= Solution
Write $\mathbf n_a=(\cos\alpha,\sin\alpha)$ and $\mathbf n_b=(\cos\beta,\sin\beta)$. Since $\mathbf b_1-\mathbf b_3=\sqrt2\hat{\mathbf x}$ and $\mathbf b_1+\mathbf b_3=\sqrt2\hat{\mathbf y}$, the <CHSH inequality> combination becomes
$$
S=-\sqrt2\cos(\alpha-\beta)
\left(\cos\alpha\cos\beta+\sin\alpha\sin\beta\right).
$$
Therefore
$$
\boxed{S=-\sqrt2\cos^2(\alpha-\beta),\qquad -\sqrt2\leq S\leq0.}
$$
Parallel or antiparallel Eve axes attain the lower endpoint; orthogonal axes attain zero. This bound is specific to the two-sided projective attack and these planar settings. It is stronger than the generic local bound $|S|\le2$ and should not be attributed to every possible eavesdropping strategy.
If Eve varies her axes independently of the users' still-secret random settings, the measured correlations average over her choices. The same linear combination obeys $S=-\sqrt2\,\mathbb E[\cos^2(\alpha-\beta)]$ and remains in the interval by <convexity>. Strategies correlated with disclosed measurement settings would violate that independence premise; the protocol must protect it.
\Image[/past-exam-of-the-quantum-information-course-of-the-university-of-cambridge/2015/paper-1-intercept-resend-bound.png]
{title=The planar two-sided intercept-resend attack cannot reproduce the singlet CHSH value}
{description=Original analytic curve as Eve changes the relative angle of her measurement axes. The dashed line is the undisturbed singlet result, outside the attack's attainable interval.}
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