For a three-act plot analysis, put on your screenwriter’s hat. Moviemakers know the formula well: at the end of Act One, the main character is drawn in completely to a conflict. During Act Two, she is farthest away from her goals. At the end of Act Three, the story is resolved.
Forgive us if we're repeating ourselves, but A Canticle for Leibowitz is both a whole novel and three very distinct stories. So you can analyze the three-act plot structure of the novel as a whole, or you can analyze each individual story's three-act plot.
But since the novel was nice enough to separate itself into three acts for us, we're going to go that route for now. Happy analyzing.
Act I
This act's job is to ensure the reader has all the information necessary to understand the story to come, and Fiat Homo performs this task quite admirably. We learn what kind of a world the Leibowitz Abbey exists in, and we learn how it got that way: by the deadly one-two punch of nuclear war and Simplification.
We also discover who Leibowitz is, and how the Leibowitz Abbey monks have carried on his tasks of maintaining the Memorabilia. Ooh, ahh.
Act II
In this act, we see the action rise, and the protagonist try to resolve whatever problem's at hand… usually through conflict with the antagonist.
Dom Paulo's struggle with Thon Taddeo in Fiat Lux seems to fit this description. Thon Taddeo is a secular scholar who owes his allegiance to Mayor Hannegan. Hannegan wishes to use the knowledge of people like Taddeo to secure his own power and further his war efforts.
Dom Paulo wishes to see Taddeo and his knowledge serve higher purposes—God for one, and humanity for another. But Taddeo despises religion and feels trapped by the demands of a state he feels is simply a necessary evil; it supports his work.
The conflict between the two builds and builds until the end of the chapter. At that point, Dom Paulo and Taddeo understand each other, but can't reach an agreement. Then Hannegan's war begins, and we enter Act III.
Act III
Or, as it's called in the novel, Fiat Voluntas Tua. This act sees the resolution to the conflicts we've been witness to so far. Be warned: a happy-go-lucky resolution this is not.
The country Hannegan started has evolved to the point of gaining nuclear power. It also controls all of the knowledge of science and technology. Yikes.
This super evil, control-freak government brings about a nuclear war. Which is pretty much the be all, end all of conflict resolution, are we right? Father Zerchi and the church attempt to counter the state's willingness for mass destruction, but they just aren't powerful enough.
The best they can do is send a group of monks into space, thereby saving the Church and the Memorabilia from total destruction. All conflicts are certainly over on Earth, as the planet is now a smoldering mass of nuclear rock. But who knows what happens to those monks in space.