Summary of the Proof of the Collatz Conjecture:

Preprint Title Proof of Collatz conjecture by formulating bottom-up m...

All natural numbers can be divided into odd and even numbers.

All even numbers become odd when repeatedly halved (divided by 2).

All odd numbers can be classified into two groups: those of the form 4n + 1 and those of the form 4n + 3.

For all numbers of the form 4n + 3, when multiplied by 3 and added 1, the result is even. When halved, it becomes either a number of the form 4n + 3 or 4n + 1, and ultimately converges to a unique odd number of the form 4n + 1.

When the unique odd number of the form 4n + 1 is tripled and added 1, it becomes even (12n + 4), which then halved becomes 6n + 2, and finally becomes the unique odd number 3n + 1.

For the unique odd number 3n + 1, if n is odd, then 3n + 1 is even, and when repeatedly halved, it converges to the unique odd number of the form 4n + 1 or 4n + 3.

By repeating this process, it eventually converges to the value n = 0, where 4n + 1 = 3n + 1. When n is 0, we have 4n + 1 = 4(0) + 1 = 1, which immediately converges to 1.

Therefore, all natural numbers, when subjected to the given operations, eventually converge to 1. This completes the proof of the Collatz Conjecture.

By presenting this detailed proof, we have shown that all natural numbers will inevitably converge to the value 1, confirming the validity of the Collatz Conjecture.