Amplitude Scaling

Amplitude Scaling is the rule for how the amplitudes of a recursive field's harmonics fall off with harmonic number n; the book requires them to decrease at least as fast as (n + 1)^(−α) with α greater than one half, so that the field's energy stays finite. In chapter 7's Resonant Field of Becoming, a collective field is built from many harmonic modes, and not every set of amplitudes gives a physically realizable field. Finite energy requires both the time-derivative and gradient contributions to integrate to finite values, ∫|∂Ψ/∂t|² dx < ∞ and ∫|∇Ψ|² dx < ∞, and the book derives from this that convergence needs the amplitude decay exponent to exceed one half. Part 7.2 gives the full form, the Amplitude Scaling Law A_n = A_0(n+1)^(-α) [1 + β cos(γ*n + φ)]: a power-law decay from a base amplitude A_0, modulated by a periodic factor of strength β, frequency γ and phase φ across the harmonic index. The book writes A_n ∝ (n + 1)^(−α) with α > 1/2, where A_n is the amplitude of the nth harmonic, n the harmonic index [∅] and α the decay exponent [∅]. Amplitude Scaling connects to Resonant Field of Becoming, Energy Density Distribution, Field Complexity Measure, Collective Field Function and Nonlinear Harmonic Spectra.