Necessity of the polarization-preservation condition for $I[S](P)$

Let $P$ be a linear multiple-access channel with associated family of subspaces $\mathcal{V}$, let $S$ be a subset of the input coordinates, and let $\operatorname{proj}_S$ denote projection onto the coordinates in $S$. The quantity $I[S](P)$ is preserved by the polarization process only if there exists a subspace $V_S$ of dimension $|S|$ such that

$$\operatorname{proj}_S(V_S)=\mathbb{F}_q^S,$$

and, for every $V\in\mathcal{V}$,

$$\operatorname{proj}_S(V_S\cap V)=\operatorname{proj}_S(V).$$

Necessity conjecture for preservation of $I[S](P)$. If $I[S](P)$ is preserved by the polarization process, then such a subspace $V_S$ must exist. The condition is presented as necessary for preservation, not as a sufficient condition. The surrounding result establishes sufficiency under the same condition, while this statement records the converse necessity.