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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.07175 |
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| _version_ | 1866911717485707264 |
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| author | Radisic, Alejandro |
| author_facet | Radisic, Alejandro |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_07175 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Optimal Equivariant Matchings on the 6-Cube with an Application to the King Wen Sequence Radisic, Alejandro General Mathematics 05C70, 05E18 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. |
| title | Optimal Equivariant Matchings on the 6-Cube with an Application to the King Wen Sequence |
| topic | General Mathematics 05C70, 05E18 |
| url | https://arxiv.org/abs/2601.07175 |