Abstract The natural world is organized through hierarchical structures extending from subatomic particles to the cosmic horizon. In this study, we present an empirical analysis investigating whether a stable statistical pattern exists in the transitions between adjacent hierarchical levels. The analysis is based on 21 independent versions of hierarchical sequences (7 per domain), comprising body diameters (83 levels), orbital distances (42 levels), and biological lengths (35 levels). Based on a total of 1,099 transition ratios, we identify a stable statistical pattern: the median transition ratios within the domains range narrowly from ×1.62 to ×1.93, while 92.3% to 100% of all observed ratios fall within the interval from ×1 to ×30. The Kolmogorov–Smirnov test (with Bonferroni correction) reveals no statistically significant differences between the distributions (p > 0.05 for all comparisons). In this study, the geometric mean is treated exclusively as a mathematical baseline, determined by the fixed dynamic range and the number of hierarchical levels, rather than as evidence of a natural law. The observed narrow dispersion and high concentration of moderate ratios define a stable candidate pattern. Final verification of this pattern requires future testing using synthetic null models to isolate the potential influence of logarithmic sampling.
Zakir Causevic (Fri,) studied this question.