I will base this analysis on what is to be observed in the most common TAR, that reflects the waked world. TAR is more than that: it certainly includes also the Gap of Infinity (where the normal dreams are); the space used by Forsaken to make their secret meetings and to move the humans they want to speak to in their dreams; maybe also the space used for Skimming; maybe the space used by Accepted test ter'angreal in the White Tower and the Apprentice test ter'angreal in Rhuidean. But I hope that the normal TAR is the most complete example and it's analysis wil also explain sufficiently all other types of Unvisible World.

Let begin with an attempt to summarize the sorts of "matter" to be found in TAR. There is plenty of different things with different matter status, and all of them are completely compatible between each other, as if they are things of the same kind. And they are apparently everything but NOT things of the same kind. Actually this matter has different origins:

1. Reflected Origin: Reflections of things from the waked world: the most common things to be found in TAR. They change when the original in the waked world changes. Their changes don't affect the original. Stabile and movable waked-world items are reflected in different way, maybe that difference is also important.

2. Dreamed Origin: Bodies of people who touch TAR in their dreams, and who use Dreamwalker Talents, and who use Dreamwalking ter'angreals. Also bodies of dreaming wild animals. Their changes reflect the original body. They do not exist without the presence of the soul at the moment there, they are soul-connected. But they allways exist, when the soul is in TAR. What happenes to these bodies affects the bodies of the same person in the waked world.

3. Fiction Origin: Objects (things, living creatures, scenes, changes in other's bodies) created by TAR visitors. It lasts only while the will of the imaginator is concentrated enough. Sometimes two TAR visitors struggle about the state and form of an fiction-origin object and it's the person with the stronger will and more TAR experience, who's variant comes to existence.

4. Imported Origin: People coming into TAR in flesh. Also things brought by them from the waked world and left there. They are the same bodies, they were in the waked world, the same state of matter.

5. Afterlife Origin: Heroes of the Horn, dead wolves, maybe also other dead wild animals. They depend on the presence of the soul in TAR and resemble very much the dreamed origin. Maybe it's the same as dreamed origin in the principles.

6. Dream-Residue Origin: The nightmares that remain in TAR after the dreaming person exited TAR. Unlike the normal dreamed origin, these objects are not dependent of the soul presence, they are only initiated by such. However, unlike Reflected Origin, once removed they don't return to existence again. Another characteristic of that sort of TAR-proper is that it is not a particular object but whole environment, totally substituting the reflected-origin landscape normal for the area. My suggestion is that this is influence of a normal dream, originally created in the Gap of Infinity, but reflected by an accident into normal TAR. This kind of TAR matter can be specified also as GoI Origin.

Maybe akin to #6 are also some examples of #3 when the Fiction-created object becomes somewhat standalone and can exist also without the maintainance of the Dreamer's mind. Maybe the unicorn Elayne and Nynaeve created and then had torubles to make it dissapear was an example of this proper (but that problem could be simply their faith in what they created, i.e. unconscious maintainance of the unicorn in existence). The fact that a dreamwalker stuck into a wandering nightmare makes the nightmare stronger means that the #6 can use the ressources of the #3 creating Force (the Dreamwalker's mind) to feed it's own force, to transform it's energy into it's own (some sort of TAR virus). I.e. the #6 and #3 have some connection.

When in TAR, objects of any kind of origin interact with each other under the same laws. They affect each other like real objects. Including affects from and on Imported Origin objects, what is the most interesting. Objects allways have effect on each other. But there are significant differences between different origins in how durable these effects are.

That makes me to suppose that everything existing in TAR has the same state. The outer origins of different kinds change it again and again, but what is created by whatever origin exists in TAR like everything else. And everything, when integrated in TAR affetcs another objects. In that sense objects affect each other like forces, but they are weaker forces then the other origins and are often just overriden by them. So maybe it can be spoken about:

7. TAR Causal Origin: effects of another TAR objects of any origin.

Objects in TAR often sway between two states. That happens commonly with reflections of movable waked-world objects, with the Fiction-Origin details of TAR-visitors without dreamwalking and concentration experience, with objects about that state two wills are in conflict in that moment, with objects balefired in TAR. In all that cases two forces are in action attempting to influence one and the same object, and in the moment of vacillation they are roughly equal in strenght.

An implication of that is, that sometimes the two processes are not equal in strenght and one of them overrides the other. So, when two dreamwalker have will conflict about the form of some detail, after some vacillation the dreamwalker with the stronger will winns and the detail becomes the form wanted by her. Sometimes one origin overrides the other without vacillation: that is IMO the case whan a dreamwalker replaces her injured body by healthy one, or when she makes herself the clothes she wants. The same case is when a dreamwalker creates a chair: her will is stronger then the common Reflected Origin and her will overrides the lack of chair at that place in the waked world. After her will ceases to act for the maintainance of the chair, only the Reflected Origin remains to act and it imposes the waked-world reflection about that place, where there is no chair, and the Fiction-chair dissapears.

Conclusion from that observation: In TAR permanently act different matter-creating (=matter-maintaining) forces. Different sorts of these forces correspond to the different origins of TAR objects and can be designated with the same names. These forces have different strenght and different permanency. The stronger force override the weaker, and the object becomes the state dictated by that Origin. When that happens the other force doesn't cease to act, it is only suppressed. After the stronger Origin ceases to act, the weaker but more persistent one resumes it's effect and the object returns to it's previous state.

So we can define two parameters of the Origins acting in TAR, that are crucial for the result:

1. Permanency

2. Intensivity (or strenght). When there is an origin conflict, the more intensive origin determines the state of the TAR object.

There are some regularities between the permanency and intensivity of different kinds of Origin. Fiction Origin can override almost everything but it's very undurable. Dreamed Origin overrides Reflected Origin but it is not so durable as the Reflectede Origin. Dreamed Origin fills the place that the Fiction Origin leaves when there is a lack of concentration. Reflected Origin is so permanent, that it can turn back even the effect of balefire, i.e. to override Causal Origin. More conscious Origins seem to be more intensive and less permanent then less conscious ones. Among the examples we have there is a regularity, that the more an origin is intensive, the less permanent is it, and vice versa: the more permanent - the less intensive.

As example the Egwene's TAR trip to Salidar is a case, where there is even coexistence of two matters. Egwene comes in TAR as Imported origin, after that she changed her appearance and her injuries dissapeare (Fiction-Origin overrides Imported Origin). While her concentration is on, she could have her Fiction-Origin replacing the injured Imported Origin. When exiting TAR she has no Fiction-Origin more and the Imported Origin resumes it's action, and when quiting TAR, Egwene returnes to her original state. If Egwene imagined something from Nynaeve's healer's arsenal and treated her injured body with it, taht would result in a Causal Origin, that would override the Imported Origin and Egwene would exit more healthy, but she actually only overrode her Imported Origin with Fiction Origin, that has weaker durability.

Here we come to the question of the interactions between TAR and the waked world.

Changes in some origins affect the corresponding waked-world objects, changes in another don't. Reflected Origin doesn't change anything in the waked world, while Dreamed Origin does. Imported Origin also can be changed. So some waked-world objects share existence with TAR proper, and some doesn't, and the effects can be monodirectional or bidirectional. Sounding of the Horn of Valere has the effect, that Afterlife-Origin has effect on waked world without the waked world to can affect them, i.e. reverse influence-situation.

So there is another parameter of the different Origins:

3. Existence-sharing

it has two sub-parameters:

3.1. Effect-direction

3.2. Presence-degree

Imported Origin has the greatest presence-degree. Dreamed origin has variable presence degree depending on how deep the dream or trance is: Perrin in WH almost losed his body-soul connection and was too much "inside", while Wise Ones when on trip in the real world use weak form of trance and appear less dense and can do less things. Also it depends on the quality of the angreal used, which presence degree one has. The greater the presence-degree the more this object can affect TAR proper and the more it can be affected by TAR proper. I.e. presence degree is factor for intensivity.

Presence degree is dependent of the effect-direction. It has no meaning for Reflected Origin, which is mono-directional. It has meaning only on bi-directional shared bodies.

Fiction-origin has special position regarding the shared existence. If a Dreamed or Imported body of a dreamwalker is overridden by Fiction-Origin, that doesn't affect the body when exiting TAR, while if it is affected by some actions made with a Fiction-Origin object, that changes the body. Egwene can imagine her healthy, but that works only while she is in TAR. But the ruffians (Fiction-Origin) Egwene created while giving lesson to Nynaeve, scratched Nynaeve and that remained on Nynaeve real body. Or an AS can come into TAR, imagine an Oath Rod, take the oaths, and they would work in the waked world.

So: direct overriding an object with Fiction-Origin doesn't effect the body (i.e. doesn't share existence with the body), while indirect usage of Fiction-Origin does effect it. Conclusion: Any origin work parallel to each other and creat any moment a result. But the result has it's own effects on all the other TAR-entries. These effects don't affect the Force of the Origins, but it effects the results. And sometimes results share existence with the waked-world base of the Origin and that causes the waked-world body to change.

The body of the dreamer is both in the waked world and TAR. That being in both places in the same moment has degrees (= the Presence-degree parameter). Theis being in both places has also direction (Reflected object is in both TAR and waked world, but the reflected original is only in waked world, not in TAR) (or the opposite example: the Heroes-presence in Falme are both in TAR and waked world, while the original existence of the Heroes in that moment is only in TAR, not in waked world).

This could be explained only if TAR is regarded as area, where different reality-states touch each other and share existence in some degree. Matter from waked world can touch TAR proper. Souls without bodies (Afterlife-Origin) can touch TAR. And Dreaming souls can touch it. The last case is the most complicite. When dreaming, one's soul remain's in the body and in the same time it is somewhere else. So the soul is some sort of link between two realities. Normal human dreaming has it's reserved area in the invisible world (GoI), where they build worlds isolated from any other kind of TAR proper, including the other dreams. But sometimes that dreames touch the main TAR space and merge with. Wild animals do not create whole worlds, they allways integrate in their environment, so they don't need Dream-area. Domestic animals are changed and don't integrate completely in the basic environment, so they have no access in TAR. Dreamwalkers can both act as normal humans and create their own environment, and as wild animals and integrate themselves in the main environment.

Sometimes normal people temporaly hit the integration state that moves them into TAR, but they keep their dream-creating activity and return to normal dreaming state (dreamwalkers can do this returning to normal dreams consciously). But in the moment of touching main TAR, these normal people are in the same time in two places - in main TAR and in their own dream. So here we have touch between TAR and a particular dream. That can be shared, also with different effect-direction. A dreamwalker of a forsaken can intervene via TAR in particular dreams, and particular dreams can leave nightmares in main TAR.

So different states of matter (bodies, bodyless souls, dreaming souls, whole particular dreams) can share existence with TAR with different sharing conditions. Depending of the effect-direction and presence-degree, TAR proper can affect non-TAR proper or it can be affected by it, or both.

By special conditions even materializations of TAR proper in the waked world can occur. Birgitte's (and her silver arrow moving) into waked world is the most complete example of such case. It is like comint into TAR in flesh, but here in reverse conditions. Moghedien used on Birgitte the TAR equivalent of Travelling to TAR, let call it Anti-Travelling.

Another examples of materialization are not so full. In one of his Baalzamon dreams in tEotW Rand gets a splinther of the Baalzamon's office in his hand, and realizes it after he wakes up, and only after he moves the splinther out of the hand, it dissapeares. Nynaeve exits the Accepted test with black thorns in her hands, they also dissapear only after she removes them from the hands. Both cases are apparently Fiction-Origin object coming to existence in waked world. Both cases show their dependence of touching the body of the TAR-visitor. So there is some degree of merging between Dreamed Origin (Imported Origin in Nynaeve's case) and Fiction Origin. As Dreamed body is shared between TAR and waked world and has some presence-degree, it can be affected by TAR proper. By some unknown conditions Fiction-Origin could also become some presence-degree in that body...