Collatz Sequence Proof (1st Way)

Abstract This paper investigates the structure of possible loops in the Collatz sequence and establishes that the only admissible cycle for all positive integers is the classical loop {4,2,1}. Using a systematic framework consisting of Taha’s Loop Table and Taha’s Cloud Table, the analysis examines all potential loop lengths r∈N+ and derives a general loop equation of the form x=3kx+S2 r−k,S=∑i=0k−13 k−1−i22i. By evaluating when this equation admits positive integer solutions, the study shows that solutions occur only when r=3k, yielding x=1, which generates the loop {1,2,4}. For all other values of r, the equation produces no positive integer solutions, excluding the existence of alternative cycles. The result is supported through algebraic derivation, geometric series evaluation of S, and induction. The paper concludes with Taha’s Collatz Sequence Lemma, demonstrating that any loop element satisfies lS(x)=1, and Taha’s Collatz Sequence Fact, showing that all Collatz loops collapse to the unique cycle {4,2,1}