Empirical Structure of Log-Space Return Times in Collatz Trajectories

This follow-up note to The Log-Space Return Lemma in Collatz Trajectories investigates the behavior of a newly introduced observable: the log-space return time T_{\text{return}}(x). This value marks when a Collatz orbit first drops below its initial logarithmic height. We compute and visualize this function for all odd integers from 3 to 1000, comparing it to classic stopping time and orbit height. The results confirm that excursion collapse occurs early and structurally, supporting modular collapse theories and extending the previous lemma into empirical analysis.