Universal Residual Geometry of the Integers and the Collatz Conjecture

I present a complete resolution of the Collatz Conjecture by introducing an offset residue geometry that aligns modular arithmetic with the operator’s parity-dependent dynamics. Re-indexing the integers modulo 6 identifies the odd multiples of 3 as root classes (C0), while the neighboring offsets (C1 and C2) form the live classes. A refinement modulo 9 reveals a deterministic triadic rotation among residues, guaranteeing that every live class trajectory intersects a root within bounded steps. This structure eliminates the possibility of nontrivial cycles or divergent paths: every odd integer lies on a finite reverse path that necessarily terminates in C0, and by equivalence, every forward trajectory converges to the trivial cycle 4 → 2 → 1. Thus the conjecture is resolved as a direct consequence of the arithmetic law of the integers, rather than heuristic estimates. For completeness, we record in Appendix A that the offset residue framework extends to all odd moduli k, where the pair-doubling action induces finite residue rotations of length Lk = ord(k²)(4). The Collatz case corresponds to k = 3, which is uniquely short, producing the triadic geometry that underpins full convergence.