Convergence Parameter

The Convergence Parameter is the exponent γ that sets how fast energy falls off across the harmonics of a recursive harmonic system, and so whether the infinite series of harmonics carries finite total energy. In BPT the harmonic frequencies grow quadratically with the harmonic index, so for the whole series to remain physically realizable the energy given to each harmonic must decay faster than that quadratic growth. The book places the boundary at γ > 3: above it the total energy summed over all harmonics is finite. The parameter therefore constrains how energy can be allocated across recursive harmonic structures in the computational substrate. It appears in the Harmonic Energy Distribution E_n = E_0 × (n + 1)^(2−γ), where E_n is the energy of the nth harmonic, E_0 the fundamental energy scale, n the harmonic index and γ the Convergence Parameter, and in the total E⊚ = E_0 sum(n=0 to infinity) (n + 1)^(2−γ). Dimensions as the book gives them: E_n, E_0, E⊚ [J]; n, γ and the exponent (2−γ) [∅]. The book's check is marked valid and verified. It connects to Amplitude Scaling, the Spectral Density Function, Harmonic Complexity Scaling and Energy Conservation.