Exact, projective, and encoded braid universality

Let $\mathrm{\Gamma }\le U(V)$ be a subgroup acting on a nonzero finite-dimensional Hilbert space $V$. It is exactly universal on $V$ when $\mathrm{\Gamma }=U(V)$, projectively exactly universal when its image in $PU(V)$ is all of $PU(V)$, and projectively dense when that image is dense in the metric $d_P$ of Projective braid gate and operator metric.

For an encoding $J:\mathcal{C}\to V$ with code projector $P=J{J}^{\ast }$, the action is approximate logical-and-leakage universal if for every $U\in U(\mathcal{C})$ and every $ϵ>0$ there is $W\in \mathrm{\Gamma }$ and a phase $z\in U(1)$ such that $\mathrm{\parallel }PWP-zJU{J}^{\ast }{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le ϵ,\mathrm{\parallel }(I-P)WP{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le ϵ\mathrm{.}$ This last notion includes a leakage bound as a separate requirement; projective density on one fusion block alone does not supply it.