$ρ_0(T) = 0$: the only $0$-dimensional subspace is $⊥$, on which $T$ vanishes identically, so every member of the defining range is $g_T(⊥) = 0$.

Consequently $χ_0 = -∞$, which is why the statements about $χ_k$ are for $k ≥ 1$; the case $k = 0$ is recorded because $Cocycle.exists_chi_ae_const$ is stated for all $k$.