Complementary-size simplicial vertex boundaries can be asymmetric ($|\partial C(63)|=36<37=|\partial C(65)|(n=7)$)

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.