= Condensate fluctuation power counting
{title2=$H_3/H_2\sim\sqrt{N_{\rm ex}/N},\quad H_4/H_2\sim N_{\rm ex}/N$}
Expand a weakly depleted <Bose gas> around its macroscopically occupied mode. Replacing a condensate operator by an amplitude of order $\sqrt N$ gives quadratic, cubic and quartic fluctuation interaction scales proportional to $NN_{\rm ex}$, $\sqrt N N_{\rm ex}^{3/2}$ and $N_{\rm ex}^2$, with the same interaction-volume factor. Thus the depletion fraction supplies formal amplitude counting. This is not a uniform bound on infrared modes or a substitute for checking the interaction regime. At fixed <particle number>, expansion of the condensate interaction must precede the final quadratic truncation; the resulting cancellation is the <number-conserving Bogoliubov quadratic Hamiltonian>.
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