Coordinate shifts of a set family ($C_{ij}\mathcal{F}$)

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.

In a set family, replace a set containing $i$ by its deletion of $i$ only when that deletion is absent. This preserves cardinality and cannot increase maximum pairwise Hamming distance. If a new set and a retained set gave a larger distance, the retained set's deletion would already have been present; that deletion and the original uncompressed set would give the same distance. Repeated downward compressions yield a down-set.

For a family in the $n$-dimensional Boolean lattice, the loss of trace cardinality on deleting coordinate $i$ equals the number of pairs $S,S\cup \{i\}$ both in the family. If every direction has at least two such edges, downward coordinate compression preserves that property. The resulting down-set contains the empty set and all $n$ singletons. Each coordinate must also occur in a two-element member. Those pairs cover all $n$ coordinates and number at least $\lceil n/2\rceil$, giving the bound. For $n\ge 2$, the empty set, all singletons and a minimum collection of pairs covering every coordinate attain it.

For a uniform set family, the lower shadow of its shifted set family is contained in the shifted lower shadow. Check the witness in the four patterns of membership in coordinates $i,j$. A surviving shadow member containing only $j$ must have both old shadow partners; a member containing only $i$ needs either old partner. Taking complements exchanges $C_{ij}$ with $C_{ji}$ and gives the same containment for the upper shadow. Thus coordinate shifts of a set family cannot increase either shadow or their disjoint union, the external vertex boundary of a uniform set family.