This is an exercise where I take a world which is otherwise the same as ours except for one physical law being changed, and try to work out what that world would look like.

This is meant to exercise my ability to work out how things are connected to other things - how does the impact of changing one physical law ripple across everything in the universe? It's also a good test of creative thinking - what technologies would people come up with in this world to exploit the new physical laws?

It was also great fun thinking about!


There are two universes, both otherwise absolutely identical to ours (and each other), except that:

  1. The sentient creatures that inhabit these universes somehow have the ability to instantaneously switch from one universe to the other, as well as to instantaneously switch anything reasonably small they're relatively close to from one universe to the other.
  2. If each universe exists for T time, and they switch from U1 at time t, they will appear in U2 at time T - t. In practice that means that time runs backwards when you switch universes. If you were to switch from U1 to U2, live in U2 for a year, and then switch back to U1, you would arrive a year earlier than you left.

 The universes both end in a big crunch after about 30 billion years.

At the very midpoint of the universe, the earths in each one look absolutely identical, and are in exactly the same location. You can safely switch from one to the other without worrying about finding yourself in outer space or buried in the earths crust.

For complicated physics reasons this binds the two earths together, such that your position relative to the center of mass of the earth stays the same when you switch even if you switch at some time other than the exact midpoint of the universe. In practice this means that switching is safe so long as the continents haven't changed to much, and gets safer the closer you are to the midpoint.

Because the effect of this is that you can drive both ways through time, but on separate lanes (universes), I'm going to call this universe pair the Highwayverse.


I'm going to list my predictions about what the Highwayverse would look like, but feel free to play the game yourself, and come up with your own predictions!


The most important fundamental law that governs the lives of the people who live in the Highwayverse is the idea of the equilibrium.

In our own universe everything has a cause (or causes), which itself has a cause, which has a cause... going all the way back to the big bang.

In the Highwayverse we can find that causes might ultimately form a loop, where something ends up causing itself. Plenty of people are their own ancestor. This is perfectly fine, so long as the result is in equilibrium.

What does equilibrium mean?

Well it's easier to start off by seeing what's not an equilibrium. In our own universe when a disease infects enough people we try and develop a vaccine for the disease, and inoculate everyone with it.

In the Highwayverse this can't happen. Because once the vaccine is developed you would go back in time and inoculate all the people before they got the disease. The natural outcome of events would stop themselves ever happening.

It's almost impossible to predict what the equilibrium solution will be, where the chain of events ends up exactly right to cause itself to happen. Indeed there might be multiple such equilibria. But whatever does end up happening has to be self consistent in this way.

Perhaps patient zero receives a message from their future self telling them to avoid a specific location where they would otherwise have got infected. They do so, then write the message to themselves and send it backwards. They may have no idea why they're doing that, but they're used to such weird events. Ignoring yourself is a really bad idea in the highwayverse (there's protocols which make this sort of thing more effective, and we'll discuss a couple of them later on).


Since the two universes that make up the Highwayverse are identical, there must be symmetry around the midpoint - so that if you switched universes at exactly the midpoint the universe would carry on exactly as if you hadn't switched.

This means that every single person has an identical copy of themselves living out the exact same life but always in the other universe, and always on the opposite end of time. You can even meet your cosmic twin (when they're younger or older than you are now), but that's not particularly interesting - people in the Highwayverse are very used to meeting themselves.

On the Origin of Species

Lets assume ordinary animals can't switch universes - only the sentient ones (henceforth called Highwayversians). Then animals evolved absolutely ordinarily.

Highwayversians however probably haven't evolved - that wouldn't be in equilibrium. Having evolved to the stage where they can go back in time, they would have gone back to where they evolved originally and lived there, which would have meant the protohighwayversians wouldn't have had an ecological niche and would never have evolved.

Perhaps the costs of travelling too far from the midpoint means this doesn't happen. Perhaps conservationists careful preserved the protohighwayversians in such a way that humans do end up evolving from them.

Or just as likely Highwayversians never evolved at all. They just exist, in a giant tangled graph where everyone has 2 parents as per standard, but it all just ends up looping back on itself such that there's no first human and half the people are their own ancestor. Plenty of people are their own parent.

In fact there might be entire species consisting of exactly 2 people who give birth to themselves, and are otherwise completely genetically unrelated to anyone else. To the citizens of the Highwayverse this is perfectly normal.

Technology level

The Highwayverse has manufacturers - your phone has to have come from somewhere, and given wear and tear with use, it can't be that it loops around in an endless cycle.

However it doesn't have inventors - why would anyone spend years working out how to manufacture a phone, when he could just ask someone coming from the other direction how they ultimately did it?

Instead knowledge exists, but has no clear originator. Knowledge is data, and it's perfectly possible for data to exist in a loop without degrading over time - the best textbooks on every subject are copied billions of times to different devices and follow every possible path through time. Ultimately though they all loop back on themselves - this file was copied from that file which was copied from that file which ultimately was copied from the original file. Since people would have fixed any errors if they noticed them, the equilibrium textbooks are of extremely high quality, without any noticeable errors.

This means that the equilibrium technology level is extremely high, at the level where somebody who spent their entire life studying technology wouldn't be able to move the bar any further. And this technology level is consistent across all points in time, or at least all those within reasonable reach of the midpoint.


You can't change the future, and you can't change the past, but you can change what the equilibrium would otherwise have been. The fundamental trick to do so is precommitments, and making them comes as naturally to the Highwayversians as walking comes to us.

I just said that the equilibrium technology level of the Highwayverse is at the level where somebody who spent their entire life studying technology wouldn't be able to move the bar any further. But this means that nobody would ever invent anything! Wouldn't that mean there are equilibria much lower than that level, as there's no incentive for anyone to study hard and invent anything?

The Highwayversians are of course perfectly aware of this, and so pay people to spend their entire life studying everything there is to know about technology, knowing they'll never achieve anything, just because doing so moves the equilibrium higher. Since they would have done the same in a hypothetical universe where the equilibrium technology level was lower, such a universe is inconsistent, and so they live in the wonderful high tech universe they find themselves.

Highwayversians use this trick all the time. If they see their future self get run over by a car they'll try their hardest to avoid the location where they get run over, not because that will help, but because it means they very rarely get run over by a car in the first place since such a world is almost always inconsistent.

Similarly they try their hardest to improve textbooks, knowing full well that they won't find anything, since this effort causes the textbooks to have been very high quality in the first place.

No-one in the Highwayverse would be confused by Newcomb's problem for a split second.


Given their high technology level, the Highwayversians have defeated aging. Due to precommitments they very rarely have accidents. One of the first thing a Highwayversian does is find out how they die, and if they don't like their death they expend enormous resources to prevent it. In practice this means that the vast vast majority of deaths are due to suicide at extremely old age once they're bored of life.

Perhaps some people even loop back on themselves, living an infinite life in an endless cycle.

Jobs in the Highwayverse are about causing the equilibrium to be as good as it is. Highwayversians are perfectly happy to pay manufacturers to produce food after they've already received and eaten the food, but also to pay the armies of people failing to improve textbooks or push science further. In fact such jobs are considered extremely important and prestigious, and the very best students are selected for them.


What happens when a Highwayversian switches from one universe to another?

Lets assume that when you switch you swap with any atoms that occupy the same space in the area you switch to. As such switching can be extremely dangerous - you could accidentally end up trapped in rock and unable to breath. Neither can you switch back, since you might not have moved in the few seconds before you switched, and now you would be switching back to a point in space you already occupied, probably splitting the version of you that's already there in half in the process.

You could of course always switch whilst running, but you'd still be switching blind and there are plenty of dangers that can kill you quickly, before you'd have a chance to react and switch back.

The obvious thing to do is to precommit that if you switch, you will check the area is safe, before switching a message back to the prior you saying that they're good to switch.

You never switch unless you receive such a message.

This isn't quite sufficient - the equilibrium solution here is that no-one ever switches because they never receive the "it's safe to switch" message, and so no-one ever writes the message.

So you also have to precommit that if you don't receive the message you'll send a probe across (e.g. a video camera which auto switches back after 1 minute) to check if it's actually safe, and if it is you'll switch, and write the message that you should have sent yourself, breaking the equilibrium in the process.

Of course it can be expected that the Highwayversians have technology that implements this (or a better) protocol seamlessly, and they don't really have to think about it at all.

Time Travel

Switching let's you travel through time in  either direction at the ordinary rate, which is fine, but a bit slow.

If you want to quickly travel to somewhere a billion years either way, the obvious solution is cyropreservation till that point in time. I would expect that this is cheap, safe, and simple in the Highwayverse.

What about communication?

If it's just a simple message this is straightforward - you just need to store the data in a medium where it can last billions of years without degrading. The Highwayversians have infrastructure to store such messages cheaply and efficiently and automatically read them and send them at the right point in time.

But what is the right point in time?

Presumably there's a target recipient, but that's not sufficient to identify someone in the Highwayverse. You also need to know when in their lifetime to send them this message. I would expect the Highwayversians have some way of identifying not just a person but a person at a particular point in time, by e.g. using the number of seconds since they were born as a timestamp.

Somebody who wants to be able to receive messages will constantly broadcast their location and timestamp. Those who want to communicate with them will send messages in both directions, saying to send this message to that person between timestamps A and B. Then the first time that a messaging station detects that person with a suitable timestamp it sends the message and deletes it. It might also send a message in the backwards direction telling you the message was received in this direction, and there's no need to send it the other way, avoiding anyone receiving the same message twice.

The recipient can then reply to the sender, and will send his current timestamp so the sender knows what timestamp to reply to.

By automating this process it's possible to to talk to somebody billions of years apart as if you were having a live phone call. Even better it's possible to pause things on your side without the other person having to wait at all, and you can speed up or slow down the other speaker whilst staying fully live.


Readers of Harry Potter and the methods of rationality will be aware that time travel lets you solve all NP problems in polynomial time. You receive a potential solution on a piece of paper, and run the computer to validate whether it is indeed the solution. If it is you send it backwards in time, if it isn't you send a different potential solution back in time. The equilibrium solution is the piece of paper contains the correct solution to the problem (at least theoretically).

This paper by Scott Aaronson uses a more complex technique to show that you can in fact solve all PSPACE problems (problems that can be solved by a classical computer in a polynomial amount of memory) in polynomial time.

The beginning and the end

Travelling forward in time to the end of the earth is relatively straightforward - you just cyropreserve yourself till you arrive.

The problem is switching once you get there. By this point the earths continents look completely different in the two universes, meaning there's nowhere safe to switch.

This could be overcome by launching yourself into space, but that's a relatively expensive solution. However since it's theoretically possible that's enough to bootstrap an equilibrium solution where specific points on the earths surface are leveled to be at exactly the same height, allowing safe and easy switching between the two universes, and allowing you to access the early earth.

The beginning and the end are still more difficult to get to and more inconvenient for switching, meaning that most people live close to the midpoint where switching is easy. The endpoints have lower tech levels and are less dense. Those who live/visit there do so for the lowered density, or may be researchers interested in the early/late earth (the researchers never discover anything new, but they're necessary to push the equilibrium further).

Going earlier than the beginning of the earth becomes way more expensive. (If they live on multiple planets, which seems likely, then going earlier than the beginning of the planet you live on). There is a certain earliest/latest point in time which anyone's ever visited. These are visited by super high tech research missions and are at the very limit of what their civilization is technologically capable of. They are presumably as close to the big bang/big crunch as it's possible for Highwayversians to survive at all.


An individual Highwayversian experiences a flow of time. They experience an individual point of time as the present. They are excited about things which will happen to them soon, but not about things which have already happened to them.

They do not however have a concept of the past and the present external to their individual flow. Time is just a direction in which you can move either way. There's no present and past, just timeleft and timeright. Similarly they don't identify other people as being at a particular time - the entire lifespan of the other person stretches out before them, and they are likely happy to communicate with them at any point along that lifespan - whenever's most convenient.

In our universe we struggle against the future, trying to make sure that our future is as pleasant as possible, but we ignore the past.

In the Highwayverse both past and future are known. Instead the main struggle of Highwayversians is against the equilibrium. They look for any point across their lifespan that could be optimized, and try their hardest to change it, knowing they will fail, but also knowing that it's only because they try that their lifespan is as good as it is.


A relationship between two people stretches across their entire lifespan, and not in any particular order. You'll speak to somebody you like whenever is most convenient for you to access them, and they may well be younger than the version you just had your previous conversation with. Somewhat like Doctor Who and River Song, but a thousand times more convoluted.

You already know which people you're going to have strong relationships with, but that doesn't mean you can't optimize it. 

The best protocol I can think of is whenever you meet someone interesting:

  1. If you receive a message from future you saying the relationship is worth investing in, and how much, invest that amount of effort into the relationship (with a minimum of a large enough effort that you would be able to guess whether it's worth investing more).
  2. If you receive a message from future you saying the relationship is not worth investing in, stay away from the relationship like the plague.
  3. If you receive no message, flip a coin to decide which to do.

If you do build a relationship then send a message back saying whether the relationship is worth investing in and how much.

It seems to me like the equilibrium is that with people you wont end up liking, the equilibrium solution is for the coin to always land tails, but for those you would end up liking it's 50/50 whether you'd end up investing in the relationship.

This hardly seems like the best possible protocol, so let me know if you can think of better!


It's of course perfectly possible for the Highwayversians to play our universe games like chess, but it would come extremely unnaturally to them. They're used to knowing both future and past, and the idea of deliberately not letting themselves check which move the other player's going to make before they make a move would seem extremely counter-intuitive. Perhaps there's a niche genre of games like that, but the majority of games are about forcing the equilibrium.

For example, Highwayversian chess would have both players fully aware of how the game plays, and even who wins and loses.

They then get a chance to find ways to improve any of their moves. They'll try their hardest to do so, but inevitably won't find any.

This might seem super weird to us, but I guess they find it fun.


I think I've tried to build up a picture for how the Highwayverse works at a super high level. I haven't gone into any details about what their society looks like because that's frankly unknowable, but I've tried to put across the meta rules that likely govern their lives.

I could carry on talking about this for days, but it gets quite repetitive. If however you do find it interesting, feel free to ask me to predict a particular aspect of Highwayverse society and I'd do my best.

Also feel free to write your own predictions, and of course to disagree with me.

If anyone can write a decent piece of short fiction set in the Highwayverse I will be incredibly impressed!

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13 comments, sorted by Click to highlight new comments since: Today at 10:33 AM

The exploration here is heavily dependent on the Equilibrium and I don't think it is a consequence of the numbered assumtions or a carry over of a similarity with the real world. It is common for physicist to assume the world timeline is stable as it makes it more manegable to imagine things but I don't think it is a result. Absent actually having time travel we don't have good way of testing this.

Note that if you take the world of Primer and consider the boxes to be in a separate (but connected throught the box opening) universse you get something that is fairly analogous to the lane setup. Primers world does not require stability and is quite different than envisioned here.

Symmetry -> Twoway traffic

Clarifying that symmetry is that all the detail being microscopically same in the central epoch and events being connected in classically mechanistc way we get that if we paced bioth timelines side by side they would be equal in all the details. A further consequence of this is that every time you switch your cosmic twin is going to switch to the place you just left. The cosmic twin is also going to have exactly the same psychological experience as you are going to have. What they are going to experience is the your past few moments just seen reversed. But this is also what you are just about to experience. Thus even if on an ontological level there is a distinction of the timelines, for a lot of practical purposes we can fuse them as a single lane that goes both ways. Then this starts to look a lot like the world of Tenet. Also even with the basics of switching twice it is really easy to make a gemini incident even without cosmic twins.

Unexperienced history

If the highwayversians are not the branches of the tree of evolution why would they be a sexual species? While there can be no chronal mechanics for equilbirium selection it doesn't mean the absence of achronal mechanisms. The idea of "pushing the equilbrium" could be turned that in a chain of unstable timelines pushing into unstable timelines if it ever (timeful word employed not in chronal sense) reaches a stable stable sense it will tend to lock there. Thus unstable timelines would not be impossible but merely ontologically transient (althought being percieving chronal time in the stable configuration could not have ever experienced them). This could be turned into a "achronal tree of life", highwayversians are sexual because the first species that caused an unstable timeline was sexual. In successive timelines the island of higwayversians became more separated from the same-timeline other chronal life. An equilbrium or stable tree of life will not push the bubbles further apart but will those that have grown going. But their states would not totally be uncorrelated, with larger differences there would be more changes needed and every change risks the transition becoming stable "too early".

Preservation of basis

In the RPG continuum the reason why the past is stable and relatively unharrased is that the Inheritors need the Hour of Inheritance to take place as it does for them to exist. For spanners having an unstable age is actively painful and lethal (frag). Being given the guaranteed gift of assured stable age would seem what would seem like a deathwish desperately sought for but never granted. I guess it does mean that a spanner has the option of healing frag while a highway versian is totally barred from making As/As nots. For highwayversians they might know that in order to exist and have bodies at all the central epoch needs to be connected to the start of the universe, so they might want to forbid or be secure in the knowledge that trying to mess up too much with the non-switching life fails. This might spring a profession / department of border security that intercepts movement away from the central epoch. For life that wishes to not switch this could mean that 3/4rds into the lifetime of the universe there are hunters that seek out to destroy life and matter living that way into the future. To parallel the other intellectual property call that socially set line as "point of entry" because it forms an event horizon that switching life can't push out of but matter makig up the bodies of the center life needs to be collected in. This line could be set to different depths in different parts. The size of inside region which is accesible to civilised switchers could be dependent on the amount of border patrolling work and having too little matter to circulate could make the world too interconnected (is there a difference betwen building a country out of 10000 atoms or 1000? How much matter is needed to hedge against radioactive decay from cosmic rays?). Maybe size of the habited area is a point of political contention for the switchers


Either a life occupies new time or it keeps turning into the same time. If the central epoch is more conductive to switching then it would seem that having the choice of moving inward or outward it would be prefererable to move inward (as outward life would mean you would live a less-switchy life). So the center is probably going to be real dense with switchers. All of the universe wants be in that tiny sphere of the central epoch which could mean that there would be natural or artifical selection on who gets to visit/utilise it. Like the result that if there was a wormhole providing time travel then a radiation could loop and pile up accumulating more and more to eventually collapse the wormhole . Like in Expanse there is a division between "inners" and "belters" there could be division of "inners" and "flatters" those who get to utilise it. Maybe if you are downthrodden it would be tempting to get the benefits of switching even if there were no guarantees of safety. Also all transmission lines of frozen people or messages also need to serve all of history at once. Solving that capacity issues could be serious if it is cheap to use for everyone (is 20 million cryo pods to cover 10 years of centralwards travel enough for everyone in that timeperiod (and all the pass throught traffic from the previous century and the previous througth traffic from the previous millenia))

Danger mitigation

If you had no preknowledge whether the switching location was safe, you would probably want to try to switch too high and then try to come down in some safe manner. Having a parachute and shooting yourself with a cannon high up could make for quite a lot of elevation difference. Or maybe it would be safer bet to first climb a mountain and then swich on the ground with parachute on? For ground known to experience changes a lot "transition stations" which by public service are designed to stay stationary could be useful from a civilization viewpoint. Having a structure that instead of laying on the ground is directly floating on the magma being able to float regardless of what the stone around it is doing. Or maybe having blips that either rely on athmosphere staying stable to maintain reliable altitude or active adjustment to actively keep at constant gps height. Maybe you want build a big incline that you can safely start rolling on from variety of heights. For actually getting the information I would think that such a boring solution of "historical" height maps would be in great demand.


Chronies use, "up" for global future, "down" for global past, "yet" for personal future and "age" for personal past. Using left and right analogs would seem to be confusing if spatial right and left are still used quite a lot. If highwayvesians do not care on which age of the target they talk to why would they target their messages to a specific timestamp. It is also somewhat trivial for a recipient to rememeber their younger selfs location and not have it automatically compromised to outsiders.


Why in universe would the higway traversian think that pushing the equibrium actually be a thing? They would seem to be like anti-inductors in doing research because it has not worked the previous times either! If there were allowed to be less than optimal people then I could suppose that there could be a statistical correlation between the luck of people that have "good character" so that people that counterfactually would have done something are well off and people that counterfactually would not have done something are not so well. However it also means that the biggest "equilibirum pushers" are almost never tested on their commitments and I could imagine some psychological hiccups on taking on herculean efforts after a lifetime of handdowns.

Interesting premise.

This universe definitely violates the second law of thermodynamics, and even our concept of probability seems like it would be something that isn't useful there. Everything is in a steady state of some acausal equilibrium that is strongly entangled with the entire universe's population of sentient minds.

Apparently if any sentient being wants the timeline changed, then that's not a stable equilibrium because essentially any of them can act on it. So it seems that the only equilibria are those in which essentially nobody ever wants it changed, at least not enough to take two flips to change it.

Initially this seems great, in that everyone gets what they want forever. However, that's not really the only type of equilibrium. One in which no sentient beings exist at all is another. A third is one in which plenty of people want things to be different but they can't flip, whether because they were born defective or through external constraints. A fourth is one in which they hate their life (or some events in it) but don't believe that they can do anything to change it. A fifth is one in which they are merely observers to things their bodies do and experience, outside of their control.

There are probably even more bizarre possibilities. We don't have any rule that tells us how the actual situation is determined from all the possible equilibria, and the rules that we have developed for investigating likelihood in our apparently one-way causal universe may have no applicability to one governed by acausal equilibrium.

That's definitely all true, but I found it more fun to go for the optimistic equilibrium!

  1. Surely there would be some contrarians who try Messing With Time (perhaps they precommit to doing something iff they learn that they didn't do it). What happens to them?

  2. Suppose someone learns how they die (say they confirm it with high accuracy with multiple self-messages and even going to the location to observe the event themselves). Wouldn't this cause them to feel comfortable taking extreme risks at other places/times since they know they'll survive?

  3. It seems like the equilibrium would push very hard for a very high population density where most people's parents wanted lots of children. You mentioned isolated self-parenting pairs. What regulates how many of these exist?

  4. I would expect that a significant fraction of the futile/necessary optimization efforts would be adversarial - analogous to your chess example but on a larger scale with higher stakes, such as political outcomes or group conflicts, with each side trying to out-terminator each other.

I think the first question is the most important.

To me the simplest solution is that such people simply don't exist. There are possible timelines in which they don't exist, and timelines that have them seem likely to be unstable, so the actual timeline will be one of the stable ones in which they don't exist.

There are other possible solutions of course, but "increasingly bizarre events occur in which their desires are thwarted" seems far more convoluted than "one of the other millions of sperm met the egg instead and that person never existed" or even a more general "this species' brains develop in such a way that they don't have such ideas".

Of course, in general this leads to the much more stable state in which although (as per premise) every sentient being potentially has the capability to flip universes, in practice none of them know that it's possible and even if they do then they don't know how to actually do it.

There are other possible solutions of course, but "increasingly bizarre events occur in which their desires are thwarted" seems far more convoluted than "one of the other millions of sperm met the egg instead and that person never existed" or even a more general "this species' brains develop in such a way that they don't have such ideas".

If we assume the universe is deterministic, that's not really an option. Which sperm entered the egg is fixed and cannot be changed. Instead to find an equilibrium something about the future has to effect the past in such a way as to make itself consistent rather than inconsistent.

I suspect that I don't know what you mean by "deterministic" here, since the meaning I have in mind can't possibly apply to such a universe. That is, that future states are completely determined by prior states. That can't possibly apply here since the universe has no global distinction between future and past. Even splitting our view into timelines within each semiverse doesn't help, since determinism is violated by the sudden appearance of sentient beings and other materials that are not in any way determined by that semiverse's prior states. So you must be using some other meaning for "deterministic".

Perhaps you just mean that the universe timeline is single-valued? That is, only one set of events actually happens at each point in spacetime in each semiverse. That is also the model I'm using, but from a different point of view. Rather than taking that a contrary person exists as a fixed event that must be worked around, I am taking a wider view of what fraction (in some sense) of possible timelines that are otherwise very similar contain that contrary person vs those that don't. Since the actual timeline has to be one of the possible timelines, this seems to be a useful consideration.

My conclusion is that contrary people drastically lower the measure of mostly-similar timelines that contain them, and so over trillions of sentient beings it seems likely that the proportion of contrary people is much more likely to be very, very small than that they are relatively frequent and cause lots of bizarre events.

My definition of deterministic is that taking one universe as fixed, there's only one possible value for the other universe.

So if there's nothing from the other universe that switches over and effects which sperm fertilizes which egg, then that is something which can't change just because it will lead to a contradiction.

Thanks, that explains why I had no idea what you meant by deterministic. It's not a meaning for the term that I would have guessed. I obviously wasn't assuming that the universe is deterministic in that sense.

It does open up more questions, and so is interesting. Let us use the function notation D2(x) to refer to the deterministically single allowed timeline of semiverse 2 given that semiverse 1 has timeline x, and similarly for D1. Such a universe is only possible if D1 and D2 satisfy certain conditions, in particular that there exists at least one pair (x,y) such that D2(x) = y and D1(y) = x.

We can eliminate the case where D1 or D2 are constant, since those correspond to causally isolated or one-way semiverses and are therefore boring.

For almost all other function pairs, almost all timelines x in one semiverse have no corresponding timeline y since in general D1(D2(x)) != x. This places drastic limitations either on what single semiverse timelines are allowed in ways that are utterly foreign to conventional causality or even continuity, or on what sorts of functions D1 and D2 are allowed. Almost all ordinary deterministic laws of physics will fail this condition.

So for this notion of determinism to be sustained, we have to consider universes in which even within a single semiverse with no flipping, the laws of physics are utterly different from our own.

That's a good way of framing it! There's some discussion in acedemia on this topic under Novikov self-consistency principle (not quite the same, since it's only in one universe, but pretty similiar, and I wouldn't be surprised if results carry over).

Note that I do actually break this rule in my protocol which involves flipping a coin to decide whether to build a relationship. However this can be fixed by flipping an "equilibrium coin". Effectively this is a device which has exactly two self consistent equilibria. For example you might have a device where either an item wont be switched at all, or it will be switched at exactly the right time to cause itself to switch in the first place. Flipping counts as heads, not flipping as tails.

Surely there would be some contrarians who try Messing With Time (perhaps they precommit to doing something iff they learn that they didn't do it). What happens to them?

What do you think happens to them?

More seriously, this is similar to a question I asked on physics stack exchange. The basic answer is there's lots of ways the universe can be self consistent, only some of which will correspond to what you want to achieve. The best you can do is make the subset of consistent outcomes that match your needs as large as possible.

If you try your hardest to make the universe inconsistent, don't be surprised if you end up killing yourself in a crazy freak accident where future you switches into you in exactly such a way as to cause you to rapidly switch in quick succession tearing apart half of past you in the process.

Suppose someone learns how they die (say they confirm it with high accuracy with multiple self-messages and even going to the location to observe the event themselves). Wouldn't this cause them to feel comfortable taking extreme risks at other places/times since they know they'll survive?

Someone who tries their hardest to avoid the death they know actually happens to them, but takes crazy risks at all other times is doing something similar to the person who's messing with time. They're making the obvious set of consistent outcomes much smaller, so who knows what kind of outcome will end up occurring.

It seems like the equilibrium would push very hard for a very high population density where most people's parents wanted lots of children.

I'm not sure why this is the case?

 You mentioned isolated self-parenting pairs. What regulates how many of these exist?

To be honest I'm doubtful these would actually exist but they're definitely a theoretical possibility.

One way of achieving them might be to precommit that you will kill your parents before you're born iff they are not you. Then one obvious equilibrium would be to have them be you. That might work but I honestly have no idea.

I would expect that a significant fraction of the futile/necessary optimization efforts would be adversarial - analogous to your chess example but on a larger scale with higher stakes, such as political outcomes or group conflicts, with each side trying to out-terminator each other.

The Highwayversians are good enough at decision theory that they might not do this, or will only do so in ways that change the point along the pareto optimal surface. However if they do there's no way in hell that anyone in this universe is going to even begin to understand the battles they're fighting!

It seems like the equilibrium would push very hard for a very high population density where most people's parents wanted lots of children.

I'm not sure why this is the case?

The average lifetime number of children would have to be exactly replacement at equilibrium, and if people are generally long-lived, this would imply an extremely low number of children per year. This seems implausible since if even a small number of people were willing/able to sustain a high fertility rate, most other people would have to have no children to keep the average low, and the few high-achievers would be the parents of most of the population.

That sounds reasonable, yes.