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We consider the operators with highest anomalous dimension $\Delta$ in thecompact rank-one sectors $\mathfrak{su}(1|1)$ and $\mathfrak{su}(2)$ of ${\calN}=4$ super Yang-Mills. We study the flow of $\Delta$ from weak to strong 'tHooft coupling $\lambda$ by solving (i) the all-loop gauge Bethe Ansatz, (ii)the quantum string Bethe Ansatz. The two calculations are carefully compared inthe strong coupling limit and exhibit different exponents $\nu$ in the leadingorder 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 notunexpected, and it provides an explicit example where the gauge Bethe Ansatzsolution cannot be trusted at large $\lambda$. Instead, the string solutionperfectly reproduces the Gubser-Klebanov-Polyakov law $\Delta = 2\sqrt{n}\lambda^{1/4}$. In particular, we provide an analytic expression for theinteger level $n$ as a function of the U(1) charge in both sectors.Comment: 42 pages, JHEP style LaTe

Bethe Ansatz solutions for highest states in Script N = 4 SYM and AdS/CFT duality