Reduced Collatz Dynamics is Periodical and the Period Equals 2 to the Power of the Count of x/2

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).

Collatz conjecture is that no matter what the number (i.e., x) is taken, the process will always eventually reach 1.
The current known integers that have been verified are about 60 bits by T.O. Silva using normal personal computers [1,2]. They verified all integers that are less than 60 bits.
Wei Ren et al. [3] verified 2 100000 − 1 can return to 1 after 481603 times of 3 * x + 1 computation, and 863323 times of x/2 computation, which is the largest integer being verified in the world. Wei Ren [4] also pointed out a new approach for the possible proof of Collatz conjecture. Wei Ren [5] proposed to use a tree-based graph to reveal two key inner properties in reduced Collatz dynamics: one is ratio of the count of x/2 over the count of 3 * x + 1 (for any reduced Collatz dynamics, the count of x/2 over the count of 3*x+1 is larger than ln3/ln2), and the other is partition (all positive integers are partitioned regularly corresponding to ongoing dynamics). Wei Ren et al. [6] also proposed an automata method for fast computing Collatz dynamics. All source code and output data by computer programs in those related papers can be accessed in public repository [8].
We thus can represent required computation as (3 * x + 1)/2 and x/2, which are denoted by I(x) and O(x), respectively. Note that, I(x) and O(x) can be simply denoted as I(·) and O(·), or I and O, respectively. Obviously, ∀x ∈ N * , That is the reason of notation -I represents "Increase" and O represents "dOwn". = 1, 2, ..., n, and " " is the concatenation of Collatz transformations. For simplicity, we just denote f i (·) as f ∈ {I, O}. ( Obviously, Collatz conjecture is held when x = 1. In the following, we mainly concern x ≥ 2, x ∈ N * . Obviously, L must be the minimal positive integer such that f L (x) < x.
(1) We call an ordered sequence f q ∈ {I, O} q in above proof as original dynamics (referring to f q (x) = 1), which consists of q occurred Collatz transformations during the computing procedure from a starting integer to 1. For example, the original dynamics of 5 is IOOO due to 5 → 16 → 8 → 4 → 2 → 1.
(2) In contrast, we call f q 0 in above proof as reduced dynamics (referring to f q 0 (x) < x), which is represented by a sequence of occurred Collatz transformations during the computing procedure from a starting integer (i.e., x) to the first transformed integer that is less than the starting integer (i.e., f q 0 (x)). For example, the reduced dynamics of 5 is IO due to 5 → 16 → 8 → 4.
(3) Obviously, reduced dynamics is more primitive than original dynamics, because original dynamics consists of reduced dynamics. Simply speaking, reduced dynamics are building blocks of original dynamics.
Due to above theorem, we concentrate on reduced dynamics. (1) Simply speaking (or recall that), RD[x] represents occurred Collatz transformations in terms of I and O during the computing process from starting integer x to the first transformed integer that is less than x.
(2) Roughly speaking, f L ∈ {I, O} L is an ordered sequence consisting of I and O.
Furthermore, this sequence implicitly matches the parity of all occurred intermediate transformed integers that are taken as input of f (·). [8]. From the data we discover the property -period and its relation to the number of computing x/2 in reduced dynamics -will be proved in the following of this paper. (6) In fact, we proved some results on RD[x] for specific x, e.g., 11] 32 ] = IIOIO, and so on [5].  (8) In fact, we formally proved that the ratio exists in any reduced Collatz dynamics. That is, the count of x/2 over the count of 3 * x + 1 is larger than log 2 3 [7]. The ratio can also be observed and verified in our proposed tree-based graph [5]. Example 1.12. RD [5] = IO. It implies following results: To better present above the implicity in reduced dynamics, we introduce two functions as follows: Remark 1.14. Simply speaking, this function checks whether the forthcoming Collatz transformation (i.e., c ∈ {I, O}) matches with the current transformed integer x.
(3) In other words, s is a selected segment in s that starts from the location i and has the length of j. Indeed, that is the reason we call this function as "Get Substring".
(4) Simply speaking, this function can obtain the Collatz transforms from i to i + j − 1 from a given inputting transform sequence (e.g., reduced dynamics) in terms of s ∈ {I, O} |s| . (5) Note that, GetS(·) itself is a function. In other words, it can be looked as GetS(·)(·). E.g., GetS(IIOO, 1, 1) (3) It is worth to stress that, although in above definition j ≥ 1, it can be extended to j ≥ 0 by assuming GetS(·, ·, 0)(x) = x.
(1) s(x) is the last transformed integer, or the first transformed integer that is less than the starting integer.

We assume
In other words, we assume the reduced dynamics of x = 1 is IO. In the following, we always concern x ≥ 2, x ∈ N * .

Period Theorem
In this section, we will formally prove RD[ RD[x] ∈ {I, O} ≥1 in this section. Note that, interestingly, L is indeed the count of x/2 computations in reduced dynamics.

Notations and Observations
Notation 2.1.

(3) We thus concentrate on x ∈ [3] 4 in the following.
For easily understanding, we point out two concerns in the forthcoming proof. Also, the parity of x + P and x are identical. That is, the parity sequence during computing for the reduced dynamics of starting integer x+P is identical with that of starting integer x, which results in the occurred Collatz transformations of both are exactly identical. Besides, P is the minimal integer to satisfy above requirements. This is one concern. 3. The other concern is to prove s(x + P ) < x + P and s( A new notation I (·) is introduced hereby to reveal the relations among I(x + P ), I(x) and I (P ).     For better presentation, we thus introduce two functions as follows: (1) Simply speaking, replacing all "I" in "s" respectively by "I " will result in "s ".

The Proof of Period Theorem
Remark 2.11. Above lemma states that if P ∈ [0] 2 , the first Collatz transformation of x + P is identical with that of x.
Proof IsEven(x + P ) = IsEven(x) because P ∈ [0] 2 , due to Lemma 2.10. Thus, the first Collatz transformation of x + P and the first Collatz transformation of x are identical.
(1) Obviously, above conclusion can be extended to include |s| = 1 by Lemma 2.10 and Lemma 2.12.
(3) Separation Lemma is general, as s could be either original dynamics or reduced dynamics of certain x ∈ N * . That is the reason we give a special name to this lemma for emphasizing its importance. (4) In above proof, we assume the number of transformations will be s. Note that, it does not influence the conclusion, as we can use condition "j = 0, 1, 2, ..." instead of "j = 0, 1, 2, ..., |s| − 1" to omit the assumption on the number of transformations.