We compare noise thresholds of the products C5⊗C5 (KCBS × KCBS) and C5⊗C7 of odd-cycle contextuality scenarios under local bit noise, for quantum realisations and for the level-1+AB (almost-quantum) relaxation. For the window class, the noncontextuality inequalities supported on the four contexts through one vertex of each cycle, we prove that the level-1+AB threshold equals Φ(1) = (2·2^(1/4) − 2)/(2·2^(1/4) − 1) ≈ 0.2745287, the maximum of the window law; the upper bound is formalised in Lean, and both ends are confirmed by exact rational certificates. Free real representations of the cycles attain Φ(1); on five further near-tight orbits of C5⊗C5 they come numerically within 7.2·10⁻⁸ of it. On the window class, the gap to the regular-ray thresholds 0.2656078 and 0.2634363 is a gap between regular and free measurements, not between quantum and almost-quantum. Pure level 1 gives about 1/2 (violation at 1/2 − 10⁻⁶). For C5⊗C7 we settle 52 of the 63 support classes of facets exactly at one noise value; 11 remain open.
We compare noise thresholds of the products C5⊗C5 (KCBS × KCBS) and C5⊗C7 of odd-cycle contextuality scenarios under local bit noise, for quantum realisations and for the level-1+AB (almost-quantum) relaxation. For the window class, the noncontextuality inequalities supported on the four contexts through one vertex of each cycle, we prove that the level-1+AB threshold equals Φ(1) = (2·2^(1/4) − 2)/(2·2^(1/4) − 1) ≈ 0.2745287, the maximum of the window law; the upper bound is formalised in Lean, and both ends are confirmed by exact rational certificates. Free real representations of the cycles attain Φ(1); on five further near-tight orbits of C5⊗C5 they come numerically within 7.2·10⁻⁸ of it. On the window class, the gap to the regular-ray thresholds 0.2656078 and 0.2634363 is a gap between regular and free measurements, not between quantum and almost-quantum. Pure level 1 gives about 1/2 (violation at 1/2 − 10⁻⁶). For C5⊗C7 we settle 52 of the 63 support classes of facets exactly at one noise value; 11 remain open.
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
For the regular odd KCBS n-cycle of qutrit projectors under white noise, E ↦ (1−η)E + ηI/3, let η_KS(n) be the least noise at which the noisy effects admit a joint measurement whose outcomes are the Kochen–Specker (KS) value assignments of the cycle, i.e. its independent sets. The dihedral symmetry reduces the semidefinite program over the ≈ 1.6^n independent sets to a second-order cone program on the orbits of arcs (runs of physically consecutive rays), whose data are Dirichlet kernels; its value η_arc(n) is an upper bound. We prove the arc lemma η_KS(n) = η_arc(n) for every odd n ≥ 5. The proof combines a combinatorial lemma — for weights b0 + b1 cos φ + b2 cos²φ with b2 ≤ 0 a minimum-weight independent set of the cycle is an arc — proved on paper and, independently, in Lean 4, with a computer-assisted certificate, in ball arithmetic, of the sign of y2 − y1 for all n. Comparing the discrete problem with its relaxation in which the arc length is a continuous parameter (value η_s(n)) we prove the sandwich η_s(n) ≤ η_KS(n) ≤ η_s(n) + C t(1−t)/n², t = frac(n ρ1(n)), for all odd n ≥ 7, with C ≤ 3.55 for n ≥ 401 (direct construction; the Lagrangian bound gives 3.59). As a consequence η_KS(n) → η_∞ = 0.42473994622455799193…, the value of an explicit continuum problem, and the rounding constant tends to c_∞ = 3.47856584115611146…, with a rate (computer-assisted): |η_KS(n) − η_s(n) − c_∞ t(1−t)/n²| ≤ 1.97/n³ for all odd n ≥ 7, and with an explicit second term. The thresholds are not monotone (computer-assisted): η_KS(n+2) > η_KS(n) and η_KS(n+2) < η_KS(n) each for infinitely many n, with approximate period 4 and a slow beat. For 5 ≤ n ≤ 401 and five larger n the thresholds are enclosed to 10^−35 as roots of explicit algebraic systems. A parameter-free form of the rounding law, fixed by a recorded hash before the corresponding discrete values were read, predicted the optimal support for 60 of 60 values 83 ≤ n ≤ 201.
A fundamental problem is to understand why quantum theory only violates some noncontextuality (NC) inequalities and identify the physical principles that prevent higher-than-quantum violations. We prove that quantum theory only violates those NC inequalities whose exclusivity graphs contain, as induced subgraphs, odd cycles of length five or more, and/or their complements. In addition, we show that odd cycles are the exclusivity graphs of a well-known family of NC inequalities and that there is also a family of NC inequalities whose exclusivity graphs are the complements of odd cycles. We characterize the maximum noncontextual and quantum values of these inequalities, and provide evidence supporting the conjecture that the maximum quantum violation of these inequalities is exactly singled out by the exclusivity principle.
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.
Contextuality is a fundamental marker of quantum non-classicality, which has emerged as a key resource for quantum computational advantage in multiple settings. Many such results hinge specifically on contextuality witnessed through Pauli measurements. In this work, we initiate a systematic study of (state-dependent) Pauli contextuality, focusing on cycle measurement scenarios, the simplest scenarios capable of exhibiting contextual behaviour. First, we study realizability of cycle scenarios with multi-qubit Pauli observables: we show that the maximum size of a cycle faithfully realizable by m-qubit Paulis is upper bounded by 3m, while we construct explicit realizations of cycles of size 2m−1 or 2m, depending on whether m≡1(mod3) or m≡1(mod3). Then, we investigate the presence of contextuality: we prove that no n-cycle Pauli realization for n>4 can witness contextuality (on any quantum state), whereas for n=4 every Pauli realization exhibits contextuality, attaining the quantum bound for all noncontextuality inequalities on some pure state. Finally, we discuss arbitrary Pauli scenarios in light of Vorob'ev's theorem, and show that, contrary to what our cycle characterization might suggest, the presence of 4-cycles is not necessary for witnessing contextuality in general Pauli scenarios.
We study the faces of the set of quantum correlations, i.e., the Bell and noncontextuality inequalities without any quantum violation. First, we investigate the question of whether every proper (facet-defining) Bell inequality for two parties, other than the trivial ones from positivity, normalization, and no-signaling, can be violated by quantum correlations, i.e., whether the classical Bell polytope or the smaller correlation polytope share any facets with their respective quantum sets. To do this, we develop a recently derived bound on the quantum value of linear games based on the norms of game matrices to give a simple sufficient condition to identify linear games with no quantum advantage. Additionally we show how this bound can be extended to the general class of unique games. We then show that the paradigmatic examples of correlation Bell inequalities with no quantum violation, namely the nonlocal computation games, do not constitute facet-defining Bell inequalities, not even for the correlation polytope. We also extend this to an arbitrary prime number of outcomes for a specific class of these games. We then study the faces in the simplest Clauser-Horne-Shimony-Holt Bell scenario of binary dichotomic measurements, and identify edges in the set of quantum correlations in this scenario. Finally, we relate the noncontextual polytope of single-party correlation inequalities with the cut polytope $\text{CUT}(\mathbf{\ensuremath{\nabla}}G)$, where G denotes the compatibility graph of observables in the contextuality scenario and $\mathbf{\ensuremath{\nabla}}G$ denotes the suspension graph of G. We observe that there exist facet-defining noncontextuality inequalities with no quantum violation, and furthermore that this set of inequalities is beyond those implied by the consistent exclusivity principle.