A Resolution of the Collatz Conjecture

This work presents a complete arithmetic framework resolving the Collatz Conjecture by decomposing the odd–to–odd dynamics into two complementary structures: a local residue–phase automaton and a global affine counting system. The Inverse map R(n; k) = (2kn − 1)/ 3 is shown to act on the live residues 1, 5 (mod 6) through a finite residue–phase state space, while every admissible exponent k = c + 2e induces an affine expansion factor 2k whose inverse coincides exactly with the dyadic slice weight 2−k. From this, every odd integer is seen to belong to a unique dyadic slice Sc,e, forming a disjoint partition of Nodd. Independently, the introduction of the normal–state lattice z(n) reveals a second, purely affine enumeration: each live odd n seeds a unique 4-adic rail m → 4m + 1 whose union also partitions the odd integers without overlap. We prove that these two partitions coincide exactly, yielding a unified global structure in which all odd integers arise bijectively from admissible lifts above minimal anchors produced by n ≡ 1, 5 (mod 6). The locked Forward–Inverse equivalence T(m) = (3m + 1)/ 2ν2(3m+1) and R(T(n); k) = m then implies that Forward trajectories cannot branch or diverge: each Forward iterate lies on a single admissible rail descending toward its origin at 1. Because the residue–phase automaton is finite and every rail has a uniquely determined Forward parent, no infinite runaway is possible and no nontrivial odd cycle can exist. Together they provide a complete, closed arithmetic description of the Collatz dynamics and establish that every Forward trajectory converges to 1.