  The Veneziano amplitude has recently been uniquely bootstrapped from crossing
symmetry, faster than power-law falloff at high energies, and a property dubbed
level truncation. In this paper we apply this bootstrap approach to fully
permutation invariant amplitudes, deriving new deformations of the
Virasoro-Shapiro amplitude for graviton scattering in string theory.
Superpolynomially soft Regge behavior yields the Virasoro-Shapiro amplitude as
the unique solution, and we find the string spectrum as an output rather than
an input of the bootstrap. While the remaining variations exhibit the same
Regge scaling as pure gravity, in the tensionless limit they reproduce
remarkable extremal amplitudes that have appeared in bottom-up studies of
positivity.
