A Deterministic Residue Framework for the Collatz Operator at q = 3

Description:
This paper presents a deterministic residue framework that resolves the Collatz conjecture at q=3q=3q=3. By classifying odd integers modulo 6 and refining their behavior modulo 18, the framework shows that every admissible reverse path rotates through the classes C0→C2→C1 in a fixed cycle. A finite-bound argument proves that within at most two increments of the admissible doubling parameter, a terminating child in C0 is always reached. This guarantees coverage of all integers and rules out nontrivial cycles. The final theorem establishes that every forward trajectory enters the 4→2→1 loop. The paper includes explicit tables, residue analysis, and annexed computations verifying the distribution of classes, offering both a complete proof structure and a transparent roadmap of the dynamics.