Golden Ratio EEG Architecture Emerges Near Schumann Frequencies but Does Not Constrain State Modulation

The golden ratio (φ) has been proposed as an organizing principle for neural oscillation frequencies, but prior work has not separated the question of where oscillatory modes exist from how cognitive state redistributes energy among them. Using 109 subjects performing motor execution, imagery, and rest (1.86 million FOOOF-detected spectral peaks from the EEGMMIDB dataset), we show these are distinct questions with different answers. A joint search over 549 candidate lattices (9 exponential bases × 61 anchor frequencies) identifies φ at f₀ = 8.5 Hz as the global optimum, with non-overlapping bootstrap confidence intervals against all alternatives — but only within a ~1 Hz anchor window (f₀ = 7.6–8.5 Hz) whose lower boundary transitions sharply near the theta-alpha transition. The dynamic reallocation of spectral energy during visual and motor engagement (r = 0.31–0.52) is real but operates identically under any exponential spacing. Beyond φ's individual dominance, irrational bases collectively outperform rational bases with large effect sizes (Cohen's d = 1.22–1.60), while non-exponential linear grids fail to detect any systematic structure — providing what may be the first formal validation that EEG spectral peak organization is logarithmic rather than linear in frequency. All findings replicate at coarser spectral resolution. The φ-lattice thus describes the fixed scaffolding of available oscillatory modes without constraining how the brain redistributes energy among them, and the anchor-dependence of its advantage has implications for principled definitions of EEG frequency bands.