The First Mathematical Law of IR: Anarchy=0

The study of international relations (IR) has long revolved around one fundamental condition: anarchy. Kenneth Waltz (1979) famously defined anarchy as the absence of a central authority above states. This definition is deceptively simple. It has structured entire schools of thought: realism, which views anarchy as the driver of self-help and competition; liberal institutionalism, which argues that cooperation can mitigate anarchy; and constructivism, which insists that the meaning of anarchy is not given but constructed by state interaction (Wendt, 1992). Niklas Luhmann (1995), writing from outside IR, conceptualized politics as an autopoietic system: self-reproducing communication without external foundation. His theory suggested that no “final center” exists to ground politics — an idea consistent with the absence of world authority in IR. Despite decades of debate, one implication of Waltz’s definition remains underdeveloped. If anarchy means the absence of a supreme central authority, then the power of that authority must be zero. And if world power is defined as the power of that central authority, then under conditions of anarchy, world power = 0. This article develops that claim rigorously. It proceeds scientifically and methodologically, offering precise definitions, hypotheses, logical derivation, and empirical illustrations. The argument is not rhetorical but deductive: a conceptual axiom that clarifies the meaning of world power in an anarchic system.