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

We study equivariant perfect matchings on the Boolean hypercube ${\text{B}}^6$ under the Klein four-group $K_4=⟨\text{comp},\text{rev}⟩$ generated by bitwise complement and reversal. Among matchings using only $\text{comp}$ or $\text{rev}$ 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 $\text{comp}\circ \text{rev}$ 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 $\alpha \mathrm{\mid }\mathrm{\Delta }w\mathrm{\mid }+\beta d_H$ throughout the open region $\alpha >\beta$. The finite orbit counts and case distinctions are checked in Lean~4.