Contextuality and nonlocality are distinct manifestations at the foundation of quantum mechanics, yet their coexistence within a single quantum state remains subtle. In a hybrid CHSH--KCBS scenario involving the entanglment of a qubit and a qutrit, the qutrit supports the KCBS contextuality test, and the CHSH nonlocality arises from correlations between the qubit and qutrit. Here, we derive the analytical closed-form expressions for both inequalities and also simulate this physics on a quantum circuit. We show that contextuality is governed solely by a population parameter
p2, associated with the occupation of the qutrit subsystem in the
∣2⟩ level, which plays a distinguished role in the KCBS structure. In contrast, nonlocality depends irreducibly on coherence, involving both amplitudes and phases encoded in parameters
(Xi,Yi). This separation of physical resources reveals parameter regimes that optimize KCBS violation while suppress CHSH violation, and vice versa. As a result, the optimal regions do not overlap, and coexistence is restricted to a narrow intermediate regime in parameter space.
We study the all-state noise threshold η_b(C_m ⊗ C_n) of the product of two regular odd n-cycle (KCBS-type) scenarios in ℝ³, with every context encoded by two bits and local white noise of the same strength on each factor. The threshold is bounded below by the single-cycle thresholds (product states) and by the maximum M_face(m,n) of the threshold over the two-qubit faces v_j^⊥ ⊗ w_k^⊥. We prove that, for all odd m ≥ 9 and n ≥ 5, the face maximum strictly exceeds the single-cycle threshold of C_m (for m ≥ 13 via the published completeness theorem of Araújo et al. for the n-cycle inequalities). Hence entanglement makes C_m ⊗ C_n strictly more robust than both factors for all odd m, n ≥ 9, and the face maximum is at least of order c/m: liminf_m m·inf_n M_face(m,n) ≥ 3√6·π/8 = 2.8857371… > 2 = lim_m m·E(m); no upper bound of order 1/m is proved. The proof uses one Clauser–Horne witness whose violation reduces to a 2×2 form. On each face a six-bit marginal, the CHSH window, is a CHSH test with visibility (1 − η′)², η′ = η/(2 − η); after this reduction Landau's bound gives the closed lower bound Φ(g_m, g_n) = 2(1 − u)/(2 − u), u = (1 S)^(−1/4). For KCBS × KCBS the face threshold equals the window value exactly, M_face(5,5) = 0.26560777045727252964 (a root of an explicit octic), while for 13 pairs (m,n) the face maximum provably exceeds the window value through a second, Clauser–Horne branch whose value stays bounded away from zero as n → ∞. Pure face dominance fails for C₅ ⊗ C₁₁; we formulate a two-regime conjecture and give the numerical evidence for it, together with the known rigorous upper bounds for all states. For KCBS × KCBS we reduce the all-state upper bound to a positivity hypothesis at noise 1/4 on the non-window facets of the noncontextual polytope, verified exactly on all known facet orbits (flip images checked for the lists of 156 and 437 orbits); under it η_b(C₅ ⊗ C₅) = M_face(5,5), the window facets being settled exactly. The stronger hypothesis at the single-cycle threshold η_b(C₅) is false: at least 25 facet orbits outside the window class are violated beyond η_b(C₅), the largest violating noise, attained by two of them, being t_max ∈ [0.2485455, 0.2485456), the lower end being the noise level up to which a planar singlet violates them (numerically its threshold). All values refer to local bit noise and the regular realisation; several of them change under other noise models. Checkers, certificates and cone covers: https://doi.org/10.5281/zenodo.23204387. Lean library (release 1.4.0): https://doi.org/10.5281/zenodo.23204263
We place several known indicators of Kochen--Specker contextuality -- the KCBS correlator
χ, the contextual fraction $\CF$, 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 $\Tcal_{ij} = d^{-1}\tr[(\hat P_i \hat Q_j)^2]$, where
P^i and
Q^j are the joint-eigenspace projectors of the two compatible observable pairs within a measurement context. The state-independent scalar content of $\Tcal$ is carried by two independent contractions: the mutual information energy $E = \sum_{ij}\Tcal_{ij}$ of Paper~I (equivalently, its logarithmic form
S2=−log2E), and the Maassen--Uffink extremal overlap $c_\MU = \max_{i,j}|\langle a_i,b_i | c_j,b_j\rangle|$. We prove that
S2 is non-increasing under coarse-graining, that $S_2(\Gcal) > 0$ is a necessary configuration-level condition for observable contextuality, and that the additive composition $S_2(\Gcal) = \sum_αS_2(\Gcal_α)$ is exact for the KCBS pentagon. We further show that in the spin-
1 realization of the KCBS pentagon, a shared
ms=0 eigenstate in each context forces $c_\MU = 1$, rendering every Maassen--Uffink-type bound trivial -- a structural mechanism that makes explicit why outcome-entropic uncertainty relations based on $c_\MU$ are silent on KCBS contextuality, while
S2≈2.7266~bits throughout. Applied to KCBS and CHSH, the framework identifies regimes in which every state-dependent witness considered here is silent yet $S_2(\Gcal) > 0$ by an amount set by the projector geometry alone.