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.

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.

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.

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.

You have a horrible, horrible problem.

Memories are formed by strengthening or creating new synapses. That is a physical change. If nothing can change, then you cannot form new memories - you will re-live the period, but arrive in the present in the exact same condition you were the first time, with all the memories and everything.

But that isn't the bad part. The bad part is this:

You start your time travel at time A1, from which you go back, let's say, 1 month. Two months of subjective time later (1 going back, 1 going forward) you are back at the same moment a second time, so let's call it A2. What exactly does "being back" mean? It means you walk through the door, sign your name on the release form, sit down in the time machine and go back in time 1 month. Another two months later, you arrive at A3, walk through the door...

There is no going forward for you from that point. Because you would have to skip over the moment when you go back in time. But since you postulated that you can change nothing, you cannot skip (because you would have had to do the skip the first time, in which case you didn't time-travel at all).

Sorry, you didn't solve time-travel. You just invented the infinite time loop.

But at least from your perspective you won't even notice it, because (see above) you can't form new memories and you will never remember that you were already here... a million times.

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?

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.

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.

Edit: As I saw many doubts about a few complications of "reliving the past", I thought about a few solutions so here gos the list.

1. 
   If you go back and go foward again in 1x speed you will neve cath up
   with people of the present moving at the 1x speed.
   Yes, thats true. But what if you actally do the rewind time quite fast thousand times the speed and YOUR perception of time is fixed at a higher level you see evrey detail while you are traveling but you are actually going really fast, you will eventually catch up with the future, and it will look like you just spaced out for a moment, for the rest of the world.


2. 
   If you go back the whole world go back with you but without memory of future events.
   That's probably bound to caos, the simples act of looking at some particules change they behavior, imagine in the future people looking at this particules in the past, changes are bound to happen, quantun changes of course but with a bit of "butterfly effect" the future can change, reverting the whole universe just for you try to find your wallet or even solve a murder is not worthy, rewinding your mind in time and going foward would be easier.


3. 
   Going back with your mind will overlap memorys and be a tremendous strain in the brain.
   Yes, it would. Just erase you memories, except for the lingering sugestion of what you are looking for, you will remake the memories along the way.

Also going back with your mind depending of the method may create a double personalite of yourself in your mind, another persona that exist just to look for your actions and will gave you a insight of what you would be looking for, when it arrive at the time of the pressing of the button. Lets say you press the button, it splits you in two minds, one go make the arduous trip in time to look for what you want, the other stays at the present, the travel goes really fast for the present one perspective and really slow for the time travelling one, this will give the present one the answer he want( and probably a enormous headache).

It is strange to think what would happen when I come back one minute and then spend a minute in the forward run. Three options come to mind, each of which somewhat problematic.

1) My own future stops and waits for me. I relive the last minute in an "observer" kind of mode, then suddenly regain control of my body. Which is probably the best option, although it will likely result in a noticeable jerk of some kind for other people. If the time is longer (say, a year), I may get out of the habit of controlling my own muscles.

2) Similar, but I want to retain the feeling that I am in control, so that the transition to the future is smooth. But then there's no space left for doing anything different. If not only my body but also my conscience relives past events as if it was there (signals to muscles etc.), I will notice nothing I had not noticed the first time. Whatever I forgot in the past will I forget again. This is technically free of paradoxes, but utterly useless, and clearly contrary to what you intended.

3) My body lives on in some kind of "automatic" regime, but as others noted, I will then never catch up. In other words, after reliving the 1 minute of my past, I will go on reliving what my body did in the meantime, during which it will keep doing more of that, always two minutes ahead. I will spend the rest of days trapped in reliving things my automatic ego thought of, effectively giving up all of my future to it. Brrr.

I am convinced you need a fast forward. Let me program the device by the amount of time I want to go back, and an amount of time I want to spend there. After doing so, I will come back to present with the newly reacquired experience. This can be instantaneous (like waking up from a dream), or can take some penalty time proportional to how far back I went, if that's a part of the mechanic you need. A rewind / fast forward at double speed might do reasonably good. But don't make it unit speed, or you'll run into 3) above.

A side point is that going back more than a day, about a third of the time relived is going to be extremely boring (except the second option, which I believe is ruled out by design). What do I see during sleep, if my conscience is detached (and thus not sleeping, at least not at the same time)? 8 hours of darkness and snoring... another good reason for a fast forward.

Another side note: assuming (1) for now, you also need a limit on how much time I can spend reliving things. Think of hopeless people who can't make even with the loss of someone and will keep coming back to see the best day again, always pressing the button over as soon as they can. People who will keep coming back to study every book until they know it bottom to top. Killers who want to commit a perfect crime, or at least learn from their past mistakes to malicious ends. The list of ideas how to abuse this goes on and on. It feels like there's no price to pay for seeing the same thing 1000 times. Nothing will be like we know it.

My point is that yes, you spend a lot of time doing all that, but always come back to a body that didn't age a second. Death is a physiological process, your conscience can buy basically unlimited amounts of time.

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...

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.

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?

Let's presume that the world 'stops' and you can actually catch up to your body (open end loop). Let us also presume that yesterday you went one day 'back', and today you want to go a week back. Your conscience will take 'the long way around', forcing you to relive yesterday's rewind as well, by your definition.

I feel like a terrible pseud saying this, but what defines the present if not that it is the point at which our consciousness resides? How would we know when we're back at "now"? What is it in the structure of the universe that makes the present moment objectively different to a moment two months in the past, or two months in the future? As you said yourself,

Imagine that your body is a four-dimensional object. It exists at every point in time simultaneously. Therefore none of your actions can ever change.

If we see time as an axis, and what-happens-within-time as a line on this axis which the consciousness can skip back and forwards along, then unless there's something objectively different about the present moment (that, perhaps, the present is the end of that line - that is, the present moment is perpetually the very end of time) wouldn't an inability to control your actions apply as much to the present as it does to the past?

This isn't time travel; it's slow memory retracing. Let's redefine this as a memory device:

At 9 AM I put the device on my head and set it for 1 hour. As the cheery green glow starts up and I fall into the memory trance, I subjectively observe everything happening around me in reverse. I subjectively have no power to do anything except watch, listen, smell, etc.

(Sidebar -- listening to everyone talk backwards for an entire hour has got to be really rough. There's no way to stop listening, since you're only recalling your own actions in reverse and you were listening then. I suggest that longer than a week be forbidden by the company providing this device; it is likely to result in an insane consumer.)

Okay, so the hour of reverse time is up and now I spend an hour subjectively experiencing everything in forward. To reiterate, I am only subjectively experiencing this. My actual body is in the present, sitting in a chair with a glowing green hat, and so far one hour has passed (my body is now at 10 AM) and another one is beginning. Since my body is in the present, I have no trouble forming new memories about what I'm seeing, so I can easily learn where I left my wallet.

The hour of repeated forward time is now up and the glowing green hat shuts off. I've spent two hours in the chair and it's now 11 AM. There's a brief "jump-shock" effect -- I go from remembering 8:59:59 AM to experiencing 11:00:00 AM; if my frat brothers have been by, I suddenly realize my pants are wet or there's a dick drawn on my face. I'll have no memory of when they actually did that, because the device had cut me off from physical sensation in favor of replaying memories from my brain.

After I hit the bathroom I go immediately to where I left my wallet, then come back and pay the man who rented me the hat.

So there's no paradox with this redefinition. However, you need to have at least potential total recall for this to be possible. Your brain must store every sensation you have had, from reading a headline to hearing a mosquito buzz past your ear, not forgetting anything at all. And it has to remember everything in perfect order. Probably there's a maximum limitation that takes effect here, and the device only can be set to an hour or five. And people who experienced brain damage in the past hour probably can't use it to revisit events that were wiped out by the injury.

A possible way out of it is that it doesn't reply your own memory; it replays the sensations recorded by a brain implant or something. But then the reason for the one-hour-per-hour becomes completely unclear; it should be random access like a DVD.

This will result in brain damage

Your brain is a physical object. Your conscious memories are recorded by making changes to that physical object. As you go backward in time, your brain must necessarily revert to the physical state it was in at the time to which the individual travels.

You have required that the consciousness of the person travels through time. For that to be the case, a copy of the traveler's brain (complete with new memories) must occupy the same location as a copy of the traveler's brain from the past (without the memories). When these two copies of the brain intertwine, any differences between the old brain and the new brain will become corrupted.

The end result, in the minor cases, would be serious confusion about what had happened and when. Perhaps you wouldn't be entirely sure whether or not you had traveled time at all, and any information about what happened in the period in which you traveled would be some scrambled mix of your thoughts in the first pass forward, the pass backward, and the second pass forward.

In the longer cases though, or the cases in which multiple passes are made through the same timespan, that confusion would become much more severe, and I would expect that time travelers would end up with amnesia or some kind of dementia, or perhaps permanent damage to memory as a whole.

This could also lead to some very interesting story-lines with brain "glitches". Perhaps the corruption in the brain's memory produces memories of bizarre things that never happened at all - maybe dreamlike, or hallucinogenic, or even mundane, but simply false.

And of course, if you go back WAY too far, your brain isn't even going to fit in your skull properly, and you'll just fall over dead. Perhaps your time-travel society has a lot of teenagers who suddenly die for "no reason" not because they're traveling time, but because their future self (alive) has traveled backward into their body and killed them. Of course, then you've got a schroedinger's cat problem - you had a living time line until it came back and killed itself. Now you've got two timelines.

This also leads us to question whether time travel of this kind would have any value. If the point is to go back and figure out what happened, but you can't retrieve reliable memories, or might even destroy the good memories that you have, or even might kill yourself to find your wallet, why go back at all (and maybe that's the moral of the story: don't mess with nature)?

The first thing to remember is that this process is not physically realizable. That is, since consciousness as we know it is a product of chemical processes within the brain, a continuous consciousness cannot operate backwards in time, since the first part of a thought cannot produce changes which occur before that thought, and so there can be no second part. If you see what I mean.

Furthermore, for a consciousness to progress forward in time without affecting the behavior of the body ("your body is locked into all the old actions") suggests that either the body is not actually being controlled by the brain, or that the consciousness is not taking place physically within the biological processes of the brain.

Let's go with the second explanation. If nothing else, it accords with the Christian concept of the immortal, unphysical soul. Note that this can only apply to the duration of the time-travelling loop. Prior to, and after the point of departure, the consciousness must be able to control the body, or else you must accept the explanation that we are all simply predestined meat-puppets being controlled by some unknown force (call it Destiny if you like), with our consciousnesses along for the ride. This, of course, has the attendent effect of eliminating free will entirely, and makes the issue of what a person gains from revisiting the past, if the newly-informed consciousness does not actually have the power to act on the knowledge, rather murky.

Under the conditions you have described, you are correct that no paradoxes are produced. The knowledge of the relative future which is brought to the relative past is unable to produce any effect whatever.

In comments on another question, it has been suggested that process must produce an infinite regress - that, in other words, your consciousness is trapped in an infinite cycle of time travel. I suggest that this is unnecessary. The point of the process is that the "traveller" consciousness has no communication with the "host" consciousness, and so there is no bleedthrough of memory. In effect, this requires that, when the time traveller "pushes the button", or whatever the trigger is, there is no merging of loop knowledge with the "original". Instead, if you like, the traveller consciousness instantaneously takes over the body which has been vacated. There is no obvious reason this cannot happen, since it (has been/will be) demonstrated that a personality can instantly be rendered powerless to affect its host body.

So, let's divide the process into three segments, A, B and C. Segment A is the life of the subject up until (let's say,) one year before time travel is triggered. Segment B is the period from one year before the trigger point until the trigger point. Segment C is the period after the trigger point.

Segment A is your bog-standard life. Your mind and your body are one.

Segment B is rather peculiar, in that your body hosts your "normal" consciousness, as well as two "parasite" minds: one travelling backwards and one travelling forwards. However, "hosting" is misleading, since the two "parasite" minds have no physically-detectable effect on the "normal" mind, and in fact they are unaware of each other as well. Both "parasites" have full access to your sensorium, but no control whatever of your physical responses. How this is managed is left as an exercise for the reader.

At the end of segment B, the "normal" mind becomes a parasite (the one moving backwards in time), and the forward parasite takes over the body and becomes "normal".

Segment C consists of a normal existence, with a consciousness which is two years (subjective) older that the physical age of the body.

In this description, there are no multiple loops of the parasite consciousness, and no recursive effects.

It also overlooks the likely effects of the reverse trip on the parasite, which might well include insanity. The reverse traveller uncontrollably experiences a year of enormously disorienting experience. With causality reversed, nothing will make sense. Any experience will have no relationship to what has (subjectively) gone before. At the same time, the traveller will spend a year without hearing intelligible speech, feeling food rising up his throat to be expelled onto his plate, etc. I expect it would be effectively a cross between sensory overload and sensory deprivation. The idea of trying to describe what sex would be like is entirely beyond me.

There's an interesting use of this device I've not seen brought up.
It sounds like however you want your device to work, you at least want the person to be able to think while they're experiencing their past, so that they can reach new conclusions.
If nothing else, this would be very useful in the middle of an exam, where you could gain extra time to think on the answer to each question.
It'd be strange, but you could also imagine squeezing tons of extra thought hours out of your employees/researchers to gain some kind of competitive edge.

And, there's one small paradox. Where does the energy for this extra thought-time come from?
And how much energy does it use? I believe the brain normally consumes 72kJ per hour (20W), so maybe that's a good number. Extreme lower bounds also exist, I believe. Computation can't take zero power without causing a paradox, I think (I believe it's even been shown that information can be turned directly into energy. Potentially, zero-power computation could be used to generate free energy).

Also, you'll have to decide when, how, and how quickly the energy actually gets spent. If we go with the 20W brain and someone travels back one month and forwards another month, that's 103.68 MJ of power consumed... instantly? Over one second? It'd have to be consumed prior to the time-travel, but where does that power actually go? I don't really have a concrete answer.

I'd try and consider how a similar device would actually have to work if you implemented it in reality. To implement it "simply" with real physics as we understand, you'd need some recording of your brain activity and a huge computer system that'll let you simulate your own brain as you replay through the footage, finally overwriting your memories in the end (or during). Such a system would require lots and lots of storage, and would use time proportional to how far back you were traveling. If someone pulls the plug in the middle of it, power stops getting consumed but you also don't get to experience your full month (or maybe not at all).

Another question to answer is why the brain's so special. Could an AI/computer also time travel? This would make abuse much easier and basically require the restraints I mentioned to prevent story-breaking possibilities (the ability to calculate anything instantly with little power is a powerful ability).