Class-Controlled Accelerated Collatz Dynamics

This article develops a fully deterministic framework for the accelerated Collatz map by separating value dynamics from the independently evolving line index. The exact index recursion shows that every transition in the accelerated map is uniquely determined by the residue class of the line index modulo 4, and that these four classes exhaust all possible dynamical behaviours. Two classes are strictly contracting, one class produces a single expansion before returning to contraction, and one class supports only finite odd segments with a uniquely determined exit point. As a consequence, every trajectory eventually enters a regime of monotone index contraction in which the line index collapses to 0. Once the index reaches 0, the corresponding value is a pure power of two, and the dynamics converges to the trivial attractor 4→2→1. Every natural number admits a unique BBF representation n= (2i+1) *2ᵃ, and strict decrease of the indices forces strict decrease of the odd parts. Thus index convergence implies value convergence, and the class‑controlled representation excludes both unbounded growth and nontrivial cycles. The accelerated Collatz dynamics is therefore reduced to a deterministic process with no remaining degrees of freedom.