Global Linear Scaling, Lineage Propagation, and Path Independence in Inverse Clotatz Subtrees

Traditional investigations into the Collatz $(3x+1)$ problem primarily target forward path trajectories or long-term statistical graph densities, which frequently behave like pseudo-random systems. This paper introduces an exact arithmetic invariant operating within the inverse Collatz map, demonstrating that the global value-sum ratio of distinct structural subtrees converges monotonically to the exact ratio of their roots. By executing an iterative deep spectral scan up to layer 65, we confirm