Assembly of four endpoint-equation charges

Let requests have two distinct natural endpoint labels $s_r,t_r$ and a natural lower-cardinality bound $m_r$. Fix $N\in\mathbb N$, $D\in\mathbb R$ and charges $H:\mathbb N\to\mathbb R$. Assume every pair of endpoint choices $a\in\{s_r,t_r\}$, $b\in\{s_u,t_u\}$ satisfies $E_N(a,b)/(m_rm_u)\le D/4+\mathbf1_{a=b}H(a)$, where $E_N$ is the source's endpoint-equation budget. Let $P_N(r,u)$ be the sum of these four budgets. Then

No positivity of the $m_r$ is required in this statement; division follows the real-field convention at zero. This collects disjoint and shared-label costs for collision estimates.