A Proof of the Collatz Conjecture Through Disjoint Partitions

This paper presents a complete proof of the Collatz conjecture through the construction of a partition of positive integers into disjoint sets.

Our proof establishes three key results: the completeness of our partition through strict monotonicity of pre-generators, the uniqueness of finite sequences converging to 1, and the identification of $1$-$4$-$1$ as the only possible cycle.

We demonstrate that our formalization is equivalent to the classical Collatz problem, thereby proving that every positive integer eventually reaches 1 under repeated application of the Collatz transformations. 

Our approach reveals the underlying structural properties that guarantee this convergence.