The Resolution of the Collatz Conjecture via a Unified Framework: Local Residue Dynamics and Global Ladder Coverage

The Collatz Conjecture asks whether repeated iteration of a simple map always reaches the trivial cycle 1 → 4 → 2 → 1. The rule is: if a number is even, divide it by two; if a number is odd, multiply it by 3 and add one. We prove that the dynamics of this map are governed by two frameworks whose union establishes a duality. The recursive local view shows that every odd integer generates a unique parent–child trajectory, ensuring complete local coverage with no gaps. The iterative global view shows that parent–child offsets extend to arithmetic ladders whose higher lifts systematically fill all odd residues, guaranteeing complete coverage of the odd integers. These structures are isomorphic: each recursive relation corresponds to an offset increment, and together they form a single unified system.This duality proves that the Collatz operator admits no gaps, cycles, or divergence. Every odd integer resolves uniquely within the recursive–iterative architecture, and every trajectory ultimately returns to the cycle 1 → 4 → 2 → 1.