Corollary 2: discarding an independent tensor factor preserves cluster mutual information

For natural numbers $n,M$ with $1\le M<n$, let $\rho$ be a complex kernel on $\mathbb N^n$ admitting an entrywise expansion $\rho(i,j)=\sum_r a_r(i)\overline{b_r(j)}$ with square-summable vectors and $\sum_r\|a_r\|\|b_r\|<\infty$, having real nonnegative finite quadratic forms $\sum_{i,j\in F}\overline{z_i}\rho(i,j)z_j$ for every finite $F$ and arbitrary complex $z$, and having summable real diagonal of sum $1$. For $C\subseteq U=\{0,\ldots,n-1\}$, define $R_C(x,y)=\sum_{z\in\mathbb N^{C^c}}\rho(x\oplus_C z,y\oplus_C z)$. Define $H(K)$ as the infimum in $[0,\infty]$ of $\sum_k\max\{-w_k\log w_k,0\}$ over representations $K(i,j)=\sum_k w_k u_k(i)\overline{u_k(j)}$ with nonnegative summable weights of sum $1$ and square-summable vectors that are unit and pairwise orthogonal at distinct positive-weight indices; an empty infimum is $+\infty$. Assume $H(R_C)\ne+\infty$ for every $C$, including the empty set, and write $S(C)=H(R_C).\mathrm{toReal}$. Put $F=\{i\in U:i<M\}$ and $T=\{i\in U:M\le i\}$, and assume, for every $x,y\in\mathbb N^n$,

For any coordinate sets $A,L,D\subseteq U$ such that $A$ and $L$ are nonempty and disjoint, $T\subseteq L$, $L\setminus T$ is nonempty, and $D\subseteq T$, the conclusion is

Here joining configurations uses their specified coordinates, and the reduction sums are total infinite sums; the discarded set $D$ may be empty or all of $T$.