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

Description:

This work gives a complete arithmetic framework for the Collatz operator in the case q=3. The main paper shows that forward and reverse steps both pass through a common residue lens modulo 18, reducing all odd integers to three middle-even residues {4,10,16} that deterministically map to the classes C0, C1, and C2. Forward and reverse steps are proven equivalent at this gate, and admissible doubling counts rotate residues in a fixed 3-cycle that guarantees eventual reach of the terminating class C0. Using the unique parent property, the reverse graph forms a rooted tree with base at 1, excluding merges, runaways, and nontrivial odd cycles. As a result, every integer is included and every trajectory ends in the loop 4-2-1.

The supplemental expands on this framework by presenting arithmetic offsets and tessellation patterns that illustrate how coverage unfolds block by block. These visual and numerical patterns complement the residue analysis, showing how higher admissible lifts fill gaps and how block boundaries are closed. Together, the two papers present both the core proof and its structural clarifications, offering a complete resolution of the Collatz conjecture at q=3.