When some states are huge and some others tiny, an “effective number” widely used for parties in political science can be applied both to state areas and populations. These numbers decrease exponentially over 5000 years, pointing to a single world state around 3700 by area and 5000 by population. Combination with estimates from top state area and population points toward a single world state by 4600, if the 5000-year trends continue. But projections are not predictions!

A square root law of effective numbers of states applies: The population-based effective number of states and tribes tends to be the square root of the area-based number. The zone of variation around the average trends is wide. Thus, even when the average millennial trends keep holding, the immediate future is wide open. The zigzags in the past curves remind us of how fleeting human history has been. Often a mere hundred years has thoroughly altered the number and size of states. We can expect similar reversals in future.

The world’s top empire area has grown in three distinct phases, reflecting shifts in message speeds. But the millennial trend has been exponential growth. If this 5000-year trend continued, a single world state would form around 4400. The most populous state’s share of world population also has increased exponentially, pointing to a single world state around 5300. The combined date is around 5000. Projections are not predictions, but still, if some people worry that the United Nations is trying to become a world government, while some others hope for it, they need not hold their breath. If the past offers any guidance whatsoever toward the future, a single world state is highly unlikely much ahead of 3000. Empires form where people are. Hence, the top shares of world population exceed the top shares of world dry land area. They do so in a logically predictable way: the square root law of people empires. The most populous state’s share of the world population tends to be the square root of its share of world dry land area. The law does not apply to “area empires” – those that are the largest but not the most populous.