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

Alejandro Radisic

TL;DR

This work analyzes $K_4$-equivariant perfect matchings on the Boolean hypercube and proves that for $n=6$ there is a unique optimal matching among those using only $comp$ or $rev$, realized by a simple reverse-priority rule and incurring total Hamming distance $120$. It further shows that the King Wen sequence of the I Ching is isomorphic to this matching, establishing a deep combinatorial connection. While allowing the cheaper $comp\circ rev$ pairings can reduce the global cost to $96$, such a construction requires per-orbit analysis and lacks the uniform simplicity of the proposed rule. All results are formally verified in Lean 4 with Mathlib, lending rigorous validation to the structural claims about orbits, distances, and the King Wen correspondence.

Abstract

We characterize perfect matchings on the Boolean hypercube {0,1}^n that are equivariant under the Klein four-group K_4 generated by bitwise complement and reversal. For n = 6, we prove there exists a unique K_4-equivariant matching minimizing total Hamming cost among matchings using only comp or rev pairings, 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. Notably, allowing comp(rev) pairings yields lower cost (96), but the King Wen sequence follows the structurally simpler rule. All results are formally verified in Lean 4 with the Mathlib library.

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

TL;DR

This work analyzes -equivariant perfect matchings on the Boolean hypercube and proves that for there is a unique optimal matching among those using only or , realized by a simple reverse-priority rule and incurring total Hamming distance . It further shows that the King Wen sequence of the I Ching is isomorphic to this matching, establishing a deep combinatorial connection. While allowing the cheaper pairings can reduce the global cost to , such a construction requires per-orbit analysis and lacks the uniform simplicity of the proposed rule. All results are formally verified in Lean 4 with Mathlib, lending rigorous validation to the structural claims about orbits, distances, and the King Wen correspondence.

Abstract

We characterize perfect matchings on the Boolean hypercube {0,1}^n that are equivariant under the Klein four-group K_4 generated by bitwise complement and reversal. For n = 6, we prove there exists a unique K_4-equivariant matching minimizing total Hamming cost among matchings using only comp or rev pairings, 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. Notably, allowing comp(rev) pairings yields lower cost (96), but the King Wen sequence follows the structurally simpler rule. All results are formally verified in Lean 4 with the Mathlib library.
Paper Structure (21 sections, 13 theorems, 8 equations)

This paper contains 21 sections, 13 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Formal Verification
  4. Preliminaries
  5. Hexagrams as Binary Vectors
  6. The Klein Four-Group Action
  7. Hamming Distances of Group Actions
  8. The King Wen Sequence
  9. Definition
  10. The Equivariance Theorem
  11. Optimality of the King Wen Matching
  12. The Reverse-Priority Rule
  13. Cost Minimization
  14. Uniqueness
  15. Discussion
  16. ...and 6 more sections

Key Result

Theorem 1.1

For $n = 6$, there exists a unique $K_4$-equivariant perfect matching on $\{0,1\}^6$ minimizing total Hamming distance among matchings where each pair is either $\{h, \mathrm{comp}(h)\}$ or $\{h, \mathrm{rev}(h)\}$. This matching is given by the reverse-priority rule: The total Hamming cost is $120$, compared to $192$ for the complement-only matching.

Theorems & Definitions (30)

  • Theorem 1.1: Uniqueness of Optimal $\mathrm{comp}/\mathrm{rev}$ Matching
  • Theorem 1.2: Isomorphism with King Wen
  • Definition 2.1: Hexagram
  • Definition 2.2: Hamming Distance
  • Definition 2.3: Total Hamming Cost of a Matching
  • Definition 2.4: Complement and Reversal
  • Proposition 2.5
  • Definition 2.6: Orbit
  • Proposition 2.7: Orbit Sizes
  • Definition 2.8: Anti-Symmetric Hexagram
  • ...and 20 more