  We compute the first curvature corrections to the spectrum of light-cone
gauge type IIA string theory that arise in the expansion of $AdS_4\times
\mathbb{CP}^3$ about a plane-wave limit. The resulting spectrum is shown to
match precisely, both in magnitude and degeneration that of the corresponding
solutions of the all-loop Gromov--Vieira Bethe Ansatz. The one-loop dispersion
relation correction is calculated for all the single oscillator states of the
theory, with the level matching condition lifted. It is shown to have all
logarithmic divergences cancelled and to leave only a finite exponentially
suppressed contribution, as shown earlier for light bosons. We argue that there
is no ambiguity in the choice of the regularization for the self-energy sum,
since the regularization applied is the only one preserving unitarity.
Interaction matrices in the full degenerate two-oscillator sector are
calculated and the spectrum of all two light magnon oscillators is completely
determined. The same finite-size corrections, at the order 1/J, where $J$ is
the length of the chain, in the two-magnon sector are calculated from the all
loop Bethe Ansatz. The corrections obtained by the two completely different
methods coincide up to the fourth order in $\lambda' =\lambda/J^2$. We
conjecture that the equivalence extends to all orders in $\lambda$ and to
higher orders in 1/J.
