The symmetry of Order 4 constrains four relational objects (a, A; b, B; c, C and d, D) to rotate through a six-membered cycle (a → B → c → A → b → C), the members of one of the four relational objects functioning as a logic mirror to keep the members of the other three pairs apart.
The three pairs form mirror opposite codons:
a B c A b C
B c A b C a
c A b C a B
The members of one side of a pairing can rotate one or two positions relative to those of another, while still conserving relational symmetry.
The three pairings correspond with:
1. a paired with A and A paired with a (same kinds of letter paired with opposite kinds signs)
2. a paired with B and C paired with a (different kinds of letter paired with opposite kinds signs)
3. a paired with c and b paired with a (different kinds of letter paired with same kinds of signs).
This can be understood visually with reference to colour pairings. In the diagram below, the three pairings constitute a triplet state and the single pairing the singlet state.
This provides a mechanism for coupling relational objects together. In effect, two relational objects exchange one of their two codons. Each can also exchange the other of their two codons with one from another relational object. One relational object can thereby become coupled with one of two other relational objects.
For example, consider the exchange of goods and services. Traders and their goods and services can each be treated as relational objects that can be coupled together, such that the offers and wants of traders can be cleared in cycles. The traders start out uncoupled. Each trader can be coupled once through an object that they want to another trader offering that object and once through what they offer to another trader wanting that object. When all traders are double coupled, a ring is formed enabling them all to be decoupled. At this point, all traders have exchanged what they offered for what they wanted.
There is a further way in which relational objects can be coupled. It is as if codons are nested. Codons can be nested across different stages of coupling. For example, a sequence of three transformations – uncoupled → singly coupled → doubly coupled → uncoupled – can itself function as a single transformation when coupled with another layer of logic. By this mechanism, couplings can be geared (hierarchically ordered in harmonic ratios).