from a reading webtoon standpoint, I've seen arch dukes mean a number of different things, it can be a direct line that is not crowned, a contingent family to the throne, a founding family to the country, someone that contributed substantially to the country, or a principality.
hope this helps or somebody else can chime in(づ ̄ ³ ̄)づ
oh okay but we've seen that just because you have a high title doesn't translate to prestige. This head of the family hasn't been doing the necessary socializing and internal upkeep it takes to run the family. he runs off and leave most of the work to Fern and just doing paperwork and a few excursions aint enough. the heir apparent is not skilled (enough) which led to his circle degrading as well.
Spoilers below:
Yurgenschmidt is the name of their country. The country is ruled by an absolute ruler called the Zent or the royal family's King (or Queen) who passed down the Zent title to their heirs since the royal family hold largest amount of mana in their whole country.
The position of Aub or Archduke is the one who rule over the duchies, granted control of the land by the authority of the Zent (somewhat like city mayor or governor equivalent in our time).
Under Aub there's Giebe (somewhat like provincial mayor or governor equivalent in our time). They are the governing noble of a local province within a duchy, granted control of the land by the authority of the Aub. Geibe can either be a Count/ Earl, a Viscount or a Baron.
The position of Aub & Zent are mostly hereditary unless they are not qualified (this is mostly true for Zent's requirement, since they upheld everything in their country through country Foundation Magic which gives mana through the land including magical tools & country gates. Aub also requires large amount of mana, chosen among the heirs to maintain the Foundation Magic of their duchy). The heirs can either be by birth or through adoption.
The Zent naturally has the right to take away the title of Aub and appoint someone else since the Zent is the one created the Foundation Magic for each duchy, it means they are the one who created duchies & draw it's borderlines.
There's a ranking system for the duchies in Yurgenschmidt. The more higher the ranking, the more influence, prestige, power, fame, cooperation & responsibility they have in helping the Zent for the sake of the country. There's also classification of duchies ranging from greater, middle & small which depends on the size of the land and population per duchy.
There are 25 duchies in Yurgenschmidt before the civil war & 21 after.
In Ehrenfest's case, (I'm not sure if their line is from royal family or branch line that will not inherit the Zent title) the new Archduke came from Central / Sovereign (country's main place of Royal governance) after the rebellion against former Zent & change of duchy name from Eisenreich to current Ehrenfest.
You could check out the maps here:
https://ascendance-of-a-bookworm.fandom.com/wiki/Every_Light_Novel_Illustration_(English)/Maps
More spoilers on world building:
I also didn't include the rules on how to become Zent because it's so meticulous and way more spoilers ( ̄∇ ̄")
There are also rules, duties, customs (names, clothing, etc.), communications, mannerisms depending per people classifications (and sometimes gender)..
Different for commoners & nobles
For commoners it depending if:
•capital citizens or rural (aso depends if direct control of Archduke or under Giebe)
•jobs like merchants, craftsmen, farmer, etc.
•location (different rules & customs under different Giebe & different Arhduke/duchy)
Also depends if you're a Clergy (mixed commoner & nobles) like grey priest, blue priest, chief priest/high priest, high bishop
(Clergy nobles are treated almost the same as commoners as they are either orphaned or abandoned, for having very low mana or other reasons)
For nobles it depends on:
•Heirarchy: Laynoble, Mednoble, Archnoble, Archducal family, Branch family of Royal family, Royal family, Zent
•Noble titles & duties: Scholar, Knight, Attendant, Giebe, Archduke candidates, & Archduke, Royal family, Zent
Sometimes the treatment depends if:
(aside for being a commoner or a noble)
•baptized
•orphaned
•gender
•married (depends on ranking, 1st wife, 2nd wife, etc.)
•slave
•namesworn
•large mana
•many elements
•legit heir
•adopted
•graduated from noble academy (graduating signifies legitimate noble by having schtappe)
•factions
•royal family or branch royal family
Extra:
•marriage compatibility (depends if commoner or noble, faction, mana, position & age)
There's so much to distinguished ( ̄∇ ̄")
Don't get me started on noble euphemisms that gives many tons of headache due to it's hard to understand manner ╮( ̄▽ ̄)╭
Off topic, do archduke means it's one of the highest title or a mistranslation or means something different in that world