Regular Triadic Coalgebras

We introduce $n$-adic cyclic coalgebras: coalgebras for the polynomial functor $F(X) = X^n$ equipped with a cyclic automorphism $\theta$ of exact order $n$ and a compatibility condition $d \circ \theta = \sigma \circ d$. Under the additional hypothesis of maximal differentiation (regular coalgebras), every $\theta$-orbit has size exactly $n$, and the structure admits a clean orbital dichotomy: orbit-closed coalgebras, where unfolding remains within each orbit, and orbit-shifting coalgebras, where it propagates between distinct orbits. We analyse chirality via anti-chiral symmetries---bijections that reverse the cyclic orientation while preserving the comultiplication. This yields a sharp chirality threshold: dyadic coalgebras are always achiral, whereas for $n \geq 3$ we prove universal chirality: every regular $n$-adic coalgebra is chiral, regardless of orbit structure. In the orbit-closed triadic case, the cyclic shift property $d(x)_0 = \theta(x)$ selects a unique local model, the cyclic triad, which is indecomposable and whose automorphism group is purely rotational. We then introduce the triadic helix, an orbit-shifting construction on $X \times \mathbb{Z}_3$ that lifts an arbitrary deterministic system $(X,f)$ to a triadic coalgebra. Abstracting its fibre permutation as a helical twist, we prove a parity constraint: such twists exist if and only if the arity is odd, making arity three the minimal arity supporting this construction. Categorically, the helix extends to a faithful functor $L : \mathbf{DetSys} \to \mathbf{TriCyc}$, which is not full; adding a canonical pointing yields an equivalence $\mathbf{DetSys} \simeq \mathbf{TriHel}_\bullet$. Thus arity three emerges as a double threshold---geometric (chirality) and arithmetic (helical twist)---for non-trivial cyclic behaviour.