Beyond the Cabello-Severini-Winter framework: Making sense of contextuality without sharpness of measurements

We develop a hypergraph-theoretic framework for Spekkens contextuality applied to Kochen-Specker (KS) type scenarios that goes beyond the Cabello-Severini-Winter (CSW) framework. To do this, we add new hypergraph-theoretic ingredients to the CSW framework. We then obtain noise-robust noncontextuality inequalities in this generalized framework by applying the assumption of (Spekkens) noncontextuality to both preparations and measurements. The resulting framework goes beyond the CSW framework, both conceptually and technically. On the conceptual level: 1) we relax the assumption of outcome determinism inherent to the Kochen-Specker theorem but retain measurement noncontextuality, besides introducing preparation noncontextuality, 2) we do not require the exclusivity principle as a fundamental constraint on measurement events, and 3) as a result, we do not need to presume that measurement events of interest are "sharp", where the notion of sharpness implies the exclusivity principle. On the technical level: 1) we introduce a source events hypergraph and define a new operational quantity ${\rm Corr}$ appearing in our inequalities, 2) we define a new hypergraph invariant -- the weighted max-predictability -- that is necessary for our analysis and appears in our inequalities, and 3) our noise-robust noncontextuality inequalities quantify tradeoff relations between three operational quantities -- ${\rm Corr}$, $R$, and $p_0$ -- only one of which (namely, $R$) corresponds to the Bell-Kochen-Specker functionals appearing in the CSW framework; when ${\rm Corr}=1$, the inequalities formally reduce to CSW type bounds on $R$. Along the way, we also consider in detail the scope of our framework vis-\`a-vis the CSW framework, particularly the role of Specker's principle in the CSW framework and its absence in ours.