A Deterministic Residue Framework for the Collatz Operator at <em>q</em> = 3

We give a deterministic, purely arithmetic framework for the Collatz operator in the classical case q = 3. The two standard views—forward n 7→ 3n + 1 followed by division by powers of two, and reverse n 7→ (2kn − 1)/3 with admissible k—are unified by a single middle-even gate modulo 18. Working with the natural odd residues {1, 3, 5} (mod 6), we show that both the forward first step 3n + 1 and any admissible reverse first step 2kn land in the same three residue classes {4, 10, 16} (mod 18), and that these residue classes alone determine the next odd class. For every live odd n (3 ∤ n) there exists an admissible k with 2kn ≡ 3n + 1 (mod 18) (forward/reverse middle-even equivalence). Incrementing k by 2 rotates the mod-18 middle-even class 10→4→16→10, making the terminating option (10, hence a C0 child) periodically available alongside the live classes C1 and C2. These ingredients give a short exclusion of nontrivial odd cycles and a global inclusion argument: every odd is live or terminating, and every even halves to the odd layer. The result is a complete residue-theoretic resolution of the 3n + 1 dynamics in the q = 3 system. For context see [1–6].