Fusion-tree task input encoding

In the unitary braided fusion category $\mathcal{C}$ fixed by Finite anyon task and promise interface, use its fixed representative set $\mathrm{Irr}(\mathcal{C})$. Fix $n\ge 1$, ordered inputs $a_1,\dots ,a_n\in \mathrm{Irr}(\mathcal{C})$, a total charge $c\in \mathrm{Irr}(\mathcal{C})$ with nonzero fusion space, and one parenthesized binary tree. Every charge label below belongs to this same representative set. An encoded fusion input consists of the finite tree shape, a simple-charge label on each internal edge, and a multiplicity-basis index at each trivalent vertex. A label is admissible exactly when every chosen vertex index belongs to the specified basis of its multiplicity space $V_{xy}^z$; the allowed local vertex data and resulting orthonormal fusion-tree vectors are those of Multiplicity-aware fusion-tree basis. The input encoding uses fixed self-delimiting binary codes for charge labels and nonnegative multiplicity indices, with out-of-range labels rejected as invalid encodings. It also records the total charge and an identifier for the selected computational subspace isometry required by Finite anyon task and promise interface, as in Fixed-charge encoding and code projector. For a fixed task family, a declared codebook assigns each such finite identifier a computational dimension and a particular isometry into the specified fusion space. The codebook and its isometries are fixed background model data; unknown identifiers are invalid. If isometries instead vary as part of the classical input, the task must declare a finite description language and its exact interpretation as isometries, and charge the full descriptions as input bits. This does not claim that every arbitrary complex isometry has such a finite description; an arbitrary abstract isometry may remain background model data only. The tree labels and isometry identifiers are finite classical data; they do not themselves specify or prepare the initial encoded quantum state, which is the separate state/preparation field in the task record.