= Random sign rectangle fourth moment
{title2=$\mathbb E\|f\|_\square^4=1/m+1/n-1/(mn)$}
Give every entry of an $m$ by $n$ matrix an independent uniformly random sign. A four-entry rectangle product has expectation zero unless its two row indices coincide or its two column indices coincide; in those cases it is one. Counting the union of these diagonal cases gives $mn^2+m^2n-mn$ before normalization. Thus some sign matrix has <box norm> fourth power at most the displayed expectation, and its <cut norm> tends to zero when both dimensions tend to infinity.
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