Glossary

Computational Load Accumulation Eq

Computational Load Accumulation is the growth of processing requirements with recursion depth that follows because each recursive step must process all previous states. Each generation inherits the computational load of the ones before it, so the load grows quadratically rather than linearly with depth. This quadratic growth drives Harmonic Complexity Scaling, shows up in the frequency domain as harmonics scaling as (n+1)² instead of with linear spacing, and links computational load to pulse density and Temporal Drag. When the initial overhead is negligible (L_0 << n(n+1)/2 for n >> 1) it reduces to the Simplified Load Function L(n) ≈ n(n+1)/2. The formal statement is L(n) = L_0 + sum(k=1 to n) k = L_0 + n(n+1)/2, where L(n) is the cumulative computational load, L_0 the initial processing overhead, sum(k=1 to n) the summation from k = 1 to n, k the processing cost at step k and n the recursion depth. Every quantity is dimensionless: L(n), L_0, k, n [∅]; the book's check is consistent. It connects to Harmonic Complexity Scaling, Pulse Density, Temporal Drag and Computational Impedance.

L(n) = L_0 + sum(k=1 to n) k = L_0 + n(n+1)/2 [∅]

Also in 7.2 , 7.4

Defined in Harmonic Complexity and the Recursive Spectrum · Calculator · Lexicon entry 8/10

Recursive Frequency Spacing Eq

Recursive Frequency Spacing is the gap between consecutive BPT harmonics, which grows linearly with the harmonic index instead of staying constant as it does for classical harmonics. Because BPT harmonics sit at quadratic positions, H_n = ν_0 × (n + 1)², the difference between neighbors increases by a fixed amount at each step. The book reads this as the computational substrate imprinting its recursive complexity accumulation on the spectrum, producing discrete lines at quadratic positions, weighted by energy through the Spectral Density Function σ(f) = sum_n δ(f − H_n) E_n, that set BPT spectra apart from classical ones. Among the Chapter 7.1 testable predictions, spacing Δf_n = ν_0 × (2n + 3) is expected in quantum dot arrays, measurable by high-resolution spectroscopy. The formal statement is Δf_n = H_(n+1) − H_n = ν_0 × [(n + 2)² − (n + 1)²] = ν_0 × (2n + 3), where Δf_n is the frequency spacing between consecutive harmonics, H_n and H_(n+1) the nth and (n+1)th harmonic frequencies, ν_0 the fundamental Pulse frequency and n the harmonic index. Dimensions: Δf_n, H_n, H_(n+1), ν_0 [𝕋⁻¹]; n [∅]; the book's check is consistent. It connects to Harmonic Complexity Scaling, the Spectral Density Function, Harmonic Clustering Coefficient, Spectral Clustering and Convergence Parameter.

Δf_n = H_(n+1) - H_n = ν_0 × [(n + 2)² - (n + 1)²] = ν_0 × (2n + 3) [𝕋⁻¹]

Defined in Harmonic Complexity and the Recursive Spectrum · Calculator · Lexicon entry 8/10