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Center the five coefficients at lags $-2,-1,0,1,2$. Their transfer function is real:Choosing consecutive causal lags instead only adds a phase and leaves the squared modulus unchanged. After $k$ applications, the white noise spectral density has becomeWrite $c=\cos \omega$. Completing the square givesFor $-1\le c\le 1$, its range is $[-1/3,7/6]$, and its absolute maximum is $7/6$, attained exactly when $c=1/2$. Consequently, on the conventional nonnegative frequency range $[0,\pi ]$,On the two-sided range $[-\pi ,\pi ]$ there is also a peak at $-\pi /3$; the ratio there is one for every $k$. This is a counterexample to the claim if read literally on a signed frequency domain. The correct two-sided exclusion is $\omega \ne \pm \pi /3$.

The result illustrates how repeated symmetric filtering can amplify an oscillation. Although $H(0)=1$ preserves a constant component, $H(\pi /3)=7/6>1$ amplifies period-six oscillations. Repetition selects an increasingly narrow frequency band around those peaks; it does not simply smooth all noise away. In fact the unnormalized output variance eventually grows without bound, since a fixed interval around either peak has $|H|>1$.