Spectral Clustering

Spectral Clustering is the compression of the BPT harmonic spectrum by which the ratio between successive harmonic frequencies falls steadily toward 1 as recursion depth increases. With harmonics at quadratic positions, the ratio of each to the one before starts with a large jump (4.000 at n = 0) and narrows quickly toward unity. The book reads the negative rate of change of that ratio as monotonic narrowing and calls the result Spectral Clustering, a convergent frequency distribution that distinguishes recursive computational systems from classical harmonic structures. It follows from the Harmonic Frequency Mapping f_n = ν_0 · P(n) = ν_0 · (n+1)², with the fundamental ν_0 equal to 1/①⥂⧗⌂, the inverse of the Local Pulse Tempo. The formal statement is R(n) = (n + 2)²/(n + 1)², with lim(n→∞) R(n) = 1 and compression rate dR/dn = -2(n+2)/(n+1)³, where R(n) is the successive term ratio and n the recursion depth index. All quantities are dimensionless [∅]; the book's checks are consistent. It connects to Logarithmic Convergence, the Successive Term Ratio, Harmonic Clustering Coefficient, Recursive Frequency Spacing and Density-Dependent Planck Time.