Minimal Lifting in Regular Triadic Coalgebras

Representing dynamical systems in structured formats often incurs overhead. We characterize exactly when this overhead vanishes for regular triadic coalgebras (RTCs)---ternary coalgebras equipped with a cyclic symmetry of order three and a regularity constraint ensuring that each state unfolds into three pairwise distinct successors. Every finite dynamical system $(C,f)$ admits a canonical lifting to an RTC of size $3|C|$. Our main result characterizes when a lifting of minimal size exists: a system with $n$ configurations admits a lifting of size $n$ if and only if $n \equiv 0 \pmod{3}$ and the maximal in-degree of $f$ is at most $n/3$. This criterion is decidable in linear time. The key tool is a dispersion lemma: cyclic equivariance forces elements within any orbit to have pairwise distinct successors, yielding a global capacity bound. Sufficiency reduces to equitable coloring of a union of cliques. The canonical lifting preserves computational complexity with constant overhead. Turing-complete RTCs exist via lifting of universal counter machines, demonstrating encodability rather than intrinsic triadic expressivity.