  The genus zero contribution to the four-point correlator $\langle {\cal
O}_{p_1}{\cal O}_{p_2}{\cal O}_{p_3}{\cal O}_{p_4}\rangle$ of half-BPS
single-particle operators ${\cal O}_p$ in $\mathcal{N}=4$ super Yang-Mills at
strong coupling computes the Virasoro-Shapiro amplitude of closed superstrings
on $AdS_5\times S^5$. Combining Mellin space techniques, the large $p$ limit,
and data about the spectrum of two-particle operators at tree level in
supergravity, we design a bootstrap algorithm which heavily constrains its
$\alpha'$ expansion. We use crossing symmetry, polynomiality in the Mellin
variables and the large $p$ limit to stratify the Virasoro-Shapiro amplitude
away from the ten-dimensional flat space limit. Then we analyse the spectrum of
exchanged two-particle operators at fixed order in the $\alpha'$ expansion. We
impose that the ten-dimensional spin of the spectrum visible at that order is
bounded above in the same way as in the flat space amplitude. This constraint
determines the Virasoro-Shapiro amplitude in $AdS_5\times S^5$ up to a small
number of ambiguities at each order and we compute it explicitly for
$(\alpha')^{5,6,7,8,9}$. As the order of $\alpha'$ grows, the ten dimensional
spin grows, and the set of visible two-particle operators opens up. Operators
illuminated for the first time receive a string correction to their anomalous
dimensions. This correction is uniquely determined and lifts the residual
degeneracy of tree level supergravity due to ten-dimensional conformal
symmetry. We encode the lifting of the residual degeneracy in a characteristic
polynomial. This object carries information about all orders in $\alpha'$.It is
analytic in the quantum numbers, symmetric under an $AdS_5 \leftrightarrow S^5$
exchange, and it enjoys intriguing properties, which we explain and detail in
various cases.
