Leading-pole consistency under coupling reparametrization ($g=g_0+a_1{g}_0^2/ϵ+b_1{g}_0^3/{ϵ}^2⟹b_1=a_1^2$)

With ${\beta }_0=-ϵg_0$, the chain rule gives $\beta (g)={\beta }_{0}dg/d{g}_0$. Inverting the displayed coupling expansion yields $g_0=g-a_1{g}^2/ϵ+(2a_1^2-b_1)g^3/{ϵ}^2$. Substitution gives $\beta =-ϵg-a_1{g}^2+2(a_1^2-b_1)g^3/ϵ+O(g^4)$, so finiteness requires the displayed double-pole condition. An omitted simple-pole term $c_1{g}_0^3/ϵ$ supplies a finite $-2c_1{g}^3$ term; higher-pole consistency does not by itself determine the finite two-loop beta function.