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From my Wikipedia research this https://en.wikipedia.org/wiki/Arnulf_of_Metz is the farthest back ancestor we can confirm of Charlemagne. His dad was either this guy https://en.wikipedia.org/wiki/Arnoald who comes from senatorial roman lineage or this guy https://en.wikipedia.org/wiki/Bodegisel who was a Frankish duke.

If we go by Arnoald's lineage the farthest back patrilineal ancestor we can confirm of his is Ferreolus, there wasn't a wiki page of him but there is of his son https://en.wikipedia.org/wiki/Tonantius_Ferreolus_(prefect) who was a Praetorian prefect. There is no information about his father so patrilineally it ends there. However the wife of the Prefect ancestor was apparently the niece of Emperor Avitus so if this line is true Charlemagne would have an illegitimate familial connection to a Roman Emperor.

Even with Bodegisel’s lineage, though not patrilineal there are still Roman connections as both his father and grandfather married daughters of Roman senators.

The farthest back patrilineal line of Bodegisel that is confirmable is https://en.wikipedia.org/wiki/Munderic. However Munderic alleged that he was the son of the son of a Frankish king this guy https://en.wikipedia.org/wiki/Sigobert_the_Lame which would be the end of the road at least patrilineally, as the only information about his father is that his name was Childebert. Not the Merovingian king btw this Childebert died in 509 while the Merovingian Childebert lived for another 50 years.

Something cool I just figured out, so the farthest back Charlemagne patrilineal ancestor of the Arnauld line was named Ferreolus
> ferreolus in latin means "iron"
> his sons name was Tonantius Ferreolus
> tonantius in Latin means "more thunderous"
> his name was literally “MORE THUNDEROUS IRON"
Especially badass considering he helped Aetius defend against the Hun invasion.