AbstractAnimal cell migration is predominantly driven by the coordinated, yet stochastic, polymerization of thousands of nanometer-scale actin filaments across micron-scale cell leading edges. It remains unclear how such inherently noisy processes generate robust cellular behavior. We employed high-speed, high-resolution imaging of migrating neutrophil-like HL-60 cells to explore the fine-scale dynamic shape fluctuations that emerge and relax throughout the process of leading edge maintenance. We then developed a minimal stochastic model of the leading edge that is able to reproduce this stable relaxation behavior. Remarkably, we find that lamellipodial stability naturally emerges from the interplay between branched actin network growth and leading edge shape – with no additional feedback required – based on a synergy between membrane-proximal branching and lateral spreading of filaments. These results thus demonstrate a novel biological noise-suppression mechanism based entirely on system geometry. Furthermore, our model suggests that the Arp2/3-mediated ∼70-80º branching angle optimally smooths lamellipodial shape, addressing its long-mysterious conservation from protists to mammals.One sentence summaryAn experimental and computational investigation of fluctuation dynamics at the leading edge of motile cells demonstrates that the specific angular geometry of Arp2/3-mediated actin network branch formation lies at the core of a successful biological noise-suppression strategy.