Digitization Can Stall Swarm Transport: Commensurability Locking in Quantized-Sensing Chains

We present a minimal model for autonomous robotic swarms in one- and higher-dimensional spaces, where identical, field-driven agents interact pairwise to self-organize spacing and independently follow local gradients sensed through quantized digital sensors. We show that the collective response of a multi-agent train amplifies sensitivity to weak gradients beyond what is achievable by a single agent. We discover a fractional transport phenomenon in which, under a uniform gradient, collective motion freezes abruptly whenever the ratio of intra-agent sensor separation to inter-agent spacing satisfies a number-theoretic commensurability condition. This commensurability locking persists even as the number of agents tends to infinity. We find that this condition is exactly solvable on the rationals -- a dense subset of real numbers -- providing analytic, testable predictions for when transport stalls. Our findings establish a surprising bridge between number theory and emergent transport in swarm robotics, informing design principles with implications for collective migration, analog computation, and even the exploration of number-theoretic structure via physical experimentation.