= Nonunique down-set extremizers for Harper theorem
Equality in <Harper theorem> need not make a <down-set> a cube-automorphic image of a simplicial <initial segment>. In $Q_4$, take all sets of size at most one and the four two-element sets forming a four-cycle. This family has size nine and <closed graph neighbourhood> size fifteen, just like the size-nine <simplicial order on the discrete cube> <initial segment>. The induced <degree of a vertex> distributions differ, so they cannot be related by a <graph automorphism>.
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