, if Rand constructed the voice, it almost implied plausible deniability. A voice is usually constructed, so that if something "Rand" can not do, "LTT" will do it, and "Rand" can claim plausible deniability, at least inside his head, as a coping method.

But that is the exactly opposite. "LTT" was not able to kill the Ashaman as LTT wished. "LTT" was not able to defeat Lanfear as "LTT" wished.

At the big battle at the end of KOD, why did "LTT" has those killer waves? If Rand constructed "LTT", why did Rand purposely surrender those knowledge to "LTT"? Because he is afraid of killing Shadowspawn? For the same lines of thought, at DW, why did LTT has the knowledge to unravel the shield? Because Rand actually wants to be captured, so he surpress those knowledge of shielding into "LTT"?

For a construct theory to work, there must be a good reason behind what parts are given to the constructed entity. The only reasonable explaination given the example in the books, is that Rand just constructed "LTT's voice" out of random bits and pieces of his past life memory. If that is the case, I would say the Wheel constructed "LTT", thus LTT is as real as Rand, which is also constructed by the Wheel.