Condensate fluctuation power counting ($H_3/H_2\sim \sqrt{N_{\mathrm{ex}}/N},H_4/H_2\sim N_{\mathrm{ex}}/N$)

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 $N{N}_{\mathrm{ex}}$, $\sqrt{N}{N}_{\mathrm{ex}}^{3/2}$ and $N_{\mathrm{ex}}^2$, 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.