I suggested that a refusal by Carl Farness to complete a causal circle might have two consequences but now I doubt the second consequence.
We are imagining a timeline in which:
in 372 AD, Carl, in the role of Odin, appears on his timecycle, a metal-boned "...skeletal horse..." (Time Patrol, p. 454), and betrays his Gothic followers;
Manson Everard, roving the period from end to end, confirms that the Volsungasaga story of Odin betraying his followers originated specifically in this action by Carl;
in 1935, Everard tells Carl that he must travel to 372 to betray his followers.
Carl complies but the question is: what would have happened if he had refused?
The Carl who had appeared and enacted the betrayal in 372 arrives in 1935, so Carl has duplicated himself. Patrol members have to avoid this kind of paradox.
2nd Consequence (which I now doubt)
Anyone traveling to a date earlier than 372 and returning futureward might arrive in a timeline in which history was different because the betrayal had not happened.
But I now think that, if someone travels pastward along a timeline in which the betrayal happened, then he should return futureward along that same timeline unless, while he is 372 or earlier, he does something (I don't know exactly what) to prevent Carl from making the betrayal. (Our hypothetical new time traveler might prevent Carl's/Odin's followers from attacking King Ermanaric so that they would not be present in Ermanaric's hall when Odin appeared in order to betray them?)
However, this is all speculative. What the Patrol definitely wants to do is, first, to prevent the paradox of Carl duplicating himself and, secondly, as far as possible, to prevent even the slightest possibility of a timeline in which the betrayal did not happen. Both these purposes are served by getting Carl to leave 1935 with the intention of appearing in 372 and betraying his followers. This completes the circle and prevents anything else from happening.
But this leads me back to the question; what does Everard mean by "...a resonance..."? (p. 449)