Gfrktyl

There are no paradox concerns.

Your system prevents modification of the past, so there is no way to cause changes which would lead you to not go back in time, or go back in time differently.

Your system cannot see into the future, so you cannot see the effects of your actions and act differently.

Really, it's more of a VCR than time travel. You can go back to see the things you previously recorded, and that's it. Also, there's no fast-forward/rewind, just play forwards and play backwards.

I think there is a major flaw:
Since there is no fast forward, you could never catch up with the present. For everyone else, time moves forward and you are always behind.
Unless triggering the time travel freezes time for everyone else.

If you do it through your mind there is no problem, that's just memory(a very good and also problematic one) or watching a movie in first person of your actions, strap a camera in your head, force yourself to follow this rules and you can do this yourself, but if you do it through external interference like magic or a machine that send you on time, that's a action that happened in the past and you stay trapped on the loop.

To break the loop paradox, you could create a "escape rope", Lets say this.
january 12 of 2019 at 13:00 you press the button to the escape rope, you keep living your life normaly, at january 13 of 2019 at 15:00 you see a murder, then at january 14 of 2019 at 12:00 you rewind time to the january 12 of 2019 at 12:59, you start the loop, then at january 12 of 2019 at 13:00 you press again the "escape rope" that have a command which says "if you are on a loop unmake the effect of the loop button when you press it" (frase it as you like, also yes, I code), when you arrive at january 14 of 2019 at 12:00 you press the button the effect of the escape rope triggers and you break out of the loop.

Musing about the concept...

Essentially, living memory becomes (potentially) more accessible. The big benefit is that those memories could contain information that was originally not really noticed, but with the hindsight of future knowledge can be 'recalled' and brought forward and acted on.

It creates an interesting new branch of criminal investigations. Witnesses can reveal forgotten details, especially form recent events, leading to more certain rulings.

It creates an interesting new branch of research, for the same reason. With proper incentive, a survivor of Event X (a Kennedy assassination, for example) can potentially bring back forgotten evidence. A WWII survivor could be 'sent back' to try to 'remember'(?) conversations that have since been lost... a conversation with Einstein, or Tesla.

Conversations with native Americans in their own language could (potentially) be recovered -- languages which are inadequately documented and have since gone extinct.

This works for any organization in fact. Crime syndicates can use it (potentially) to better learn the identity of informants. Or, to more easily remember (or notice!) the location of valuables.

While I do not see any issues with paradoxes with this "type" of time-travelling, there are some odd things that I'd question.

First of all, your example of losing your wallet. While it's great to "re-live" your past and remembering that you put it in your couch, how would going back help if you were not conscious that you lost your wallet, by dropping it for example? Your original body would not realize that you dropped it, so even if you go back a second time, you'd still not find it.

That goes the same for a pickpocketer. How would you know if someone is doing that to you when you go back? Does your consciousness have an out-of-the-body experience and see what you couldn't see before? Probably not, based on your explanation of doing all the actions and experiencing everything all over again. So, if you get pickpocketed without realizing it, you wouldn't find out even if you go back.

Then, there's things like sleep. Will your consciousness perceive your original body sleeping? Or will it fall asleep the same as you did before? Would you remember or experience past dreams? Would you still retain information about dreams when you wake up? Will your consciousness stay "awake" during sleep instead? If I relive the moment my house got robbed while I was asleep, would I be able to hear the robbers even though I was asleep before? There are a lot of questions to be answered here..

If your consciousness relives everything, does that include things that do affect consciousness like drinking alcohol? What if you relive the moment wherein you were drunk? Will the consciousness that went back in the past also become drunk? Wouldn't that mean that you won't be able to retain information during this state because the consciousness that went back is also drunk?

And finally, I'd worry about mental health. What if someone relives a very emotional moment in their life? The feeling of success, joy of being accepted by your partner, winning, pleasure, getting "high", etc. What if someone keeps reliving those moments everyday? If you go back to the present, you'd lose those feelings, and might want to do that all over again. That could stagnate someone or even depress them, as they try to compare their past from the present.

While these questions do not refer to having a paradox, it's curious how would this system work for an everyday person.

You have a whomping big paradox

You stipulate that the traveller "cannot change anything that happened to [he/she]." That's a problem. You can jump back and relive time, but you can't change the fact that you jumped back to relive time. You'll never live another second of free will at all. Let's investigate this.

Assumption

Let's ignore the entire is-there-free-will question by simply assuming that time becomes locked into place once experienced. From this perspective, you can jump into the past and "relive" it until the "present" while meeting your stipulation.

There is, of course, a problem with this. Nothing can change. You can't change the focus of your eyes, their placement, the attention you're giving to what you're listening to, nothing at all. Your conscious self is "reliving" the time from the perspective that it's storing the info into memory, but even if you had the ability to analyze what was happening — you can't do anything, change anything, nothing. It would be maddening, using your example, to jump back in time and not be able to actually focus on where you lost your wallet because you can't change what your physical body is doing: and what it's doing is not paying attention to the wallet.

But, let's return to the paradox...

You jump back and back and back... what happens to everyone else?

Because you can't change a darn thing, you're stuck. You made the mistake of choosing to jump back in time and you'll make that same choice, step into that same machine, watch Dr. Naidoo push that button to send you back, forever. Because you can't change anything.

What's the paradox? What's happening with everyone else? Theoretically, you took off to the past and your wife, grateful that you did, is now eyeballing the proverbial post man thinking that she has all the time in the world.

Or would she?

Does time become locked into place once a traveler makes that first jump? Or does it continue? The question can't be answered from the traveler's perspective. It can only be answered by you, the author, choosing how to resolve the paradox.

Does time stop for everyone when someone jumps backward, or not?

The reason this is a paradox is because everyone else is involved when someone jumps back in time. Fundamentally, everyone jumps back in time with him (unless you want to declare that there are two independent time streams: one for the traveler and one for everyone else). If the traveler can't get out of the time loop, neither can anyone else. Time stops moving forward for everyone.

Yes and no....

Part of time is that not only does the past affect the future, but the future also affects the past.

By going back in time and taking note of the pickpocket, for example, you learn enough about him to file a police report, so you do so.

However, had you not gone into the past to discover what happened to your wallet, you have still changed the future, as you would not have reported it stolen had you not discovered it was stolen and not lost.

So, say instead, you were going to meet a friend for lunch, and in the future, that friend would have dropped his wallet, and lost it at your lunch. Now, he doesn't lose it...

You go back by rewinding, but your conscience does not go backwards? It is actually remembering future events relative to the time you return to you. It's like information travels backwards with you. In the first scenario you can act to undo mistakes and therefore the mistake was never done and there was no reason for the trip - That's already a paradox. In the second scenario, the same things will happen again. That is, you do the same mistakes even though you know it. What prevents you from taking corrective measures? Why, in your example, you would see the pickpocketer and could do nothing about it? Would you be able to see him from a different angle and still be unable to warn your past-yourself? Suppose you stopped your time-rewind and returned to "replay", will you be the "same one", or will another observer see you and your past-you?

Your system cannot work with our understanding of physics.

Think of Schrödinger' cat. The effect most people associate to it cannot be reproduced on a macroscopic object such as a cat, but it is reproducible with particles. It is happening at every moment to every subatomic particle. But for the sake of argument, let's call some paint particles on a brick of a wall our cat particles. They can cause the paint to peel off in various patterns; You will only know when you observe them.

You passed by your backyard yesterday without paying attention to a certain brick. Today you notice some paint is peeling off that brick, forming a pattern.

You go back to the "past" to do the wallet finding thing. You think you lost it close to the wall, so now you pay attention to the brick.

The fact that you are observing them on the past means they have to collapse their wave function at that moment, so the universe decides whether and how the paint peels off at that moment. You just changed the past.

You have a paradox now. If the paint must always off to be just as you remember from the "present", then the wave function collapse in the "present" extended to the "past", meaning anything happening in the present affects and changes the "past" retroactively. Otherwise, wave function collapsing from direct "past" observations may cause changes to events anyway, and those changed will accumulate through time a la butterfly effect.

You're missing something... you are re-living things, but without being able to control the body you must somehow also be aware of the thoughts of your past-tense self. So you're not just sensing it, you're also sensing yourself! If this were not so, then as you were living "forwards from the past" how could you make an informed judgment as you live out the predetermined history again?

Also, if you were to re-live the same thing twice, you would necessarily become aware of the you that was re-living it the first time, since that, too, was you. Otherwise you could never re-live what you had been thinking when you re-saw what you saw!

Then if you re-lived the same thing many times, your experience would be crowded with your own consciousness and it would be overwhelming.

This is not a coherent concept. Time traveling in this way would be indistinguishable from not time traveling at all.

Imagine you give me something that is either this time travel device or a placebo. It's 5:00, and I intent to travel back to 5:00 at 6:00.

Here's my reasoning at 5:00: "Hmm, this could be the me that hasn't yet pushed the button. Or this could be the me from 6:00 coming back to 5:00. I can't tell any difference. If I could, I could act differently and change the past. Seems to be working."

Here's my reasoning at 6:00: "Okay, I'm ready to push the button. Let's see what happens next. Well, I guess, I sort of do. I mean, I already pushed the button in some sense."

Here's my reasoning at 6:01: "Cool. I pushed the button. I remember living from 5:00 to 6:00. I remember pushing the button. It all seems cool to me. I guess it worked. Maybe."

There is no conceivable way anyone could tell the difference between having this device and not. Consciousness always implies the ability to use your memory to make decisions. If you could ever remember the future, you would encounter paradoxes because with that memory, you could act on future knowledge and without it, you could not.

If you mean time travel where I go back to the past, remember the future, but cannot base my decisions on what I know, you're talking about a torture device that turns us into watchers rather than deciders and actors. If I go back in time, can I act based on my memories or not?