Authors - Jose R. Rosas-Bustos, Jesse Van Griensven The, Roydon Andrew Fraser, Sebastian Ratto Valderrama, Nadeem Said, Andy Thanos Abstract - CHSH tests are increasingly used as integrity probes for quantum devices and services: when a verifier observes a value S > 2, it is tempting to treat this as a compact certificate of nonclassical correlations. In operational deployments, however, the meaning of a small violation depends on how accurately measurement settings are implemented and how uncertainty is budgeted. We develop a resolution matched comparison between (i) a finite-precision quantum description based on smeared POVMs and (ii) a coarse-grained Bell-local baseline built from deterministic response functions. Using total-variation distance and witness-level tolerances, we define convergence vicinities where the two descriptions become indistinguishable for a realistic verifier. We then interpret weak violations through relaxed Bell bounds under measurement dependence, highlighting that controller leakage can compromise CHSH-only acceptance rules even when the device itself is locally causal. Angle-scan simulations map where these vicinities arise, and IBM Quantum demonstrations illustrate that hardware noise further narrows the practical margin above S = 2. We conclude with controller-aware guidance for integrity pipelines and DIQKD-oriented deployments, emphasizing diversified tests and explicit uncertainty budgets.
