Equation (19): the six directed SSA excesses agree in three pairs

Let $\rho$ be a complex kernel on $\mathbb N^3$ that admits an entrywise expansion $\rho(x,y)=\sum_r u_r(x)\overline{v_r(y)}$ with square-summable vectors and $\sum_r\|u_r\|\|v_r\|<\infty$, whose every finite quadratic form is real and nonnegative, and whose real diagonal is summable with sum $1$. For each $C\subseteq\{0,1,2\}$, define $\rho_C(a,b)=\sum_{z:C^\complement\to\mathbb N}\rho(a\sqcup z,b\sqcup z)$, where $a,b:C\to\mathbb N$ and $a\sqcup z$ combines their coordinates. For any kernel $Y$, define $H(Y)\in[0,\infty]$ as the infimum of $\sum_r[-a_r\log a_r]_+$ over all entrywise expansions $Y=\sum_r a_rq_rq_r^*$ with nonnegative summable weights of sum $1$ and square-summable vectors whose positive-weight members are unit and pairwise orthogonal; zero-weight vectors are unrestricted, $[t]_+=\max(t,0)$, and an empty infimum is $+\infty$. Assume $H(\rho_C)\ne+\infty$ for every $C$, including the empty set. Set $S(C)=\operatorname{toReal}H(\rho_C)$ and $J(A,B)=S(A)+S(B)-S(A\cup B)$, with the conversion convention $\operatorname{toReal}(+\infty)=0$. For every permutation $p$ of $\{0,1,2\}$,