Proof of the Collatz Conjecture via mod 6 geometry Classification and Function of Odd Integers and mod 9 Deterministic Residue Transformation Mapping (Collatz Operator)

This paper presents a complete resolution of the Collatz conjecture by establishing a residue–invariant framework based on mod 6 classification and mod 9 deterministic residue rotation. Odd integers are partitioned into live and root classes, and the triad rotation lemma shows that residues advance in cycles of exact order three. The bounded-hit corollary guarantees that every live class reaches a root within a finite, bounded number of steps. Combined with the reverse–forward equivalence lemma, this enforces global convergence of all trajectories to the trivial cycle . The approach differs from earlier modular analyses (Terras, Everett, Crandall) by closing the system under iteration and eliminating the possibility of cycles or divergence. The manuscript is self-contained, fully formalized through lemmas, corollaries, and theorems, and demonstrates that looping is impossible by design.