 Solution

For predetermined outcomes, the four products multiply to one:The four desired signs are $(+1,+1,+1,-1)$, whose product is minus one. Thus at least one winning condition must fail for every deterministic assignment. Uniform independent settings give success at most $3/4$ per assignment, and averaging over hidden assignments preserves the bound.

An observed rate near $0.85$, with enough trials to exclude statistical fluctuation, therefore contradicts Eve's predetermined-local-outcome claim, provided the settings are independent of her devices and the devices cannot communicate during a run. It violates the local bound of the CHSH inequality while remaining consistent with the quantum optimum ${\cos }^2(\pi /8)\approx 0.853553$.

The observation is not by itself exact certification of a perfect two-qubit state. Maximal quantum CHSH correlations support CHSH self-testing only with the appropriate quantum/locality assumptions and up to local changes of representation and unused ancillary systems; a near-maximal finite data set requires a quantitative error analysis. An unrestricted communicating pair of black boxes can instead win every run, so the quantum bound is not a bound on arbitrary devices without these assumptions.