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In the setting for a story I'm planning, superpowers of a great many varieties can be gained by earning them, as judged by an impartial magical force that is effectively part of the setting's physics. Effort and achievements can help earn a power, but the most important factors are usually creativity, knowledge, and understanding, especially for the stronger and more esoteric powers.

These superpowers can be improved after they are earned, and creativity and understanding are even more important for improving a power than for acquiring it in the first place. Practice and experience with a superpower can strengthen it, but improvements gained only from that are usually minor. Really significant improvements come from improving your understanding of how your power works, from using your power in new ways, and most especially from creatively applying your power in a non-obvious way to achieve a result that is beyond what obvious and straightforward use of your power can do. If you manage to pull off a roundabout convoluted trick with a weird combination of unorthodox applications of your superpower, that "convoluted trick" might suddenly become a standard part of your power that is simple and straightforward to repeat in the future, and the strength of all of your power's uses will also increase. You might also use that new part of your power later as a building block for another convoluted trick to develop your power even further.

The main character will have a superpower based on his understanding of the Novikov self-consistency principle. He can send information back in time to himself, and whenever he receives information from his future self, it always works out such that when he reaches that time in the future he voluntarily sends exactly the same information to his past self, so that the two ends of the message's time travel are consistent with each other. Conversely, whenever he chooses to send information back in time, it is always consistent that he received that exact information earlier.

Early in the story, when he first gains this power, it will have some major limitations. By the time the story reaches its eventual climax near the end, he will have developed this power to remove or bypass most or all of those limitations, making it incredibly powerful, possibly even the most potent superpower in the setting.

The part I'm having difficulty with is coming up with explanations for how he achieves the developments needed to reach that ultimate height of the power's capabilities. The importance of creativity and understanding in the mechanics of such developments is a core part of what makes the story concept interesting to me, and developments in the main character's special power are too central to the story to easily gloss over or handwave.

The power's initial state:

  • It can only send information several seconds into the past.
  • If anything is about to happen within that time limit that he would want to have been warned about, he receives a warning before it happens. This might not apply for events that will kill or incapacitate him faster than he can react.
  • He can plan in advance to use it, attempting to do something and then sending back advice on how to do it better. He receives (and later sends) advice so good that he can't think of any better advice to give his past self.
  • Receiving advice from his future self does not give any special improvement in his ability to follow that advice.
  • He only receives information good enough that he can't come up with any definite improvement of that information to send back instead. He will not receive advice that is beyond his ability to devise in hindsight, nor advice that he would only ever send after having successfully followed it.

An example improvement:

  • If he receives information from the future that would be better to have known even farther in the past, he would of course react to that by immediately sending that same information back in time again.
  • With that idea in mind, he can send information back in time that's intended for farther back than his power can actually send it, relying on his past self to repeat the signal to send it farther back.
  • By using his past selves as a series of signal repeaters in this way, he can bootstrap into sending information many times farther into the past.
  • After he realizes this and intentionally makes use of it, his power will develop to send information much farther into the past, directly in a single step.

The power's ultimate state:

  • The idea for this power is inspired by the character Contessa from Worm, and in particular her power, called Path to Victory. This is one of the most potent superpowers I have ever read about, short of literal omnipotence. It is so powerful and so versatile that Contessa once said, as a summarized description of her power, "I win."
  • It can operate over time spans of at least a few years, and possibly arbitrarily long.
  • When used to plan a course of action, the power gives literally the best possible plan; the global optimum.
  • Even if the best possible way to achieve the goal involves actions that he would never even think of in a million years, the power still produces that best possible plan.
  • If the goal is to gain certain information, the power may simply produce the desired information. Information gained in this way is reliably accurate, even if the character never verifies it.
  • The only limit on what goals the power can produce plans for is that the chosen goal must be physically possible for the character to achieve.
  • He can follow advice and plans received through this power perfectly, with absolute precision. If a plan from this power involves ricocheting a bullet off of a garbage can to hit someone in the eye around a corner, in a trick shot that would be one-in-a-million even for the best sharpshooter in the world, the power's plan will include the exact position, angle, and timing of the shot, and he can reliably follow that to successfully perform the absurd trick shot.
  • The plans this power produces can rely on that perfect precision, finding courses of action to achieve otherwise impossible things in ways that would normally be doomed to fail.

What creative tricks, and what expansions of understanding, could I reasonably use to explain to the readers the various steps on the way from that initial state to the climactic final state of what this superpower can do?

The specific aspects of it that I'm having the hardest time with are:

  • How to justify having the power not require eventual verification of information it produces, and yet still be reliably correct.
  • How to justify having the power provide advice/plans that the character would never think of, even in hindsight.
  • How to make the leap from merely "the best the character would think of with hindsight" to "the actual best possible."

I think that explaining these developments in the power satisfactorily will likely require breaking each one into a series of developments that build on each other, but I don't have any ideas yet for a good breakdown.

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  • $\begingroup$ "it always works out such that when he reaches that time in the future he voluntarily sends exactly the same information to his past self, so that the two ends of the message's time travel are consistent with each other" Be careful about interlacing plot rules with handwavium. A clever character would realize that after receiving a message from e.g. 5 years into the future, that they therefore cannot die between now and that point in time. When you're dealing with undefined creativity, which this question clearly is, you're going to have to actually define the boundaries and edge cases here. $\endgroup$
    – Flater
    Commented Apr 16 at 3:47
  • $\begingroup$ Could totally die. The point of sending info to the past is to change it. Which means you're trying to get yourself to do something you've never done before. Which means if you want to know the results of an action 5 years down the line you're going to have to burn at least 10 years of your life over it. 5 to get to where you send the info back. 5 to wait to see what the results are of what you changed. Remember, when you change your past, it's not your past anymore. $\endgroup$ Commented Apr 16 at 17:11
  • $\begingroup$ "When used to plan a course of action, the power gives literally the best possible plan; the global optimum." Other than making sure the plan succeeds in achieving specified goals, this is inherently "paradoxical": Imagine a character with this power thinking "I want only ice cream every day for the rest of my life". After a few months of this, his future version will be perfectly willing to feed his past self wrong information to avert this fate. The "out" would be 100% fixed views/values, and this seems to be decidedly non-optimal - hence the power would actively prevent global optimality. $\endgroup$
    – sh4dow
    Commented Apr 19 at 7:24

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Yeah... I've already worked this one out for my own story.

If you can send information back in time, that means that you can send information about the consequences of your own actions.

Let's say you're trying to do the million-to-one trick shot. It's going to depend upon a whole lot of factors, so you point your gun in more or less the right direction, and just sort of... wave it around. Sooner or later, when the shot lines up just right, you receive the instruction from the future to fire... Bang! ping! spang! whop! 'Gotcha!'

Sooner or later, the power will develop to the point where you can effectively look into the future and see the consequences of acting as you intend. Sometimes it won't work out, and you have to come up with something else, the rest of the time, you can do the things that will improve your probabilities, and wait for just the right moment to act.

Of course, it takes about as long to look into the future as the interval into the future over which you want to look, so the more time you have, the further you can look and the better you can optimise your solution.

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    $\begingroup$ Probably the easiest way to use such power is to not try to perfect plans, but dozens of mutually redundant plans. Rather than try for a million-to-one trick shot to kill someone, use 50 bullets at a close, not-tricky distance, an hour AFTER you already poisoned your target, and a second before a bomb under their feet goes off. Even if the target somehow survives, and is sent to a hospital, you already bribed their doctor to inject them with the Ebola virus, and someone else to place a morrocan death scorpion under their pillow. $\endgroup$ Commented Apr 15 at 9:02
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You created a character that would have to be ab Omniscient, Omnipotent God to do those things, AND your universe would have to be completely preordained.

Some problems:

  • the universe is inherently chaotic, due to the practical randomness of particle physics, and ontological randomness of quantum physics. Therefore no tricks based on perfect information are possible. For example, sending back information about the exact parameters of a trick sniper shot need to eliminate a target is pretty pointless, since things like Brownian motion in the air, chaotic oxygenation in the explosive load that pushes the bullet, and the nature of the light that hits your sniper scope are random, impossible to predict, and impossible to control.

  • as far as we can say from available data, human cognition also involves quantum effects in the electrical synapses. Therefore, it is impossible to predict another human's thinking, or even your own. Any information sent back in time would snowball changes in human cognition that are inherently impossible to predict

  • if there are any other prescient/time traveling characters in the world, you just created impossible paradoxes, or split timelines (which kinda defeats the purpose of the character's power).

  • Finally, on meta-level, a story where all characters are completely deterministic, and there is not even an illusion of free will are not particularily entertaining. If the Character can send perfect information about other people back in time, then this means other people are just automatons, not "people" in any emotionally compelling sense. This is, at best, a Video-Game like universe with one player and billions of NPCs.

How to get around that? Easily, do not focus on precision, focus on redundancy. Your character does not need to try to come up with perfect plans, just enough of redundant, pragmatic, and strategically robust plans that the chance of total failure is close to zero. For example, a plan to assassinate someone should not involve calculating the absolute perfect way to ricochet a bullet, but rather 27 different mini-plans that cover all likely options, from different guns, different sniper positions, the use of explosives, poison, paid traitors, arson, etc etc. Even if the assassination target changes their behavior etc, there is still 26 other ways to kill them, all already prepared. You cannot kill someone with a trick ricochet shot at a 1 km distance, but if you shoot them at 30 meters distance with a full clip from a semiautomatic, right after poisoning them, and a second before their car explodes, AND a hijacked dump-truck slams into them at full speed - the target is going to be as dead as anyone can be.

So consider, that if you focus on extreme redundancy, not on precision of the plan/information, all your stated problems disappear:

How to justify having the power not require eventual verification of information it produces, and yet still be reliably correct.

No need to verify it as long as each option in the plan is at least plausible. If Action A gives you only about 13% chances of success, it does not matter, since you are going to to everything from Action A to Z anyway, so SOMETHING is likely to work

How to justify having the power provide advice/plans that the character would never think of, even in hindsight.

The ability to send information back in time means you can ask other people to devise plans for you, you can toss a coin or a dice over particular choices, or even use AI to assist you in coming up with "out of the box" solutions.

How to make the leap from merely "the best the character would think of with hindsight" to "the actual best possible."

Again, no need to at some level of redundancy, their plans simply become impossible to foil due to their sheer volume and the fact that foiling Plan A simply activates Plan B, etc, etc until something works.

The actual best plan possible, is most often something completely mundane but prepared so thoroughly that there is no realistic option for another person to counter it, even if they knew in advance. Think less "Path of Victory" and more "Campaign of Victory". Rather than finding one golden path to winning, it is more about following so many paths to victory at the same time that it is impossible to lose. Your Character would thus be even more powerful than Contessa, since even Contessa could not beat plans that left no options for her to pursue. Your Character could beat someone with Path to Victory by thoroughly booby-trapping every Path that exists.

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    $\begingroup$ Counterexample to your final bulletpoint: Groundhog Day (or any other story based around a time loop that includes the trope of the looping person learning how to perfect their interactions with particular people). Myself at least, I still care about Angie McDowell in that movie just as much as I care about Bill Murray. $\endgroup$
    – Idran
    Commented Apr 15 at 21:07
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    $\begingroup$ As to the last bullet point: two of the biggest box office hits ever (Marvel Avengers: Infinity War and the subsequent Endgame movies) rely on precisely that notion by having Dr Strange evaluate 14,000,605 possible timelines and being able to reproduce his actions in a cherrypicked one of them to achieve the previously evaluated outcome. It is also an incredibly common theme in nearly all time travel narratives that undoing an alteration to the timeline restores the original events; again suggesting that the same arrangement leads to the same conclusion. $\endgroup$
    – Flater
    Commented Apr 16 at 3:55
  • $\begingroup$ Overall, this answer's bullet points come across as one interpretation of a timeline gimmick, while at the same time presenting it as if it were the better interpretation. There's inconsistency in first dismissing other interpretations by harping on about quantum randomness leading to unpredictable outcomes (without quantifying any scale to this randomness), only to then continue with an answer that relies on knowing that your multiple redundancies will reliably work the way you foresee them to. The same caveat applies to this answer as it tries to apply to every other possible answer. $\endgroup$
    – Flater
    Commented Apr 16 at 4:01
  • $\begingroup$ @Flater my assumption is that OP was working on world-building for a textual form of fiction, which is significantly less forgiving than movies. Avengers and Groundhog Day can afford to be built on plot holes and contrived coincidences, because they need to pack the entire plot in about 2 hours, so mistakes are forgiven. A piece of written fiction is far easier for the readers/players/whoever to nitpick. Multiple redundancies do not work reliably, hence why they are multiple and redundant in the first place. High redundancy of both plans and GOALS means something will work eventually. $\endgroup$ Commented Apr 16 at 10:45
  • $\begingroup$ "Avengers and Groundhog Day can afford to be built on plot holes" is just further proof of that dismissal of anything that doesn't fit your interpretation of time travel concepts in fiction. You're genuinely arguing that a different interpretation is inherently a plot hole because it doesn't subscribe to your opinion on determinism and quantum randomness. Since time travel is not a thing, it is patently unspecified how seeming randomness behaves when a timeline is repeated. I'm not saying your interpretation is wrong, I'm saying it's wrong to pretend it's the only one that can be right. $\endgroup$
    – Flater
    Commented Apr 16 at 23:37
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The main character is wrong. While he always experiences reality consistent with the Novikov self-consistency principle, what actually happens is that any universe where he fails to send back the same information is pruned (destroyed) and reset.

So the main character believes that the message going back is pre-ordained, as this is what every message he has ever gotten acts like. If he ever chose to experiment with not sending the message back, all that happened was his reality was erased, and a new reality where no message arrived is created, and whatever happens in that reality plays out.

This power is insanely stronger than the one you describe, because you can pull off some crazy stunts with it, so long as you are ok with being a pantovore.

As an information theoretic example, suppose you have a problem in NP. These are problems whose answers can be verified relatively cheaply, but no known ways to find the answer exists.

You attempt to solve the problem by picking one random branch of possible answers, and if it fails you send a message back to try the next branch. If you are right, you send a message back to try the branch that worked.

All of the universes where you picked wrong are eaten by your power. The remaining universe contains the one with the branch that works. To your perspective, you came up with this plan, got sent a message with the correct answer, you verified the answer and sent the correct answer back in time.

Note that if you fail to verify the answer and send it back in time, the answer may not be correct. But if failure to verify the answer is because you sent a message from further in the future telling you not to bother and you trust your future self, you can do this as well.

So:

List all of the possible answers (0 through N).
If you don't get a message, try answer 0.  Otherwise, try the answer you are told to try.  Call this K.
If you are told to not bother checking, just trust K is right.  Send back "don't bother checking".  Exit.

If your check of the answer fails, send back K+1.  Universe destroyed, as this isn't consistent.
If your check of the answer succeeds, send back K.
Now send back "don't bother checking".  Universe destroyed, as this isn't consistent.

This meta algorithm lets us find the answer to any question we can check without having to check it, so long as we can survive the checking process.

Things get weird when you start doing this to an extreme degree. Like, imagine walking up to a bridge and tapping it. If it fails to implode, you send back a message to tap it at a different spot.

Under modern understanding of physics, there is a non-zero chance that the bridge will spontaneously implode even without you tapping it. And if you keep on randomly rolling the dice of the universe, eventually it happens.

Your "send back a message to tap the bridge" is just a way to reset reality, not actually important here.

But while it is true that the bridge randomly imploding is an eventual result, so is your random implosion.

In short, you can use your this power to cause infinitesimally unlikely events to occur by eating a near infinite number of universes, but this is limited by the fact that if it is more likely that some other event occurs and you get the message consistent with your past, that other event happens instead.

For example, you suffer a stroke and randomly send back the same tapping location might be far, far, far more likely than the bridge spontaneously imploding.

So, once you set up the process for an infinitesimal event to occur, the most likely infinitesimal event consistent with your power happens. And that is very unlikely to be one that you desire.

The next question is the ability to bootstrap messages out of nothing. You have noticed the ability to relay messages from the deep future: but you can also orphan such messages.

Imagine getting a message that says it is from the deep future saying to do X. The good result of doing X is apparent in one time step. When you expect to get a message from the deep future saying "relay back to do X", you don't get it. And you send back "a message from the deep future, do X" anyhow.

Pulling this off is similar to the infinitesimal event generator, as any the failure of the "deep future do X" chain would have caused realities where "do X" is sent in turn to collapse. Except by using infinitesimal event generation, together with sometimes sending random messages to the past (and only repeating them if the random message is useful) you can spontaneously generate useful messages.

None of these random messages that aren't sent are ever sent or received in the final narrative. But understanding the power and setting up plans is enough to allow uncaused cause messages to be generated.

You can even set up a planned trajectory towards victory, and plan to randomize and reset time until you are along the path. The randomized messages can be in the form of seemingly useful advice, or random noise; what matters is resetting the dice.

The character may or may not come to the horrible realization of what the power is actually doing. Understanding the raw scale of the destruction caused by the power may eventually lead to the character stopping and no longer being willing to use it.

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  • $\begingroup$ Instead of destruction, the universes may never happen in the first place because the time loop is not stable. Either way, you can use this to break RSA and solve the halting problem, so seems like a good compromise to me. $\endgroup$
    – Seggan
    Commented Apr 15 at 18:40
  • $\begingroup$ @Seggan While the destroyed time loops "do not exist", their impact on the events that do exist are measurable, in that they change the distribution of messages that form closed time loops (and are hence observable). For example, "The virtual time loops exist" is a message you can send yourself from such a loop, and you can resend it to yourself having already received it. In fact, whenever you set up a reliable message to send that message if you don't get it, then flip a coin to see if you send it if you do get it, the coin always lands "send it again". $\endgroup$
    – Yakk
    Commented Apr 15 at 20:46
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    $\begingroup$ @Seggan This setup doesn't let you solve the halting problem, because it still requires that an algorithm has reasonable bounds on memory required to evaluate an individual calculation step. I can't find the reference now, but I remember seeing a paper on the arXiv once analyzing computational complexity + retrocausal loops, and from what I remember adding in time loops only lets you collapse the hierarchy up to PSPACE because of that issue. Having an infinite amount of time doesn't help if you need 10^100 bits to complete a single step of computation. $\endgroup$
    – Idran
    Commented Apr 15 at 21:19
  • $\begingroup$ @Seggan Plus even if this did work for true hypercomputation, that would just bump up the Halting Problem one stage; it's already a proven result in Computation Theory that adding a Halting Problem oracle doesn't allow you to evaluate all functions. The set of functions computable with a Halting Problem oracle isn't the set of all functions, it's just a larger set than the set of computable functions; there's an infinite chain of computability in that sense. See cs.stackexchange.com/questions/96794/… $\endgroup$
    – Idran
    Commented Apr 15 at 21:25
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    $\begingroup$ @Idran ah oops, i see. at least RSA is still breakable :P $\endgroup$
    – Seggan
    Commented Apr 15 at 22:14
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The main character's success becomes self-causing

Meet Mr. Winmore. Mr. Winmore has your character's power, at its maximum potential, and is also God King of the entire world. He has a host of loyal advisors (disloyal advisors don't last long against a precog), access to all of human knowledge, and can direct colossal R&D budgets toward anything that strikes his fancy.

Mr. Winmore took over the world when he was 28. Ever since then, he's had only one project: To change the past such that he took over the world when he was 27 instead. The problem is, his best plans for world domination take 10 years to execute, and his 17-year-old self hadn't yet figured out how to send a message more than a few seconds into the past (along with a host of other limitations, described in the Question).

Small Victories Compound

Anyone with your character's power will eventually become Mr. Winmore, unless they don't want to. Even if he starts from a single penny, he can precog the results of sporting events and gamble on them to build a small fortune, then use that small fortune to research his past and find opportunities that he missed, to tell his past self about them.

The next iteration will be stronger. The next iteration will be stronger than that. And so on, until you get Mr. Winmore.

The Journey

Mr. Winmore will be trying to guide your protagonist from the start. Mr. Winmore will be using whatever means he has to push them into the realizations that will cause them to break their power's limits sooner, etc.

We'll start with autohypnosis. Mr. Winmore is an expert in it (because it was worth learning). Your character has weird prophetic dreams, because Mr. Winmore is able to get hypnotize him to break his power's limits in his sleep even before he's learned to do it while he's awake. He dismisses them (along with that persistent sense of deja-vu) as "just dreams," or a weird side-effect of his power.

However, those dreams have been teaching him valuable skills - so maybe he gets into a fistfight and discovers that somehow, his body already moves like a trained martial artist. They've also been coaxing him into situations where he'll break the limits of his power in the waking world.

The moment of Apotheosis

Eventually, your character would become aware of Mr. Winmore's existence. He now knows what his power is capable of, even if he doesn't know how - that's half the battle. After that, the realistic-but-boring option would be that he starts keeping a dream journal so that he can better absorb what he's learning in his sleep. The more narratively exciting option would be that he suddenly makes contact with Mr. Winmore in a desperate situation - maybe entering a time out of time where Mr. Winmore drags him into something resembling a two-way conversation, and gives him a well-thought-out plan for how to escape his predicament.

How it works

  • It can operate over time spans of at least a few years, and possibly arbitrarily long.

Mr. Winmore sends a message to his past self, who sends a message to his past self, and so on. By using autohypnosis, Mr. Winmore can transmit his skills to his past self in dreams, and completely hijack his past selves' abilities while they're asleep to use them as unthinking relays.

  • When used to plan a course of action, the power gives literally the best possible plan; the global optimum.

If Mr. Winmore ever, at any point in the past, escaped from a bad situation with a "just barely good enough plan," he then spends the rest of his life and all of his eventual resources studying the circumstances of that situation, and formulating a better plan. He then sends that plan to his past self. It won't always be the global optimum, but it would certainly seem like it.

  • Even if the best possible way to achieve the goal involves actions that he would never even think of in a million years, the power still produces that best possible plan.

Mr. Winmore has a cabinet of loyal advisors, a lifetime of experience and knowledge, and years of time to consider every possible plan, which would expand the search space.

  • If the goal is to gain certain information, the power may simply produce the desired information. Information gained in this way is reliably accurate, even if the character never verifies it.

Mr. Winmore can send scouting directives back in time, by giving his past-self instructions like "Surrender, allow yourself to get captured, then send back the names of the people guarding your cell."

  • The only limit on what goals the power can produce plans for is that the chosen goal must be physically possible for the character to achieve.

Technically, we don't have this, but we have a close-enough approximation. We only get the best possible plan that Mr. Winmore is capable of coming up with, given a lifetime of effort.

  • He can follow advice and plans received through this power perfectly, with absolute precision. If a plan from this power involves ricocheting a bullet off of a garbage can to hit someone in the eye around a corner, in a trick shot that would be one-in-a-million even for the best sharpshooter in the world, the power's plan will include the exact position, angle, and timing of the shot, and he can reliably follow that to successfully perform the absurd trick shot.

"Just try it. This is your 1,722,486th attempt. If it doesn't work, add one to this number and send it back in time. Different inputs result in different outputs. You'll get it eventually." - Mr. Winmore

  • The plans this power produces can rely on that perfect precision, finding courses of action to achieve otherwise impossible things in ways that would normally be doomed to fail.

This is a combination of all of the above. Due to Mr. Winmore's scheming, your main character is constantly getting into situation where things they did before just happen to be relevant. For example, they help an old man cross the street, and three months later they're imprisoned and the prison guard "just happens" to be that old man's grandson.

Narrative Bonuses

The Shadow of Mr. Winmore also implies a couple more phenomena, which I think would make it easier to write an interesting story.

First, the protagonist (who experiences his power and its limits in first-person perspective) would be much less impressed by it than the people around him. To the protagonist, the "coincidences" created by Mr. Winmore's scheming look like coincidences - his power grants him very-limited precognition, and he knows what it feels like to use it, so everything else must just be luck. Furthermore, because he's been experiencing these sorts of one-in-a-million coincidences his entire life, he doesn't find them nearly as weird as his companions.

To his companions, though, his levels of good fortune cannot possibly be explained by coincidence. His scientifically minded friends might believe that he's hiding the extent of his powers, while the less scientific ones might think he's the beneficiary of some supernatural force like Fate or God. (Both of them are half right - and the only person who's wrong is the future Mr. Winmore himself, who thinks its all just luck.)

Second, it lets you foreshadow his abilities before he masters them. For example, suppose that at some point, the protagonists enemies attempt to drug him with a poison that suppresses higher brain functions. However, instead of incapacitating him, the drug allows Mr. Winmore to hijack his body through autohypnosis. In his drugged stupor, he demonstrates a bunch of powers he doesn't have yet. After he recovers, he remembers none of it. (The drug would have to have some long-term side effects that Mr. Winmore doesn't want if taken in excess, though, otherwise it would become a "power-up pill" that might not be good for the story.)

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Here's a really cool trick that's possible with Novikov style time travel, and it can easily be adapted to your case, where you only send information back in time. Gonna be entilrely honest, the only place I've ever seen it described is in this Harry Potter fanfic (of all things) so you might want to check that out: https://hpmor.com/chapter/17

I'm not gonna be able to explain exactly how it works, so you'll have to read that chapter. But basically, it's an algorithm to break RSA instantly. You do this by creating a time loop who's only stable state is sending yourself the solution to your problem.

When your character realizes they're capable of this, they'll start getting better and better at creating these situations where the only time-loop-stable information they can send themselves is the "optimal plan" for whatever they want to do.

EDIT Ok, you win, here is a condenced version of the chapter:

Have a friend pick two 3 digit prime numbers, mutiply them together and tell you the result, let's call it X. Isolate yourself in a room with a pen and paper and shut your eyes. Around thirty seconds later, you'll heard the sound of steps coming in and then out of the room. Open your eyes, you'll see a folded piece of paper left on the floor, a gift from your future self. Call that piece of paper "Paper-2". Call the paper you came in the room with "Paper-1". These are, of course, the same piece of paper.

Reviewed in your mind the algorithm that you will follow:

  • If you open up Paper-2 and it is blank, then write "101 x 101" down on Paper-1, fold it up, go outside and back in time, drop off Paper-1 (which would thereby become Paper-2) in the room while past you has their eyes closed, and head on out.
  • If you open up Paper-2 and it had two numbers written on it, multiply those numbers together.
  • If their product equals X, write down those two numbers on Paper-1 and send Paper-1 back in time. Otherwise, add 2 to the number on the right and write down the new pair of numbers on Paper-1. Unless that made the number on the right greater than 997, in which case add 2 to the number on the left and write down 101 on the right.
  • And if Paper-2 sais 997 x 997, leave Paper-1 blank. Which means that the only possible stable time loop is the one in which Paper-2 contains the two prime factors of X.

If this worked, you can use it to recover any sort of answer that is easy to check but hard to find. You can use it to find the combinations on combination locks, or passwords of every sort.

That is, of course, until you take Paper-2 in your trembling hands, unfolded it, and it reads: DO NOT MESS WITH TIME

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    $\begingroup$ Yep, that ended pretty much as I remembered... Harry receives a message saying, "DO NOT MESS WITH TIME". $\endgroup$
    – Monty Wild
    Commented Apr 15 at 15:52
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How would you deal with the obvious case of "Endgame Character" not sending all his knowledge to "Starting Character" and "Starting Character" being just as powerful as "Endgame Character"? (I assume he can only send information as far back to the point of acquiring their powers) Does this character never suffer any kind of loss? Or is it not relevant enough to send the information of how to avoid that loss to the past?

After he realizes this and intentionally makes use of it, his power will develop to send information much farther into the past, directly in a single step.

Doesn't this mean they instantly can send as far back as they wish? Or how do you explain finding out you can chain sending a message 10 minutes back + 10 minutes back + 10 minutes back, the power evolves to send it 30 minutes back in a single step, and not going:

  • 30 min + 30 min + 30 min + 30 min = 2 hours
  • 2 hours + 2 hours + 2 hours + 2 hours + 2 hours = 10 hours
  • 10 hours + 10 hours + 10 hours + 10 hours + 10 hours = 50 hours

Where does it stop evolving and why wouldn't the character just realize this goes on infinitely? If he has to intentionally makes use of it for the power to evolve it's probably a question of how many of these loops he makes (Is there a limit to power usage (Like mana, energy, headaches...)).

Say I'm the character, and I send the message "We can overcome the time limit by re-sending a message to the past as soon as we receive it from the future, then the power evolves to be able to send messages just as far back without chaining. Once it´s evolved rinse and repeat to evolve it again" to my past self and past self sends it back to past past self

Whose power evolves? Mine? past self? past past self? All of them?

Even if the best possible way to achieve the goal involves actions that he would never even think of in a million years, the power still produces that best possible plan.

If the goal is to gain certain information, the power may simply produce the desired information. Information gained in this way is reliably accurate, even if the character never verifies it.

The only limit on what goals the power can produce plans for is that the chosen goal must be physically possible for the character to achieve.

He can follow advice and plans received through this power perfectly, with absolute precision. If a plan from this power involves ricocheting a bullet off of a garbage can to hit someone in the eye around a corner, in a trick shot that would be one-in-a-million even for the best sharpshooter in the world, the power's plan will include the exact position, angle, and timing of the shot, and he can reliably follow that to successfully perform the absurd trick shot.

The plans this power produces can rely on that perfect precision, finding courses of action to achieve otherwise impossible things in ways that would normally be doomed to fail.

I think at this point you've change the power from MC relaying information from the future to the past to Deus ex machina things work just because

Keep in mind that according to your own explanation this is not some kind of unknown entity sending the information but the main character themselves:

He can send information back in time to himself, and whenever he receives information from his future self, it always works out such that when he reaches that time in the future he voluntarily sends exactly the same information to his past self, so that the two ends of the message's time travel are consistent with each other. Conversely, whenever he chooses to send information back in time, it is always consistent that he received that exact information earlier.

How to justify having the power provide advice/plans that the character would never think of, even in hindsight.

You can't unless you change the fundamentals of the power (Turns out it was not the main character sending the info but some other being, but then you have to explain the power's improvement if it turns out the MC didn't actually understand it)

These superpowers can be improved after they are earned, and creativity and understanding are even more important for improving a power than for acquiring it in the first place. Practice and experience with a superpower can strengthen it, but improvements gained only from that are usually minor. Really significant improvements come from improving your understanding of how your power works, .....

How to make the leap from merely "the best the character would think of with hindsight" to "the actual best possible."

Same as before, you can´t with these constraints.

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  • $\begingroup$ Relaying the message could be error-prone. Either the channel itself or the character not being able to remember very complicated message with 100% precision even for a few seconds. $\endgroup$
    – Sebastian
    Commented Apr 16 at 13:35
  • $\begingroup$ PS: Nicolas Cage's Next 2007 $\endgroup$
    – Daviid
    Commented Apr 16 at 14:04

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