Of those temporal anomalies recognized under the replacement theory, the N-jump is the good outcome, the temporal event that allows history to continue.

What identifies an N-jump is that all causes of all effects are found in a single timeline.  In this, the final history in an N-jump is indistinguishable from fixed time; the difference is how it is achieved.

Image © E. R. Jones and M. Joseph Young.  Used courtesy Valdron Inc

Under replacement theory, it is assumed that no one can arrive in the past before he departs from the future, in a sequential sense; and that this sequential sequence is tied in the original history to temporal sequence.  Thus there is always an original history in which no time traveler ever arrived from the future.

To understand this, assume that in 2000 our traveler is ten years old, and that later in 2020 when he is thirty years old he travels back to 2000.  In the sequence of events in his own life, the thirty year old version of him that arrives in 2000 must have been twenty-nine in 2019 before he left for the past.  Yet if he has not yet left for the past, he cannot be living in a world in which he already arrived in the past.  Thus there is an original history in which traveler grew from ten to thirty years old, and then at thirty years old traveled back to when he was ten, changing the history of the world at the very least by virtue of his own arrival within it.  There is then a new history of the world in which the young traveler ages from ten to thirty, unless his older self does something which interferes with that.

Since the time traveler has erased the original history and created a new version, it is logical that he has also erased his own history and thus his own existence.  Since, however, his presence in the past is dependent on his departure from the future, he will vanish from the past--unless something is done which will preserve his existence in the past.  An N-jump means that whatever the time traveler does in the past leads ultimately to his unaltered identical younger self making the same trip to the past, and so doing the same things.  This confirms the altered history as the stable history of the world, since at the end point of the anomaly the "same person" will make the "same trip" to the "same time and place" and do the "same things".

If the traveler's duplicate self fails to confirm his actions in the past, the result will be that history has to repeat itself until it is stable.  If the failure is major, such as that the traveler does not make the trip at all, this will create an infinity loop; a lesser failure may result in a sawtooth snap of some form, delaying the resolution for some uncertain number of repetitions of history.

Elaborate schemes have been devised by which a time traveler could change the past but ensure that he does not undo the information which led to his actions.  Some of these involve temporal protections which isolate some person or group from the impact of such changes, but the better methods rely on delivering accurate information concerning the original history to the people responsible to oversee it.

Ultimately, unless the author intends to create a temporal disaster, the N-jump is the desired outcome of all time travel stories.  It is the only outcome of time travel which allows a future under the replacement theory.

Next time we will examine sideways time.