Is string theory uniquely determined by self-consistency? Causality and unitarity seemingly permit a multitude of putative deformations, at least at the level of two-to-two scattering. Motivated by this question, we initiate a systematic exploration of the constraints on scattering from higher-point factorization, which imposes extraordinarily restrictive sum rules on the residues and spectra defined by a given amplitude. These bounds handily exclude several proposed deformations of the string: the simplest "bespoke" amplitudes with tunable masses and a family of modified string integrands from "binary geometry." While the string itself passes all tests, our formalism directly extracts the three-point amplitudes for the low-lying string modes without the aid of worldsheet vertex operators.