We have codified the algebraic fundamental group of anabelian geometry as a multi-sorted logical structure so as to use model-theoretic ideas, analogies, and language to go further with the study of hyperbolic curves over number fields. Consequently, a definability analysis is now possible on smooth quasi-projective schemes and their algebraic fundamental groups in characteristic zero. We form a connection between the algebraic fundamental group and the Lascar group of a stable first-order theory of covering spaces. We then provide a formulation of a Grothendieck-type section conjecture in terms of pure stability. One such use-case, for finitely generated k, is the application of geometric stability theory to use elimination of imaginaries to construct k-rational points on hyperbolic k-curves.