  A new inequality for a nonlinear surface layer integral is proved for
minimizers of causal variational principles. This inequality is applied to
obtain a new proof of the positive mass theorem with volume constraint. Next, a
positive mass theorem without volume constraint is stated and proved by
introducing and using the concept of asymptotic alignment. Moreover, a positive
quasi-local mass and a synthetic definition of scalar curvature are introduced
in the setting of causal variational principles. Our notions and results are
illustrated by the explicit examples of causal fermion systems constructed in
ultrastatic spacetimes and the Schwarzschild spacetime. In these examples, the
correspondence to the ADM mass and similarities to the Brown-York mass are
worked out.
