Does lavud create a mechitza? If it does why can one not sleep under a space of less than three t'fochim"? If not how does it complete a mechitza of four t'fochim?
Although Lavud creates a Halachic Mechitzah, that is only so for Halachos that require no more than a Mechitzah. The Sechach of a Sukah is different. It is not sufficient to merely view the open space in the Sechach as filled in through the principle of "Lavud." Rather, that open area (or any other form of invalid Sechach) must be physical, valid, shade-providing Sechach in order to be permitted to sleep underneath it. (This requirement is a Halachah l'Moshe mi'Sinai; see the ROSH (2:3), who says that Lavud will not help for the Sechach because the Sechach must provide more shade than sunlight, and Lavud cannot provide shade.)
A more perplexing question is why one may not sleep under a break in the Sechach ofless than three Tefachim, yet such a break does not invalidate a 7x7 Tefach Sukah, nor does it split a larger Sukah in two. If we are applying Lavud, let the person sleep under it as well!
The answer is that Lavud joins an object on one side of the open space with what is on the other side. But to sleep underneath that area, it must be physically closed, and not just Halachically so. Why, though, may it be counted towards the 7x7 minimum size of a Sukah, if it is not actual Sechach? The answer is that we are looking at the entire Sukah as a whole. It is not necessary for every bit of a Sukah's roof to be covered with valid Sechach; only most of it must have Sechach (Tzelsah Merubah). Since the break of less than three Tefachim does not cut the Sukah in two, it is like any other part of the minority of the seven by seven Tefachim of roof, which need not be covered with kosher Sechach. On the other hand, when discussing a person sleeping under a less than three-Tefach break, we mean that most of his body and his head is exclusively under the open break in the Sechach (Ran, bottom of 9a in the Dafei ha'Rif). In such a case, by his choice of sleeping under the open space alone the person has isolated the open space; we cann from the rest of the Sukah. We cannot take the rest of the Sukah into consideration since he is sleeping exclusively (or primarily) under the space. This is why one may not sleep under a space of less than three Tefachim, although the same amount of space in a Sukah of 7x7 Tefachim is Mitztaref.
(See also the TAZ as cited by the MISHNAH BERURAH 632:10, who says that the invalid Sechach is "less Pasul" than open space, and one may sleep under it, since it is not as distinguishable from the valid Sechach surrounding it as open space -- that is, it too provides shade.)
I hope that this clears things up a bit.