A Dyadic Dynamical System Generated by x+B and the Partition of Odd Residues into Terminal Basins

For a fixed odd positive integer B, this paper studies the arithmetic dynamical system on odd positive integers defined by adding B to x and then removing the full power of 2 that divides x + B. The map sends odd integers to odd integers and generates a simple but rigid dyadic dynamics. The main result proves that every orbit eventually enters the finite terminal region formed by the odd numbers from 1 to B. Inside this region, the map becomes a permutation, and its terminal cycles can be completely classified through the multiplicative action of 2 modulo B. Equivalently, the eventual terminal cycle of any odd integer is determined only by its residue modulo B. The paper presents a self-contained proof of this basin decomposition and includes explicit examples for B = 15, 21, and 25. The contribution is framed not as a strong historical novelty claim, but as a clear finite-state dyadic automaton model attached to the parameter B, useful for understanding how dyadic valuation and modular arithmetic interact in elementary arithmetic dynamics.