We study genuine multi-entropy as a diagnostic of multipartite entanglement in the toric code, which provides a controlled setting for probing multipartite structures in topologically ordered states. Our main question is whether genuine multi-entropy captures information that is not reducible to conventional lower-party entropic data, such as topological entanglement entropy. We first analyze toric-code ground states that admit a stabilizer-state description, where the relevant quantities can be evaluated exactly. In this sector, genuine multi-entropy reflects the topological structure and symmetries of the toric code, while exhibiting highly constrained relations to lower-party multientropies. We conjecture that, for stabilizer states and q >= 4, the q-partite genuine multi-entropy at replica index n < q collapses to a linear combination of multi-entropies involving at most q-2 parties. We establish this pattern explicitly for q = 4 in the toric code stabilizer sector: for n = 2, 3, the genuine multi-entropy is proportional to the tripartite information I3 and, for the Kitaev-Preskill partition, contains no independent genuine four-partite information beyond that captured by the topological entanglement entropy. At n = 4, however, this reduction breaks down: the genuine multi-entropy is no longer proportional to I3, but remains a topological invariant of the toric-code stabilizer ground states. For generic non-stabilizer superpositions within the ground-state manifold and for coherent superpositions of local excitations, the low-n reduction also fails. These results show that genuine multi-entropy probes multipartite entanglement structure beyond the tripartite information, and hence beyond the topological entanglement entropy in the Kitaev-Preskill partition, whereas for stabilizer states at low replica index it reduces to lower-partite entropic data.