Coordinate shifts of a set family
= Coordinate shifts of a set family
{title2=$C_{ij}\mathcal F$}
For distinct coordinates $i,j$, replace a member containing $j$ but not $i$ by its partner with $j$ exchanged for $i$, provided that partner is absent. This <set family> operation preserves <cardinality>. Repeating shifts with $i<j$ terminates because every change decreases $\sum_{A\in\mathcal F}\sum_{a\in A}a$. Its effect on shadows is described by <coordinate-shift shadow containment>.