EPR Revisited: Context-Indexed Elements of Reality and Operational Completeness

Abstract: We reframe the EPR argument through an operational lens, replacing the notion of fixed "elements of reality" with context-indexed conditional states - what's often referred to as a measurement assemblage. This move deliberately sidesteps the assumption of context-independent values for incompatible observables. Our updated version of the Reality Criterion works like this: if Alice measures observable x and obtains outcome a, then Bob's system must adopt a conditional state that ensures the corresponding outcome for that specific context. Crucially, we also assume operational completeness - a condition that quantum mechanics satisfies when we're dealing with quantum-reachable assemblages. Now, in any theory where one party cannot signal to the other (so-called one-sided no-signaling theories), perfect predictions do support drawing context-indexed inferences. But - and this is key - they don't legitimize assigning fixed values across all contexts. We rigorously demonstrate this distinction. To ground the argument, we offer examples: the qubit singlet scenario using Pauli settings and CJWR thresholds, a continuous-variable case based on the Reid criteria, and a counterexample in the spirit of the PR box, which highlights the boundaries of what quantum theory can actually reach.