Fusion-channel measurement access

Let $H$ be a fixed finite fusion space with an orthonormal fusion-tree basis in the category and simple-representative domain of Fusion-tree task input encoding. For a colored task, $H$ is the currently occupied ordered-list summand tracked by the task record. For an adjacent pair of leaves or a declared adjacent pair of subtrees, recouple the tree by its specified unitary $F$-moves so that the pair fuses first. For each allowed simple total charge $a$ of that pair, let $P_a$ be the orthogonal projection onto the span of basis vectors whose pair label is $a$, transported back by the inverse recoupling unitary. Distinct channel subspaces are orthogonal and their direct sum is $H$, so $(P_a{)}_a$ is a finite projective measurement in the sense of Finite projective measurements. Its outcome is $a$, its unnormalized branch vector on input $\psi$ is $P_a\psi$, and the recoupling follows the inverse-precomposition F convention of F-symbol as a recoupling matrix, whose unitarity is proved in F-symbol unitarity. The branch vector is unnormalized until its probability is checked.