In terms of sheer geographic extent, the largest imperial state in ancient history was established under:

Under what? I think you need to add a few things.

To determine the largest imperial state in ancient history in terms of sheer geographic extent, you can follow these steps:

1. Compile a list of ancient empires: Start by researching and listing all the major ancient empires that existed throughout history. These could include well-known empires such as the Roman Empire, Persian Empire, Han Dynasty of China, or the Maurya Empire in India.

2. Research the size of each empire: Once you have your list of empires, research the extent of each empire's territory at its peak. Look for historical records, maps, and scholarly sources that provide information on the size and boundaries of the respective empires.

3. Compare the size of each empire: Analyze the data you have gathered and compare the sizes of the empires. Look for information on the total land area or the number of regions and provinces under each empire's control.

4. Determine the largest empire: Based on your research and comparison, identify the empire with the largest geographic extent. This empire, in terms of land area or control over regions, would be considered the largest imperial state in ancient history.

Keep in mind that determining the largest empire can be subjective, as different historical sources may provide varying and sometimes contradictory information. It is important to consider multiple sources and interpretations to get the most accurate understanding.