The first challenge is that the story is not presented in either chronological order (that is, following the calendar) nor sequential order (following the events of the time traveler's experience), but in an order organized around events while maintaining the surprises until later.  Thus it may help if we begin by putting all the major events in the order in which they occurred, complete with such dates as are provided.  However, it is still convoluted and extensive, so we are going to do it in two parts.

There is also a problem of names.  The main character telling most of the story never actually gives us his, although we eventually deduce it; in the subtitles he is identified as "Barkeep", which is as good a name as we are likely to get.  The second critical character gives his name as "John", so although he is identified in the subtitles as "The Unmarried Mother" (his nom de plume) we will use "John".  The girl, from baby until motherhood, is always "Jane", although sometimes it will be necessary to distinguish the "Baby", also named "Jane".  There is also an important figure known as the "Fizzle Bomber" whom we will call "Bomber" even though eventually we do learn his name.

On September 13th 1945 someone registered at a hotel as "Gregory Johnson" but who is actually Barkeep delivers a two-week old baby, Jane, to the Cleveland City Orphanage.  Jane grows up there wishing she had a normal family, but is introduced to Mr. Robertson, loosely connected with Space Corp, and begins training for that program.  She is disqualified after a fight when it is discovered--but not revealed to her--that she is a hermaphrodite.  Then, on April 3rd, 1963, she meets John, and quickly falls in love.  This whirlwind romance lasts a little less than three months, as on June 24th, 1963, Barkeep comes and takes John away to enter the Temporal Bureau.

Robertson manages to get Jane back into Space Corp training, but she then discovers that she is pregnant and is dropped from the program.  The delivery goes badly at the charity ward, turning into an emergency Caesarean Section and total hysterectomy--but this time when they discover that she is a hermaphrodite with hidden immature male genitalia, they work on reconstructive surgery and hormone therapy to transform her into a man.

Oddly she takes the name "John", which is the more odd because she hates John for ruining her life.  As the therapy has its effect she looks more and more like John, but does not realize that she is becoming him.

Meanwhile, on March 2nd 1964, Barkeep kidnaps Jane's two-week-old baby Jane from the hospital nursery, to take her to the orphanage (in 1945).

We have covered almost twenty years of Jane's life to the point that she has had a baby and started to become John, who will be the father of her baby, who is herself.  That is the first of the several surprises in the movie; there are more in the later years, after a four-year break about which we are told little.

We began reconstructing all the events of the film in the order in which they occurred temporally, that is, by dates on the calendar.  We reached March 2nd, 1964, when Barkeep kidnapped baby Jane to take to the orphanage.

Sometime in 1968 the Fizzle Bomber destroys the Hardshaw Weapons Factory; this apparently prevents a terrorist occupation of that building that killed at least three hundred forty-six people.  It is the earliest attack by Bomber reported to us.

At 8:45 on March 2nd, 1970, Barkeep arrives in the area of one of Bomber's attempted bombings, and goes to the location.  Bomber is setting an eighteen-minute timer, and gets in a gun battle with Barkeep, which takes them out of the area.  Bomber leaves Barkeep stunned, but by this point the timer is down to a couple of minutes and John has arrived to prevent this bombing.  He is about to remove the bomb when he hears Bomber returning; they get into a gun battle as well, but Bomber flees in time for John to remove the bomb and put it into a containment unit--but not in time for him to seal the containment, with the result that he is badly burned by the explosion.  Barkeep walks over and puts John's USFF Coordinates Transformer Field Kit (disguised as a violin case) within reach, and John retreats to the future.  Barkeep finds a piece of the bomb timer, and leaves for 1964 to kidnap the baby.

On November 6th, 1970, John walks into the bar where Barkeep is working, and tells his story, and Barkeep offers to give him the opportunity to kill the guy who ruined Jane's life if John will then consider working for his boss, who is the same Robertson who tried to recruit Jane earlier.  John agrees, and is removed from that timeline.  A short while later, Barkeep returns to the bar and quits his job, then again leaves for his retirement in or about 1975 by way of 1985.

Sometime in 1974 there is a major Chicago chemical spill which kills 324 people; Bomber sets a bomb which prevents the driver from getting to work that day, preventing this disaster.

Sometime very early in 1975, possibly very late in 1974, Barkeep arrives in New York intending to retire.  His USFF Coordinates Transformer Field Kit shifts to "DECOMMISION" but then a moment later gives the message "FAIL ERROR FAIL".  He opens the envelope he got moments (or years) before from Robertson, and finds information on the timer he found which enables him to track Bomber to a laundromat at one in the morning on a particular unidentified day in March.

Before that day he meets Alice at an antiques shop; this apparently develops into some kind of relationship, because Bomber ridicules it when they meet.

At that meeting, Barkeep realizes that he is--or will be--Bomber.  Bomber wants to restore their past loving relationship and be together; Barkeep is horrified at the explosion that has not yet happened.  Barkeep kills Bomber.

A few days later, eleven thousand people are killed when the Fizzle Bomber levels ten blocks of New York City--or that is the history Barkeep was trying to prevent and should have prevented by killing Bomber.

The rest of the chronology is something of a denouement.  In 1981 time travel is invented, with a range of plus or minus fifty-three years from the moment of its invention.  1985 is in some way the official "headquarters" of the Temporal Bureau, and on August 12 of that year, at 11:01 in the evening, Barkeep arrives with John, who is about to become a temporal agent.  On April 3rd, 1991 Bomber destroys something in Hamburg, Germany, preventing a history in which one thousand eight hundred sixty-one lives were lost.  On February 21st, 1992, a burnt John arrives from 1970, is treated, undergoes plastic reconstructive surgery, and becomes Barkeep.

So if you were keeping score, Jane, John, Jane and John's baby Jane, the Barkeep, and the Bomber are all the same person interacting with himself over a span of about half a century.  Next we will attempt to put it together in the order in which he experienced all this.

We have reconstructed the temporal order of events in two parts, up through 1964 and 1968 and after.  It may, however, help if we also reconstruct events in the order in which they occur in the life of the lead character, who as we saw changes identities several times through the film.

Jane is born on or about February 17th, 1964, given that she was two weeks old when she was kidnapped on March 2nd, and 1964 is a leap year.  Two weeks later she is kidnapped and taken to September 13th, 1945, where she is left at the orphanage.  She is largely undisturbed by time travel until June 24th, 1963, when she is almost eighteen years old and meets John.  She gets pregnant and he vanishes, and she is forced to drop out of her training program and have the baby in the charity ward.  The baby is kidnapped, and she is badly harmed by the caesarean section; she is informed of her hermaphroditism and surgically and hormonally altered into a man, becoming John; although she looks familiar to herself, she does not realize that she is the same person who becomes the father of her baby.

She, now he, becomes the moderately successful writer of Confessions stories under the nom de plume "An Unmarried Mother".  On November 6th, 1970, about the time the doctors announce that his male genitalia are fully functional, he meets Barkeep at Pop's Place and tells him his story.  Barkeep offers him the chance to kill the guy who did this to him, claiming that he suspects that guy might be Bomber, who has by now begun to make New York City people very nervous.  John accepts the offer, and winds up back on June 24th, 1963, where he discovers that he is the guy who ruined his life, and falls in love with himself, that is, Jane, staying with her for most of three months and impregnating her before Barkeep returns on June 24th, 1963 and takes him to August 12th, 1985, to become a temporal agent.  At this point Barkeep tells John that he, Barkeep, is John's future self.  John apparently has several successful missions which are not detailed, and then is sent to March 2nd, 1970, to prevent a bombing.  He is attacked by the Bomber, but manages to remove the bomb, almost containing it in the containment unit but being badly burned by the blast when it detonates.  Barkeep appears, pushes the USFF Coordinates Transformer Field Kit into John's grasp, and John leaps forward to February 21st 1992, where he is immediately hospitalized and undergoes treatment and surgery to rebuild his badly burned appearance.  When he is fully recovered, he is Barkeep.

As Barkeep, he now has his final, and rather complicated, mission.  He takes a job as a bartender at Pop's Place a couple weeks before November 6th, 1970, so that he will be there to meet John, to take him to June 24th, 1963, so he will meet Jane and fall in love.  He makes an unauthorized side trip to March 2nd, 1970, to try to stop Bomber before the bomb harms him, but only succeeds in helping John reach the Field Kit.  Then he goes to March 2nd, 1964, to kidnap baby Jane and give the piece of the bomb timer to Robertson, and begins recording instructions for John.  He takes baby Jane to the orphanage on September 13th, 1945, and calls the orphanage, and then goes to June 24, 1963, to reveal his identity to John and take him to August 12, 1985, to become a temporal agent.

At this point he officially retires, choosing to live in New York City in 1975 just before the huge explosion.  Robertson gives him an envelope with information about the piece of the timer he found.  Several things happen in what is a somewhat overlapping order, including that the Field Kit fails to decommission, Barkeep meets Alice at the antiques store, he studies the information in the envelope, and he identifies a time and place where he can catch Bomber.

He discovers that Bomber is an older version of himself, but is so horrified at what he apparently became he shoots himself, that is, Barkeep shoots Bomber, several times.  However, Bomber is clearly considerably older than Barkeep, and still has access to a time machine.  We know that he made quite a few trips, in no identifiable sequence, including one to prevent a 1974 chemical spill in Chicago, one to prevent an unexplained disaster in Hamburg in 1991, and one to destroy a weapons factory in 1968 so that terrorists could not take it.

It also seems that he caused the destruction of ten blocks of New York City a few days after he is shot; that is one of the problems that must be addressed in resolving this film.

I was tempted to reserve the problem of the baby for last, but it really is in the way of everything else.  The story is built on the fundamental predestination paradox that Jane is her own child.  That is, in order for Jane to be born, Jane must give birth to herself.  As we have said, anything that can only happen if it happens will never happen.  However, such paradoxes can be resolved in Replacement Theory by extrapolating an original cause.  Following Sherlock Holmes' advice, then, once the impossible is eliminated, whatever remains, no matter how improbable, must be the truth.

In this case, we simply need an original baby.  Someone had a child and abandoned it at the orphanage late in 1945.  The child need not have been a hermaphrodite; we will get to that.  This is our original Jane, and she excelled in many of the skills we see in the Jane we meet and pursued a similar course in life.

It then comes to the point where the child, presumably named Jane, falls in love and gets pregnant.  There is a narcissistic element in the relationship we see, as she falls in love with her older male self, but as we saw in About Time, there may well be a sense in which people fall in love with someone when they reach a moment in their lives when they are ready to fall in love with someone, so without John, Jane very likely would have met someone else right around that time, maybe at that spot.  She is actually a very pretty girl, and one who is unaware of how pretty she is and suffers from self-esteem issues so would be easily flattered by the first person who actually notices.

From this point forward, it is difficult to reconstruct with any level of certainty, but since time travel will not be invented until 1981, no one from the future can tamper with events in 1963 and 1964.  We presume that she gets pregnant, because that is a necessary event to a later history which will not to this point have been altered.  Maybe the father leaves her before she knows she is pregnant, or after she tells him, or when the difficult cesarean section leaves her sterile.  She winds up a single mother.  Perhaps she becomes a writer.  In any case, Robertson, probably part of the space program, will see that someone with great potential got a lot of bad breaks, and will remember her.

It then all changes in, or shortly after, 1981, when Robertson takes charge of the new time travel program.  He needs agents, and he remembers one person who might have been very good at this had her life not taken such bad turns.  Of course, in 1981 that person turns thirty-six years old, and a younger agent would be a better recruit--but then, the life of this particular individual is such that it would be difficult to find the right time.  By the time she was eighteen she was pregnant; by the time she was twenty she was a single mother trying to raise a daughter who would undoubtedly face difficulties herself.  Of course, the daughter also shows promise, but is sixteen going on seventeen in 1981, so it would mean waiting a couple years for her.  The ideal would be to bring Jane into the program a few years after the birth, when she is about twenty-five, but the baby Jane is in the way.  So Robertson has an idea, and this changes everything.

Whatever happened to baby Jane?

We observed that Robertson, head of the new time travel program, is aware of an excellent candidate whose life is scrambled in ways that make it difficult to recruit her.  Before Jane has the baby, she is too young; after the baby is born, she is occupied with raising the child.  The child would also be a good candidate, but is also too young.

The interesting solution would be to remove the baby from the picture.  Don't kill her--she has potential, too, as daughter of the promising Jane.  It might be interesting to take the baby back in time, so that she will be just the right age to be an agent around the same time that Jane is the right age.  Then Robertson might get two promising agents from the same orphanage, who knew each other and knew how to work together.  So Robertson goes to 1964 and kidnaps baby Jane.

This creates an anomaly because history has been altered, but before that can resolve Robertson has left 1964 and arrived in 1945, creating another anomaly:  Now there are two foundlings at the Cleveland Orphanage.

This is a problem.  We assume that the original Jane was delivered there, and now the replacement Jane is also delivered there.  Where is the original Jane?

The answer is rather simple, really.  Jane tells us that she was one of several.  If baby Jane was delivered first and given the name Jane, then mother Jane would have been given a different name--there is no reason for an orphanage naming its babies to give the same name to two children, as that only confuses matters.  Neither of them has any reason to think they are related, let alone mother and daughter; they look similar, but not identical, because baby Jane is the product of a relationship with our unknown father, and so carries his genes.  If baby Jane excels beyond what her mother had accomplished, then the overshadowed original is likely to withdraw from the competition at an early age and become considerably more "normal", less impressive, and perhaps more adoptable.  The original Jane vanishes into the population, replaced by her own temporally-displaced daughter.

Ah, but is Jane not a hermaphrodite?  We noted that problem, and it is answered in the identity of the mystery father, addressed next.  Thus the original Jane can plausibly be replaced by her own daughter.  However, things are about to become considerably more complicated.

To reach the point at which Jane is replaced by her own daughter, we had to extrapolate that she fell in love with and got pregant by someone else before meeting John, and that Robertson kidnapped the baby in order to make Jane available for the time travel program.  That gives us several problems related to this original father.

Our second Jane, daughter of our original Jane, takes over that history, undoubtedly with some changes but still excelling at everything.  She becomes the space program candidate, and she meets and falls in love with the man whom she does not know is her own father.  She also gets pregnant, he again leaves, and Robertson again kidnaps the baby to take to 1945.  Since now the previous replacement baby was never born, this baby replaces the second baby Jane and in turn displaces the original Jane--but we hit a problem, a serious genetic issue that results in a cycling causality, a sawtooth snap.

Here is the problem.  The original Jane had a mother and a father, whom we will call Mom and Pop, each contributing 50% of Jane's DNA.  When she has baby Jane, that child's DNA is 50% hers, which means 25% Mom and 25% Pop, plus 50% that of the father, whom we will call Dad.  That Jane has just met Dad and had a replacement baby Jane, whose DNA must logically be 12.5% Mom, 12.5% Pop, and now 75% Dad.  (This is why incestuous relationships are genetically dangerous.)  That child replaces the mother, in turn meets Dad, and has the next iteration of Jane, whose original DNA from Mom and Pop has dropped to 6.25% each, the remaining 87.5% coming from Dad.  This shift away from the DNA of Mom and Pop in favor of Dad continues, each iteration reducing the original parental contribution by half and increasing that of Dad to provide the difference.  Gradually Jane becomes her own father.  Eventually, as her DNA approaches 100% Dad, she has to become male.

Perhaps, though, there is a solution to this that also resolves the other problem.  What if the man with whom she falls in love is the hermaphrodite?  While it is normal for hermaphrodites to mature as female first, it is not always the case, and environmental factors (such as exposure to testosterone products at an early age) can tip the balance the other direction.  This simplifies the story:  the original Jane was an extraordinary ordinary girl, and at some point as her original parents' DNA is replaced by that of her lover she becomes a hermaphrodite.  Not maturing in the same environment she presents originally as female, and so still makes the same connection, and the child of two hermaphrodites is far more likely to be a hermaphrodite than otherwise.

Mathematically we still have the problem that no matter how many times we reduce the original parental DNA by half it is never zero; functionally, though, we reach a point at which the last gene has been replaced, and Jane and her lover are the same person, one presenting as female and the other as male.

If the conception process were entirely random, this would still be a problem.  After all, if you have a hundred pennies and a hundred dimes and you randomly select fifty of each, you will not get the same set of pennies and dimes twice in a row.  However, genes are arranged in chromosomes, which come in pairs, and the genes that can fall in any particular point in the code can only come from the equivalent position in the parental code.  We are dealing with an extreme improbability--but once we reach the point at which Jane and her lover are the same person, we have at least the chance that they might produce a child who is a genetic match to themselves, and if they happen (against tremendous odds) to do that, history stabilizes into an N-jump as the exact same choices will lead to the birth of the exact same child, and history can continue--at least for the moment.

We have managed to reach the point at which Jane is her own baby, and a hermaphrodite, and the baby was kidnapped and taken to 1945 to become her.  At this point, the delivery is such that she becomes that man, and Robertson recruits him into the time travel service.

John, however, is pretty upset about that guy, whom we named Dad, who ruined Jane's life.  There are rules, of course, and there is some way for those who enforce the rules to be aware of infractions (somehow Robertson knows that Barkeep made an unauthorized trip to try to catch the bomber in 1970), but that probably will not prevent John from trying to remove this pain from his past.  He travels to April 3rd, 1963, and waits at the place where Jane met Dad.  He intends to kill his former lover.

So what happens to the lover?  Obviously, Jane encounters John first, and John being genetically exactly the same person as Dad and the future version of Jane, he becomes the replacement lover.  In this first encounter he may know, or at least think, that he is not the original villain of this story, but suppose that his attentions toward Jane will guard her from that scoundrel--and in some sense correctly so, but that now he is forced to leave.  Probably Robertson realizes what is happening, and comes to remove John from the situation.  Robertson will have to point out to him that if he does not leave, he will never become the person who arrived, so his presence here now is dependent on his departure.

This is of course problematic as well.  Once it has happened, it would not be a matter of preventing John from meeting Jane but of undoing that encounter, and there is no way that Robertson can as much as identify the original Dad, let alone arrange that meeting with Jane, at that point.  On the other hand, if he waits for the baby to be born John will undoubtedly prevent the kidnapping.  The only solution at this point is to pull John out of time after Jane is pregnant but before the baby is born, probably before Jane knows she is pregnant.  Robertson has to allow John to become his own father.

This is again complicated by that same genetic problem we faced with Dad:  just because two people are genetically identical does not mean that they will produce a child genetically identical to them.  The odds are perhaps twice as good as the odds that two siblings who are not identical twins would be genetically identical (twice as good because this child draws DNA from two identical parents, but still drawing only half the DNA from each and so not necessarily getting the correct matching set).  So again we have the extremely improbable circumstance that the baby must, against the odds, be a perfect copy of the parents.  This time, too, it is made less likely by the fact that the situation changes, as we saw in About Time when Tim changed the identity of his daughter Posy to become his son.  The first time John encounters Jane it will not be the meeting he remembered having as Jane because it is a different meeting; it is not until the next time through that John is the Jane who met John.  That changes their relationship ever so slightly, and makes this unlikely in the extreme, because Jane and John must give birth to themselves in the person of the child that follows both from the new relationship they create in their first time through and in the replacement relationship they create when John potentially remembers having lived through all this as Jane.

Again, though, it is not impossible, only improbable in the extreme.  Thus we could reach the point at which temporal agent John becomes his own father through his relationship with his own younger female self Jane, and we get an N-jump termination on a sawtooth snap that allows us to move forward a bit further.

We now come to an enigma.  Somehow, John becomes Barkeep, and in doing so becomes someone that John, who already had the experience as Jane of meeting a future version of himself, does not even suspect might be a future version of himself.  In the film, this is accomplished by introducing Bomber, and having one of the bombs burn John so severely that complete reconstructive surgery is needed.  However, Bomber is a future version of Barkeep, starting to lose his mind from too many trips through time.  If John never becomes Barkeep, Barkeep never becomes Bomber; and if Barkeep never becomes Bomber, John is not burned by the bomb and does not become Barkeep.

I have not read the book, but am reasonably reliably informed that Bomber was invented for the movie.  As a footnote to this, it is suggested that without Bomber, this works as a fixed time story.  I am not entirely persuaded; the very concept of a temporal police agency preventing crimes before they happen implies that history can be altered, as there would be no motivation to attempt to change events that were already right or could not be changed.  That is, either everything the temporal police want to prevent never happened, or the temporal police have a one hundred percent failure rate, and either way they would not exist.  However, that does not impact the movie.  There is a clear statement in the movie that Bomber, at least, has changed history, and therefore fixed time does not work for the movie version.

What matters to us, though, is that there must have been some reason in the book for John to become Barkeep.  It is rather unlikely that this was done to disguise his identity--at this point, Robertson has done everything that needed to be done.  Robertson might assign John the task of taking baby Jane to 1945, but probably he does not need to be disguised for that--the hospital staff have never seen him, and Jane did not recognize him as herself.  Evidently something happened which led to a decision that John needed to have reconstructive surgery.  He becomes Barkeep, and (foolishly, perhaps, given the dangers of meeting yourself which we discussed recently in Mr. Peabody & Sherman and previously in Timecop and in Back to the Future II) is given those assignments related to his younger selves.  He is now the one who recruits John, and who kidnaps the baby and takes her to 1945.  It is probably his idea to suggest to John that he can use this opportunity to kill the original father, and thus he sets up his own meeting with himself.  There are a thousand tweaks to history, a thousand things that can go wrong particularly in that way that butterfly effects do, but the remote possibility that everything will fall into place as we see it.  Then Barkeep retires and chooses to go to 1970, for no particular reason.  His Coordinates Transformer Kit fails to decommission, and he gets the idea that he can continue preventing disasters as a freelance time traveler.  He becomes Bomber.

We have no notion of the sequence in which he does his bombings, nor even whether his first attempt to prevent a disaster was done with a bomb.  Our best candidates for his first bombing are the 1974 chemical spill, where a bomb prevented a driver from getting to work and causing an accident fatal for three hundred twenty-four people (we do not know how many people were killed in that bombing), or the rather vaguely described 1991 Hamburg, Germany bombing in which almost two thousand lives were saved by a bombing about which we know nothing.  If Bomber regarded these as successful, it would encourage him to keep going, and perhaps to accept greater casualty rates as successes.  He becomes a target of the temporal enforcement agency, and John is assigned to identifying and stopping him.

There are still some problems, though.

Barkeep is trying to catch Bomber, but our instinct is that this cannot be done, because Bomber already knows everything Barkeep is going to do--after all, he was Barkeep, and he has a good memory.  But it does not work quite like that.  As we saw in Looper, the notion of the future self knowing what the past self does has an inherent flaw, which here is that the past self is reacting to the future self and so the information is changing.

Let us look at the complicated one, the thwarted bombing.  In the original history there was no bombing, because Bomber did not exist; we go through quite a bit of alteration and reconstruction before we get Bomber, but eventually Bomber sets that bomb.

We see only one version of those events, although we see it from two different perspectives; there are quite a few other versions of them.  As noted, in the original history there was no bomb, no explosion; in the history we see (twice), John arrives to disarm the bomb, gets in a firefight with Bomber, prevents the bomb from doing its intended damage but suffers the injuries from being caught in its incendiary blast.  We ask ourselves how John knew there was a bomb there, and there is really only one possible answer:  there was a history of the world in which the bomb exploded, killing some unreported number of people.  In that sense, John is not preventing a bombing but rather undoing one.  This is perhaps the inherent problem with police time travel stories:  if the crime did not happen, it cannot be prevented, and if it did happen and you prevent it, it did not happen.  Barring some application of the misunderstanding of Niven's Law, temporal police agencies fail.

There is, though, another problem.  There is probably a twenty to thirty year age difference between John and Bomber.  We know that John was two weeks old in 1945 and was removed from time in 1970, making him twenty-five years old.  We know that he had an illustrious career.  It is not clear when the mission to prevent the bombing occurred.  On the one hand it seems as if it was the first mission on which he was sent, while on the other hand it seems that after the reconstructive surgery which followed it there was much talk about how many trips he had already made and that he was about to launch his final mission.  Yet the last time we see Barkeep he was certain he would never become Bomber, and Bomber had a significant career, probably spanning quite a few years itself.  The minimum gap we could imagine is about five years, and twenty is considerably more plausible.  Let us settle for ten, for the sake of discussion.

Bomber sets the bomb in 1970, and it destroys whatever was targeted.  No one arrives to stop it.  Then at some point John is detailed to attempt to prevent this bombing, and he arrives.  There is no reason to suppose that Bomber waited for him, and therefore John will successfully disarm the bomb.  The Bomber who planted this bomb did not know John would do this, because he never was that version of John.  It will take this John ten years to become that Bomber--then that Bomber will know that that John defused that bomb, and will stay to prevent that.  He will get in a gun battle with John, and flee just before the bomb detonates, giving John sufficient time to remove it from the target but insufficient to contain it.  John is caught in the blast, and badly burned.  He finds his violin case, and returns to headquarters.

Note that Barkeep is not there.  Barkeep has no reason to be there--in his memory, he removed the bomb and contained it, and there was no sign of Barkeep.  It will be a couple of years before the version of John who remembers being caught in the blast becomes Barkeep, and makes the trip to attempt to catch Bomber at the scene.  That is also when Barkeep gives the violin case to John, changing another detail--but it must be a token gesture, because John must have reached it on his own in order for Barkeep to be there at all.  That means that that Barkeep will now become the future Bomber, who will also remember that Barkeep was there.

Note that this is unlike Next, in which Cadillac's knowledge enabled him to dodge bullets and blows and evade pursuit:  Bomber does not know what will happen, only what happened when he was that age, and as he changes his actions in response to his memory, his younger selves change what they do to counter the new actions.

It is remarkable that he does not get caught--but perhaps he does, because in the next iteration of history he would know what he did wrong and be able to avoid making that mistake again.  On the other hand, doing it right erases the knowledge of what happened when he did it wrong, so ultimately it has to stabilize into one version of events that are good enough, the version we see.

The exact date of the bombing in New York kept changing.  This was probably because Bomber kept adjusting his schedule in response to his memory of events--even just to avoid his memory being correct.  That is, if he remembers the bomb detonating on March 10th, he knows that his younger self will be expecting the bomb to detonate on March 10th, so he moves it to March 8th, or March 15th, or to another day when it will not be expected.

This automatically gives us a sawtooth snap:  whatever the history was in the previous version of time, someone is intentionally altering it in this version.  The only question is whether there would be method to his madness, some pattern which means that if last time it was the 10th, this time it is the 8th, and if it was the 8th it becomes the 15th, and if it was the 15th it becomes the 10th.  That would create an infinity loop termination.  Either way, though, as long as Bomber keeps changing the date of the bombing, the future is destroyed.

That is problematic, though.  We have several times mentioned that if you erase an event, you also erase all knowledge of that event, because you made it such that it never happened.  Further, as we noted in Back to the Future II, if you cause an event never to have happened, you also prevent the existence of any newspaper clippings about it--Bomber's scrapbook is undone, and the wall of cuttings about the bombings maintained by Barkeep can only include such bombings as were successful.  Thus Bomber can only remember when the bombing occurred in the history he knew as Barkeep, and Barkeep can only know when the bombing occurred in the only history that existed, and when Bomber changes the date the result is that the old date is forgotten, and the new history includes only people who know the new date.  It is a foolish impossibility--Bomber can change the date, but Barkeep cannot know that it keeps changing, only whether it happened on the date anticipated when he traveled there to stop it, and then all the information changes and no one knows the date was changed.

The other problem, though, which confronts the viewer is that a few days before the bombing, this time, Barkeep tracks Bomber to the laundromat and kills him.  This should, we think, prevent the destruction of ten blocks of New York City.  Here, though, we are failing to think fourth-dimensionally.  The simple solution is that Bomber has already destroyed New York City several days in the future, and traveled back to do his laundry, perhaps because this is laundry day and his decaying intellectual abilities force him to keep to the routine, perhaps because he wants to meet the younger self with whom he is still in love, perhaps merely to establish an alibi for the bombing.  In any case, Barkeep fails to recognize that this has already happened, and so as he kills Bomber he thinks he has prevented a bombing which already exists in the future, already was accomplished by his older self.

It is from there a simple matter to suppose that Barkeep gradually becomes Bomber.  He perceives that he can still prevent disasters, and begins doing so, and as his mind deteriorates from excessive time travel he starts to see the merit in using bombs, preventing a greater disaster by causing a lesser one.  The climax of his career is, of course, the destruction of ten blocks of New York, and we have no idea what disaster he was attempting to prevent, but perhaps that one did not work as planned--and then he kills himself in a way that is not viewed as suicide, and his story ends.

Ultimately, there are some problems that are sticking points--notably the moving date of the New York bombing, also the aspect of preventing crimes or disasters before they happen based on the knowledge that they happened when they were not prevented--but overall it was a pretty decent time travel film, and what were presented as the big problems proved on consideration to be soluable, if perhaps highly improbable.  It was also a fascinating movie to watch, well worth the time.