Take a break, Davian...I'm pretty sure I can explain this better than you can. ;) Perhaps even better than Callandor...Callandor is the one that convinced me, but there are several aspects of his arguments that I disagree with, and his terminology I think is the worst aspect of it. He uses quite a few too many terms that are interpreted differently by different people, and I agree with Ares that the "summation of memories" definition of personality isn't sufficient. It's Callandor's idea that, for a personality to be "real" (which is the foremost of those ambiguous terms), then all of the memories must be present. This seems to me to be blatantly false - even normal folks with only one set of memories are constantly learning and forgetting things. What, then, is a "summation" of memories? How can you describe any set of memories as being "complete"? You simply can't. Rand's Lews Therin memories are certainly spotty - not a set of memories that any person would consider normal in its scope. The memories that come to Rand are usually the memories that he needs, but not always. For instance, the memory of the plum orchard wasn't particularly vital, nor is the tendency to hum and thumb his earlobe when he sees a pretty woman. But when Rand spoke to Taim about the Forsaken and their deeds, he certainly seemed to be drawing on a very comprehensive knowledge of the Forsaken. A "summation" of Lews Therin's memories about them (this is perhaps the best example of a "summation" of memories from Lews Therin in the books). But Rand thought of it afterward as though he consciously drew on these memories - certainly as if he made a conscious choice to speak the words. Other things, such as Rand having to be told what certain Old Tongue words mean, show that the memories are limited, not only by the bits and pieces that every person forgets, but by whole large swaths of knowledge. A normal person simply does not up and forget a language in which they are fluent.
There are a few other disagreements that I have with Callandor's theory - one of them is his method of "demonstrating" that Semirhage was lying, with the Graendal quote, which I believe can be interpreted a couple of different ways. Semirhage's comments aren't particularly troubling to be, because I don't believe that, when she says Lews Therin's voice is "real", it necessarily means that she's implying that Lews Therin is a being with a mind of its own. Yes, it's quite possible that she's lying, and it's pretty certain that she has a limited understanding of what's going on inside Rand's head, and also quite possible that even Graendal had a limited understanding of such things. But Semirhage told them that Rand knew who she was because Lews Therin told him, which isn't exactly true. Rand recognized Semirhage. He recognized her because of Lews Therin's memories, but Lews Therin was silent.
The distinction that I feel is ultimately important in the Rand/Lews Therin situation is that Lews Therin is not a being with a mind of its own. The speaking, acting Lews Therin is an illusion created by Rand because he refused to accept the memories, in much the same way as the body will sometimes reject an alien organ after a transplant operation, and for similar and yet completely different reasons, if that makes any sense. :) Rand wanted not only to disassociate himself from Lews Therin's legacy as much as possible, but he also hates the knowledge of what he might become. This is evidence in his paranoia concerning the girls (how he's always trying to send them away, and taking great care to protect them), and it is especially evidenced in his excessive self-flagellation with the litany of dead women. It is evidenced in his thoughts when he was first told that he was the Dragon Reborn, and it is evidenced in his extreme denial (I am not Lews Therin! I am Rand al'Thor!) when he first realizes the source of his alien memories.
The issue of graduality (I think I just made that word up) isn't really one that can be claimed by either side as an indicator of the correctness of their theory, but I believe that the specific manner in which Rand developed the ability to actually converse with Lews Therin is pretty telling. The Fires of Heaven is pretty much the exposition of Lews Therin (Rand didn't realize, or at least didn't admit to himself where the memories were coming from before Chapter 6 of that book), but that ability to converse was developed in Lord of Chaos. Those two books in particular are, I feel, vital to understanding the nature of the relationship between Rand and Lews Therin.
This debate usually does get that way, but it doesn't have to stay that way - I'm sure that everyone here (including Ares and Davian) are capable of discussing this rationally and politely.
Yes, this is the first time that Rand and Lews Therin really have what might be considered a conversation, and also one of the only times that Lews Therin has actually "told" Rand how to do something. Usually, important memories come to Rand without commentary, and the commentary is generally arbitrary.
Lews Therin is an aspect of Rand - or an aspect of his soul, if you prefer. He's always "with" Rand. But between Cadsuane's comment about voices and madness, and Rand's first meeting with Torval, Lews Therin apparently wasn't "with" Rand. Do you know why? Do you know where he might have been?
Whether you believe that Lews Therin has a mind of his own or not, Lews Therin didn't "show" Rand how to weave Deathgates. Rand (or Lews Therin) just wove them, and as they were forming, Rand knew what those weaves were called, with no commentary.
I'm of the opinion that Lews Therin, the apparently sentient personality, is born of stressful moments - in particular the stress that the memories cause Rand, and his desire to disassociate himself from those memories by denying them.
Have you ever read Calvin and Hobbes?
It's undeniable that Rand has gained a great deal of vital knowledge from Lews Therin's memories, but most of the important stuff comes to Rand without commentary. Lews Therin's "presence" is generally very arbitrary, and as far as I can see, it only serves one real purpose. With Lews Therin's "presence", Rand can pretend that the memories have nothing to do with him (especially those memories of his final mad act, and what we saw in the prologue of The Eye of the World). Also, it is a convenient outlet not only for that guilt but also for other thoughts and emotions that Rand finds inconvenient. Lews Therin's wild reactions to Taim and Asha'man in general are a good example of this - when Lews Therin is "gone" in the period I mentioned before, those thoughts and emotions remain, but Rand can no longer pretend that they are not his own thoughts.
This is a rather involved subject, though I feel that the construct theory satisfies Occam's Razor far better than the assumption that Lews Therin is a sentient entity separate from Rand (why some supernatural being whose "presence" isn't quite logical when there is a much simpler and logical psychological explanation?) and what I've said here isn't anywhere near a comprehensive defense of this theory - I'll probably never have time to present a comprehensive defense on this board - but I hope I was able to elucidate what the construct theory actually implies.