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

Abstract

We study equivariant perfect matchings on the Boolean hypercube $\B^6$ under the Klein four-group $K_4 = \langle \comp, \rev \rangle$ generated by bitwise complement and reversal. Among matchings using only $\comp$ or $\rev$ pairings, there is a unique Hamming-cost minimizer, given by a simple ``reverse-priority rule'': pair each element with its reversal unless it is a palindrome, in which case pair it with its complement. This matching has total Hamming cost 120, compared to 192 for the complement-only matching. The historically significant King Wen sequence of the I Ching realizes precisely this matching. Pure Hamming minimization over the full $K_4$ action is different: allowing $\comp \circ \rev$ lowers the cost to 96. The King Wen rule is recovered, however, as the unique Hamming-weight-preserving optimum: it minimizes failures of Hamming-weight preservation before Hamming distance, and it is stable for the weighted energy $α|Δw|+βd_H$ throughout the open region $α>β$. The finite orbit counts and case distinctions are checked in Lean~4.