= Complementary-size simplicial vertex boundaries can be asymmetric
{title2=$|\partial C(63)|=36<37=|\partial C(65)|\quad(n=7)$}
In the seven-dimensional <Boolean hypercube>, $C(63)$ contains every set of size at most three except $\{5,6,7\}$. Its external <vertex boundary> consists of that missing triple and all 35 four-sets. The complementary-size <initial segment> $C(65)$ contains every set of size at most three and $\{1,2,3,4\}$. Its external <vertex boundary> has the other 34 four-sets and three five-sets. Thus complementary family sizes do not impose a symmetric boundary profile.
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