Arbitrary-n triple-anyon fixed-charge code

Let $n\ge 1$ and $c\in \{1,\tau \}$ with $V({\tau }^n\to c)\mathrm{\ne }0$, using the left-associated tree for both fixed-charge spaces. In the ordered list of $3n$ Fibonacci $\tau$ charges, group consecutive leaves into $n$ triples; within each triple fuse its first two charges and then its third, and fuse the $n$ triple outputs by the left-associated $n$-leaf tree to total $c$. Restrict each triple output to $\tau$; its two local paths are indexed by the charge $x\in \{1,\tau \}$ after the first two leaves. Fix a rebracketing sequence of associators from this grouped tree to the fixed left-associated $3n$-leaf tree, and let $R_{n,c}$ be the induced composite unitary. For each tensor basis vector $e_{x_1}\otimes \cdots \otimes e_{x_n}\otimes v_{\lambda }$, where $e_1,e_{\tau }$ are the local triple paths and $v_{\lambda }$ is a basis vector of $V({\tau }^n\to c)$, map it to its grouped fusion-tree basis vector, then apply $R_{n,c}$. Extend linearly and call the resulting map $J_{n,c}$. Its image is the code ${\mathcal{C}}_{n,c}$ and it is an isometry; set $P_{n,c}=J_{n,c}{J}_{n,c}^{\ast }$ for the code projector. The factor $V({\tau }^n\to c)$ retains all spectator fusion paths and is not a physical preparation at zero cost.