Flatland: abelian extensions of the Standard Model with semi-simple completions

We parametrise the space of all possible flavour non-universal $\mathfrak{u}(1)_X$ extensions Standard Model that embed inside anomaly-free semi-simple gauge theories, including up to three right-handed neutrinos. More generally, we abelian (i.e.) by any number $\mathfrak{u}(1)$'s) SM with such completions. The resulting is a collection planes dimensions $\leq 6$. Numerically, find roughly $2.5\%$ maximum charge ratio $\pm 10$ can be embedded in theories. Any vector-like extension embeds (at least) $\mathfrak{g} = \mathfrak{su}(12)\oplus \mathfrak{su}(2)_L\oplus \mathfrak{su}(2)_R$. also provide simple computer program tests whether given $\mathfrak{u}(1)_{X^1}\oplus \mathfrak{u}(1)_{X^2}\oplus \dots$ assignment has completion and, if it does, outputs set maximal algebras which $\mathfrak{sm}\oplus\mathfrak{u}(1)_{X^1}\oplus model may embedded. hope this useful tool pointing way from $\mathfrak{sm} \oplus\mathfrak{u}(1)_{X^1}\oplus models, have many phenomenological uses, their unified completions ultraviolet.