Genuine Multi-Entropy in Abelian Chern--Simons Theory: Exact Key-Ring Collapse and Its Breakdown for Generic Link States
We study genuine multi-entropy in Abelian $U(1)_k$ Chern-Simons theory. For key-ring link states, where only the linking numbers between one distinguished component $K$ and the remaining $\mathtt{q}-1$ components are nonzero, we derive an exact closed-form expression for the $\mathtt{q}$-partite Rényi multi-entropy for general $\mathtt{q}$, level $k$, and Rényi index $n$. For $\mathtt{q}=4$, this shows that the genuine multi-entropy $\mathrm{GM}^{(4)}_n$ collapses exactly onto the tripartite information $I_{3,n}$ for all $n$, while for $\mathtt{q}=5$ it is likewise completely determined, for all $n$, by a linear combination of tripartite and bipartite Rényi multi-entropies. We then go beyond the key-ring class and study general four-component link states with arbitrary pairwise linking numbers. A numerical scan over Chern--Simons levels $2\leq k\leq24$ shows that the all-$n$ collapse found analytically for key-ring states does not survive for generic link states. Remarkably, the collapse remains exact at $n=2$ for every level examined. At $n=3$, violations occur, within the scanned range, only when $3\mid k$, with a further dependence on the $3$-adic valuation of $k$. At $n=4$ and $n=5$, violations occur for every level examined, with rates that vary strongly with $k$. These results show that the breakdown is not controlled simply by the zero-divisor structure of composite $\mathbb Z_k$, but instead exhibits a nontrivial joint dependence on the Rényi index and the arithmetic structure of the Chern--Simons level.