Lemma: Collective chain

Each $B_{m,k}$ is an isometry. Distinct $(m,k)$ images are orthogonal, and

Absent endpoint terms are zero.

Proof. Each partition has a unique union; distinct partition words are orthogonal, and there are $\left(\genfrac{}{}{0ex}{}{m}{k}\right)$ terms per word. Hopping moves one occupied pair. Counting moves and comparing the binomial normalizations gives the coefficients with the CAR signs already included in $B_{m,k}$.