Bit-Level Finiteness, Pre-Root Architecture and Framework for Collatz Trajectories

Abstract

This paper establishes a rigorous bit-level and modular framework for resolving the Collatz conjecture. By bypassing intermediate even division steps via a generalized shortcut operator, we prove that infinite climbing ($k=1$ states) leads to a structural finiteness contradiction for positive integers. Furthermore, we formalize the Pre-Root transition architecture and Root Number sinks ($R_m$), demonstrating via a \textit{reductio ad absurdum} framework that all trajectories are deterministically funneled into absolute convergence to $1$.