Might as well throw a hat in the ring. There is definitely a general consensus forming, but one problem is how to have the desired outcome (AP in Rome/West, HS in Constantinople/East) occur without the human player. Here is a perspective on how we could do it.
600 AD: Catholicism holy city = Rome, Apostolic Palace prebuilt in Rome, St. Peter's Basilica prebuilt in Rome, Orthodoxy holy city = Constantinople, Hagia Sophia prebuilt in Constantinople, Church of the Holy Sepulcher prebuilt in Jerusalem
3000 BC:
- Christianity is founded by first to discover Theology. If no civilization has discovered Theology by 40 AD, it is automatically founded in Jerusalem.
- Church of the Holy Sepulcher is automatically built in Jerusalem in 325 AD. Church of the Holy Sepulcher acts as a shrine for all three Christian religions; controlling it grants +2 happiness to each city depending on the state religion of the civ that controls it (no happiness if Islam, Secularism, etc.; +2 happiness to all cities with Orthodoxy for civs that have Orthodoxy as a state religion if controlled by a civ holding Orthodoxy as state religion; same as Orthodoxy for Catholicism and Protestantism). The bonus itself could change, but this retains Jerusalem as the holy city of Christianity despite the shrines of individual sects being in different cities.
- Once one of these three conditions is met, Catholicism is founded:
- Apostolic Palace is built. If built by Byzantium, it means the dominant form of political Christianity is Orthodoxy, in an alternate history scenario. If built by Rome, it means dominant political form of Christianity in Middle Ages is Catholicism, per history.
- Western Roman Empire collapses. Once this happens, all non-core possessions of Western Rome flip to Byzantines, and southern Italy (Neapolis, Rhegium, Pompeii, etc.) flips to Byzantines. Large stack of longbowmen spawn in Rome. Catholicism is automatically founded in Rome, Apostolic Palace is automatically built in Rome, St. Peter's Basilica is automatically built in Rome. Christianity is renamed to Orthodoxy, and the holy city moves to Constantinople. Hagia Sophia is automatically built in Constantinople. Church of the Holy Sepulcher stays in Jerusalem.
- In 553 AD (Second Council of Constantinople), Christianity is renamed to Orthodoxy and the holy city is moved to Constantinople. The Hagia Sophia is automatically built in Constantinople. In 1014 AD (Great Schism), Catholicism is founded in Rome. St. Peter's Basilica is automatically built in Rome. All Western cities and civs automatically convert from Orthodoxy to Catholicism.