Characteristic solution of the multiplicative Callan-Symanzik equation ($C(e^{2t}r,g)=\exp (2{\int }_0^t\gamma (g(u))du)C(r,g(t))$)

For a dimensionless two-point factor obeying $(-2r{\partial }_r+\beta {\partial }_g+2\gamma )C=0$, evolve $g^{\prime }=\beta (g)$ and accumulate the displayed anomalous-dimension multiplier. The flow property verifies the equation: its transport generator is $\beta {\partial }_g+2\gamma$. This fixes a frequent sign ambiguity between raising the physical momentum and changing the reference scale.