Background-Free Device-Independent Violations of Causal Inequalities
The process-matrix framework describes quantum correlations without presupposing a global causal order, yet its standard formulation implicitly relies on background structure through a fixed Choi-Jamiolkowski identification of local input-output spaces. We analyze how such background assumptions can be treated operationally relative to a fixed device-independent interface defined by a causal game. We impose local-frame covariance, requiring invariance under independent actions of a physical symmetry group $G$ on each laboratory, thereby excluding symmetry-breaking background resources. Covariance induces a representation-theoretic decomposition into symmetry sectors and symmetry-invariant multiplicity subsystems, introducing physical degrees of freedom that lie outside the declared device-independent interface. We then analyze causal-inequality signatures at the level of interface-observable statistics and identify when symmetry-induced, interface-inaccessible degrees of freedom undermine device-independent certification. A certification is called background-free if it arises from a locally covariant implementation and does not rely on hidden control mediated by interface-excluded degrees of freedom. We prove that background-free certifications cannot yield device-independent violations of bipartite causal inequalities in the multiplicity-free regime or when all multiplicity subsystems are classical-classical (CC). Such violations necessarily require non-CC multiplicity, with a concrete sufficient route provided by input-output embeddability of an effective process-matrix structure into non-CC blocks. These results delineate which device-independent causal signatures remain certifiable once both symmetry-breaking background structure and interface-level hidden control are excluded.