Nonconvexity as the Analytic Shadow of Contextuality

We develop a convex-analytic framework that turns algebraic obstructions to global sections in sheaf models of quantum contextuality into certifiable signatures on quantum state space. For a finite family of projective measurement contexts we consider, for each context, the entropy increase incurred by dephasing in that context and the lower envelope of these associated classicalisation costs. Mathematically, we show that this envelope is convex exactly when a single classical frame can explain all statistics; genuine contextuality generically produces kinks and strictly positive Jensen gaps along chords, an epigraph-union manifestation of the failure of a global section. We obtain dimension- and overlap-dependent lower and upper bounds on these gaps using relative-entropy inequalities, and show that the resulting convexity defects are monotone under dephasing-covariant quantum channels. Operationally, the framework requires only context-wise frequency tables, yields finite-sample-robust witnesses and visual diagnostics, and is directly implementable on cloud quantum devices. A contextual Landauer principle links algebraic incompatibility of contexts to a nonzero minimal erasure work in branched-then-erase protocols. Taken together, these results provide algebraic and geometric invariants of measurement covers that translate foundational features of contextuality into experiment-ready benchmarks for contemporary quantum hardware.