= Solution
Introduce the <renormalization-group beta functions> and field anomalous dimension
$$
\beta_i=\Lambda\frac{dg_i}{d\Lambda},
\qquad
\beta_{m^2}=\Lambda\frac{dm^2}{d\Lambda},
\qquad
\gamma_\phi=\frac12\Lambda\frac{d\log Z_\Lambda}{d\Lambda}.
$$
Independence of physics from the arbitrary sliding scale gives the functional <Callan-Symanzik equation>
$$
\boxed{\left[
\Lambda\partial_\Lambda+\beta_{m^2}\partial_{m^2}
+\sum_i\beta_i\partial_{g_i}
-\gamma_\phi\int d^4x\,\phi(x)\frac{\delta}{\delta\phi(x)}
\right]S_\Lambda^{\rm eff}=0,}
$$
up to the equivalent sign convention obtained by defining $\gamma_\phi$ with a minus sign. For an operator $O_i$ of dimension $d_i$ containing $n_i$ fields, differentiating its coefficient shows the canonical and wave-function pieces
$$
\beta_i=(d_i-4+n_i\gamma_\phi)g_i+\beta_i^{\rm vertex},
$$
where $\beta_i^{\rm vertex}$ contains mixing among <local field operators> and genuine corrections at higher <loop order>. Thus $d_i<4$, $d_i=4$, and $d_i>4$ canonically produce <relevant operators>, <marginal operators>, and <irrelevant operators>, respectively, before anomalous corrections.
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