= Planar two-sided intercept-resend CHSH bound
{c}
{title2=$S=-\sqrt2\cos^2(\alpha-\beta)$}
Use Alice's CHSH axes $\hat x,\hat y$ and Bob's $(\hat x+\hat y)/\sqrt2,(-\hat x+\hat y)/\sqrt2$, with sign pattern $S=C_{11}-C_{13}+C_{31}+C_{33}$. If Eve measures both singlet particles along planar angles $\alpha,\beta$, the <intercept-resend attack> correlation gives $S=-\sqrt2\cos^2(\alpha-\beta)$. Consequently $-\sqrt2\le S\le0$. Randomly mixing axes independently of the users' secret settings preserves this interval by averaging. The bound is specific to these settings and this attack, rather than a general security theorem.
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