Squared derivative correlations force frequency-graph energy (${\sum }_{k\in B}|\widehat{{\Delta }_{k}f}(\varphi (k)){|}^2\ge \alpha N^3⟹E(\mathrm{graph}\varphi )\ge {\alpha }^4{N}^3$)

Use the unnormalized Fourier transform and $|f|\le 1$. Normalize each derivative correlation by $N$ and align its phase. Their absolute sum is at least $\alpha N$. One Cauchy-Schwarz inequality, expansion by shift/frequency differences, and a second Cauchy-Schwarz inequality bound its fourth power by $N$ times the additive energy of a frequency graph. Parseval identity on a finite group supplies the factor $N$; exact equality of both shift sums and frequency sums is retained.