Resolution of the Collatz Conjecture: A Rigorous Analysis of Collatz Sequences and their Unique Cycle

This article presents a rigorous approach to the Collatz Conjecture, focusing on fundamental properties of Collatz sequences. We establish key properties of the Collatz function and its inverse, including surjectivity and injectivity. The structure of Collatz sequences is analyzed in depth, proving important results such as the Bounded Subsequence Property and the uniqueness of cycles. Central theorems on the properties of Collatz sequences, including the boundedness of all sequences and the nature of the unique cycle, are presented and proved. These results culminate in a complete resolution of the Collatz Conjecture, demonstrating that all Collatz sequences eventually reach the cycle 1, 4, 2. We provide a rigorous proof of the conjecture, while emphasizing the need for thorough peer review and verification by the mathematical community given the significance of this long-standing problem.