Helical Triadic Coalgebras Final Coalgebras for F(X) = Z3 × X3 in Z3-Set

This paper investigates the final coalgebra for the endofunctor ( F(X) = mathbb{Z}_3 times X^3 ) on the category Z3-Set of sets with a Z3-action. We call the resulting F-coalgebras Helical Triadic Coalgebras (HTCs). The factor Z3 records an observable phase that makes distinct cyclic positions distinguishable. We develop the notion of Z3-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 S to the final coalgebra Ω is not injective: translations induce a bisimulation collapsing S onto a 12-element quotient ( mathcal{Q} cong K_4 times mathbb{Z}_3 ). The ( V_4 )-symmetry of srs further collapses Q onto a 3-element image I. A symmetry analysis reveals that I is symmetric while Ω 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.