Poul Anderson, The Shield Of Time (New York, 2006).
In 1137, Time Patrol milieu HQ (due to move to Rome in 1189) is in Mainz, a major city of the Holy Roman Empire. The milieu director is Otto Koch, born in Germany in 1891, but in 1137 playing the role of a rich merchant in Mainz. Manson Everard arrives by timecycle in the hidden part of the HQ and informs Koch of a "'...turning-point moment, sometime later this year.'" (p. 316)
After that moment, there is an alternative history containing no Time Patrol bases and referred to by chapter headings as 1137alpha, 1138alpha etc. The few Patrol members who know of the alpha timeline are trying to prevent it.
Everard and Koch discuss two coherent possibilities:
at the turning-point moment, an agent arrives and tells Koch that the alpha timeline has been prevented;
the Patrol fails, in which case its surviving members, living into 1137alpha and subsequently, meet to decide what to do.
Everard and Koch also discuss what I think is an incoherent option: after the turning-point moment, Koch lives in the alpha timeline knowing that, if the Patrol succeeds, "'...everything I do and see and think will become nothing?'" (p. 316)
It will not become nothing at any particular moment - call such a moment "T" - within the alpha timeline itself. Koch will not exist until T, then at that moment become nothing. If he lives in the timeline of 1137alpha, followed by 1138alpha etc, then he should live the rest of his lifespan in that timeline. When, two chapters later, Everard visits Anagni in 1138alpha, then Koch should still be living in Mainz in 1138alpha - and should continue to live after that.
The only way to give meaning to the proposition that the alpha timeline ceases to exist, becomes nothing, is to say that the entire four dimensional space-time continuum of the alpha timeline ceases to exist at some moment along a second temporal axis at right angles to the first. But that axis is a fifth dimension and is not Koch's temporal axis. One Koch lives in the original timeline guarded by the Patrol. A second Koch lives in the alpha timeline. Perhaps a third Koch lives in the restored timeline after ("after" along the second temporal dimension) the Patrol has caused the unwanted timeline to go back out of existence.
The inhabitants of the restored timeline can, from their point of view, speak of the entire alpha timeline as having ceased to exist but the inhabitants of the alpha timeline will, from their equally valid point of view, have lived their full lifespans within that timeline.
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