A Resolution of the Collatz Conjecture

This work establishes a complete arithmetic resolution of 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 reverse map R(n;k)=(2^{k}n-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 $2^{k}$ 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 {S}{c,e}, forming a disjoint partition of {N}odd. Independently, the introduction of the zero–state index Z(n) reveals a second, purely affine enumeration: each live odd n seeds a unique 4-adic ladder 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 from admissible lifts above anchors {1,5}.The locked forward–reverse equivalence T(n)=(3n+1)/2^{\nu_2(3n+1)} and R(T(n);k)=n then implies that forward trajectories cannot branch or diverge: each forward iterate lies on a single admissible ladder descending toward its zero–state origin at 1. Because the residue–phase automaton is finite and every ladder has a uniquely determined forward parent, no infinite runaway is possible and no nontrivial odd cycle can exist. All constructions, residue frameworks, and affine decompositions used in this paper are original to this work. Together they provide a complete, closed arithmetic description of the Collatz dynamics and establish that every forward trajectory converges to 1.