Gaussian fourth moment ($\mathbb{E}{Z}^4=3v^2$)

For a centered normal random variable $Z\sim N(0,v)$, $\mathbb{E}{Z}^4=3v^2$ and $\mathrm{Var}(Z^2)=2v^2$. This converts squared normal increments into exact variance calculations, including the degenerate case $v=0$.