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

We prove the Collatz Conjecture by establishing a complete arithmetic and dynamical closure of the map that sends: even n to n/2, and an odd n to 3n+1. Odd integers are classified by their residues modulo 18, and the least-admissible reverse lift determines a unique parent for each non-multiple of 3. This defines a deterministic, non-branching reverse graph. The only descending reverse case k=1 in residue class C₁ is arithmetically bounded by 3-adic valuation, eliminating the sole infinite descent corridor. All other admissible lifts strictly ascend, and no nontrivial odd cycles exist. Ternary cylinder sets encode every reverse path, and their affine recursion partitions Nₒdd without overlap. Finite reverse depth implies forward convergence to the trivial cycle 1 to 4 to 2 to 1 for every positive integer n.