= Leading-pole consistency under coupling reparametrization
{title2=$g=g_0+a_1g_0^2/\epsilon+b_1g_0^3/\epsilon^2\quad\Longrightarrow\quad b_1=a_1^2$}
With $\beta_0=-\epsilon g_0$, the <chain rule> gives $\beta(g)=\beta_0\,dg/dg_0$. Inverting the displayed <coupling> expansion yields $g_0=g-a_1g^2/\epsilon+(2a_1^2-b_1)g^3/\epsilon^2$. Substitution gives $\beta=-\epsilon g-a_1g^2+2(a_1^2-b_1)g^3/\epsilon+O(g^4)$, so finiteness requires the displayed double-pole condition. An omitted simple-pole term $c_1g_0^3/\epsilon$ supplies a finite $-2c_1g^3$ term; higher-pole consistency does not by itself determine the finite two-loop <beta function>.
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