only found in a narrow gap which closes as the number of iterations increase. I
further generalize to all types of odd-even and even-odd iterations. Given that
previous work has shown that only very large non-trivial cycles are feasible
during the Collatz iteration and this study shows the low probability of large
simple cycles, leads us to the conclusion most likely cycles other than the
trivial cycle are not possible during the Collatz iteration.
