Lemma: Odd compression and reached transitions

For every $j$, $V^{†}{c}_{j}V=V^{†}{c_j}^{†}V=0$. Nevertheless even joint hopping can act nontrivially. Also,

A nonzero boundary subspace invariant under all $c_j,{c_j}^{†}$ is the full Fock space.

Proof. Parity makes odd compression its negative. Joint hopping changes both parities together. Insert $P+(I-P)$ for the composite identity. An occupation projector selects a nonzero word in an invariant vector; annihilations give vacuum and creations give every word.