Radial-ordering sign in commuting beta-gamma contractions ($\beta (z)\gamma (w)\sim -(z-w{)}^{-1},\gamma (z)\beta (w)\sim +(z-w{)}^{-1}$)

Commutation allows exchanging the factors at their existing arguments; it does not allow exchanging their arguments while retaining the same pole sign. With the displayed convention, differentiation produces $\partial \beta (z)\gamma (w)\sim (z-w{)}^{-2}$ and $\partial \gamma (z)\beta (w)\sim -(z-w{)}^{-2}$. These signs are essential for the conformal weights and central charge of a commuting beta-gamma system.