The thesis of this library
Why apoteichisis is heavenly communion
The wall is not a departure from communion. It is the guarding of communion with God when earthly shepherds betray it.
Begin with the word this whole library serves: communion. It is the most beautiful word in the vocabulary of the Church, and it carries two meanings that belong together.
Communion means, first, union with God: the life of the Holy Trinity given to man, sealed in Baptism, fed at the chalice, perfected in the age to come. And communion means, second, union with one another in the Church: one faith, one baptism, one altar, one visible fellowship of believers under their shepherds. In the healthy life of the Church these two are one thing, as a branch's union with the vine and its union with the other branches are one life. The second exists for the sake of the first, and the first flows through the second.
The choice that heresy forces
But what happens when a shepherd begins to preach, openly, a teaching the Church has already condemned as false? Then the two meanings of communion, which God joined, are forced apart by men, and the Christian is placed before a terrible choice.
If he keeps the second communion, the visible fellowship, by naming and following the false teacher as his own in the faith, he begins to lose the first; for no man is united to the Truth while confessing a lie at the altar. If, to keep the first, he simply flees the Church altogether, he loses both; for there is no communion with the Head apart from the Body.
The narrow way between
The saints found the narrow way between. They kept the faith and they kept the Church. They ceased to commemorate the false teacher, while remaining, in faith and love and Mysteries, exactly where they had always been. That narrow way is apoteichisis. It is not a departure from communion. It is the guarding of heavenly communion when earthly shepherds betray it.
For they have defied, not bishops, but pseudo bishops and pseudo teachers; and they have not sundered the union of the Church with any schism, but, on the contrary, have been sedulous to rescue the Church from schisms and divisions.
What a wall is, and what it is not
The word itself teaches the manner. Teichos is a city wall; apoteichisis is the raising of a wall against a besieger. Consider what a wall is, and what it is not. A wall is not a sword: it attacks no one. A wall is not a road out of the city: the man who builds it stays inside. The wall is raised against the heresy, never against the Church. The builder remains a citizen, keeps the city's law, and waits upon the city's courts, which in the Church are her councils.
Whoever turns the wall into a sword, or into a road out, has left this teaching on one side or the other. Much of the labour of this library is to show, from the history of the saints and of the schisms, exactly where each side lies.
The ground beneath it
All of this rests on what the Creed confesses the Church to be: one, holy, catholic and apostolic. Because she is one and her mark is the one faith, communion is confession; the four marks are the ground on which the wall is built.
Where it all ends
At the marriage supper of the Lamb. The wall is not the goal; the goal is the unbroken communion of the soul with Christ, kept safe through the storm until the truth is vindicated, as it always has been, and the faithful come home to the banquet that has no end. Everything in this library exists for that, and for nothing else.
Read it at length
This thesis is argued in full, with the Fathers and the canons, in the book hosted on this site: five parts and sixteen chapters, each closing in plain words.