Bell scenarios encode classical non-causality

Bell and contextuality scenarios are unified within the Acín-Fritz-Leverrier-Sainz hypergraph formalism, in which measurements are hyperedges. Multipartite measurements are obtained from product rules that combine independent contextuality scenarios into a single hypergraph. Several such rules exist: they are structurally distinct, yet yield the same classical, quantum, and boxworld correlations. Here, we show that this difference is not incidental, unveiling an unexpected connection. The measurements of composite Bell scenarios are exactly the process functions, the deterministic classical communication protocols that may be non-causal, i.e., lack a definite causal order without generating paradoxes. Validity of a hyperedge and logical consistency of a process function are thus one and the same condition. This allows us to characterize process functions by normalization and no-signaling alone, providing transformations that generate non-causal process functions from no-communication ones. The product rules, in turn, order process functions by causal structure, from no-communication to non-causality, which yields a graphical criterion for the latter: a process function is non-causal precisely when its hyperedge is generated by the disjunctive but not the maximal product. We thereby obtain the first joint hypergraph representation of (non-causal) process functions and (nonlocal-without-entanglement) measurements, recasting classical non-causality as a non-standard way of composing black boxes.