Contextuality from the Projector Overlap Matrix

We place several known indicators of Kochen–Specker contextuality – the KCBS correlator $\chi$, the contextual fraction $\text{CF}$, the Shannon-entropic $n$-cycle inequality of Chaves and Fritz, and the operational commutator witness $D$ of Paper~I – into a single projector-geometric framework organized around the overlap matrix ${\text{Tcal}}_{ij}=d^{-1}\text{tr}[({\widehat{P}}_i{\widehat{Q}}_j{)}^2]$, where ${\widehat{P}}_i$ and ${\widehat{Q}}_j$ are the joint-eigenspace projectors of the two compatible observable pairs within a measurement context. The state-independent scalar content of $\text{Tcal}$ is carried by two independent contractions: the mutual information energy $E={\sum }_{ij}{\text{Tcal}}_{ij}$ of Paper~I (equivalently, its logarithmic form $S_2=-{\log }_{2}E$), and the Maassen–Uffink extremal overlap $c_{\text{MU}}={\max }_{i,j}\mathrm{\mid }⟨a_i,b_i\mathrm{\mid }{c}_j,b_j⟩\mathrm{\mid }$. We prove that $S_2$ is non-increasing under coarse-graining, that $S_2(\text{Gcal})>0$ is a necessary configuration-level condition for observable contextuality, and that the additive composition $S_2(\text{Gcal})={\sum }_{\alpha }{S}_2({\text{Gcal}}_{\alpha })$ is exact for the KCBS pentagon. We further show that in the spin-$1$ realization of the KCBS pentagon, a shared $m_s=0$ eigenstate in each context forces $c_{\text{MU}}=1$, rendering every Maassen–Uffink-type bound trivial – a structural mechanism that makes explicit why outcome-entropic uncertainty relations based on $c_{\text{MU}}$ are silent on KCBS contextuality, while $S_2\approx 2.7266$~bits throughout. Applied to KCBS and CHSH, the framework identifies regimes in which every state-dependent witness considered here is silent yet $S_2(\text{Gcal})>0$ by an amount set by the projector geometry alone.