摘要
We study genuine multi-entropy in AbelianU(1)_(k)Chern-Simons theory. For key-ring link states, where only the linking numbers between one distinguished componentKand the remaining𝚚-1components are nonzero, we derive an exact closed-form expression for the𝚚 -partite Rényi multi-entropy for general𝚚 , levelk , and Rényi indexn . For𝚚=4 , this shows that the genuine multi-entropy\mathrm{GM}{⁽⁴⁾}{_(n)}collapses exactly onto the tripartite informationI_(3,n)for alln , while for𝚚=5it is likewise completely determined, for alln , 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 levels2≤ k≤24shows that the all- ncollapse found analytically for key-ring states does not survive for generic link states. Remarkably, the collapse remains exact atn=2for every level examined. Atn=3 , violations occur, within the scanned range, only when3∣ k , with a further dependence on the3 -adic valuation ofk . Atn=4andn=5 , violations occur for every level examined, with rates that vary strongly withk . These results show that the breakdown is not controlled simply by the zero-divisor structure of compositeℤ_(k) , but instead exhibits a nontrivial joint dependence on the Rényi index and the arithmetic structure of the Chern–Simons level.