  We consider the operators with highest anomalous dimension $\Delta$ in the
compact rank-one sectors $\mathfrak{su}(1|1)$ and $\mathfrak{su}(2)$ of ${\cal
N}=4$ super Yang-Mills. We study the flow of $\Delta$ from weak to strong 't
Hooft coupling $\lambda$ by solving (i) the all-loop gauge Bethe Ansatz, (ii)
the quantum string Bethe Ansatz. The two calculations are carefully compared in
the strong coupling limit and exhibit different exponents $\nu$ in the leading
order expansion $\Delta\sim \lambda^{\nu}$. We find $\nu = 1/2$ and $\nu = 1/4$
for the gauge or string solution. This strong coupling discrepancy is not
unexpected, and it provides an explicit example where the gauge Bethe Ansatz
solution cannot be trusted at large $\lambda$. Instead, the string solution
perfectly reproduces the Gubser-Klebanov-Polyakov law $\Delta = 2\sqrt{n}
\lambda^{1/4}$. In particular, we provide an analytic expression for the
integer level $n$ as a function of the U(1) charge in both sectors.
