Theorem 1: correlation information is additive over every cluster partition

For every natural number $n\ge1$ and complex kernel $\rho$ on $\mathbb N^n$, assume that it admits an entrywise representation $\rho(i,j)=\sum_r a_r(i)\overline{b_r(j)}$ with square-summable vectors and $\sum_r\|a_r\|\|b_r\|<\infty$, that all its finite quadratic forms $\sum_{i,j\in F}\overline{z_i}\rho(i,j)z_j$ are real and nonnegative for arbitrary finite $F$ and arbitrary complex functions $z$, and that its real diagonal is summable with sum $1$. For $C\subseteq U=\{0,\ldots,n-1\}$, define $R_C(x,y)=\sum_{z\in\mathbb N^{C^c}}\rho(x\oplus_C z,y\oplus_C z)$. For any kernel $K$, define $H(K)$ to be the infimum of $\sum_k\max\{-w_k\log w_k,0\}$ over entrywise representations $K(i,j)=\sum_k w_k u_k(i)\overline{u_k(j)}$, where the nonnegative real weights are summable with sum $1$, the vectors are square-summable, and vectors with positive weights are unit vectors and mutually orthogonal at distinct indices; the infimum is taken in $[0,\infty]$, with empty infimum $+\infty$. Assume $H(R_C)\ne+\infty$ for every coordinate set $C$, including the empty set, and put $S(C)=H(R_C).\mathrm{toReal}$. For every finite collection $P$ of nonempty coordinate sets that are pairwise disjoint and cover $U$, the asserted equality is

Joining configurations in $R_C$ means using the first configuration on $C$ and the second on $C^c$, and the kernel sums are total infinite sums.