= Solution
The quarter-rate limit is the strong-elasticity limit $\lambda/\alpha\ll1$, not the weak-elasticity limit. The original PDF repeats $\alpha/\lambda\ll1$ here; that is inconsistent with the preceding small-$\alpha$ result. In fact at $\alpha=0$ the maximum is $\lambda/4$, directly contradicting a quarter of that value in the same limit. The converted TeX repairs this particular ratio.
For $\alpha\gg\lambda$ and <wavenumbers> near the maximum, $S=\alpha s\gg L$. In the <quadratic equation>, the growing root is $O(\lambda)$, so its square and the $-L\beta$ term are small compared with $S\beta$. The leading balance is $S\beta-LS/4=0$. Consequently
$$
\beta\sim\frac\lambda4s(1-s),\qquad
\boxed{s_*\to\frac12,\qquad\beta_{\max}\sim\frac\lambda{16}.}
$$
The second root is a rapidly damped <surfactant>-relaxation mode. Thus the stated quarter-rate result is correct after reversing the printed small-parameter ratio.
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