Zero-mode criterion for fusion of boundary defects

Let $\mathcal{A}_i$ and $\mathcal{A}_j$ be Lagrangian algebras, and let $\tau_{ij}$ denote a boundary defect between them. Let $t_i$ be a simple object of the boundary fusion category $\operatorname{Rep}(\mathcal{A}_i)$, and write the lifting of $t_i$ in terms of bulk anyons $a$ using the fusion coefficients $n_a^{t_i}$, so that $n_a^{t_i}\neq 0$ precisely when $a$ occurs in the lifting of $t_i$. A bulk anyon $a$ is a zero mode of $\tau_{ij}$ when it belongs to $\mathcal{A}_i\otimes\mathcal{A}_j$, meaning that there exist $a_i\in\mathcal{A}_i$ and $a_j\in\mathcal{A}_j$ with $a\in a_i\otimes a_j$. Zero-mode criterion. The simple object $t_i$ appears in the fusion product

$$\tau_{ij}\otimes\tau_{ji}$$

if and only if every bulk anyon $a$ in its lifting, equivalently every $a$ with $n_a^{t_i}\neq 0$, is a zero mode of $\tau_{ij}$ or $\tau_{ji}$. This criterion is a heuristic proposed for determining the coefficients in the defect fusion decomposition. It concerns boundary defects between distinct Lagrangian algebras and is intended to describe which boundary excitations occur when the intermediate boundary segment is shrunk; the source does not establish its general validity.