Helical Triadic Coalgebras

This paper investigates the final coalgebra for the endofunctor $F(X) = \mathbb{Z}_3 \times X^3$ on the category $\mathbb{Z}_3\text{-}\mathbf{Set}$ of sets with a $\mathbb{Z}_3$-action. We call the resulting $F$-coalgebras Helical Triadic Coalgebras (HTCs). The factor $\mathbb{Z}_3$ records an observable phase that makes distinct cyclic positions distinguishable. We develop the notion of $\mathbb{Z}_3$-bisimulation, which generalizes standard bisimulation by allowing cyclic shifts. Our main results concern a natural HTC structure on the srs lattice (Laves graph). The canonical morphism from the srs coalgebra $\mathcal{S}$ to the final coalgebra $\Omega$ is not injective: translations induce a bisimulation collapsing $\mathcal{S}$ onto a 12-element quotient $\mathcal{Q} \cong K_4 \times \mathbb{Z}_3$. The $V_4$-symmetry of srs further collapses $\mathcal{Q}$ onto a 3-element image $\mathcal{I}$. A symmetry analysis reveals that $\mathcal{I}$ is symmetric while $\Omega$ is not. We also define orbital invariants (binding index, degeneracy, multiplicity) and establish that every regular coalgebra is chiral. Finally, we prove that among sub-coalgebras of the final coalgebra, symmetry and connectivity alone characterize srs uniquely (up to chirality). These results bridge coalgebraic methods with graph theory and crystallography.