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.

The proof is based on the recursive definition of sets $U_i$ that alternate between even and odd numbers, connected by bijective transformations.

We prove three fundamental results: first, that this partition covers all positive integers through a strict monotonicity property of pre-generators; second, that every number has a unique finite sequence converging to 1; and third, that $1$-$4$-$1$ is the only possible cycle in the system.

We then demonstrate that our formalization is equivalent to the classical Collatz problem, thereby extending our proof to the original conjecture.

The proof resolves the longstanding question of whether every positive integer, under repeated application of the Collatz transformations, eventually reaches 1.

Our approach through disjoint partitions reveals the underlying structural properties that guarantee this convergence, providing a complete resolution to this fundamental problem in number theory.