Section compression in binary order ($C_i$)

Fix a coordinate of the Boolean hypercube, delete it in each of the two sections, and replace each section by a binary initial segment of the same size. This preserves the total size. Mathematical induction on dimension proves that it does not decrease contained edge counts; the crossing contribution is the size of the section intersection, maximized by nested initial segments.

If both sections for every coordinate are binary initial segments, then the family itself is a binary initial segment, apart from the possible family $(\mathcal{P}([n-1])\setminus \{[n-1]\})\cup \{\{n\}\}$. Any missing vertex earlier than a present vertex must disagree with it in every coordinate; the two are therefore complementary and consecutive in binary order on the discrete cube, forcing this sole exceptional shape.