We show that the Veneziano amplitude of string theory is the unique solution to an analytically solvable bootstrap problem. Uniqueness follows from two assumptions: faster than power-law falloff in high-energy scattering and the existence of some infinite sequence in momentum transfer at which higher-spin exchanges cancel. The string amplitude-including the mass spectrum-is an output of this bootstrap. If the amplitude merely vanishes at high energies, the solution is a three-parameter family containing the Veneziano, Coon, and hypergeometric amplitudes, and more.
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