The Proof of the Collatz Conjecture

In this paper, I intend to prove the Collatz conjecture. In the following, I will prove that the Collatz conjecture is true for any positive integer and also for natural numbers. That is, after sufficiently using the rules of the Collatz conjecture, we must reach 1. In fact, I aim to give a definite answer to the question that has not been answered since 1937: Does the Collatz sequence eventually reach 1 for all-natural or positive integer initial values? My answer is yes.


Introduction
The Collatz conjecture proposed by Lothar Collatz, a German mathematician at the Humboldt University of Berlin, in 1937, is one of the unsolved problems in number theory in mathematics.
According to the Collatz conjecture, n is an arbitrary positive integer or a natural number (n ≠ 0, n ∈ ℤ or n ∈ ℕ).If n is an even number, divide it by 2; otherwise, multiply it by 3, add it to 1, and divide it by 2, and then, repeat the operations and processes, i.e. taking the result of each step as the input for the next step.In this way, one can create a sequence of numbers, beginning with a positive integer regardless of the n value, which will always reach 1.
X: The number of even and odd numbers in the closed interval of [1,2, …,nxk] ˄ ∀ nxk ∈ ℕ ∨ Z + , x, k ∈ ℕ x, n, k ≠ 0, where nek is even and nok is odd, is calculated as follows: Here, nq is the number of even and odd numbers in the above mentioned closed  . 15, pp.97-115 (2023) ISSN: 2668-778X www.techniumscience.comDue to the above, we must find a relationship between the even and odd numbers of the selected or desired nxk, which is obtained by the number of consecutive divisions by 2 to reach 1 in the Collatz conjecture sequence, Scc(n), because of the following facts.
[1] In each of the Collatz conjecture sequences, it can be observed that the number of consecutive divisions necessary to reach the final and desired number 1 is equal to the number of even numbers existing in that particular Collatz conjecture sequence, Scc(n).
[2] Therefore, by finding the number of even numbers in each Collatz conjecture sequence, Scc(n), in the closed interval [1,2, …, nxk], we can certainly determine the number of consecutive divisions necessary to reach 1 in the Collatz conjecture.However, finding the above mentioned number is very complicated because of the operation of (3nok + 1), with nok being odd, that repeatedly changes numbers and also changes the order and position of numbers in the sets of ℕ or Z + as well as their evenness or oddness.Self _Shrunken Relation = Ʀ Ə (x).

All examples provided in
And I define it as a relation that, by its inherent definition, is "recursive.".
It repeatedly and successively until reaching a fixed number according to a specific definition for it, dividing by a real number R, gets small and smaller.
And the Collatz conjecture relation can be considered as an example of a Self _Shrunken Relation = Ʀ Ə (x).
More precisely, due to the definition of the Collatz conjecture relation; if the starting or initial number n, is even after dividing by 2 will get smaller or if n is odd, although multiplied by 3, but, after adding with 1, it becomes an even number and therefore is divisible by 2, which becomes smaller.
Those processes are repeated and for each time, with fluctuations and jumps similar to the damping oscillations of a spring, after a slow or fast process, reaches a maximum value (identical to probability distribution diagrams) from which, due to the consecutive divisions by 2 that mentioned above, it will finally and definitely reach the number 2 and then the desired and final number 1; that usually occurs after the two consecutive and immediate divisions by 2.

Table 1 .
Table 1 indicate that the members of the Collatz conjecture sequences,Scc(n), easily show the number of divisions by 2 required to reach the desired number 1 in the Collatz conjecture, Scc(n), which is equal to the number of even numbers in that Collatz conjecture sequence, Scc(n), i.e.; The characteristics of the Scc(n) members.

The Mechanism of the Collatz conjecture relation and Conclusion Now
, I would like to introduce a new relation in mathematics that I called it;