the 13-entry of (\[perturb\]) and perturbations $\epsilon_{ii}$ to the diagonal entries. To leading order, $\delta$ only shifts the atmospheric mixing from maximal (to $\theta_{12}\sim \pi/4-\delta/2$). $\epsilon_{22}$ and $\epsilon_{33}$ large enough to affect the Solar mixing tend to give too large contributions to $|\theta_{13}|$, so we will ignore them (their inclusion would merely lead to additional fine-tuned parameter ranges). One then finds $$%
\frac{\pi}{4} - \theta_{12} \simeq 0.19 \simeq
\frac{1}{4}(\epsilon - \epsilon_{11}) ,
%
\label{solardev}$$ whereas the solar mass difference is $$%
\frac{\Delta m^2_{12}}{\sqrt{2}|\Delta m_{\rm atm}^2|} \simeq
\frac{1}{43} \simeq (\epsilon + \epsilon_{11}) .
%
\label{ratio}$$ Satisfying these constraints would require a moderate tuning of $\epsilon$ and $\epsilon_{11}$. Moreover, they would each have to be of order 0.4 in magnitude, somewhat large to be considered perturbations.

On the other hand, a simple and realistic pattern emerges when we instead allow for small departures from $U_e \propto
\mathbf{1}$ [@Frampton:2004ud; @Rodejohann:2003sc; @Petcov:2004rk; @Altarelli:2004jb; @Romanino:2004ww; @Datta:2005ci], and for the general superpotential in (\[Wfull\]) there is no reason for such mixings to be absent.[^3] For example, starting from (\[perturb\]) a Cabibbo-sized 12-entry in the charged lepton mixing matrix $$%
U_e^{\dagger} \sim \left( \begin{array}{ccc} 1 &
-\sin\theta_{12}^e & 0
\\ \sin\theta_{12}^e & 1 & 0 \\ 0 