 Solution

Suppose local elements of reality predetermined outputs $x(a,\lambda ),x(a^{\prime },\lambda ),y(b,\lambda ),y(b^{\prime },\lambda )\in \{-1,1\}$. For every hidden state $\lambda$,equals $2$ or $-2$, because exactly one bracket vanishes and the other equals $\pm 2$. 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.