Common Relative-Phase Structure of C₃ Rotation and C₂ Oscillation

This paper reformulates the coupling mechanism between longitudinal C₂ oscillation and the two transverse C₃ closures located on its two sides in a tetrahedral–octahedral composite cell as a common relative-phase structure. In a previous study, a mutual-induction sequence was introduced in which a change in C₂ induces C₃ circulation and the C₃ response induces the next C₂ update; however, the directional relation and the conversion mechanism themselves remained assumptions. We first show that, for a single closed three-state C₃ circulation, the first-order in-plane displacement cancels over one cycle, whereas an oriented quantity constructed from cross products of successive displacements remains along the normal to the rotation plane. This normal coincides with the C₂ long axis of the composite cell and geometrically establishes a correspondence between C₃ chirality and the signed direction of the C₂ axis, i.e., a right-handed directional relation. A rigid rotation of a single C₃, however, does not change the relevant connection lengths and therefore does not by itself require a C₂ displacement. We therefore introduce the relative phase Δφ of the two C₃ structures. From two three-component C₃ states separated internally by 120°, a symmetric coupling proportional to cosΔφ and an antisymmetric coupling proportional to sinΔφ can be constructed. Under a minimal mapping of these quantities to the position and update-direction components of C₂, the C₂ reciprocating oscillation and the relative rotation of the two C₃ structures are represented as different geometrical realizations of the same phase circle, and an inverse mapping also exists. C₂→C₃ and C₃→C₂ may therefore be interpreted not as two independent induction laws but as two representations of a single common relative-phase consistency condition. Finally, symmetry breaking is placed in the same framework: one chirality is selected from left–right symmetric alternatives, and the resulting closed update sequence is retained through Persistence.