Corollary 1: evaluation along every successive binary partition

For every natural number $n\ge2$ and complex kernel $\rho$ on $\mathbb N^n$, assume that it admits an entrywise expansion $\rho(i,j)=\sum_r a_r(i)\overline{b_r(j)}$ with square-summable vectors and summable products $\|a_r\|\|b_r\|$, that every finite quadratic form $\sum_{i,j\in F}\overline{z_i}\rho(i,j)z_j$ is real and nonnegative for arbitrary finite $F$ and arbitrary complex $z$, and that its real diagonal is summable with sum $1$. For $C\subseteq U=\{0,\ldots,n-1\}$, let $R_C(x,y)=\sum_{z\in\mathbb N^{C^c}}\rho(x\oplus_C z,y\oplus_C z)$, with configurations joined according to $C$ and its complement. 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; the empty infimum is $+\infty$. Assume $H(R_C)\ne+\infty$ for every $C$, including the empty set, and put $S(C)=H(R_C).\mathrm{toReal}$. Let $t$ be any binary tree labelled by $U$, with each leaf valid and each internal node having valid children with disjoint leaf-label sets, and assume its root label set is all of $U$. Define $J(\operatorname{leaf}(i))=0$ and $J(\operatorname{split}(l,r))=S(A_l)+S(A_r)-S(A_l\cup A_r)+J(l)+J(r)$, where $A_l,A_r$ are the children's label sets. Then