⭐⭐ The first break on this thread, and it was already laid in the 3rd century — Cyprian said the bishop of Rome was only first among equals, only a title of honor. The East kept to those words all along; the West step by step turned them into jurisdiction.
The issue
Is the bishop of Rome first among equals (an honor), or the head of the whole church (with jurisdiction)?
Claim
The West: by apostolic succession, the bishop of Rome is Peter's successor (church tradition says Peter was the first apostle to reach Rome). Jesus gave Peter the keys of the kingdom of heaven and promised to build his church on this rock, and the Roman church read this rock as Peter himself. So, as Olson sets out the Western view, until the Lord returns, all the bishops of Rome hold first place over the whole church of Jesus Christ. ⚠ The Western bishops also used this to resist any emperor's authority over the church and theology. They insisted, Olson notes, that even Charlemagne must submit to the pope's authority — at least in spiritual things, if not in worldly ones.
Counterclaim
The East: from Origen on, the East always read the rock as Peter's faith, not Peter the man — let alone, Olson adds, the bishop of Rome. So, as Olson puts it, every Orthodox bishop is a true successor of Peter, and the five great sees of Rome, Constantinople, Antioch, Alexandria and Jerusalem all have equal dignity, power and authority. The East was willing to grant that the patriarch of Rome is first among equals — but, Olson notes, they held that this was only a title of honor, and the patriarch of Rome of course refused that view. ⚠ The East was mocked by the West for caesaropapism (Caesar as pope); looking back at the West, the East saw the Latin church trying to force on Christendom a papocaesarism (the pope as Caesar).
The ruling
⚠ Not a ruling, but mutual excommunication. On 16 July 1054 the representatives of Pope Leo IX laid a bull of excommunication on the altar of the cathedral of Hagia Sophia, excommunicating Michael Cerularius, patriarch of Constantinople; Cerularius answered at once with an excommunication of his own (it was the legates he excommunicated, not the pope himself).
What was at stake for salvation
⚠ This break bears directly on salvation, because Cyprian's system still held: the church is the indispensable ark of salvation, and valid acts of salvation can come only from communion with a true bishop. The moment the two sides excommunicated each other, each was declaring that the Christians of the other half stood outside the ark. ⭐ We do not judge who was right — but we ask readers to notice this: after that day, the saying outside the church there is no salvation was no longer a comfort. It had become a weapon.
Aftermath
⭐ Olson gives the outcome in a single sentence: the rift has never healed.
⚠ And Meyendorff's judgment explains why. This, he says, was the basic cause of the trouble, and it could never be healed: the root of it was that the two sides took different attitudes to the government of the church. On questions of doctrine, he goes on, the idea of Roman primacy as it developed in the Middle Ages was far apart from the view of the church current in the East; so as long as their views of church authority differed, no agreement could be reached on these questions or on how to solve them.
⭐⭐ Look back to the 3rd century. Cyprian plainly held that no one bishop was head of all the bishops, and that the primacy of the bishop of Rome was only a title of honor; and when Stephen, bishop of Rome, tried to overturn the decisions of a synod he had called in Carthage, he simply ignored him. González calls this system collegial episcopacy. Olson goes straight on: the Latin tradition of Western Roman Catholicism later raised the bishop of Rome to a supreme position and finally declared him infallible, while the Eastern churches kept to collegial episcopacy. — The schism of 1054 had already been prepared in the third century.