Empirical Structure of Normalized Cocycle Accumulation in the Collatz Iteration

This work presents an empirical analysis of normalized cocycle accumulation in the Collatz iteration. Rather than following long and irregular trajectories, the dynamics are reformulated in cumulative cocycle coordinates, which eliminate trajectory-level degrees of freedom and retain only irreversible accumulated constraints. Numerical experiments reveal a sharply bounded admissible region in the normalized cocycle space. This bounded structure persists across large ranges of odd-step depth k, in clear contrast with heuristic growth models that predict unbounded divergence. The observed envelopes indicate that the Collatz iteration is governed not by trajectory complexity, but by global cocycle constraints imposed through cumulative valuation imbalance. This empirical study complements a previously introduced normal form and trace-compressed coordinate system, together forming a unified framework for analyzing irreversible arithmetic iteration via accumulated cocycle bounds. The Collatz system is treated as a minimal nontrivial test case, and the methodology is intended to be portable to other iterative systems where trajectory-based analysis fails to capture global structural constraints.