  We consider $\alpha'$ corrections to four-point correlators of half-BPS
operators in $\mathcal{N}=4$ super Yang-Mills theory in the supergravity limit.
By demanding the correct behaviour in the flat space limit, we find that the
leading $(\alpha')^3$ correction to the Mellin amplitude is fixed for arbitrary
charges of the external operators. By considering the mixing of double-trace
operators we can find the $(\alpha')^3$ corrections to the double-trace
spectrum which we give explicitly for $su(4)$-singlet operators. We observe
striking patterns in the corrections to the spectra which hint at their common
ten-dimensional origin. By extending the observed patterns and imposing them at
order $(\alpha')^5$ we are able to reproduce the recently found result for the
correction to the Mellin amplitude for $\langle \mathcal{O}_2 \mathcal{O}_2
\mathcal{O}_p \mathcal{O}_p \rangle$ correlators. By applying a similar logic
to the $[0,1,0]$ channel of $su(4)$ we are able to deduce new results for the
correlators of the form $\langle \mathcal{O}_2 \mathcal{O}_3 \mathcal{O}_{p-1}
\mathcal{O}_p \rangle$.
