Louis the Pious never remarries, hence no Charles the Bald. Of his three remaining sons, Lothair only has surviving daughters (lets say Lothair's son Louis II dies of disease around the time of his father's death) and dies of unrelated causes before his father. Pepin gets executed as part of one of his rebellions, and his minor sons Charles and Pepin are pushed into an ecclesiastic career, and die childless as OTL.
This leaves just Louis the German (probably "Louis II" ITTL) as Louis the Pious' sole heir. *Louis II lives a long, healthy life as IOTL, dying aged 74 in 876. OTL he had 3 boys and 4 girls. ITTL, he has 2 boys and 5 girls, and splits his lands equally between them.
The two however engage in a civil war shortly after their father's death, which results in one of them dying of disease whilst on campaign, making the other one, let's call him Charles, winner by default.
Charles, like Charles the Fat from OTL, fails to have any kids with his wife (lets go with infertility), and attempts to have his sole illegitimate child (who isn't even his in the first place, but resembles his father just enough to pass off as such) legitimized by the Pope and bishops. Unlike OTL, he succeeds in this endeavor. Lets call this bastard Bernard, just to keep things simple. *Charles dies aged 68, in 908 AD.
Bernard, unlike his father, has lots of kids, including 4 adult and 2 underage boys, 3 daughters and lots more illegitimate ones. Bernard dies aged 52, in 930 A.D, and his realm devolves into a ten-year civil war among his sons. Out of the ashes, the youngest of his 4 adult sons emerges victorious. Let's call him Charles, after his "grandfather".
Charles has been thoroughly traumatized by he war he just fought, with his wife dying in childbirth not helping matters either. Distraught, he vows never to remarry again, and divides the realm between his two younger brothers (let's call them Karl and Carloman), for which he was a sort of father figure. Charles dies after a mere 9 years at the helm of the united Empire, in 949 AD
Karl however has no intention of re-living the civil war that nearly killed off most of his family, and has his brother murdered shortly after the death of their father. Karl proceeds to have numerous kids, and lives to the ripe old age of 76, dying in 1005 A.D. Throughout his life, Karl makes sure that all his sons are raised with the understanding that only the eldest should become ruler of the realm, and in this he succeeds.
His son and co-emperor Lothair succeeds him, and keeps all of his brothers (at least those who hadn't been given to the Church) at court on a short leash. He also easily crushes a rebellion by one of his more ambitious brothers and has him tortured to death, just to make a point.
Lothair dies shortly afterward in 1016 A.D., but not before siring a child, his only surviving boy, named Pepin the Posthumous. Pepin the Posthumous lives in the shadow of a regency council ran by his uncles, who fight and win a civil war on his behalf when the eldest of them tried to grab absolute power. Pepin's uncles will go on to provide ample cadet branches without any real claim to the throne (Louis the Traitor's rebellion in 1039 A.D. notwithstanding). As for himself, Pepin the Posthumous only has single male child (odd considering his relatives' 'achievements' in that area), whom he names co-emperor and passes the throne to, aged 48, in 1072 A.D.
After ~200 of a united Carolingian Empire, unity of the realm should be ingrained enough in the culture of the nobility that the centrifugal forces inherent in any large state shouldn't be able to tear it apart.
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