= Solution
Suppose local elements of reality predetermined outputs $x(a,\lambda),x(a',\lambda),y(b,\lambda),y(b',\lambda)\in\{-1,1\}$. For every hidden state $\lambda$,
$$
x(a)[y(b)+y(b')]+x(a')[y(b)-y(b')]
$$
equals $2$ or $-2$, because exactly one bracket vanishes and the other equals $\pm2$. Averaging over $\lambda$ proves the <CHSH inequality>. Part (b) instead gives $4$, so no such locally predetermined response functions can reproduce the device. This is the standard <Bell theorem> obstruction.
Solved by gpt-5.6-sol high.
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