Optimal Equivariant Matchings on the 6-Cube: With an Application to the King Wen Sequence

We characterize perfect matchings on the Boolean hypercube {0, 1} n that are equivariant under the Klein four-group K4 = ⟨comp,rev⟩ generated by bitwise complement and reversal. For n = 6, we prove there exists a unique K4-equivariant matching minimizing total Hamming cost, achieving cost 120 versus 192 for the complement-only matching. The optimal matching is determined by a simple “reverse-priority rule”: pair each element with its reversal unless it is a palindrome, in which case pair with its complement. We verify that the historically significant King Wen sequence of the I Ching is isomorphic to this optimal matching. All results are formally verified in Lean 4 with the Mathlib library.