For starters this obviously has many variables so it would be difficult to provide concrete answers on the level of 1+1=2. Thus I'm including our historical precedent element so it won't be total chaos.
The context is I have this one particular massive empire with all sorts of good stuff that tragically fell one day. And having read like at least 2 Wikipedia pages on history it was not difficult to figure out the blood and terrible power vacuum...etc.
The end result was many many small states all claiming to be the true heirs to the empires.
Yes it is not lost on me that perhaps people are not truly interested in restoring the empire with all its complexity but I digress.
I want all the states on the continent to have a strong connection to the previous empire and be greatly influenced by them.
We know from history that Byzantium claimed to be Romans even after more than a 1000 years. But I see that as complete PR move. They were no more Romans than the later Ottomans. Speaking of which Ottomans also claimed to be the rightful caliphate despite the impossibility of that in technical terms, it requires the holder to be a male with clear heritage to a tribe, but they did not care.
So. The claiming parts is easy. I want the actual influence part.
That is why I'm asking about what elements would be needed for the overall society to be influenced by the original state?
The time part I think is integral to the answers below. So. I really hope it is not as a second question because it's like if you do X then you get effect that lasts X.
This, time and elements, will be important as I want the story to happen in a time where several hundred years, at least, had passed and the people are very slowly forgetting.
The setting is very inconsistent to our own.
For example legal representation, investigation, burden of prove. right of silence...etc are all parts of the legal systems of the vast majority of state now. This is directly inherited from that after mentioned empire. However nobility has many exceptions.
Similarly there are many places with a more "Greek" style free citizens kind of mentality while there are absolute monarchy similar to older Japanese stories, I'm merely judging from movies and mean no offense.
But basically there are not French revolution or printing press or Newtonian physics or gunpowder or electricity.
So. Just to make it clear. I want elements I can incorporate, say awesome buildings, in the story that keeps most people in the setting truly believing that the old empire was awesome and they need to copy it as much as possible. The more subtle stuff is easy to figure out, I not easy but that is my job.
What are those elements and for how long can they keep working?
Edit: Sorry if it is confusing. As "JBH" said it needs clarifying. What I'm looking for is "elements" that exist in the world so that the later states are still influenced but the older state.
Like what are the fundamental principles that I need to follow so that kingdom A after like 2000 years is still influenced by Empire B?
I tried giving an example of building but guess it was stupid.
Now let me give you an obvious example. If we have something like say god-king who created an empire and made a detailed list of laws, best music types, and the best architect. And then he dies and his empire fall.
Now later kingdoms would believe that he is like the best dude ever and would go around making their laws as close as possible to his laws and making their music and architect as close as possible, or just copy it, because he was a god-king.
The fundamental principle here is that he was a god-king.
So. I'm looking for such fundamental principles.
Like would it be martial success, or would it be scientific progress, or economical power, a combination of both...etc.
And I know people and culture are different. But we in our history had a lot of that so there are definite and clear rules to such things even if they are different from place to place.