The Resolution of the Collatz Conjecture: A Unified Arithmetic and Dynamical Framework

This paper presents a complete arithmetic proof of the Collatz Conjecture. The approach is based on a deterministic classification of odd integers by their residues modulo 18, a unique least-admissible reverse lift that defines a non-branching reverse graph, and an explicit ternary cylinder coding of all reverse trajectories. The only descending case, corresponding to k=1 in residue class C1, is shown to be finitely bounded by 3-adic valuation, eliminating the sole infinite descent corridor. All other lifts strictly ascend, and no nontrivial odd cycles exist. The affine recursion determined by ternary cylinders partitions the set of odd positive integers without overlap, ensuring full coverage and finite reverse depth. It follows that every forward Collatz orbit reaches the trivial cycle 1→4→2→1.