= Local constancy principle
Fourier localization controls the variation of a demodulated envelope at reciprocal scales. Suppose the <Fourier support> lies in a rectangular box centered at $\xi_0$, with widths of order $N$ in spatial-frequency coordinates and $N^2$ in a time-frequency coordinate. Set $g(x,t)=e^{-i\xi_0\cdot(x,t)}f(x,t)$. Choose a <Schwartz function> whose transform is one on the centered box and rescale it to obtain a kernel $K_N$. Then
$$
g=g*K_N,\qquad K_N(x,t)=N^{d+2}K(Nx,N^2t)
$$
for $d$ spatial coordinates. The <convolution> averages on spatial scale $N^{-1}$ and temporal scale $N^{-2}$. For bounded $g$, differentiating the kernel gives $\|\partial_{x_j}g\|_\infty\leq C N\|g\|_\infty$ and $\|\partial_tg\|_\infty\leq C N^2\|g\|_\infty$.
These are quantitative variation bounds, not literal constancy on boxes. Removing the center frequency is essential: multiplying a slowly varying envelope by $e^{iMx}$ can make its phase rotate rapidly while leaving its Fourier-support width unchanged. The principle controls a localized envelope or a smoothed magnitude, rather than forbidding carrier oscillations.
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