= Density fourth-power functional
{title2=$\psi(f)=\int f^4$}
Along <bounded density tilts>, this functional has <derivative> $4\int f^4g=P_f(4f^3g)$. At a bounded baseline density its <canonical gradient>, relative to these regular paths, is $4(f^3-\int f^4)$. A common local bound on nearby densities suffices to obtain the same <derivative> along arbitrary <differentiable-in-quadratic-mean paths> in that bounded neighborhood. A bounded baseline alone is insufficient, because of the <spike obstruction to density-power differentiability>.
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