Radial selection rule for the breathing mode of a harmonically trapped gas

Within a fixed hyperangular channel $s>0$ of a harmonically trapped system, the $1/R^2$ perturbation is absorbed exactly into a shift of the channel parameter, $s\to s_\eta$, so the single-channel model remains a harmonic oscillator with a shifted inverse-square term: radial gaps stay at $2\hbar\omega$ exactly and no monopole spectral weight appears at forbidden frequencies at any order. The first-order cancellation is also proved independently by a compact algebraic argument in which the ket and bra contributions cancel pairwise; this is the main new result. Substituting single-channel quantities into the established $m_1/m_{-1}$ sum-rule bound yields $Q^{-1}$ scaling of the sum-rule estimate ($Q\equiv 2q+s+1$, $q$ the radial quantum number) with an explicit coefficient; its finite-temperature average has a low-$T$ plateau and a $1/T$ high-$T$ tail. All results hold for any real $s>0$. The Laguerre polynomial identities extend formally to three dimensions, but exact 3D results show $q$-dependent contact corrections along $SO(2,1)$ ladders, so the physical interpretation there requires a separate derivation.