Proposition: Finite-bank nuisance separation

Let a declared response bank be $r\in {\mathbb{R}}^J$, and let the complete permitted nuisance class be

A linear readout $w^{\mathsf{T}}r$ is invariant under every permitted nuisance exactly when

There exists such a readout with nonzero response if and only if

Proof. Invariance means $w^{\mathsf{T}}Ba=0$ for every $a$, which is equivalent to $B^{\mathsf{T}}w=0$. The orthogonal complement of $\mathrm{range}(B)$ is $\ker B^{\mathsf{T}}$. If $r$ is outside the range, choose $w=P_{\ker B^{\mathsf{T}}}r$; then $w^{\mathsf{T}}r=\text{norm}{w}_2^2>0$. If $r$ is in the range, every invariant readout annihilates it.