= Solution
Let $v_H=\langle H\rangle$ be the <vacuum expectation value> in the convention $M=v_H\lambda$. For canonically normalized <fermion> kinetic terms, physical <charged lepton> masses are the nonnegative values in the <singular value decomposition> of $M$, equivalently the square roots of the <eigenvalues> of $M^\dagger M$.
In case (i),
$$
M^\dagger M=|v_H|^2\operatorname{diag}(|y|^2,|y|^2,|y_3|^2),\qquad
\boxed{\{m_1,m_2,m_3\}=|v_H|\{|y|,|y|,|y_3|\}.}
$$
The opposite algebraic <eigenvalues> $y$ and $-y$ of the doublet block do not represent different physical masses. The unavoidable twofold degeneracy conflicts with the three distinct nonzero measured <charged lepton> masses, for any assignment of the three families.
In case (ii),
$$
\boxed{\{m_1,m_2,m_3\}=\{0,0,|v_Hy_3|\}.}
$$
One can fit the <tau lepton> mass by choosing $|v_Hy_3|$, but the <electron> and <muon> remain massless. Thus neither exact <dihedral group> assignment with this single singlet <complex scalar field> reproduces the observed spectrum. Additional symmetry-breaking <Yukawa interactions> or other sources of mass are required; a basis change cannot remove the degeneracy or change the <matrix rank>.
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