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Is string theory uniquely determined by self-consistency? Causality andunitarity seemingly permit a multitude of putative deformations, at least atthe level of two-to-two scattering. Motivated by this question, we initiate asystematic exploration of the constraints on scattering from higher-pointfactorization, which imposes extraordinarily restrictive sum rules on theresidues and spectra defined by a given amplitude. These bounds handily excludeseveral proposed deformations of the string: the simplest "bespoke" amplitudeswith tunable masses and a family of modified string integrands from "binarygeometry." While the string itself passes all tests, our formalism directlyextracts the three-point amplitudes for the low-lying string modes without theaid of worldsheet vertex operators.

Multiparticle Factorization and the Rigidity of String Theory