Data-Driven EEG Band Boundaries Converge Near Euler's Number: A Multi-Method Analysis Across 244 Subjects

Conventional EEG frequency band boundaries (e.g., theta 4–8 Hz, alpha 8–13 Hz) were established by visual inspection decades ago and have never been systematically tested against mathematical organizing principles. We applied three independent boundary detection methods—spectral parameterization (FOOOF/specparam), spectral derivative analysis, and cross-frequency topographic correlation—to resting-state EEG from N = 244 subjects across three public datasets. All three methods placed the alpha–beta to theta–alpha boundary ratio nearest to e − 1 = 1.718 among the tested constants (φ, e − 1, 2:1, √2): FOOOF 1.853 ± 0.045 (N = 55), derivative 1.826 ± 0.039 (N = 158), correlation 1.685 ± 0.031 (N = 171, TOST-equivalent at ε = 0.10). A mixed-effects model yielded a population-level intercept of 1.787 (95% CI: 1.717–1.857), with e − 1 falling within the confidence interval. Permutation testing confirmed this clustering as non-random (p < 0.002). The boundary ratio was statistically equivalent to the spectral centroid ratio from our companion paper (paired t: p = 0.35, Cohen’s d = 0.075, TOST-equivalent at ε = 0.15), confirming self-similar organization across spectral description levels. Empirical boundaries diverged substantially from convention: the beta–gamma boundary averaged 25.3 Hz (vs. conventional 30 Hz), while the theta–alpha boundary (7.73 Hz) was consistent with Klimesch’s (2013) theoretical prediction of 7.5 Hz. Power analysis revealed that studies with N < 30 systematically favor φ, while adequately powered samples converge toward e − 1—suggesting that some published reports of golden-ratio organization may reflect insufficient statistical power.