Light minicharged particles are well-motivated targets in searches for physics beyond the Standard Model, but their masses are often introduced as free parameters. In this talk, I will discuss a chiral hidden-sector framework in which the light masses are protected by a spontaneously broken gauge symmetry. Requiring quantum consistency then imposes anomaly cancellation conditions on the chiral charge assignments. I will show that these conditions are exactly equivalent to the degree-three Prouhet–Tarry–Escott problem in number theory. This correspondence turns anomaly cancellation into a predictive organizing principle for the particle spectrum: the minimal consistent sector contains at least four minicharged mass eigenstates, and minimal solutions typically contain partner states with the same minicharge and nearby masses. I will explain the origin of this structure and discuss its implications for laboratory, astrophysical, and cosmological searches.