Logarithmic Coherence Time Scaling in Collatz Parity Ensembles

This work investigates the temporal scaling behavior of a wavelet-based coherence observable constructed from ensemble Collatz parity dynamics. For system sizes N=2kN = 2^kN=2k, with 16≤k≤2216 \le k \le 2216≤k≤22, characteristic coherence times are defined using three independent estimators: Peak location (argmax) Half-maximum crossing Temporal centroid After correcting finite-time truncation effects (Tmax = 350) and performing replicate-resampled bootstrap regression (R = 40, B = 500), we observe consistent logarithmic scaling of the form: t(N)=αlog⁡2N+ct(N) = \alpha \log_2 N + ct(N)=αlog2N+c Across estimators, the measured coefficient is: α≈4.3\alpha \approx 4.3α≈4.3 Bootstrap 95% confidence intervals for the half-maximum estimator are approximately [3.5, 4.9], with comparable ranges for peak- and centroid-based estimators. The agreement across independent estimators and replicate dispersion analysis indicates stable logarithmic scaling within the tested range N≤222N \le 2^{22}N≤222, though finite-size corrections cannot be excluded. This study provides the first quantitative measurement of a characteristic temporal scaling law for parity coherence in ensemble Collatz dynamics.