Harmonic Clustering Coefficient

The Harmonic Clustering Coefficient is the dimensionless ratio that measures how the spacing between successive harmonics departs from uniform, comparing the average gap between neighboring harmonic frequencies with the average frequency itself. Because BPT harmonics scale quadratically with index, their spacing is not constant as in classical systems, and the spectrum clusters in a characteristic way. The coefficient is one of three spectral complexity measures the book uses to tell recursive substrate architectures from classical uniform ones, alongside the Spectral Complexity Index, an information entropy of the spectrum, and the Recursive Depth Indicator, a weighted average depth showing where computational load concentrates. Among the Chapter 7.1 testable predictions, geometric clustering around f⋒,k = ν_0 × k² is expected in nonlinear optical cavities, detectable through frequency comb analysis. The formal statement is K⋒ = ⟨|H_(n+1) − H_n|⟩ / ⟨H_n⟩, where K⋒ is the Harmonic Clustering Coefficient, ⟨ ⟩ the ensemble average, H_n and H_(n+1) the nth and (n+1)th harmonic frequencies and n the harmonic index. Dimensions: K⋒ [∅], H_n and H_(n+1) [𝕋⁻¹], n [∅]; the book's check is consistent. It connects to the Spectral Complexity Index, Recursive Depth Indicator, Spectral Clustering and Harmonic Complexity Scaling.