Pairwise Fisher graphs capture local covariance information, but they cannot distinguish an irreducible multi-observable radiation pattern from a collection of ordinary pairwise correlations. We show that this missing structure is naturally supplied by higher-order Fisher tensors. In a finite basis of binned EECs, ECFs, or EFPs, and in the natural exponential-family coordinates generated by that basis, the same local tensor has three equivalent interpretations: a coefficient in the local Kullback-Leibler expansion, a connected cumulant of the chosen correlator observables, and a signed weight on a hyperedge linking those observables. This gives an exact Fisher-correlator-hypergraph triality in the local exponential-family embedding. The triality provides a direct construction of physics-informed hypergraphs from correlator data. Extending the quadratic Fisher matrix to the first non-trivial higher tensor identifies genuinely connected multi-observable radiation patterns, supplies hyperedge weights for higher-order Laplacians and message passing, and gives a principled criterion for compressing observable bases beyond pairwise information. We develop these constructions and spell out why the exact cumulant interpretation is special to natural exponential-family coordinates. We illustrate the framework in four applications. In a minimal local-KL study, the cubic Fisher tensor reduces the KL truncation error and isolates the dominant triplet structure. In a two-versus-three prong jet substructure benchmark, the hypergraph selector improves compressed-basis classification. In a 33-observable basis-design problem, the Fisher hypergraph retains more third-order local response at twelve observables. A low-capacity learning benchmark then shows how the same Fisher hyperedges can be used as an interpretable inductive bias for message passing on correlator observables.