  The $ \beta $-functions of marginal couplings are known to be closely related
to the $ A $-function through Osborn's equation, derived using the local
renormalization group. It is possible to derive strong constraints on the
$\beta$-functions by parametrizing the terms in Osborn's equation as
polynomials in the couplings, then eliminating unknown $\tilde{A}$ and $T_{IJ}$
coefficients. In this paper we extend this program to completely general gauge
theories with arbitrarily many Abelian and non-Abelian factors. We detail the
computational strategy used to extract consistency conditions on $ \beta
$-functions, and discuss our automation of the procedure. Finally, we implement
the procedure up to 4-, 3-, and 2-loops for the gauge, Yukawa and quartic
couplings respectively, corresponding to the present forefront of general $
\beta $-function computations. We find an extensive collection of highly
non-trivial constraints, and argue that they constitute an useful supplement to
traditional perturbative computations; as a corollary, we present the complete
3-loop gauge $\beta$-function of a general QFT in the $\bar{\text{MS}}$ scheme,
including kinetic mixing.
