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 $\alpha|\Delta w|+\beta d_H$ throughout the open region $\alpha>\beta$. The finite orbit counts and case distinctions are checked in Lean~4.
| Subjects: | General Mathematics (math.GM) |
| MSC classes: | 05C70, 05E18 |
| Cite as: | arXiv:2601.07175 [math.GM] |
| (or arXiv:2601.07175v3 [math.GM] for this version) | |
| https://doi.org/10.48550/arXiv.2601.07175 arXiv-issued DOI via DataCite |
From: Alejandro Radisic [view email]
[v1]
Mon, 12 Jan 2026 03:50:11 UTC (11 KB)
[v2]
Wed, 14 Jan 2026 13:04:24 UTC (13 KB)
[v3]
Mon, 25 May 2026 18:10:04 UTC (26 KB)
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