Harmonic Origin Sequence

The Harmonic Origin Sequence is the sequence of perfect squares {1, 4, 9, 16, 25, 36, …} produced by the Recursive Growth Function P(n) = (n+1)² as recursion depth n runs through 0, 1, 2, …. It counts the nonlinear accumulation of pulse iterations, and the energy density that builds up with them, at each recursive step through Prime Pulse Bifurcation. The book derives the quadratic law from computation: each recursive step must process all previous binary decisions, so the load L(n) = L_0 + sum(k=1 to n) k = L_0 + n(n+1)/2 grows at the rate dL/dn = n + 1, and this growth, amplified through Resonance Coupling, gives the square law. The same sequence extends the harmonic complexity framework of Chapter 7.1, where nonlinear spectral patterns arise from substrate dynamics. The generator is the Quadratic Complexity Generator P(n) = n² + 2n + 1 = (n+1)², where P(n) is the pulse density function and n the recursion depth index, both [∅]; the book's check is valid and verified. It connects to the Harmonic Origin Pulse, Pulse Density, Computational Load Accumulation, Resonance Coupling and Prime Pulse Bifurcation.