Noise thresholds of products of odd-cycle contextuality scenarios: a CHSH window, an exact face threshold for KCBS × KCBS, and two regimes

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