= Solution
The <lepton mixing matrix> is <unitary>, so each row has unit norm. Explicitly its electron row obeys
$$
|U_{e1}|^2+|U_{e2}|^2+|U_{e3}|^2
=c_{13}^2(c_{12}^2+s_{12}^2)+s_{13}^2=1,
$$
where $c_{ij}=\cos\theta_{ij}$ and $s_{ij}=\sin\theta_{ij}$. This is also the normalization needed for unit <electron neutrino> survival at $L=0$.
Use $\operatorname{Re}(1-e^{ix})=1-\cos x=2\sin^2(x/2)$ and define $\Delta_{ij}=(\phi_i-\phi_j)/2=\Delta m_{ij}^2L/(4E)$ in <natural units>. The exact vacuum <neutrino oscillation> result becomes
$$
P_{ee}=1-4\sum_{i<j}|U_{ei}|^2|U_{ej}|^2\sin^2\Delta_{ij}.
$$
The given <mass-squared differences> satisfy $|\Delta m_{12}^2/\Delta m_{23}^2|\simeq0.027\ll1$, so the two fast <oscillation> frequencies are close: $\Delta_{13}=\Delta_{23}+\Delta_{12}$. Approximating their phases as equal combines their coefficients into
$$
4s_{13}^2c_{13}^2(c_{12}^2+s_{12}^2)=\sin^22\theta_{13},
$$
while $4c_{12}^2s_{12}^2c_{13}^4=c_{13}^4\sin^22\theta_{12}$. Hence
$$
\boxed{P_{ee}\simeq1-\cos^4\theta_{13}\sin^22\theta_{12}\sin^2\Delta_{12}
-\sin^22\theta_{13}\sin^2\Delta_{23}.}
$$
The complex <CP> phase cancels from every $|U_{ei}|^2$. The phase-replacement approximation is uniformly reliable for $|\Delta_{12}|\ll1$, especially at short reactor baselines; closeness of the <mass-squared differences> alone does not guarantee close sine values at arbitrarily large $L/E$. At medium baselines the distinct fast frequencies can beat, so the exact three-frequency expression is preferable if their detailed pattern matters.
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