Ordering and positivity of statistical strengths

For every Bell experiment $S$ with $k>0$ parties, positive finite setting counts, and positive finite outcome counts at every local setting, let $\mathcal A$ be its joint settings, $\mathcal X_a$ its joint outcomes at $a$, and $\mathcal V$ its assignments of outcomes to every local setting, with $v(a)$ denoting the selected joint outcome. Let $Q(a,x)$ be a real table satisfying all three following hypotheses: for each $a$, its entries are nonnegative and $\sum_{x\in\mathcal X_a}Q(a,x)=1$; for every subset $J$ of parties, every pair $a,b$ of joint settings agreeing on $J$, and every $v\in\mathcal V$, the sums of $Q(a,x)$ over outcomes satisfying $x_i=v_i(a_i)$ for all $i\in J$ and of $Q(b,x)$ over outcomes satisfying $x_i=v_i(b_i)$ for all $i\in J$ are equal; and there exist positive integers $d_i$, a positive semidefinite complex matrix $\rho$ of trace one indexed by $\prod_i\{0,\ldots,d_i-1\}$, and, for every party $i$ and local setting $c$, positive semidefinite matrices $M_{i,c,z}$ indexed by $\{0,\ldots,d_i-1\}$, summing to the identity over its local outcomes $z$, such that

For a probability distribution $p$ on $\mathcal V$, put $L_p(a,x)=\sum_{v:\,v(a)=x}p(v)$. For a probability distribution $s$ on $\mathcal A$, define $D_{Q,s}(p)=+\infty$ if some trial has $s(a)Q(a,x)>0$ and $s(a)L_p(a,x)=0$; otherwise define it as the nonnegative truncation of

Let $F_Q(s)=\inf_{p\ge0,\ \sum_vp(v)=1}D_{Q,s}(p)$, let $u(a)=1/|\mathcal A|$, let $U_Q=F_Q(u)$, let $I_Q$ be the supremum of $F_Q(s)$ over all distributions of the form $s(a)=\prod_i s_i(a_i)$ with each $s_i$ a local setting probability distribution, and let $C_Q$ be the supremum of $F_Q(s)$ over all joint setting probability distributions. Then, as extended nonnegative real inequalities and a logical equivalence,

All setting and potential distributions may have zero coordinates, the marginal-consistency hypothesis includes the empty subset of parties, and the infima and suprema do not assert attainment.