Spectral suppression of composite harmonics via prime operator cascades H_p(ω)
Fourier spectral analysis converts discrete arithmetic functions into continuous frequency domain power spectra. By applying prime notch filter operators $\widehat{H}_p(\omega) = 1 - \frac{1}{p} \sum_{r=0}^{p-1} e^{-2\pi i r \omega / p}$, resonance spikes at composite frequencies $\omega \equiv 0 \pmod p$ are extinguished.
As additional prime notch operators are engaged in cascade ($T(\omega) = \prod_{p \le p_{\text{max}}} \widehat{H}_p(\omega)$), background composite spectral energy decays deterministically. Clean prime impulses stand out above the suppressed harmonic background, demonstrating how wave-mechanical operators sieve numbers in frequency space.
This tool plots the power spectral density $|S(\omega)|^2$ across the fundamental frequency band. Increasing $p_{\text{max}}$ demonstrates the progressive clearing of noise channels, illustrating the exact mathematical mechanism underlying phase cancellation.