Bonus: The athority in a state

I believe the most important characteristic of a state is the state authority. Not only is this one of the best predictors of a complex state, it is also absolutely an essential characteristic to the success of a complex state.
A community can have agriculture and even intensive agriculture. Without out state authority though, there would be no way to distribute the goods, oversee the labor if necessary, or provide the authority for projects like irrigation. Even if there is specialization in a society, some labor in the fields while others make leather goods, for example, this does not make a state. If housemates agree that one will mow the grass and clean the pool and the other will keep track of bills and maintain the inside lighting and plumbing fixtures according to their skills, it is not a state. If there is someone to govern how and when these chores are done is starts to sound more like a state. A complex economy can’t possibly reach a level of complexity without someone to pull strings. Even the freest of markets have to be made free by laws, which the state creates and enforces. State authority goes hand in hand with elites. While a state creates specialized labor (in agriculture, in religion, in public works, etc.), it is at the same time creating an elite that prospers of the work done by others. Stratification is important to state authority as it often keeps the state’s power from being challenged. Stratification alone will not lead to a state. Just because one person has more wealth or power than another does not give them the ability to govern; that is the job of authority. A society must have a need to be governed for it to be a complex state. Otherwise there is no mandate for state authority (because it would lack legitimacy and result in tyranny).
State authority is what makes a state a state, otherwise it is simply a group of people maybe with agriculture, maybe with some trade networks or a social hierarchy. Even with these state characteristics, this society really is not a state until it is governed.
It goes without saying that secondary state characteristics are very secondary to primary ones like state authority. In fact, most of these secondary characteristics (state religion, monumental public works, state art, tribute, development of a written language, and mass production) would be possible without state authority.