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The plot of Deja Vu movie plays a lot with backward time travel theme.

Backward time travel in science fiction can work in one of two ways: immutable timeline or mutable timeline. The first one implies single, totally fixed timeline. This mean that any time travel is part of this timeline, so it is impossible for a time traveler to "change" history in any way. The classic example is Robert A. Heinlein's story "By His Bootstraps" which gave its name to the well-known temporal paradox. Unlike the above mutable timeline mechanism allows a time traveler to create altered timeline which is the result of his/her actions starting from the moment of his/her appearance in the past.

On the one hand Deja Vu pretends that it implement immutable timeline scenario, giving us some hints such as "U can save her" clue and Claire's call to Doug's office. However all these hints don't actually verify immutable timeline case since there is one compelling evidence of mutable timeline: prevention of the ferry explosion and Claire's death - the key events from the "source" (first shown in the movie) timeline which don't occur in the altered timeline.

So, as mutable timeline mechanism is proven, Doug of the "source" timeline couldn't cause the above hints himself, i.e. he couldn't leave the clue from the altered timeline to the "source" timeline. This mean that there "was" at least one more time travel and hence at least one more (most likely dead) Doug in the "source" timeline.

Why "at least"? Cause there really could be many "extra" (loser) Dougs in the "source" timeline. This depends on time offset when each "subsequent" Doug appears in the past in relation to the "previous" one. If he comes "earlier" than the "previous" one, he replaces (rewrites) the latter (the previous timeline) with himself (new timeline). It seems that this is exactly what happened to the "source" timeline's Doug. Otherwise, if the "current" Doug comes "later" than the "previous" one, they both share the same timeline which is the result of merging their actions. The example of such merging is Marty's second trip to 1955 in the BttF II movie. In this case the second Marty is the same as "subsequent" Doug and it's obvious that the second Marty comes into 1955 later than the "previous" one which result in the shared timeline.

So is there any reference (or hint) to how many Dougs were in the "source" timeline as well how many total Dougs were in all timelines?

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Thanks for the nice analysis of the mix of "seeming immutability" with mutability, and how that implies that more time travels happened than shown in the movie.

The following diagram shows a way to explain the events using four timelines (thus four Dougs), three time-traveling notes, two time-traveling laser signals, and two of the four Dougs time-traveling (the diagram refers to each of the four Dougs as Doug1 and renames two of them to Doug2 after they time-travel):

enter image description here

Source: 2007, 2008, 2013

The first timeline shown in the movie is referred to as "Timeline 3" in the diagram and as the "source" timeline in this question. With this way to explain the events, two Dougs spend some time in the "source" timeline.

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