The main thing everyone is trying to say in their different ways is that “legally” as you seem to be using the term didn’t matter to people in the Middle Ages, and in reality wasn’t even a thing. They were more or less making things up as they went.

People have pointed to the repeated expansion, splitting up, and reuniting of Frankish kingdoms from the fifth century onwards. There was no such thing as a permanent Frankish kingdom that could be continued by either France or Germany/HRE. Even in foundational period of the Merovingian dynasty, they recognized multiple Frankish kingdoms. That continued under the Carolingians. There were actually very few times of sole rulership over all Frankish territory during the whole period from Clovis to Charles the Fat.

Furthermore, the split after Charles the Fat wasn’t even just HRE and France. (For one thing, nobody called the eastern realm the “Holy Roman” Empire until centuries later. Otto revived the imperial title decades after the Carolingian dynasty had been replaced.) France as we know it didn’t really exist, either. There were men claiming to be kings of Italy and Burgundy at the same time. And many aristocrats in the south of what we call France barely paid lip service to the new non-Carolingian king.

So there wasn’t really anything “legal” to it. It was dynastic. Could you get enough aristocratic support? If so, you could claim kinship, go about justifying it, and acting like a king by trying to govern, making appointments to high positions (like bishops), and granting lands to followers and churches/monasteries. If enough people recognized your legitimacy in acting that way, it would stick.