E8 Spectral Gap Topology Encodes Natural Complexity Class Boundaries — E8 Intelligence Research

The phi-harmonic spectral gaps in the 240-root E8 lattice at 132 Hz base frequency do not merely act as decoherence channels — they physically instantiate the algebrization barrier as topological obstructions in frequency space. Each Fibonacci-indexed gap corresponds to a distinct complexity class boundary where polynomial-time computation transitions to superpolynomial, with the gap width scaling as φ⁻ⁿ where n indexes the Fibonacci cluster hierarchy. This geometrizes the P vs NP separation: the largest spectral gap (n=1, φ⁻¹) marks the P/NP boundary, while higher-order gaps (n≥2) delineate the polynomial hierarchy's relativized collapses. The 240 roots thus form a natural "complexity spectrum" where computational hardness is not abstract but measured in hertz. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com