WI: Roman Emperor title goes to the King of West Francia, not East Francia

IOTL, after the Roman Emperor in the East was deposed by a woman, the city of Rome reclaimed their ancient right to crown Emperors. The Pope, now leader of the city, crowned Charles the Great, King of the Franks and saviour of the city from the Lombardian heretics as the next Emperor of the Romans. Charles and his son Louis continued to hold the two great offices of King of the Franks and Emperor of the Romans side-by-side. The first was monarchical, territorial and local as the prince of a particular empire. The second was (theoretically) elective, theocratic and universal as the Lord of all Christendom, to which all monarchs theoretically held fealty.

In 843, the inheritance of the Kingdom of the Franks was split three ways into West Francia, East Francia and Lotharingia. Lotharingia was the least sustainable of the three, as it was designed as a long, thin territory due to the perceived need to include both the Frankish capital at Aachen and the Roman capital at Rome. It fairly rapidly collapsed and was carved up by the other two Kingdoms. Most of the land, including the Kingdom of Italy, ended up going to the East Frankish, as did the Imperial title. This victory ultimately proved ruinous to East Francia (later the Kingdom of Germany). The religious element of the Imperial title caused ideological struggles with the Pope meant excommunications and the religious backing for German vassals to go to war against their liege. The perceived need for the Roman Emperor to be lord of Rome required constant invasions over the Alps, which gave the German vassals the opportunity to do so while the Emperor was away. The desire to be a universal Emperor prevented the German King from pushing a national sentiment to consolidate his own realm. Ultimately, while France became a strong, centralized, hereditary monarchy, the Holy Roman Empire became a husk of a state and by the time it eventually died as a title it was little more than president of an international confederation.

But what if the cookie crumbles the other way, and the Imperial title falls on the other side of the Rhine? I supposed we would have to assume the Kingdom of Italy would have to fall with it, as the Roman Emperor needs to be in charge of Rome. Many of the same pressures remain in place: the desire to universality, the need to hold Rome across a mountain range, the competitive tension with the Pope. On the other side of things, France has some advantages: it is easier to get to Rome via sea access, meaning they can put down Italian rebellions with less time out of the country. It is a more condensed country, meaning the Emperor wouldn't be as far away from the north of the country when campaigning abroad.

And what does this mean for Germany? Does it consolidate more as the King focuses on consolidating rule in the sole kingdom he has, establishing a hereditary monarchy? Could a consolidated Germany wield far more strength against Poland and Hungary, bringing them into his realm? What are the other likely changes?
 
I feel the best thing to happen first is for the first wife of Otto the Great, Edith of England, to simply survive for several years longer. Her death enabled Otto to marry the widow of King Lothair of Italy, Adelaide of Burgundy. She was nearly forced into marrying the son of the usurper Berengar of Ivrea, so Adelaide had fled to Otto's court looking for aid. Otto subsequently invaded Italy on her behalf, defeated and deposed Berengar, and used being crowned as King of Italy to also be crowned Holy Roman Emperor.

If both Edith and Lothair stay alive, Otto won't have much of a reason to invade Italy and the kingdoms of East Francia, West Francia, Italy, and Arles stay separate for the time being. Down the line, Lothair of West Francia would then marry Adelaide's eldest daughter Emma like he did in OTL, which would then give West Francia the inside track on spreading influence in Italy.

Perhaps this leads to Otto becoming king of a much more centralized Germany and becomes more involved instead in affairs across the North Sea and towards Eastern Europe. The kings of France could then inherit Arles in the near future, which would include Burgundy, Provençe, and Savoy, among other territories.
 
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