Empirical Analysis of Record Collatz Peaks for Prime Initial Values

This work presents an empirical study of Collatz trajectory peak values for prime initial numbers.All Collatz trajectories for primes up to 10610^6106 were computed exactly, and their maximum excursions were analyzed using logarithmic normalization. The results reveal a clear separation between typical behavior and rare extreme events. While large Collatz peaks do occur, they form a sparse record process whose growth slows with scale and exhibits no periodic or oscillatory structure. The bulk distribution of normalized peak values remains stable across the entire investigated range. The main article is provided as a PDF file.Additional figures illustrating the distributions for p≤105p \le 10^5p≤105 and p≤106p \le 10^6p≤106, as well as the corresponding record envelopes, are included as separate image files. Exploratory attempts to extend the computation beyond 10610^6106 highlight significant computational and methodological challenges. Full brute-force verification at larger scales requires lower-level implementations and is left for future work.