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.