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// Theory of the Time-Loop 
// Story: A Mathematical Theory of ‬Hard Reset 
// by furrypony 
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Several days later at Lyra's home,‭ ‬Twilight was teaching Lyra the basics of time-traveling,‭ ‬drawing diagrams on a board while doing so.

‭“Let's call 'soul' that which is preserved every time I die, and not be concerned about what exactly it's made of. It could be simply be a part of my brain made of physical material, or some special nonphysical thing that interacts with my brain in some way, or just a special mathematical pattern of my brain.

Now,‭ ‬when I'm still alive in a world,‭ ‬my soul is connected to the world,‭ ‬experiencing it through a body.‭ ‬But‭ ‬every time I die,‭ ‬my soul is disconnected from this world,‭ ‬and immediately connected with a world that,‭ ‬other than my soul,‭ ‬is exactly the same as the world that I was first connected to,‭ ‬up to the‭ ‬reset point.

Just how many worlds are there‭? ‬Possibility one:‭ ‬every time I disconnect from a world,‭ ‬that world stops existing.‭ ‬This way,‭ ‬there is no simultaneous existence of worlds.‭ ‬There is always at most one world existing at any moment.‭”

Twilight drew a diagram on the board.





“But there's a problem with saying‭ ‬'two worlds exist at the same time‭'‬,‭ ‬since it seems to require a way to measure time outside of the worlds.‭ ‬We have to imagine all the worlds as floating around in some hyper-world,‭ ‬and that there is a hyper-world time,‭ ‬with which we can meaningfully say something like‭ ‬'This‭ ‬4‭ ‬o'clock happened before that‭ ‬4‭ ‬o'clock.‭'‬ That sentence would mean,‭ ‬'In hyper-world,‭ ‬this world's‭ ‬4‭'‬clock happened at the hyper-world's‭ ‬14‭ ‬o'clock,‭ ‬and that world's‭ ‬4‭ ‬o'clock happened at the hyper-world's‭ ‬19‭ ‬o'clock.‭'” 

Twilight ran her hoof along the‭ “‬Time‭” ‬arrow on the top of the diagram,‭ “‬Here,‭ ‬we are showing the time in hyper-world as a straight line,‭ ‬along which the many worlds are arranged.‭ ‬And we see that in this model,‭ ‬no two worlds exist at the same time.

Now,‭ ‬consider the following question.‭ ‬When I die in world A,‭ ‬does the magic create a world B that is an exact copy of the world‭ ‬at‭ ‬reset point,‭ ‬and put my soul into that world‭? ‬If that is so,‭ ‬what happens to world A‭? ‬Does it get immediately destroyed,‭ ‬so that when seen in the hyper-world,‭ ‬there is no world A beyond that time‭? ‬Or it just goes on without me,‭ ‬and my friends mourn my death in world A‭ ‬while I keep on living in world B‭?”





“‬Either way,‭ ‬what about the other souls‭? ‬They have no continuity,‭ ‬unlike my soul.‭ ‬Anypony other than me does not remember anything from the previous worlds.‭ ‬They always begin exactly as the ponies in the first world I was in,‭ ‬and no change in one world would carry over to the next world.

Suppose other souls exist and are made in the same way as my soul.‭ ‬My soul might be unique‭ – ‬or copied,‭ ‬but there's no way for me to tell unless I could see it in hyper-world‭ – ‬but their souls must be copied.‭”

Lyra asked,‭ “‬How did you know that‭?”

Twilight replied,‭ “‬I have made experiments to systematically probe the extent of soul-copying,‭ ‬in the following manner:‭ ‬I took a map of Equestria and chose by random twenty cities.‭ ‬I then traveled to these cities and made friends and acquaintances in these cities,‭ ‬of all species‭ – ‬pony,‭ ‬griffin,‭ ‬and others‭ – ‬then I revisited them in the next‭ ‬lives to see whether there is any recollection from their past life.‭ ‬There was no‭ ‬retention‭ ‬of‭ ‬events beyond the reset point.‭ ‬I also checked if they would have memories and personalities that are different with their previous life,‭ ‬for example,‭ ‬I would ask them if they went to a certain school when they were‭ ‬6‭ ‬years old,‭ ‬or ask about the details of some pivotal event in their life that occurred before the reset point.‭ ‬There were almost no‭ ‬inconsistencies.‭ ‬The‭ ‬inconsistencies that did occur were minor‭ ‬and‭ ‬could be attributed to memory errors.

Thus,‭ ‬I determined that the soul-copying applies to all souls everywhere,‭ ‬and the copying is practically perfect,‭ ‬and they are always copied from the original‭ ‬at the reset point.

This seems extremely unlikely,‭ ‬for such a spell I made to always copy all the souls in the world repeatedly.‭ ‬It would be far more likely for the spell to affect only me personally.‭”

Lyra asked,‭ “'‬Unlikely‭'‬...‭ ‬Why is that‭?”

Twilight replied,‭ ‬“It's a principle of magic theory,‭ ‬that is hard to experimentally verify,‭ ‬but makes intuitive sense,‭ ‬and explains a lot of phenomena.‭ ‬It's called‭ '‬The Locality-Energy Correspondence‭'‬,‭ ‬and it states that the‭ '‬extent‭' ‬affected by a spell is proportional to the‭ '‬effort‭' ‬required to cast it.‭ ‬For a spell to affect a large area,‭ ‬such as all of Equestria,‭ ‬requires the power of at least a thousand ponies.‭ ‬I performed the spell alone,‭ ‬so I have reason to believe it only affected me specifically.

So let's assume that the spell does not affect any souls except my own.‭ ‬Then where did all the extra worlds come from‭? ‬Maybe in the hyper-world,‭ ‬all possible worlds already exist.‭ ‬Like in a giant garden,‭ ‬all the worlds are trees that grow higher and higher as their history unfolds.‭ ‬The spell simply transports my soul across them.‭”

Twilight drew another diagram on the board.





“But then,‭ ‬suppose upon the death in world A,‭ ‬my soul is transported to world B,‭ ‬at the reset point,‭ ‬then what happened to world-B-Twilight's soul‭? ‬Did her soul get destroyed‭? ‬Lost in the hyper-vacuum between the worlds‭? ‬And if that were the case I must have killed hundreds of alternative-Twilight's souls by now,‭ ‬that means the spell can kill a very large number of souls across the hyper-world.‭ ‬Thus the spell has an extremely powerful and non-local effect,‭ ‬which as I said,‭ ‬is very unlikely...‭ ‬Or maybe not.‭ ‬Metaphorically speaking,‭ ‬maybe this spell has made my soul a kind of mobile parasite.‭”

Twilight dragged her hoof along the arrows on the diagram.

‭“After I experience death in world A, I split free, swim across the hyper-vacuum to an uninfected world B. Then I find the reset point in world B, kill the oblivious world-B-Twilight, then attach myself. This is certainly too much metaphor, but at least it makes it at least possible to think of this as a model of how the spell works while also assuming that the spell can affect only my soul, and maybe a little bit of space and time around me.

But this can't be true either,‭ ‬for,‭ ‬suppose I died in world A at‭ ‬2‭ ‬years after reset time,‭ ‬then I must swim to world B,‭ ‬then crawl backwards in time for‭ ‬2‭ ‬years before injecting myself into world B at reset point.‭ ‬Now,‭ ‬what would happen to the‭ ‬2-year-long chuck of world B's history after the reset point‭? ‬Did it just get chopped off,‭ ‬since,‭ ‬after my injection,‭ ‬world B's history after reset point will unfold in a different way‭?”





“Maybe there is only one world-tree in the hyper-world that has branches everywhere,‭ ‬each branch starts a different way the future can unfold.‭ ‬Yesterday,‭ ‬I bet a coin throw with a pony,‭ ‬and at that place,‭ ‬the world-tree branched into many branches.‭ ‬In some branches,‭ ‬I win the coin,‭ ‬in others,‭ ‬they win the coin,‭ ‬and in yet others,‭ ‬the coin rolls away.‭”

Twilight drew another diagram.





“In this case,‭ ‬the world-tree must have the ability to constantly generate new branches.‭ ‬A simple coin throw makes many branches,‭ ‬and the tree must branch out enough to allow them all to happen in some branch.‭ ‬The world is copied all the time,‭ ‬the souls are copied all the time.‭ ‬It's now a thing that the tree does,‭ ‬not my spell.

Every time I experience death,‭ ‬I detach from the branch I'm one and attach myself at the reset point.‭ ‬The tree then grows a new branch at the reset point,‭ ‬and I would kill the alternative-Twilight,‭ ‬which was just copied in that branch,‭ ‬and inject myself.‭”





“This‭ ‬is where I have gotten so far.‭ ‬It stands better than all the other‭ ‬hypotheses.‭ ‬It involves only local effects,‭ ‬and explains copying as an inherent ability of the world itself.‭”

After Twilight gave Lyra a rough sketch of the theory and practice of time-loop traveling,‭ ‬to Twilight's satisfaction,‭ ‬Lyra enthusiastically agreed to become a time loop traveler,‭ ‬so she took her under her tutorship.



One day,‭ ‬a few months into the tutelage,‭ ‬Twilight was lecturing to Lyra about implications of the spell on Lyra.

‭“Suppose Twilight and Lyra are currently in world K, at time '+5 years after reset point', and Lyra casts the spell on herself. Then what will happen? Lyra will live on for a while, then she would die, and her soul would go back to world K, at her reset point here, and get a branch of the world, leap onto the newly-copied Lyra's soul, kill her, and inject herself into there.

What would‭ ‬Twilight‭ ‬observe‭? ‬Well,‭ ‬the moment after‭ ‬Lyra's reset point,‭ ‬the world splits into many branches,‭ ‬and‭ ‬it would depend on luck what future branch‭ ‬this Twilight‭ ‬would‭ ‬go down.‭ ‬Assume that there is equal probability that‭ ‬Twilight end up on a certain branch,‭ ‬then the probability that‭ ‬she end up in a branch that contains‭ ‬'time-traveler Lyra‭'‬ is N1/‭(‬N1‭ ‬+‭ ‬N2)‬,‭ ‬where‭ ‬N1 ‬equals number of‭ ‬new‭ ‬branches of the universe‭ ‬created by Lyra's time-traveling,‭ ‬and N2 ‬equals number of universes that would have branched at that moment if Lyra doesn't time-travel.‭”

Lyra asked,‭ “‬Aren't N1,‭ ‬N2 ‬both infinite‭?”

Twilight replied,‭ “‬Let's ignore time-traveling for a moment,‭ ‬and just consider how the world should branch.‭ ‬From each‭ ‬state of the world,‭ ‬shouldn't there are an‭ ‬uncountably‭ ‬infinite number of possible worlds that could follow‭? ‬For,‭ ‬if a coin can land‭ ‬1‭ ‬mm to the right or‭ ‬2‭ ‬mm to the right,‭ ‬shouldn't it also be that it can also land with any distance in between‭? ‬Thus,‭ ‬the number of possible worlds should be at least the number of real numbers,‭ ‬so N2 ‬is uncountably infinite.‭ ‬But Lyra,‭ ‬since she can only travel back in time for a countably many number of times,‭ ‬N1 ‬is countably infinite.‭ ‬So...


N1/‭(‬N1 ‬+‭ ‬N2) = ‬0


In other words,‭ ‬Twilight almost certainly will‭ ‬not meet a time-traveler Lyra.‭ ‬She‭ ‬will find‭ ‬herself going into a universe where Lyra has time-traveled‭ ‬exactly‭ ‬0‭ ‬times.‭”

Lyra asked,‭ “‬So...‭ ‬you think that,‭ ‬just after I cast the spell,‭ ‬you will find yourself meeting a Lyra that is quite clueless and unsure if the spell had worked‭?”

Twilight nodded,‭ “‬Exactly.‭”

Lyra looked at her suspiciously,‭ “‬But I thought you said you wanted to teach me because you wanted to find somepony that can truly understand you.‭”

Twilight replied,‭ “‬Yes,‭ ‬but I also want to help you find your adventure.‭ ‬Besides,‭ ‬maybe my calculations were wrong and I could meet you with positive probability.‭ ‬We have to give it a try to know.‭”