Successive Term Ratio

The Successive Term Ratio is the ratio of each pulse-density value to the one before it, used to measure how the BPT harmonic spectrum compresses as recursion depth increases. With pulse density P(n) = (n+1)², the ratio begins with a large jump and narrows quickly: R(0) = 4.000, R(1) = 2.250, R(2) ≈ 1.778, R(3) = 1.563, R(4) = 1.440. It approaches 1 as n grows, which the book calls Logarithmic Convergence, and its steadily negative rate of change is what the book calls Spectral Clustering. Among the Chapter 7.2 predictions, this convergence should be observable in nonlinear acoustic systems through successive harmonic measurements. The formal statement is R(n) = P(n+1)/P(n) = (n + 2)²/(n + 1)², with lim(n→∞) R(n) = 1 and dR/dn = -2(n+2)/(n+1)³, where R(n) is the Successive Term Ratio, P(n) and P(n+1) the pulse density at levels n and n+1 and n the recursion depth index. All quantities are dimensionless [∅]; the book's check is valid and verified. It connects to Logarithmic Convergence, Spectral Clustering, Pulse Density, the Harmonic Origin Pulse and Recursive Frequency Spacing.