On the impossibility of non-trivial cycles and divergence in the Collatz Conjecture

Independent research on the Collatz Conjecture (3x+1 problem) utilizing fundamental arithmetic tools and logical frameworks. Developed by an 18-year-old student (started at 16), this paper explores the impossibility of non-trivial cycles and divergence through 2-adic valuation and modular arithmetic. The current version introduces the variable R (m-excess) to definitively rule out the existence of non-trivial cycles by demonstrating an algebraic non-divisibility invariant. seeking to provide a fresh perspective for further guidance and peer discussion.