1. In fact, the reason why Rebbi Meir says that a plank of four Tefachim is unfit is not because of Lavud. Lavud is a Halachah which depends on three Tefachim, not four. Rather, the reason why Rebbi Meir disqualifies the plank of four Tefachim (according to the way Rav explains the dispute between Rebbi Meir and Rebbi Yehudah) is that the ceilings of most houses are made of planks of four or more Tefachim. Therefore, Rebbi Meir makes a Gezeiras Tikrah; he says that if we allow people to make the roof of a Sukah out of planks which are similar to the roofs of their homes, then they will say, "What is the difference between my house and my Sukah? I may as well live in my house during Sukos."

2. However, there is a connection in the Sugya between Lavud and Gezeiras Tikrah. This is evident in the way that Rav Papa explains the dispute between Rebbi Meir and Rebbi Yehudah, according to Shmuel. Everyone agrees that planks of four Tefachim are unfit and, conversely, everyone agrees that planks of less than three Tefachim are fit. The dispute involves only planks between three and four. Rebbi Meir maintains that since these planks exceed Lavud we make a Gezeiras Tikrah.

3. Rashi (DH Keivan) explains what the connection is between Lavud and Gezeiras Tikrah. Since the planks are three Tefachim, they exceed Lavud and, consequently, they possess a degree of importance ("Chashivei"). Being "Chashivei," they are similar to the ceiling of a house which is also important. Therefore, Rebbi Meir maintains that the Rabanan made a Gezeiras Tikrah that if one uses planks of three or more, he might come to think that this is just the same as living in his house during Sukos.

Kol Tuv,

Dovid Bloom