Products of contextuality scenarios: where the product law holds and where entanglement breaks it

The noncontextual fraction of the product of two empirical models is the product of their noncontextual fractions (Abramsky, Barbosa and Mansfield). We ask which parts of this product law survive when a product of contextuality scenarios is realised quantumly and probed with all states, entangled ones included, and when noise is added. Infimal quantities obey it: for quantum realisations of arbitrary scenarios the minimal contextual fraction over all states satisfies 1 − a = ∏ᵢ (1 − aᵢ), and under noise acting locally on the factors the state-independent noise threshold of a product is the largest threshold of its factors; for Yu–Oh ⊗ Yu–Oh, a(η) = 1 − (4/5 + 83η/40)² up to η = 8/83. Joint noise on the pair outcomes breaks the law: the threshold becomes 64/289. Supremal quantities do not obey it. Entangled states activate strong contextuality: for the regular (KCBS-optimal) realisations of odd cycles this happens for KCBS ⊗ KCBS, in exactly ten two-qutrit states with noncontextual marginals; among planar singlets of Cₙ ⊗ Cₙ with odd n ≤ 15 it happens only for n = 5, and over all two-qutrit states it does not happen for C₅ ⊗ C₇, C₇ ⊗ C₇ or C₅ ⊗ C₉; no two-qutrit state is strongly contextual for Yu–Oh ⊗ KCBS. We give exact noise thresholds of individual entangled states, including the window in which the planar singlet of KCBS ⊗ KCBS is Kochen–Specker contextual but Bell local, and we summarise from a companion paper that entanglement strictly raises the all-state noise thresholds of products of cycles. The proofs consist of short arguments, a Lean 4 formalisation of the product laws and activation criteria, and exact certificates with checking scripts; we state which computations have been reproduced by an independent implementation. Checkers and certificates: https://doi.org/10.5281/zenodo.23196178. Lean library (release 1.3.0): https://doi.org/10.5281/zenodo.23099654