= Monopole cancellation of internal cubic-estimator contractions
{title2=$\sum_m(-1)^m\mathcal G^{\ell\ell L}_{m,-m,M}=0\quad(L>0)$}
In a <full-sky cubic bispectrum estimator>, pairwise Gaussian covariance within one triple gives a sum of <Gaunt integrals> with opposite $m$ indices. The <spherical harmonic addition theorem> turns their pair into the constant $(2\ell+1)/(4\pi)$; its integral against $Y_{LM}$ vanishes for $L>0$. A zero temperature monopole removes $L=0$. Thus only the six pairings connecting the two triples survive in the Gaussian variance; isotropic weights are essential for the cancellation.
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