= Solution
Use $\operatorname{Re}(e^{ia}-1)=\cos a-1=-2\sin^2(a/2)$. With $k=1/4$, the source's $\Delta_{ij}=2k(\phi_i-\phi_j)$ is precisely $(\phi_i-\phi_j)/2=(m_i^2-m_j^2)L/(4E)$. The electron-row <probabilities> are $c_{12}^2c_{13}^2$, $s_{12}^2c_{13}^2$, and $s_{13}^2$. Therefore
$$
\boxed{P_{ee}=1-c_{13}^4\sin^2(2\theta_{12})\sin^2\Delta_{12}
-\sin^2(2\theta_{13})\left[c_{12}^2\sin^2\Delta_{13}+s_{12}^2\sin^2\Delta_{23}\right].}
$$
Neither $\theta_{23}$ nor the <CP violation>[Dirac phase] appears in vacuum <Neutrino oscillation>[electron-flavour survival]. The same vacuum <Neutrino oscillation>[survival expression] applies to <Nuclear reactor>[reactor] <electron> <antineutrinos>. A sufficiently large <Neutrino matter potential>[matter effect] would require a different propagation calculation, but is negligible at the short <Nuclear reactor>[reactor] baselines used for measuring $\theta_{13}$.
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