 Coordinate shifts of a set family

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.