"Time stops moving forward for everyone." Sure. Right up until the instant the traveller's consciousness arrives back at the starting point. Then, since no time has passed for the stay-at-homes, everything picks up where it left off, with no one (except the now-mentally-older traveller) aware that anything has happened.

@WhatRoughBeast, remember, the "starting point" was the moment when the the traveller went back in time - and now goes back in time again. How do you break the loop?

This needs more upvotes. This paradox is real.

@JBH - No paradox. The subject's consciousness forms an open loop, not a closed one. For a one-year loop, the traveller consciousness is, let's say, 21 years old at the instant of travel. At the end of the loop, the traveller consciousness is 23 years old (subjective), and takes over the body. The traveller was not the operator of the body during the loop - that was done by the original consciousness, or his earlier version, if you prefer. The early version was untouched and unchanged by the traveller hitchhiking. However, this also implies that free will is an illusion.

This is only a paradox if the time-travel technology is flawed, causing a, infinitely repeating closed loop instead of a single open-ended loop, releasing the time traveller upon return from the trip to the past. Otherwise it'd just be an infinite loop - why would anyone choose to use that technology except perhaps by mistake or as a doomed experiment?

Think of it as a GoTo / Return without branches - you resume from Next Step, and if your mental time-travel intersects with another instance of mental time-travel, you don't also travel that path. e.g. if you go "back and forth" by a 2 hours, then an hour later you go "back and forth" by 2 hours again, you experience 4 hours - there and back - not "gone" for 12 hours (1 hour back in line B, 2 hours back in line A, 2 hours forward in line A, 1 hour back in line B, 1 hour forward in line B, 2 hours back in line A, 2 hours forward in line A, 1 hour forward in line B)

@Gertsen and there isn't a solution here. If the technology allows for "breaking out of the loop", then it could allow it at any point in time. At the very least the following: travel back in time from point A -> return back to point A -> time passes -> travel back in time from point B -> return. You now have a "junction" at A before you reach B which means that you can break out early and affect the time. This is a paradox. There are potential solutions but they would all involve basically the author's word, same with other time travel mechanics that could cause a paradox but don't.

@WhatRoughBeast, I'm bound by the OP's rules, which states that the traveler cannot change anything that happened in the past (You experience everything in perfect, exquisite detail just as you did the first time around.) - including the act of traveling to the past. You're describing the consciousness being forced to remember everything - not actually traveling to the past. Therein is the paradox. If actual time travel took place, then the act of traveling to the past was included with it and must be relived - over and over. Only the OP can resolve the paradox.

What's the paradox? I'm looking for some sort of contradiction or seemingly contradictory bit in the consequences you bring up, but not seeing it. A major problem, yeah, but just because there's time travel involved in your problem, doesn't mean it's a paradox.

This paradox depends on the mechanism (on how the consciousness' travel through time is altered) and on the nature of consciousness & the body. For example, this paradox is avoided if the body contains multiple consciousnesses (the original, and any that are going on a time journey, whether or not any are aware of the others), as when the moment that the button is pressed is reached a second time, the original consciousness is sent on the journey and the consciousness that just finished the journey regains control of the body.

... If the mechanism is in any way analogous to riding on a bus (or train or ferry or blimp or whatever mass transit you like), there's no paradox. The vehicle loops, the consciousness gets on and off at the station (the button press).

I'm reminded of this hacker koan.

There isn't a paradox. This model of time travel is explicitly deterministic. Not only does the time traveller not change anything in the past nor can anyone else. The only thing that goes back in time is the consciousness of the traveller. It can only observe the past. First, backwards & then forwards. It's pure determinism. No paradoxes.

@vlaz: "I described it in my previous comment but if the time travelling consciousness can "hop back in" to the body, then it can do it at a previous point in time" - there's an easy explanation for that. At the earlier point in time the body already had a consciousness. In the "present" the original consciousness just left the body to go back in time so the returning one can take over. Also I'll agree with others that say this isn't a paradox you are describing. There is no logical inconsistency, just an infinite loop which isn't a paradox.

What if the machine is actually sending the current consciousness back in time by supplanting it with the current consciousness returning from the past? After all, if the consciousness keeps going back and it is an entity then there would be infinitely accumulating consciousnesses which wouldn't make sense given the description of the machine. Furthermore it doesn't allow the person to push back in, because they can only reenter when the consciousness is removed.

@TheGreatDuck, that's a decision the OP would need to make. As defined when I posted my answer, there's no way out of the loop because the final mandatory action is always being sent back again.

@JBH you're being pedantic. It's clear from their description that an infinite loop does not occur. Whatever the mechanism is, the fact that they say "With luck they can observe themselves losing the wallet but of course they can't do anything about it. However at least they know precisely where and when it happened and this will help them find it." already makes it clear that on the return trip your observing mind re-takes over the body. However, I do question how it can be re-inserted if it isn't detectable and therefore alter the past by being detected.

@TheGreatDuck, You just quoted the inherent problem in the OP's question. Right along side of You cannot change the past. You just watch history unfold as a spectator locked in your own body. You experience everything in perfect, exquisite detail just as you did the first time around. Which includes the event that sent you back in time. (And you should go look up the word pedantic. Unless you really think I'm either showing off my vast knowledge or being unimaginative.)

@JBH "pedant - a person who is excessively concerned with minor details" You're taking their words so literally and so rigidly that your aren't understanding what they described. They already made it clear that the mind on the return trip does not get sent back. It doesn't matter how that occurs. It doesn't. This is not answering the question. Also time can't get "stuck". There's no such thing as 5th dimensional time-time.

@TheGreatDuck, Are you the OP? Nope, it's Chasly. Chasly can make that determination. You can't. I've pointed out a flaw as requested. If you don't like it, you're free to downvote or, better still, post your own answer defending the "minor details" you think support your perspective and weaken mine. (You actually know enough about time to know it can't get stuck? Go look up "arrogant.") Have a good night!