= Solution
After the <scalar field> obtains a <vacuum expectation value> $h$, the <fermion mass matrix> in a canonically normalized <basis> is $M=h\lambda$. Physical <masses> are its <singular values>, not arbitrary entries or signed <eigenvalues>. Thus
$$
\boxed{m_e=m_\mu=|ha|,\qquad m_\tau=|hb|.}
$$
The independent <Yukawa coupling> $b$ can fit the <tau lepton> <mass> relative to the doublet scale, but the <electron> and <muon> are predicted to have equal <masses>. The values supplied by the paper instead give $m_\mu/m_e\simeq105.7/0.511\simeq207$. Consequently exact unbroken $S_3$ with this <representation> assignment cannot reproduce all three <masses>. Complex phases of $a,b$, or changes of <basis>, cannot split the two equal <singular values>. Additional breaking of the <symmetry> or a different field content would be needed; the discrepancy cannot be removed merely by choosing different invariant <Yukawa couplings>.
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