A principal-stress rule for cell division in epithelia

Dense active materials must continuously relieve internal mechanical stress to remain structurally stable as they are driven far from equilibrium. In epithelial tissues, this relief occurs through cell division, where, according to Hertwig's century-old rule, cells divide along their long axis. We show that as epithelia densify, this geometric rule is superseded by a mechanical one: cells divide along the principal axis of anisotropic stress, independently of the isotropic stress state. Using direct force measurements and stress inference, we demonstrate that stress orientation governs division across mechanically distinct systems, from fluid-like to jammed monolayers and structurally heterogeneous organoids, remaining predictive where cell shape fails. This stress-oriented remodeling is reciprocally coupled to the material's mechanical state: anisotropic stress accelerates cell-cycle progression, while division locally dissipates it, closing a negative feedback loop. This "principal-stress rule" provides a general mechanical framework linking internal stress, structural remodeling and homeostasis in living materials.