= Solution
{c}
For predetermined outcomes, the four products multiply to one:
$$
(A_1B_1)(A_1B_2)(A_2B_1)(A_2B_2)=A_1^2A_2^2B_1^2B_2^2=1.
$$
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 \b[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 $\cos^2(\pi/8)\approx0.853553$.
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.
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