  In this paper, we prove that reduced dynamics on Collatz conjecture is
periodical, and its period equals 2 to the power of the count of x/2
computation in the reduced dynamics. More specifically, if there exists reduced
dynamics of x (that is, start from an integer x and the computation will go to
an integer less than x), then there must also exist reduced dynamics of x+P
(that is, if starting from an integer x+P, then computation will go to an
integer less than x+P), where P equals 2 to the power of L, and L is the total
count of x/2 computations (i.e., computational times) in the reduced dynamics
of x (note that, equivalently, L is also the length of the reduced dynamics of
x). Therefore, the power (or output) of this period property, which is
discovered and proved in this paper, is - the study of the existence of reduced
dynamics of x will result in the existence of reduced dynamics of x+P (and
iteratively x+n*P, n is a positive integer). Hence, only partition of integers
needs to be verified for the existence of their reduced dynamics. Finally, if
any starting integer x can be verified for the existence of its reduced
dynamics, then Collatz Conjecture will be True (due to our proposed Reduced
Collatz Conjecture).
