[2025-09-09v4]A Positive Proof of the Collatz Conjecture using Collatz Phase Representation on figshare.pdf

After a long history of frustrating mathematicians, we present a solution to the Collatz Conjecture [1,2,3,4]. Previous research has primarily focused on a number-by-number approach [5]. In this study, however, we move away from that framework and develop a new method that models positive natural numbers as a unique structural system called "Collatz Phase Representation." This approach reveals that the seemingly random Collatz operation can be described as a unified transformation process based on the number's internal structure. This proof is built upon the sole prerequisite of basic knowledge of binary numbers. By using our newly developed perspective on numbers—the "Collatz Phase Representation"—anyone can follow the rigorous logic of this proof. This research not only provides a conclusion to the difficult problem of the Collatz Conjecture but also has the potential to offer a new perspective for future number theory research by using the Collatz Phase Representation to capture the essence of numbers from multiple angles.<br>