Scale Induction in the 3n+1 System

This paper develops a complete inverse-word and affine-scale framework for the 3n+1 system. Using the coordinates S, j, and B, it establishes the concatenation calculus, credit-before-debt identity, ordered B structure, affine rails, ternary-cell decomposition, and compatible dependency trees arising across the systems 3n+3^a. A global scale-induction theorem then proves that every positive odd integer belongs to the rooted component generated by 1. Universal rootedness is combined with exact origin-relative positivity and the trichotomy of odd trajectories. Nontrivial cycles are excluded because they require zero origin-relative displacement, while every nonempty admissible rooted word has strictly positive displacement except for the trivial return at 1. Divergent trajectories are excluded because a runaway tail would require an inverse descent below its least tail value, contradicting the rooted credit-before-debt bound. Consequently, every positive odd integer reaches 1, and division by powers of two extends the conclusion to every positive integer.